142 Propellants, Explosives, Pyrotechnics 27, 142 ą 149 (2002)
Burn Rate Models for Gun Propellants
Norbert Eisenreich*, Thomas S. Fischer, Gesa Langer, Stefan Kelzenberg, and Volker Weiser
Fraunhofer Institut f�r Chemische Technologie ICT, Joseph-von-Fraunhoferstr. 7, D-76327 Pfinztal (Germany)
Dedicated to Professor Dr. Hiltmar Schubert on the Occasion of his 75th Birthday
Summary these new propellants cannot be described correctly so that
an improvement of Vieille�s law or completely new ap-
In the past, Vieille�s law and minor modifications of it described
proaches have to be taken into account. In the case of solid
sufficiently the linear burning rate of gun propellants which
rocket propellants the modelling of the burning rate of
governs the design of charges by interior ballistic simulations.
composites(2,3) and modified double base formulations(4ą6)
Recent developments to increase the performance led to new gun
resulted also in new descriptions.
concepts and innovative propellants. These are the electrother-
mal-chemical gun, porous and foamed charges as well as This paper reports some modifications to Vieille�s law to
formulations exhibiting a temperature independent burning.
account for the temperature dependence of the burning rate
Vieille�s law cannot fully meet experimental results in these
or the combustion in porous propellants by a simplified
cases. Approaches based on the heat flow equation in the solid
analysis of the heat flow in the solid phase. In addition,
energetic material give simplified formulas to extend the validity.
These burning rate models have the ability to describe the results of applying more detailed reaction models, heat flow
experimentally determined burning behavior at least in a sim-
and diffusion in the gaseous phase are briefly outlined.
plified or qualitative way. More sophisticated methods consider
complex geometrical structures in the solid or take into account
the actual progress in phase behavior and reaction kinetics of
heterogeneous combustion. The dependence of the burning rate
on initial temperature, on phase transitions, porous structure and
2 Temperature Dependence of the Burning Rate
gaseous reactions can be described.
The transition of the condensed phase to the gaseous
phase dominates the combustion of solid energetic materi-
1 Introduction
als. This means that the cold solid is heated up to the
temperature of the burning surface caused by energy
The simulation of the interior ballistic behavior of gun
transfer from the flame. It possibly undergoes phase
propellants requires as a major input the linear burning rate
transitions e.g. to a liquid. The conversion to the gaseous
and its dependence on pressure. It is obtained by closed
phase can occur by endothermic evaporation, exothermic
vessel experiments or gun firings(1). The dependence of the
pyrolysis or heterogeneous reactions induced by some
burning rate on pressure is usually described by Vieille�s law,
�
unspecified energy sources (Q[x, t] from the flame) in the
Eq. (1), which is then incorporated into these ballistic
gaseous phase. These effects can be included in the heat flow
simulations. The parameters of this law are derived by
equation, Eq. (3), whereas diffusion of species can be
analyzing experimental pressure records according to the
neglected (the following outline is described in more detail
simplified relation of Eq. (2).
in Refs. 6 ą 7).
rpą � a pn 1ą
: X
@T @2T @ci
1 cp l � Q �x; t� � qi 3ą
dp
@t @x2 @t
i
/ Apąrpą2ą
dt
Ei;j
@ci X
RT
� Ai;j e f ci; cj 4ą
Vieille�s law describes the burning rate for a broad variety
@t
j
of solid propellant types with sufficient accuracy if minor
modifications are introduced. Recent developments of new
For an inert solid without phase transition energy the heat
classes of propellants concern temperature insensitive
flow equation can be solved by use of the Green�s function of
propellants, radiation absorbing formulations or compact
the heat flow equation(8)
charges with porous structures. The burning behavior of
x x0
ą2
4 t t0
ą
e
0 0
* Corresponding author; e-mail: [e-mail] GU x; x ; t; t � p������������������� 5ą
4pt t0ą
ą WILEY-VCH Verlag GmbH, 69469 Weinheim, Germany, 2002 0721-3113/02/2701-0142 $ 17.50+.50/0
Propellants, Explosives, Pyrotechnics 27, 142 ą 149 (2002) Burn Rate Models for Gun Propellants 143
which enables the construction of any solution for an inert (Note the sign convention: endotherm phase transition
�
solid (dci /dt � 0) for an arbitrary heat source Q[x, t]. L > 0 (Eq. (10), exothermal chemical reaction q > 0). They
A simplified formula for the ignition delay is derived show that conductive and radiative heat transfer influences
when calculating the time needed to increase the temper- the burning rate in the same way and that endothermic
ature of the propellant to an unspecified pyrolysis temper- phase transitions decrease and exothermic chemical reac-
ature Tp by the intensity of an external heat transfer I. tions increase it. Formulas like Eq. (11) were derived for
ablating surfaces when exposed to heat transfer(9,10) and
s�����������������������
discussed by Glick(11) and Ewing and Osborn(12) for rocket
t
T�0; t� � 2 I 6ą
propellants to describe the dependence on initial temper-
p l 1 cp
ature. Crow and Grimshow proposed a similar law for gun
2 propellants(13) although the parameters were assigned
p l 1 cp Tp
tign � apą 7ą different.
