Full Paper

DOI: 10.1002/prep.201400023

Modeling Ignition and Thermal Wave Progression in Binary

Granular Pyrotechnic Compositions
Sebastian Knapp,*[a] Volker Weiser,[a] Stefan Kelzenberg,[a] and Norbert Eisenreich[a]
Dedicated to Beat Berger (1949–2014)

Abstract: Oxidizer and fuel particles are the ingredients of dizer and fuel particles and their size for various concentra-
classical pyrotechnics. Particle concentration, size, melting, tions influence the burning rate beneath the reaction kinet-
evaporation, and decomposition of the particles, heat and ic parameters. The computational results were compared
mass transfer, reaction kinetics, and heat of reaction control with experimental progression rates and temperatures
the burning behavior of these mixtures. A hot-spot ap- measured for an example system composed of various Al/
proach models the reaction progress in three dimensions CuO-thermite mixtures with aluminum contents from 8 %
taking into consideration the particulate nature of pyro- to 70 %. The particle sizes were fixed to micrometer-scale.
technic compositions. The governing reaction is assumed The curve of progression rate calculations depending on
to be the oxidizer decomposition described by an Avrami- the aluminium particle concentration and their distribution
Erofeev model. Predominantly, the distribution of the oxi- show the same shape as the experimental results.
Keywords: Thermite reaction

1 Introduction

Typical representatives of binary particle reactions are ther- ments with pure oxygen [7]. Important fuels are aluminum
mite reactions. They were first described in 1898 by H. and magnesium.
Goldschmidt as a “carbon-free” method to extract metals Currently, numerous investigations on thermite reactions
by reducing their oxides with aluminum [1]. Still this type are done using different metals and metal oxides [8] study-
of reaction is of widespread interest to rapidly produce ing burning rates [9], influence of particle size [10] and
large amounts of heat at high temperature levels and has pressure [11, 12]. However, the physico-chemical mecha-
found many applications [2]. So the thermite welding pro- nisms of thermite type reactions are still far from being
cess is still the most frequently used method for welding of completely understood. Therefore, the modeling is also of
railroad tracks [3], it is also used to purify ores of some particular interest [13] and there are some preliminary ap-
metals. The preparation of ceramics is also a well-known proaches for it [14], which describe particle ignition and
domain for thermite reactions [4]. Due to the large heat re- propagation of reaction fronts in porous energetic materi-
lease and its self-sustaining nature, thermites have been als [15–17]. In these studies, we only used the heat flow
used in warheads as incendiary devices [5]. As a big ad- equation with a zero order reaction and obtained the result
vantage most microscopic thermite compositions are not by application of the Green‘s function mirrored at the
very sensitive towards friction, impact and shock. boundaries. In principle, any reaction mechanism might be
In general, thermite reactions can be described as exo- included. We baptized this approach Hot-Spot modeling. It
thermic chemical reactions, where a metal-oxide (M2x2Ox3) is is very suitable for granular systems since it contains a parti-
reduced by a less noble metal (M1) to form the more stable cle-like character by itself. This simplified model – we will
metal-oxide (M1x4Ox5) and resulting metal (M2). The overall discuss the model and its simplifications below – only re-
reaction equation is given as [6]: quires reduced numerical expense and computational time.
The investigations with the Hot-Spot model were not only
performed to learn more about the chemical behavior but
also about the influence of physical properties. Especially

[a] S. Knapp, V. Weiser, S. Kelzenberg, N. Eisenreich

Couples of M1 and M2-oxides can be chosen on the basis Fraunhofer Institut fr Chemische Technologie ICT
of their position in the electrochemical series or their Gibbs 76327 Pfinztal, Germany
energy from the Ellingham diagram of the oxidation of ele- *e-mail: sebastian.knapp@ict.fraunhofer.de

