ANALYSIS OF A TWO-STAGE COMPETITIVE ENDOTHERMICEXOTHERMIC REACTION SCHEME
W.Y.S. Wee1, J.J. Sharples1, H.S. Sidhu1, V.V.Gubernov2
1
Applied and Industrial Mathematics Research Group
School of Physical Environmental and Mathematical Sciences
University of New South Wales at the Australian Defence Force Academy
Canberra, ACT 2600
Australia
[email protected], [email protected], [email protected]
2
I.E. Tamm Theory Department
P.N. Lebedev Physical Institute of Russian Academy of Sciences
53 Leninskii Prospect, Moscow 119991
Russian Federation
[email protected]
ABSTRACT
In this paper we consider a reactant which can follow either of two competing chemical
reaction pathways: one exothermic and one endothermic. Properties of the reaction
wave fronts arising in a general two-stage competitive endothermic-exothermic reaction
scheme are derived numerically over a range of different parameter values, such as
those describing the relative enthalpies, rate constants and activation energies of the
endothermic and exothermic reactions. We focus on an adiabatic reaction scheme with a
more dominant endothermic reaction pathway as compared to previous studies, and
show the existence and behaviour of stable and pulsating front propagation in specific
parametric regions. In particular, we demonstrate the transition in reaction front
behaviour from the steady speed regime to different pulsating regimes.
INTRODUCTION
Combustion processes have played a significant part in the advance of technology and
civilisation throughout human history. The multiple applications of combustion
processes in today’s industries such as the synthesis of ceramics (Subrahmanyam and
Vijayakumar, 1992), combustion engines, and power generation are the result of
numerous studies and experiments conducted by past researchers (Merzhanov and
Rumanov, 1999). In particular, flames that propagate through a reactive media, not
unlike the dust-filled atmosphere within a coalmine shaft, are of interest to industry,
especially combustion waves that reflect travelling wave solutions in such systems. A
combustion wave is essentially a propagating reaction front delineating the change from
the initial chemical reactants to the reaction products. Whilst most observed combustion
processes consist of multiple (perhaps tens or hundreds) chemical reactions occurring
consecutively or simultaneously, it is often useful to ‘group’ reactions into simpler
models describing the dominant reaction kinetics. The reaction-diffusion systems that
arise from these reduced kinetic schemes are more tractable to detailed mathematical
analysis, which can provide important insights into the overall processes. For example,
where thermal effects are prominent, the detailed kinetics can be reduced to a simple
useful model consisting of one exothermic and one endothermic reaction. Such systems
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can be classified as: ‘sequential’ where both reactions occur consecutively (Please et al.,
2003, Liu et al., 1998); ‘parallel’ where both reactions occur at the same time whilst
consuming different reactants (Ball et al., 1999); and ‘competitive’ where both reactions
are simultaneously occurring and feeding on the same reactant.
While parallel and sequential schemes have been relatively well studied, there have
been few studies focussed on kinetic schemes with competing endothermic and
exothermic reactions, despite such schemes being relevant to decomposition and
pyrolysis processes (Antal and Varhegyi, 1995) and in applications to ammonium
nitrate explosives (Sinditskii et al., 2005). Recent studies include (Hmaidi et al., 2010,
Gubernov et al., 2012) and (Sharples et al., 2012). (Hmaidi et al., 2010) investigated the
existence of propagating combustion waves in a one-dimensional reactive solid slab
(infinite Lewis number), where the reactive system lost heat via a competing
endothermic pathway. The study yielded asymptotic solutions for the reaction front, but
restricted the analysis to solid reactants and for an exothermic-dominated scheme.
(Sharples et al., 2012) considered non-infinite Lewis numbers, thus accommodating
non-solid reactants within the analysis and also determined parameter ranges
corresponding to pulsating reaction fronts using stability analyses based on direct
numerical integration and Evans function techniques. (Gubernov et al., 2012)
investigated the properties of combustion waves and the travelling wave solutions in a
similar competitive system applying asymptotic analysis and establishing the existence
of endothermic, exothermic and mixed endo-exo parameter regimes. Notable
behaviours such as the loss of stability of the reaction front in such reaction schemes
can occur for certain parameter ranges, and have been observed in experiments resulting
in possibly undesired outcomes in industrial applications (Makino, 2001).