4 I2
Eqs. (7) and (11) enable to analyze the effects of physical
and chemical parameters of solid propellants on ignition
Eq. (7) indicates that the ignition delay depends on the
delay and linear burning rate. As an example: the higher
square of the pyrolysis temperature and on the inverse of the
pyrolysis temperature and the additional melting of HMX
square of the transferred energy intensity. Here, the second
or RDX which requires a latent heat L causes the strongly
term a(p) on the right hand side accounts for the initiation
increased ignition delays and the low burning rates of
and stabilization of gas phase reactions and depends there-
nitramine propellants when compared to double base
fore on pressure. It dominates only at high intensities I. The
propellants (L � 0) at low pressures.
high energy transfer induced by plasma ignition is the reason
In general the temperature sensitivity P is given by
for the short and reproducible ignition delay in this case.
dln[r]/dT1. Using Eq. 11 for the burning velocity this results
Also laser ignition data can be described by Eq. (7).
in:
Provided that a stable combustion has developed, Eq. (3)
can be transferred to the coordinate frame of the moving dr=
1
dT1
P � � 12ą
flame front where chemical reactions are taken into account.
L P qi
r�T1�
T1 �
cp i cp
@T @T @2T
1 cp r 1 cp l
Ignoring phase transition energies and chemical reactions,
@t @x @x2
Eq. (13) allows to obtain the unknown Q0 (I � 0) and Ts
: X
@ci @ci
when fitting it to r-values measured at various T1(6,15).
� Q �x; t� � qi r qi 8ą
@t @x
i
Qo � I
r � 13ą
After cancelling the time dependent terms:
1 c Ts T1ą
X
:
dT d2T dci
It was found that Eq. (13) represents the temperature
r 1 cp�T� � l � Q �x� � r qi 9ą
dx dx2 dx
i dependence of the linear burning rate of many solid
propellants very well. The fit parameter Ts is systematically
This equation can be integrated. A phase transition is
higher than the pyrolysis temperature obtained in thermal
described by a singular point in the specific heat cp:
analytical experiments (e.g. TG or DSC). Q0 turns out to
correspond to the pressure dependence of the burning rate
cp�T� � cp � L d�T TL� 10ą
reproducing the pressure exponent n(6,14). Eq. (13) can
therefore be reformulated to modify Vieille�s law for taking
Heating by radiation is taken into account: into account the radiative heat transfer and the dependence
on initial temperature:
:
Q �x� � b I e b x
a pn � I
r � 14ą
1 cp Ts T1ą
and using the boundary conditions:
T[0] � Ts T[ 1 ] � T1 Figure 1 shows the least squares fit of Eq. (13) to the
burning rates of the gun propellant JA2 at one temperature
(see details in Ref. (15)).
ldT/dx � Q0, (heat conduction from the flame) at x � 0
The temperature dependence of the burning rate is
and ldT/dx � 0 at x !1
influenced by phase transitions and chemical reactions in
the condensed phase as obvious from Eq. (11). Investiga-
ci[0] � 1 and ci[ 1 ] � 0; (complete conversion)
tions using the various phases of HMX confirmed the
influence of the latent heat of a structural phase transition
Qo � I
r � 11ą
on the burning rate(16) at low pressures, especially when
P
1 cp Ts T1ą � L qi
testing the phase d. In addition, the results of plasma pulse
i
144 N. Eisenreich, T. Fischer, G. Langer, C. Kelzenberg, and V. Weiser Propellants, Explosives, Pyrotechnics 27, 142 ą 149 (2002)
phase to inert evaporation. In the mesa range the burning
rate(6) is given by a combination of Eqs. (11) and (15):
s��������������������������
Ts
R
E
R T
l Z e dT
Qo � I T0
r � � 17ą
1 cp Ts T1ą � L Qo � I
r depending on Q0 � I exhibiting a minimum which is given
by Eq. (15).