Propellants Explos. Pyrotech. 2014, 39, 423 – 433  2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim 423
Full Paper S. Knapp, V. Weiser, S. Kelzenberg, N. Eisenreich

the influence of fuel and oxidizer densities at same particle reality very fast at the conditions of the thermite reac-
size and thereby the number of fuel and oxidizer particles tion), with kinetics governed by the kinetic constants
will be discussed. In this way the particle distribution and of the thermal decomposition of the oxidizer.
their arrangement dependent on the fuel concentration is (vi) The heat generated by the reaction is distributed ac-
of special interest. cording to the heat flow equation.
For a comparison of the modeling results with a real ex- (vii) The transport coefficients are assumed to be inde-
ample system a series of aluminum/copper(II)oxide ther- pendent of temperatures and species concentrations.
mite formulations was studied experimentally. The temper-
ature and propagation rates at different fuel/oxide concen- The main physical processes in the combustion of con-
tration ratios were determined including the necessary densed energetic materials like pyrotechnic mixtures are
input parameters for the model. heat and mass transfer. This means heat and species are
generated and consumed by various processes at the com-
bustion front and distributed in the material and the burn-
out zone. Transferring the hot spot model to the descrip-
2 Theory: The Hot-Spot-Model for Two-
tion of heterogeneous reactions the heat flow equation has
component Pyrotechnic Materials
to be solved simultaneously with the mass transfer equa-
Modeling of the ignition and combustion of an energetic tion. As a consequence, to model such processes, the relat-
material composed of fuel and oxidizer particles is difficult ed partial differential equations have to be solved in three
to achieve. In the case of a thermite reaction it is still not dimensions for temperature and at least three species ci :
sufficiently explored how the reaction starts and proceeds.
Depending on the melting point of the considered fuel and
oxidizer and the decomposition temperature of the oxide it
may happen that one or both ingredients are molten
before ignition. As first chemical reaction step it can be as-
sumed that oxygen is released during the thermal decom-
position of the involved oxide. The released oxygen would
diffuse outwards and react immediately with the fuel. The
thermal decomposition is being considered the rate-deter-
mining step. This assumption would then nicely explain the where 1 is the density, cp the specific heat capacity, l the
high, thermal stability, and the high ignition energy re- heat conductivity, Di the diffusion coefficient for the i-th
!
quired to set off thermites. This model assumption is not in species and ci[ x ,t] the concentration of component i. The
contrast to the widely accepted picture of the aluminum equations describe the propagation of the scalar tempera-
oxidation starting with the melting of aluminum and frac- ture field and species fields in space and time in dependen-
turing of the alumina shell. These processes can be neglect- cy of the physical properties of the material and of the
ed because they take place before the decomposition of source terms on the right hand side. The source terms
the oxide and release of oxygen. The oxidation reaction comprised the chemical reactions, the related heat and
cannot start until both components oxygen and fuel are species generation and consumption.
available. Therefore the decomposition of the oxide is the In the case of the combustion of pyrotechnic mixtures,
rate-determining step and used here as the starting point. the chemical reaction is exothermic and consequently
The model described herein for the reaction of thermites a heat release occurs. The source term of the energy re-
induces substantial simplifications with respect to the “real” lease by the chemical reaction is given for a general reac-
process and uses the following assumptions: tion scheme by:

(i) Oxidizer and fuel are present in separated particles of

defined sizes distributed randomly or regularly accord-
ing to predefined stoichiometry. On reaction they
form a single homogeneous condensed material.
(ii) The oxidizer particles decompose to release oxygen where qi are the heats of reaction. The rates of reaction are
due to an initial local heat pulse. The decomposition given by:
of the oxidizer is the rate-determining step.
(iii) The oxygen is distributed by diffusion (diffusion equa-
tion) to the metal component to enable a contact for
the reaction.
(iv) Phase transitions are neglected.
(v) On contact fuel and oxidizer react to the final product with the rate constant ki,j is given by the Arrhenius Equa-
in an overall second order reaction (this reaction is in tion:

424 www.pep.wiley-vch.de  2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim Propellants Explos. Pyrotech. 2014, 39, 423 – 433
Ignition and Thermal Wave Progression in Binary Granular Pyrotechnic Compositions

possible. Therefore, only a numerical method can be ap-

plied. In order to avoid the current cumbersome methods
of numerically solving differential equations the method of
Here, Zi,j is the pre-exponential factor, R the universal gas Green‘s function is used where a numerical integration is
constant, T the temperature, and EA the activation energy. performed which is a faster and a more stable process and
The Arrhenius Equation gives the dependence of the rate enables to involve complicated geometries generated by
constant ki,j of the chemical reaction on the temperature T the reaction of the individual particles. If the appropriate
and the activation energy EA. Green’s function for the homogeneous problem is known,
The function f[ci,cj] in the equation above depends on it only has to be convolved with the source term of the dif-
the reaction scheme to be modeled. In the case of a chemi- ferential equation in case of infinite space. The heat source
cal reaction of a fuel with an oxidizer generating the heat and chemical reaction are calculated by simultaneously
of reaction qreac like A + B!C (i = A, B, C) an overall second solving the related differential equation at the selected
order reaction is assumed. The heat output of the chemical time steps Dt.
reaction and therefore the source term for the chemical re-
action energy release in the equation of heat transfer is
given by:

The Green’s function in infinite space for the differential

equation above in three dimensions is a Gaussian-like func-
tion (k = l/cp1) [18].