The aim of the present paper is to investigate the behaviour and stability of the reaction
fronts resulting from a competitive scheme with a more balanced contribution from the
endothermic and exothermic reactions for generic reactants or fuel, thus extending past
research (Sharples et al., 2012, Hmaidi et al., 2010, Gubernov et al., 2012) on
competitive endothermic-exothermic schemes.
PROBLEM DESCRIPTION AND MATHEMATICAL MODEL
We consider a reaction scheme with two-stage kinetics for combustion wave
propagation in a pre-mixed one-dimensional reactive medium under adiabatic
conditions. The model is similar to the set-up used by (Turcotte et al., 2008), where
combustion is initiated via a hot-wire ignition. We neglect hydrodynamic effects, as
well as physical heat loss processes such as radiative cooling or the conduction of heat
away from the system, as with (Sharples et al., 2012), but will inherently have an aspect
of heat loss through the endothermic reaction. The model will allow Lewis number
variations so as to include non-solid reactants and the parameter values we assume will
allow for more contribution from the endothermic reaction and remove the reliance of
large activation energies.
It is assumed that the reactant undergoes one of two possible competing reaction
pathways: one endothermic and one exothermic, and that the reaction products are
chemically inert and do not change the physical properties such as the heat capacities,
density or diffusivity of the system. A real-world example of such a configuration
would be a long insulated cylinder containing a fuel undergoing decomposition, with an
appropriate a priori averaging over the transverse spatial dimension of the reaction
front. Such a set-up has been used in laboratory experiments studying the burning of
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ammonium nitrate (NH4NO3), in the context of emulsion explosives (Turcotte et al.,
2008, Chan and Turcotte, 2009).
With the aim of analysing the properties of a combustion wave front in a competitive
reaction-diffusion scheme in one spatial dimension, we propose a generalised
description of the scheme’s reaction pathways, assuming Arrhenius kinetics:
Reactant → Product 1 − heat (Q1 ),
k1 (T ) = A1e - E1 / RT
Reactant → Product 2 + heat (Q 2 ),
k 2 (T ) = A2e - E2 / RT (exothermic )
(endothermic)
(1)
(2)
where the exothermic reaction drives the combustion wave front and is characterised by
its activation energy E2 (J mol-1), pre-exponential rate constant A2 (s-1), and heat release
Q2 (J kg-1). Heat from the system is lost via the endothermic reaction with activation
energy E1 (J mol-1), pre-exponential rate constant A1 (s-1). The temperature and reaction
rates of the endothermic and exothermic reactions are represented by T (K), k1 (s-1) and
k2 (s-1) respectively.
For instance, a decomposition reaction of ammonium nitrate modelled by two ‘lumped’
competing endothermic and exothermic reactions might be represented as (Sinditskii et
al., 2005):
NH 4 NO 3 → NH 3 + HNO 3
NH 4 NO 3 → N 2O + 2H 2O
(endothermic)
(3)
(exothermic )
(4)
By applying heat and mass balance to the reaction and diffusion of reactant and heat, the
governing equations for the competitive system are obtained. Similar equations can be
found in (Weber et al., 1997) and (Clavin et al., 1987):
(
)
∂T
∂2T
− E / RT
− E / RT
ρc p
= k 2 + ρ −Q1 A1e 1 + Q2 A2e 2 C
∂t
∂x
(
)
∂C
∂2C
− E / RT
− E / RT
ρ
= ρ D 2 − ρ A1e 1 + A2 e 2 C
∂t
∂x
(5)
(6)
where the reactant mass fraction and temperature are denoted by C and T respectively
and the time and space coordinates are t and x respectively. ρ (kg m-3) is the density, k
(J s-1 m-1 K-1) is the thermal conductivity, D (m2 s-1) is the coefficient of mass diffusion,
R is the universal gas constant (equal to 8.314 J mol-1 K-1) and cp (J kg-1 K-1) is the
specific heat capacity at constant pressure of the reactant.