4 Porous Propellants
Figure 1. Least squares fit of Q0 versus p curve; Q0 derived from
The burning behavior of porous and foamed propellants
analysing JA2 closed vessel test results using Eq. (13).
deviates also from Vieille�s law. Current predictions of
interior ballistics simulations fail when based on a straight
forward use of it. The mass conversion rates lie essentially
exposure of JA2 with intensities proportional to I can be above those obtained by the linear burning of compact
described by Eq. (11) as shown in Ref. (44). materials. Some theoretical approaches assume hot gases of
the flame to penetrate the porous solid according to Darcy�s
law, its velocity being proportional to the pressure gradient
3 Reactions in the Condensed Phase and the permeability of the material(17ą19).
rp � kd vhg 18ą
If the chemical reactions are incomplete in the condensed
phase Sqi is unknown and a numerical solution has to be
found for the burning rate. The gases generate hot spots in the propellant pores which
A limiting situation is encountered if the energy transfer evolve to (quasi) spherical burning zones. This leads to
from the gaseous phase can be neglected. Then, the temper- increased burning surfaces and in consequence to a higher
ature gradient is small in the zone where the chemical mass conversion rate. This porous burning occurs when the
reactions take place and the highest temperatures are stand-off distance of the flame which depends on pressure
reached. The burning rate does not depend on the heat (see e.g. Refs. (15, 20, 21)) is lower than the pore radius. The
release of the reaction but on chemical kinetics(6,11,12). For a total burning surface of the propellant consists of the sum of
0th order reaction the Zeldovich formula can be derived all expanding (by the burning rate) pore surfaces within the
(details also for nth order reactions see Ref. (6)) if dT/dx � 0 penetration depth where a flame could develop. The mass
is assumed: conversion or the pressure is given by:
v���������������������������������������������������
X
u
dp dm
Ts
u R
E / � 4 p 1 x2 �t� r 19ą
i;j;k
u
R T i;j;k
2 l Z e dT
dt dt
u
T1
t
r � 15ą
cp 1 Ts T1 � L cp
When the pores unite, small propellant residues still burn
resulting in a degressive burning behavior at the end of the
The dependence on T1 of the integral in Eq. (15) can conversion. The behavior of the pores is illustrated by Fig. 2.
usually be ignored because the Arrhenius term is 0 A surface increase can be up to a factor of 20.
compared to it at Ts. The temperature sensitivity is now The implementation of these effects to interior ballistic
given by: calculations at Fraunhofer ICT is obtained by two different
models: A phenomenological model using the concept of
1
cellular automates and a hot spot model using a simplified
P � 16ą
2 Ts � L cp numerical solution of the heat flow equation.
The phenomenological model(22) enables to apply the
Such a behavior was first found for modified double base linear burning rate to the enlarging pores of burning
rocket propellants at low pressures in the super rate burning energetic materials. Two-dimensional form functions are
regime. Some modified double base propellants even exhibit obtained by a formal procedure. In addition, on the basis of
a mesa effect which means that the burning rate decreases the Noble-Abel-equation the adiabatic pressure rise in a
with increasing pressure for some pressure range. Within this closed vessel is simulated. Closed vessel tests with non-
pressure interval the temperature sensitivity is low due to a porous and porous propellants were used to modify the
change of the mechanisms from reactions in the condensed model parameters.
Propellants, Explosives, Pyrotechnics 27, 142 ą 149 (2002) Burn Rate Models for Gun Propellants 145
Figure 5. Influence of the interaction penetration depth on the
pressure rise.
Figure 2. Conversion of multiple regularily set pores if the
Explicit consideration of the internal structure of the
flames can stabilise in their initial size. An increase of the burning
surface of a factor 8 is obtained here. porous charges enables the qualitative description of the
burning phenomena found in experiments(23,24). These
comprise changes of density, formulation, pore size and
pore distribution but also influences of changed experimen-
tal parameters like loading density.
Figure 3 presents the calculated influence of the internal
structure changes on the pressure rise in a closed vessel. In
this example the total pore volume respectively the overall
density of the porous energetic material was constant (60%
of the theoretical density). The propellants had a cube
geometry with 1 cm3 volume and the porosity was generated
by pores with identical diameters. The calculations were
made for three different pore sizes and a nonporous charge,
all with loading density 0.2 g/cm3.
In Figure 4 the theoretical calculations of the linear
burning rate are presented in dependence on the loading
density for porous and nonporous charges. In experiments
Figure 3. Influence of the pore size (D � diameter) on the the same behavior is found.
pressure rise.
A factor that makes quantitative predictions complicate is
the strong influence of penetration depth of interaction on
the pressure rise (see Fig. 5).
Another approach to describe porous systems uses a
simplified model of the heat flow equation. A three-
dimensional calculation describes the conversion of the
solid based on overall chemical kinetics and heat of
reaction(25ą28).