The solution of the equation of heat transfer is than

given by:
To start the combustion process of a pyrotechnic mixture
it must be ignited. In the heat transfer equation a second
source term on the right hand side must provide an energy
input at a defined point in time and space. This is called
a Hot-Spot. For example it can be described by a 3D Gaus-
sian function at position (x0, y0, z0) multiplied by a Dirac dis- The profiles of oxidizer, fuel, and reaction product pro-
tribution in time, which ensures that it occurs only at a de- ceed in an analogue way:
fined time t0 :

In Equation (15) the space integral reaches over the infin-

Including the initiating hot spot at t0 and the heat of re- ite space. According to the boundary conditions of the lim-
action, the equation of heat transfer results in: ited dimensions of a real sample the energy will be reflect-
ed at the boundaries of the sample (mirror sources) and
the integral reaches only over the sample dimensions. In
case of fast reactions and steep profiles of temperature and
species, the width of GU is small compared with the size of
the included space. This means, in a numerical solution of
Equation (15) the integration has to include only a small
section of the total space neighboring the maximum of GU
(GU ¼
6 0). On the other hand this prevents conservation of
! mass and energy. But the error can be neglected for a limit-
Without a chemical reaction &Qdot;reac[ x 0 ,t] the problem ed number of time steps. Without interdiffusion both com-
can be solved analytically. Therefore several analytical and ponents fuel and oxidizer at all positions x would develop
numerical methods are known [18]. However the non-line- according to the solution of Equation (5) for the proposed
arity of the Arrhenius term makes an analytical solution im- case of a 2nd order reaction:

Propellants Explos. Pyrotech. 2014, 39, 423 – 433  2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim www.pep.wiley-vch.de 425
Full Paper S. Knapp, V. Weiser, S. Kelzenberg, N. Eisenreich

The integral via dt’ results in the integrand itself because

of the Delta function. The numerical procedure e.g. 3D
Gaussian quadrature performs the spatial integral on a 3D
lattice [19]. In case of steep gradients and short time inter-
For the numerical solution of the integral Equation (15) vals and correlated squares of the lattice node distances
above, the algorithm mainly consists of three steps. The Dx2 the Green’s function is of small width and strongly re-
first step generates initial temperature and species profiles duces computing time because of reducing the integral in-
resulting from the n initially given hot spots and the parti- tervals to that sections where the Green’s function GU ¼ 6 0.
cles of both types A and B also approximated by 3D Gaus- T3 !T1 and c3.i !c1,i close the loop to start now iteration
sians: with a further step according to Equation (22) by including
contribution of the chemical reaction terms.
Steps 2 and 3 are repeated iteratively. Step 1 can be in-
cluded as often as new hot spots occur from an external
heat source, which is not the case here. Figure 1 shows
a schematic view of the modeling steps.

For further calculations, one Gaussian hot spot initiates

the thermite; however, n fuel and oxidizer particles are dis-
tributed according to a plan representing the experimental
configuration. The initial condition for the reaction product
is c1,C = 0.

In the second step the progress of the chemical reaction

(17), (18), and (19) for a small time step Dt is calculated
generating: Figure 1. Schematic view of the model steps.