For convenience, we introduce the dimensionless temperature and space and time
coordinates (e.g. (Sharples et al., 2012)):
u=
RT
,
E2
x' = x
ρQ2 Α 2 R
kE2
,
t'= t
Q2 A2 R
.
c p E2
(7)
The dimensionless system of equations with the primes omitted is thus obtained:
∂u ∂2 u
= 2 + Ce−1/u − qrCe− f /u
∂t ∂x
(8)
∂C 1 ∂2C
=
− βCe−1/u − β rCe− f /u
2
∂t Le ∂x
(9)
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where the new parameters q, f, r, Le and β are defined by:
q=
Q1
,
Q2
f=
E1
A
, r= 1,
E2
A2
Le =
k
,
ρcp D
β=
c p E2
RQ2
(10)
Following the parameterisation in (Sharples et al., 2012) we refer to q as the ratio of the
reaction enthalpies, f as the ratio of activation energies, r as the ratio of pre-exponential
rate constants, Le as the Lewis number describing the ratio of the thermal and molecular
diffusivities in the system and β as a parameter describing the energy balance of the
exothermic reaction within the system (known as the exothermicity parameter). The
Lewis number, Le, has two significant values: Le → +∞ for solid fuels and Le = 1 for
gaseous fuels.
The boundary conditions for the system are:
x → +∞ :
x → −∞ :
u = 0,
C = 1,
(11)
∂u
∂C
= 0,
= 0,
∂x
∂x
(12)
corresponding to a reaction front propagating in the positive x direction. On the right
boundary ( x → +∞ ), we have the cold ( u = 0 ) and unburnt state (C = 1). On the left
boundary ( x → −∞ ) corresponding to the burnt product mixture however, both the
concentration and temperature of the product mixture cannot be explicitly stated. We
know that there are no reactions occurring on the left boundary where the wave front
has already passed, thus setting the derivatives of u and C to zero for ( x → −∞ ).
NUMERICAL SCHEME
The partial differential equations (8) and (9) subject to the boundary conditions (11) and
(12) were solved numerically using FlexPDETM (http://www.pdesolutions.com), a
commercial finite-element package for obtaining numerical solutions to partial
differential equations in one, two and three dimensions for both steady state and time
dependent solutions. As a space and time adaptive finite element package, FlexPDETM
minimises errors to relative error tolerance levels (known as the variable ‘errlim’ within
the code) by estimating the relative uncertainty of the variables of the partial differential
equation integrals and comparing them to the error tolerance level. Where this tolerance
level has been reached or exceeded, the mesh size is refined and the solution is reiterated till the estimated errors in the variables are less than the set tolerance level. All
solutions to the partial differential equations in this paper were obtained using this
package, with most of the error tolerance limits set to errlim = 0.001.
For the numerical integration of the governing partial differential equations, the initial
condition for the reactant mass fraction was taken to be C = 1, while the initial
temperature was taken in the form of a Gaussian pulse, simulating a hot wire ignition at
the starting end of the domain (Sharples et al., 2012). This is as seen in Fig. 1 showing
the initial condition for the temperature for 0 ≤ x ≤1000 . This initial condition was
applied throughout all solutions found in this paper. The integration domain was set as
0 ≤ x ≤ 5×10 4 in order to accommodate the travelling wave profile and trace the
dynamics over a time interval of the order 106. As in previous studies (Sharples et al.,
2012, Gubernov et al., 2012), we will assume q = r = 1 throughout this paper, focusing
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on the behaviour of system as Le and β is varied. The front speed was determined by
tracing the progression of the point of half-height of the reactant mass fraction profile.
0.12
0.1
u
0.08
0.06
0.04
0.02
0
200
400
600
800
1000
x
Fig. 1: Initial conditions for temperature for the domain 0 ≤ x ≤1000 showing the
Gaussian pulse ‘ignition’ for the initial temperature profile. The temperature profile
remains flat for x ≥1000 .
RESULTS
10
9
8
B
Lewis Number
7
C
6
5
4
A
3
Point 'H'
2
1
0
5
6
7
8
9
10
11
12
13
14
Exothermicity Parameter β
Fig. 2: Stability diagram showing Hopf (green) and extinction (blue) loci for f = 1.5.
The triangles and blue curve denote the values of β, beyond which only the trivial
solution was obtained by FlexPDETM. The circles and green curve show the Hopf point
values using FlexPDETM, with the parameter region between the green and blue curves
generating pulsating fronts in the reaction scheme.