The calculations result in three-dimensional temperature

and: and concentration profiles for each time step, and in heat
output and position of the reaction front over time, plotted
are mainly two-dimensional cross sections.
Summing up all lattice nodes after calibration where con-
version is completed (e.g. for stoichiometric conditions
!
c3,A,B[ x ,t] ffi 0 or half of the conversion, maximum tempera-
ture reached) defines a conversion rate, which in linear pro-
gression can be considered as burning rate.
In pyrotechnic mixtures fuel and oxidizer particles are
These temperature and species profiles are assumed to
the sources of reacting fuel and oxidizer. Including diffusion
be instantaneously inserted, therefore being multiplied
by the same way like the heat transfer equation the parti-
each by the Dirac Delta function d(t). The third step is to
cles can react, as soon as gaseous fuel and oxidizer get into
calculate the solution of the heat transfer equation. It calcu-
contact. The model does not include any convection or ra-
lates heat and mass diffusion for the same time step Dt by
diation.
convolution of the related profiles and the Green’s func-
In conclusion three types of parameters are necessary to
tions.
run the Hot-Spot model:

(i) Initial distributions, sizes and numbers of hot spots,

fuel and oxidizer particles.

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Ignition and Thermal Wave Progression in Binary Granular Pyrotechnic Compositions

(ii) Material parameters: density, heat capacity, heat con- bulk density of about 1 g cm3 was produced. Ignition was
ductivity, diffusion coefficient. performed using a melting wire enhanced with a pellet of
(iii) Reaction parameters: heat of reaction, Arrhenius-pa- pyrotechnic igniter compound.
rameters (pre-exponential factor, activation energy). The visible flame front was observed with a high-speed
video camera (Redlake Motion Pro X3) using 1000 fps and
a spatial picture resolution of 9 pixels mm1. From the re-
cords, the flame propagation was derived using a software
3 Experimental code called AVICOR developed at Fraunhofer ICT for analyz-
ing video samples of combustion processes [21]. Addition-
3.1 Samples
ally, a macro lens was available to produce movies of
Samples with different compositions of aluminium/copper(- 5000 fps with a limited field of view of 2  6 mm2.
II)oxide (Toyal Europe Alcan 400, 99.7 %min., aluminum
powder atomized; Alpha Aesar Copper(II)oxide ACS,
99.0 %min., powder CuO) were milled using an automatic
3.3 Temperature Measurement Setup
mortar. The stoichiometries of the compositions and the
particle diameters are summarized in Table 1. A wide range By using test tubes a problem occurs when condensed ma-
of stoichiometry was realized. The particle size was deter- terial covers the inner wall of the glass tube immediately
mined using a laser diffraction device (Malvern Instruments behind the reaction front. So the tube got opaque and
Mastersizer 2000) reaching from 2 mm to 30 mm for both only emission from the colder tube wall was detected by
types of particles with mean diameters (d50) of 6 mm (Al) the spectrometer. Therefore, temperatures were measured
and 9 mm (CuO). So the particles of the two components in free air on bulk material combustion. The bulk material
can be regarded to have the same order of size. Figure 2 was placed in a row of about 40 mm length and 10 mm
exhibits SEM images of characteristic particles (Zeiss-SEM width on a SCHOTT CERAN plate in air. The igniter was
Supra 55 VP, Carl Zeiss SMT AG, Germany). placed at one end and obscured in a way that it did not in-
terfere with optical measurements. For temperature mea-
surement NIR spectra from 1.0 mm to 2.2 mm were mea-
3.2 Burning Rate Measurement Setup
sured using an MCS 611 PGS-NIR 2.2 spectrometer from
The experiments were performed in a nitrogen atmosphere Carl Zeiss AG, Germany equipped with an optical fibre. The
in a chimney-type window bomb under constant pressure spectra were calibrated using a black body radiator in in-
of 0.1 MPa N2. To measure the progression rate 2 g of the tensity per unit wavelength. The fibre optics of the Zeiss
sample mixture were filled into a test tube (Rotilabo test spectrometers used in the experiments make a calibration
tube, beaded rim, borosilic glass 5.1, thin walled, 2 mL, in units of spectral radiance difficult. The spectra were ana-
70 mm long, inner diameter of 6 mm). By tapping, an initial lyzed with ICT-BaM code.

Table 1. Composition of investigated mixtures.

Material Concentration [ma%]
Al (Alcan 400) 5 8 13 19 24 30 40 50 55 63 70
CuO (Alfa Aesar 325mesh) 95 92 87 81 76 70 60 50 45 37 30
Mass equivalence ratio Ø 0.22 0.37 0.64 1 1.35 1.83 2.84 4.26 5.21 7.26 9.95
Superstoichiometric (fuel lean) Stoichiometric. Substoichiometric (fuel rich)

Figure 2. SEM pictures of aluminum (left) and copper(II)oxide (right) particles.

Propellants Explos. Pyrotech. 2014, 39, 423 – 433  2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim www.pep.wiley-vch.de 427
Full Paper S. Knapp, V. Weiser, S. Kelzenberg, N. Eisenreich

ICT-BaM does not require spectral radiance to derive ric analysis (TG) and differential scanning calorimetry (DSC)
temperatures from NIR because it models spectra of gas- were carried out simultaneously with a heating rate of
eous reaction products, soot and continuum radiation 10 K min1 (Netzsch STA 449C Jupiter, Netzsch GmbH & Co.
based on their band shapes. The evaluation procedure is KG, Selb, Germany) in synthetic air atmosphere.
based on a Least Squares Fit of calculated spectra to the
measured ones. It is described in more detail in Ref. [22].
For the recent application only the model of grey-body 4 Results
emitter was needed which shows an excellent agreement
4.1 Burning Rate
between experimental and calculated spectra to determine
temperature of the emitting particles. The analysis resulted The burning experiments with test tubes result in linear
in intensity, emission temperature and a signal proportional progression rates. Figure 3 shows screen shots of the reac-
to the emissivity with a temporal resolution up to 70 Hz. tion zone at different mixture ratios.
Resulting values for the different mixtures are plotted in
Figure 4 on logarithmic scale as a function of aluminum
3.4 Thermogravimetric Measurement Setup
content (open circles). The values correspond to the right
For the determination of the chemical reaction kinetic pa- axis of ordinates. Although the reproducibility of the single
rameters, a thermogravimetric measurement of copper par- measurements is weak, clear trends can be observed. It is
ticles with the same particle size as the copper(II)oxide par- well known, that the progression rate of particle mixtures
ticles in the burning experiment were performed. The depends not only on mixture ratio, particle size, tempera-
copper particles were used to study the oxidation process ture and pressure but also on the experimental conditions
and the following spit-off of oxygen. The thermogravimet- like orientation of the flame front, diameter of the speci-

Figure 3. Screen shots of the reaction zone at different mixture ratios.

Figure 4. Modeled (left axis) and measured (right axis) propagation rates of Al/CuO thermite in dependency of the fuel concentration.

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Ignition and Thermal Wave Progression in Binary Granular Pyrotechnic Compositions

men, bulk density etc. During the recent experiments all of maximum temperature, 2800 K, is close to stoichiometric
these parameters were held constant, so that the relations ratio of 19 ma% Al. But also from 16 % to 50 % tempera-
can be compared. tures higher than 2500 K are expected. Beyond this range
The progression velocity varies by more than one order adiabatic temperatures decrease rapidly. Between 28–
of magnitude from 50 mm s1 at fuel-lean conditions (< 50 ma% aluminum the combustion temperature is deter-
19 ma% Al) to 1000 mm s1 at about stoichiometric condi- mined by the dissociation temperature of Al2O3 and hence
tions and decreases to 10 mm s1 at fuel-rich conditions (@ yields a plateau. Beyond 50 ma% Al the exothermicity falls
30 ma% Al). The highest progression rate is found near stoi- and hence the temperature drops.
chiometric conditions in the substoichiometric field at The experimentally determined temperature values were
30 ma% Al. also charted in Figure 5 as open diamonds. The absolute
values are in a good agreement with the adiabatic flame
temperatures calculated with EKVI-Code for the fuel lean
4.2 Temperature of the Reaction Front
compositions. Especially the dependence on fuel concen-
tration corresponds well with these theoretical values.
In Figure 5 the triangles indicate the adiabatic flame tem- However at fuel rich conditions (> 30 % Al) measured
peratures calculated with EKVI-Code [20] at 0.1 MPa. The values are significantly higher than the theoretical ones. In
this range, the progression velocity and with it the reaction
rate is relative low so entrained air might also burn with
excess aluminum. In the fuel-lean area the effect of ambi-
ent air is negligible. However in the fuel-rich area the alu-
minum would also react with the air oxygen and the tem-
perature could be higher than the calculated adiabatic tem-
perature.