Fig.2 illustrates the Hopf and extinction loci for f = 1.5 for 1 ≤ Le ≤10 , showing the
different schematic behaviour for differing regions of parameter space. The green curve
and the blue curve demarcate the Hopf and ‘extinction’ loci respectively. The Hopf
locus is the series of points beyond which pulsations in the wave front occur, while the
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extinction locus is the series of points beyond which non-propagation or extinction of
the wave occur. In our numerical integrations, we obtained a stable regime of
propagating waves with unique speeds in region A, a steady pulsating regime with
oscillatory wave speeds in region B and the effective cessation of a propagating wave in
region C which we refer to as ‘extinction’.
At a value of Lewis number less than that of Point ‘H’, for instance at Le = 2 and β = 7
(region A), a stable and constant wave speed v = 0.0152 was obtained, with the
temperature and reactant mass fraction profile represented in Fig.3a. As the value of β
was increased and approached the extinction locus (found to be βextinct = 10.89), the
temperature and reactant mass fraction profile as shown in Fig.3b were notably less
sharp and also possessed a lower burnt temperature as compared to the previous case
corresponding to β = 7.
(a)
(b)
1
0.8
0.8
0.6
0.6
u,C
u,C
1
0.4
0.4
0.2
0.2
0
0
0
0.5
1
1.5
2
2.5
x
3
3.5
4
4.5
5
4
x 10
0
0.5
1
1.5
2
2.5
x
3
3.5
4
4.5
5
4
x 10
Fig. 3: Temperature and reactant mass fraction profile of the propagating reaction front
arising in the competitive endothermic-exothermic reaction scheme for (a) f = 1.5, Le =
2 and β = 7 and (b) f = 1.5, Le = 2 and β = 10.88. The red curve shows the profile of the
dimensionless temperature u, while the blue curve shows the profile of the reactant
mass fraction C.
As β was increased beyond the extinction locus, i.e. β ≥ βextinct where βextinct =10.89 for
Le = 2, the temperature and reactant mass fraction profiles separated from each other
after several pulsations and tended to the equilibrium values u = 0 and C = 1. The
FlexPDETM package was not able to distinguish the propagating reaction (with low
average speed) front from the trivial solutions of the system. We will refer to this
phenomenon as a kind of numerical ‘extinction’.
It is worth noting, however, that from a mathematical perspective, the ‘extinction’ does
not imply the non-existence of non-trivial solutions to the initial boundary value
problem (8) and (9); a stable travelling wave solution will still exist, albeit with a very
low average speed. From a practical point of view, however, such solutions can
effectively be treated as an extinguished flame front.
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Fig. 4: Schematic of bifurcation diagram for f = 1.5, q = r = 1, Le = 6 showing
approximate wave speeds. Extinction of the wave front occurs for β ≥ 9.142 .
(a)
(b)
0.05
0.028
0.045
0.026
Dimensionless Wave Speed, ν
Dimensionless Wave Speed, ν
0.04
0.024
0.022
0.02
0.018
0.016
0.014
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
0.012
3.8
3.9
4
4.1
4.2
4.3
Dimensionless time, t
4.4
4.5
3.3
3.4
3.5
5
x 10
3.6
3.7
3.8
3.9
Dimensionless time, t
4
4.1
6
x 10
Fig. 5: Wave speed derived using FlexPDETM for f = 1. 5, r = q = 1, Le = 8, for
(a) β = 7.4 and (b) β = 8.98
Where the Lewis number is greater than that of Point ’H’, however, a different
behaviour was observed as the bifurcation parameter β varied. As β was increased past
the Hopf locus, the solution was observed to lose stability, manifesting as pulsations and
local maxima in the temperature profile as well as cyclical oscillations of the wave
speed. As β was increased beyond β = 7.23 (Hopf point) for Le = 8, the travelling
combustion wave fronts were found to possess oscillatory behaviour, characterised by a
constant amplitude and period in the wave speed profile. This is as seen in Fig. 5a,
showing the pulsating wave speed profile for β =7.4. Increasing the value of β further
within region B (found to be β ≥ 8.4), we detected a transition to period-2 type
oscillatory behaviour, illustrated in Fig. 5b showing β = 8.98. A bifurcation diagram of
the wave speed is provided in Fig. 4, denoting the thresholds of the steady speed,
period-1 and period-2 regimes before extinction occurred. Unlike (Gubernov et al.,
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2010), however, we were not able to detect a regime of chaotic flame propagation
before numerical ‘extinction’ occurred; the system exhibited ‘extinction’ at β ≥ 8.99,
beyond which only the trivial solution was found via the numerical integration. The
precise nature of the unstable front propagation beyond such values remains open to
further inquiry.