4.3 Thermogravimetric Results

The Arrhenius-Parameters were determined by thermogra-
vimetric analysis of copper in air. Under the assumption
that the decomposition of copper oxide determines the
speed of reaction the obtained data are used for parameter
determination. The TG, DTG, and DSC curves for copper at
a heating rate of b = 10 K min1 are shown in Figure 6
which contains already CuO and Cu2O. A two-step oxida-
Figure 5. Calculated adiabatic temperatures and measured tem- tion between 523 K and 650 K of copper leads to the
peratures in dependency of fuel (aluminum) concentration. copper oxide. At a temperature of 1280 K the copper oxide

Figure 6. Example of a thermal gravimetric measurement of copper in air.

Propellants Explos. Pyrotech. 2014, 39, 423 – 433  2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim www.pep.wiley-vch.de 429
Full Paper S. Knapp, V. Weiser, S. Kelzenberg, N. Eisenreich

begins to split-off oxygen till a temperature of 1340 K, ac- Table 2. Material parameters of fuel (aluminium) and oxidizer
cording to the reaction scheme: (copper oxide) particles found in literature [23].
Parameter Fuel [Al] Oxidizer [CuO]
3
Density 1 [g cm ] 2.7 6.48
2CuO ! Cu2 O þ O Heat capacity cp [J kg1 K1] 897 63.68
Heat conductivity l [W m1 K1] 237 27

5 Modelling Results
To start the calculations with the Hot-Spot model the nec-
essary input parameters must be defined. Therefore, first of And the number of oxidizer particles by:
all the number of fuel particles dependent on total number
of particles, fuel concentration, particles densities and parti-
cles size was calculated as follows:
where nTot is the total number of particles, nx the number
of particles, cx the concentration, 1x the density, and dx the
diameter of particle of type x (x = fuel or oxidizer). The total
number of particles was fixed to 196. Fuel and oxidizer
densities are from the literature (see Table 2). The particle
size was experimentally determined and in a first approxi-
mation a diameter of 10 mm was chosen for both types.

The particle distribution was calculated by two different

ways: a regular and a random method. The two different
distributions are shown in Figure 7.
Last but not least the reaction parameters are necessary
for the model. Due to the fact that the measured tempera-
tures in the fuel-lean area are in good agreement with the
adiabatic temperatures, the heat of reaction for each fuel
Figure 7. Particle distributions calculated for a fuel concentration concentration was determined from the adiabatic tempera-
of 24 mass-% and same particle size of fuel and oxidizer particles. tures of the EKVI-Code calculations (Figure 5).
At the left-hand side a regular distribution is shown and at the To determine the Arrhenius-Parameters, pre-exponential
right-hand side a random distribution. factor and activation energy, a 3rd order Avrami-Erofeev
mechanism was fitted to the experimental thermal gravi-

Figure 8. Simultaneous non-linear least-squares fit of a 3rd order Avrami-Erofeev mechanism to two normalized TG curves (symbols,
5 K min1 and 10 K min1) using “FindMinimum” of Mathmatica8 of WolframResearch.

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Ignition and Thermal Wave Progression in Binary Granular Pyrotechnic Compositions

metric data of the decomposition of the copper oxide. A

direct least-squares fit was applied (see [24]) to the sigmoid
curve normalized to 1 and 0 encircled in the diagram
Figure 8, using the inverse function a(T) in Equation (31):
Two thermogravimetric measurements of copper were
done and the curve of the oxygen split-off analyzed by
a non-linear least-squares fit (Figure 8) [24]. The results are
shown in Table 3 in comparison to literature values [25].
They are in good agreement with each other and the

Figure 9. Fuel, oxidizer, product concentration, and temperature profile calculated by the Hot-Spot model for a random particle distribu-
tion with 30 ma% initial fuel concentration. The space units are arbitrary. The concentration reach from 0 to 1 and the temperature from 0
to 4000 K.

Propellants Explos. Pyrotech. 2014, 39, 423 – 433  2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim www.pep.wiley-vch.de 431
Full Paper S. Knapp, V. Weiser, S. Kelzenberg, N. Eisenreich