Comparing the stability diagrams for f = 2 in black (Sharples et al., 2012), f = 1.75 in
red and f = 1.5 in green in Fig. 6, we note small changes in the Hopf loci as f was varied
from f = 2 to f = 1.5. This implies that the onset of pulsating reaction fronts as β is
varied does not change noticeably in the range of f considered. However, it was found
that the extinction loci shifted, albeit slightly to the left as f, the ratio of the activation
energies was decreased. As denoted in (10), the smaller the value of the exothermicity
parameter β, the greater the influence of the exothermic reaction. In addition, the
smaller the value of f, the greater the contribution by the endothermic reaction in the
model. Thus, such a leftward shift of the extinction loci is not unexpected, since
‘extinction’ would be expected for smaller β values where the endothermic reaction is
more dominant within the system.
Fig. 6: Stability diagram showing a zoom-in of Hopf and extinction loci for f = 2 (black
with triangular markers), f = 1.75 (red with square markers) and f = 1.5 (green with
circular markers).
CONCLUSIONS
We have investigated the behaviour of the travelling combustion wave solutions arising
in a competitive endothermic-exothermic reaction scheme under adiabatic conditions
and in one spatial dimension. The reaction-diffusion model was analysed numerically
for f = 1.75, q = r = 1 and f = 1.5, q = r = 1 using the finite-element package FlexPDETM
for the numerical integration of the governing partial differential equations. The analysis
demonstrated the existence of Hopf bifurcations and ‘extinction’ values not dissimilar to
those found by (Sharples et al., 2012).
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Values of β above the Hopf point corresponded to oscillatory unstable solutions, with
the solutions evolving to period-2 type behaviour as β was increased. No further period
doubling was observed before the numerical solutions devolved to the trivial solutions.
Whilst previous restrictions (Sharples et al., 2012, Hmaidi et al., 2010, Gubernov et al.,
2012) on the ratio of activation energies have been relaxed, thereby permitting a greater
contribution from the endothermic reaction, there are regions of the system’s parameter
space that remain unexplored. In particular, the stability properties of the competitive
system over a wider parameter range will be analysed more comprehensively in future
work.
The present study has only considered the effects of chemical kinetics on the
characteristics of the resulting flame propagation. Other effects, such as heat loss and
hydrodynamic interactions have been ignored in this simplified analysis. However, the
results obtained pose some interesting questions about the influence that pulsating
reaction fronts arising, for example, under two-stage kinetic schemes might have on the
dynamics associated with flame acceleration and transitions from deflagration to
detonation (Kessler et al., 2010). Such questions are particularly pertinent given the
relevance of the competitive endothermic-exothermic reaction scheme to ammonium
nitrate combustion. These questions will be considered in our ongoing research.
ACKNOWLEDGEMENTS
This work was supported by the Australian Research Council Grant No. DP 0878146.
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Appendix 1 – Nomenclature
Term
Definition
Units
x
Space coordinate
m
t
Time coordinate
s
ρ
Density
kg m-3
T
Temperature
K
R
Universal gas constant
J mol-1 K-1
C
Reactant mass fraction
-
D
Diffusivity
m2 s-1
k
Thermal conductivity
J s-1 m-1 K-1
cp
Specific heat capacity at constant pressure
J kg-1 K-1
E1
Activation energy (endothermic)
J mol-1
E2
Activation energy (exothermic)
J mol-1
A1
Pre-exponential factor (endothermic)
s-1
A2
Pre-exponential factor (exothermic)
s-1
Q1
Heat absorbed (endothermic)
J kg-1
Q2
Heat released (exothermic)
J kg-1
x’
Space coordinate (dimensionless)
-
t’
Time coordinate (dimensionless)
-
u
Temperature (dimensionless)
-
q
Q1/Q2
-
f
E1/E2
-
r
A1/A2
-
Le
Lewis Number: the ratio of thermal conductivity
and molecular diffusivities
-
β
Exothermicity Parameter
-
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REFERENCES
ANTAL, M. J. & VARHEGYI, G. 1995. CELLULOSE PYROLYSIS KINETICS THE CURRENT STATE KNOWLEDGE. Industrial & Engineering Chemistry
Research, 34, 703-717.
BALL, R., MCINTOSH, A. C. & BRINDLEY, J. 1999. Thermokinetic models for
simultaneous reactions: a comparative study. Combustion Theory and
Modelling, 3, 447-468.
CHAN, S. K. & TURCOTTE, R. 2009. Onset Temperatures in Hot Wire Ignition of
AN-Based Emulsions. Propellants Explosives Pyrotechnics, 34, 41-49.
CLAVIN, P., FIFE, P. & NICOLAENKO, B. 1987. MULTIPLICITY AND RELATED
PHENOMENA IN COMPETING REACTION FLAMES. Siam Journal on
Applied Mathematics, 47, 296-331.
GUBERNOV, V., KOLOBOV, A., POLEZHAEV, A., SIDHU, H. & MERCER, G.
2010. Period doubling and chaotic transient in a model of chain-branching
combustion wave propagation. Proceedings of the Royal Society a-Mathematical
Physical and Engineering Sciences, 466, 2747-2769.
GUBERNOV, V., SHARPLES, J., SIDHU, H., MCINTOSH, A. & BRINDLEY, J.
2012. Properties of combustion waves in the model with competitive exo- and
endothermic reactions. Journal of Mathematical Chemistry, 1-11.
HMAIDI, A., MCINTOSH, A. C. & BRINDLEY, J. 2010. A mathematical model of
hotspot condensed phase ignition in the presence of a competitive endothermic
reaction. Combustion Theory and Modelling, 14, 893-920.
KESSLER, D. A., GAMEZO, V. N. & ORAN, E. S. 2010. Simulations of flame
acceleration and deflagration-to-detonation transitions in methane-air systems.
Combustion and Flame, 157, 2063-2077.
LIU, F., MCELWAIN, D. L. S. & PLEASE, C. P. 1998. Simulation of combustion
waves for two-stage reactions.
MAKINO, A. 2001. Fundamental aspects of the heterogeneous flame in the selfpropagating high-temperature synthesis (SHS) process. Progress in Energy and
Combustion Science, 27, 1-74.
MERZHANOV, A. G. & RUMANOV, E. N. 1999. Physics of reaction waves. Reviews
of Modern Physics, 71, 1173-1211.
PLEASE, C. P., LIU, F. & MCELWAIN, D. L. S. 2003. Condensed phase combustion
travelling waves with sequential exothermic or endothermic reactions.
Combustion Theory and Modelling, 7, 129-143.
SHARPLES, J. J., SIDHU, H. S., MCINTOSH, A. C., BRINDLEY, J. & GUBERNOV,
V. V. 2012. Analysis of combustion waves arising in the presence of a
competitive endothermic reaction. Ima Journal of Applied Mathematics, 77, 1831.
SINDITSKII, V. P., EGORSHEV, V. Y., LEVSHENKOV, A. I. & SERUSHKIN, V. V.
2005. Ammonium nitrate: Combustion mechanism and the role of additives.
Propellants Explosives Pyrotechnics, 30, 269-280.
SUBRAHMANYAM, J. & VIJAYAKUMAR, M. 1992. SELF-PROPAGATING
HIGH-TEMPERATURE SYNTHESIS. Journal of Materials Science, 27, 62496273.
TURCOTTE, R., GOLDTHORP, S., BADEEN, C. M. & CHAN, S. K. 2008. Hot-Wire
Ignition of AN-Based Emulsions. Propellants Explosives Pyrotechnics, 33, 472481.
WEBER, R. O., MERCER, G. N., SIDHU, H. S. & GRAY, B. F. 1997. Combustion
waves for gases (Le=1) and solids (L->infinity). Proceedings of the Royal
11
Wee et al
Society of London Series a-Mathematical Physical and Engineering Sciences,
453, 1105-1118.
12