Table 3. Obtained Arrhenius-Parameters by fitting thermal gravi- tions of the input parameters. For the kinetics model the
metric analysis data with a 3rd order Avrami-Erofeev model in parameters has been derived from TG/DSC measurements,
comparison with literature values [25]. which results in similar values as from the literature. Much
Data set Z [s1] Ea [K] more difficult is the situation for parameters like the heat
1 1014.75 51539 capacity, the heat conductivity and the diffusion coefficient.
2 1016.58 57494 In the model, they are taken temperature-independent,
Literature 1013.3 31994.23 (266 kJ mol1) which is problematic for a wide temperature range. The
first choice we did, aimed to get the model running. This
may explain the great absolute difference between the cal-
culated values for the propagation rates and the measured
values from dataset one were used for the Hot-Spot calcu- ones.
lations. The Hot-Spot model calculations were performed at
With these input parameters of Table 2 and Table 3 the a particle size ratio of fuel to oxidizer particle of 1 : 1. From
Hot-Spot model calculations were performed with regular that data, the propagation rates in dependency of the fuel
and random particle distributions for the same fuel concen- concentration were calculated and shown in Figure 4.
trations as the experiments were done. Fuel and oxidizer In comparison to the experimentally determined propa-
particle size were identical. An example output of gation rates the shapes of the functions are in good agree-
a random distribution with a fuel concentration of 30 ma% ment. The modeling results show as well as the experimen-
is shown in Figure 9. tal data the speed maximum to be in the same substoichio-
In Figure 9 in the first row the fuel concentration is metric range. Also the sharp increase in superstoichiometric
shown, below the oxidizer and product concentrations and range and the flatter slope in substoichiometric range are
in the bottom row the temperature profile. The columns in good agreement with the experimental results. From
show four time steps. In the first column the initial distribu- a fuel concentration of nearly 55 ma% the propagation
tion is shown. The fuel and oxidizer concentrations show rates have only a poor dependency on the fuel concentra-
the Gaussian-function shaped particles. The product con- tion. The sharp increase in the superstoichiometric range
centration is zero and the temperature profile shows the and the difference between the regular and random parti-
single Hot-Spot to ignite the particle mixture. At this geo- cle distribution show the strong dependency of the propa-
metrical point the fuel and oxidizer concentration shows gation rate in this range on the fuel and oxidizer particle ar-
ten time steps later (column two in Figure 9) decreasing rangement in space. The accordance between the regular
concentration and in contrast increasing product concen- and random particle distribution in the substoichiometric
tration and temperature profile. This continues at later time range shows that this dependency is substantially weaker
steps (column three and four) and the development of in that range.
product concentration and temperature is shown. In conclusion, it was shown that the Hot-Spot model
To determine the rate of combustion the output data of qualitatively mimics the dependency of the propagation
the Hot-Spot model were analyzed. Therefore, in the tem- rate on the fuel concentration. This suggests that heat and
perature profile the position of the moving temperature mass transfer are the main physical processes in granular
front was determined over time. With a linear least-square reactions, like pyrotechnic mixtures. It describes also ade-
fit the propagation rate was determined. This was done for quately that the effect of fuel and particle configuration is
all fuel concentrations. The resulting propagation rates are not negligible particularly in the superstoichiometric range.
shown in Figure 4. The black dots and line shows the prop-
agation rate of the regular particle distribution. The grey di-
amonds and line shows the propagation rate for the Symbols and Abbreviations
random particle distribution. Both lines correspond to the
left axis of ordinates. 1 Density [kg m3]
cp Specific heat capacity at constant pressure [J kg1 K1]
l Thermal conductivity [J s1 m1 K1]
ci Concentration of the i-th component [mol m3]
6 Discussion and Conclusions
Di Diffusion coefficient for the i-th component [m2 s1]
First of all the Hot-Spot model described herein is an at- qi Heat of reaction [J mol1]
tempt to investigate the influence of certain parameters on Zij Pre-exponential factor [s1]
the combustion process of granular systems. As it is work EA Activation energy [J mol1]
in progress it cannot be expected that all the results are in R Gas constant [J mol1 K1]
perfect agreement with the example system of Al/CuO- Q0 Energy of a Hot-Spot [J]
thermite. The first aim was to reproduce the course of the Q̇ Heat flow [J m3 s1]
propagation rates but not the absolute values. Because of T Temperature [K]
the strong simplifications it is difficult to get useful estima- kij Rate constant [s1]

432 www.pep.wiley-vch.de  2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim Propellants Explos. Pyrotech. 2014, 39, 423 – 433
Ignition and Thermal Wave Progression in Binary Granular Pyrotechnic Compositions

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ing Particle Size and Composition; EuroPyro 2007 (9ime Con-
The authors feel grateful to intensive and fruitful discussions with grs International de Pyrotechnie du GPTS) and the 34th Interna-
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