The hot spots are approximated by a sum of Gaussian
curves; the temperature development is given by
2
~ ~
x xi;j;k
ą
I;J;K 4 t ti;j;k
ą
X
Qi;j;k e
Ths~ tą � ti;j;k > t 20ą
x;
3=2
4 p t ti;j;k
i;j;k�1
A propagating hot gas flow with speed vhs which could
depend on the conversion initiates hot spots at time tn, j,k
Figure 4. Influence of the loading density on the linear burning
rate. possibly including a response time tR (� 0, here):
146 N. Eisenreich, T. Fischer, G. Langer, C. Kelzenberg, and V. Weiser Propellants, Explosives, Pyrotechnics 27, 142 ą 149 (2002)
Figure 6. Temperature distribution of a porous reaction propagating in z-direction.
Figure 7. The conversion of a porous propellant block and the conversion rate for a different speed to set hot spots and propagation in
one direction and continuous increasing speed:
a: hot spots close together
b: hot spots set with moderate speed and larger distances, a nearly constant conversion rate can be obtained
xn;j;k xn 1;j;k
tn;j;k � tn 1;j;k � � tR 21ą
solve the 3-D heat flow equation and examples using
vhS
nitrocellulose decomposition kinetics is described in
Refs. 25 ą 28. The model could also qualitatively describe
In addition, it is possible to initiate the hot spots depend- the burning behavior of transparent JA2 after plasma
ing on an achieved propellant conversion or after contact fragmentation(44).
with the flame front. Chemical reactions of Arrhenius type The propagation of a reaction front in a porous medium
lead to nonlinear behavior of the heat flow equation which with statistically distributed pores is shown in Figure 6,
can no longer be solved analytically. An initial temperature where the decomposition kinetics of nitrocellulose is used.
distribution is converted to an instantaneous heat source The conversion rate continuously increases with rising
that would provide this temperature distribution. Chemical penetration depth of the reaction front when the initiation
reactions take place which, in addition, contribute to of the pores already occurs before the flame front arrives.
instantaneous heat sources. The procedure to numerically (see Fig. 7a). As an alternative, below a conversion (pres-
Propellants, Explosives, Pyrotechnics 27, 142 ą 149 (2002) Burn Rate Models for Gun Propellants 147
Figure 8. Transient burning rates (rb) of nitromethane at ambient temperature and pressure after a "!strong+" ignition pulse followed by
a constant burning rate of 0.02 ą 0.03 cm/s
sure) limit, the pores are only initiated when the flame front propellant nitromethane assumes a reaction of two consec-
passes and the conversion rate remains at a moderate value, utive steps of 1st order: A ) B ) P, the differential equation
then rising to a high, more or less constant level when the of which can be solved analytically.
reaction penetrates to a certain depth above this limit (see To compare the measured burning rates with calculations
Fig. 7b). the method of Zarko and Rychkov(40,41) was applied. Zarko
Both methods result in the increased mass conversion et al. developed a one-dimensional computer code CTEM
rates which are found also experimentally. (Combustion Transients of Energetic Materials), which
simulates the transient burning of heterogeneous energetic
materials(40,41). The calculations confirm the difficult ignition
5 Calculation of Burning Rates from Physical and of nitromethane and the low pressure burning rate of 0.02 to
Chemical Data 0.03 cm/s(42,43). In Figure 8 an ignition to a transient and
subsequent static burning rate is plotted.
Various approaches are made to simplify the complete set
of equations for describing the combustion of solid propel-
lants. Meanwhile, numerical methods and physico-chemical
data are available to calculate a priori burning rates of 6 Conclusions
energetic materials based upon detailed reaction schemes
and physical and chemical kinetic data. It was shown for Vieille�s burning rate law is of a surprising simplicity and
nitromethane that the results correlate to experimental describes experiments with good agreement in the range of
measurements(29,30). three orders of magnitudes of pressure. It uses two
These investigations proceed from the transport equa- parameters where the pressure exponent is of predominant
tions for the various phases and elementary reaction importance.
schemes for a gas phase consisting of NOx, O2, N2 and CHx. The temperature dependence of the burning rate r is
They are described in detail by Miller and Bowman(31). The derived by analyzing the heat flow in the solid and can be
most important radicals of such flames are CN, NH, NO and considered as a modification of Vieille�s law:
OH. They were experimentally investigated for CH4/NO2/ Radiation can strongly influence the burning rate of solid
O2 flames(32ą34) and nitromethane(29,35) and nitramine pro- gun propellants and acts in addition to the heat conduction
pellants(36ą39). The results are in good agreement with of the flame to the solid.
fundamental mechanisms. The CHEMKIN II code enables The burning behavior of porous and foamed propellants
the calculation of detailed profiles of species and temper- deviates also from current predictions of interior ballistics
atures on adiabatic conditions which correlate to measured simulations when based on a straight forward use of Vieille�s
profiles(30). law. Hot gases of the flame penetrate the porous solid
An application to models of heterogeneous gun propel- according to Darcy�s law and produce an enlarged con-
lant combustion requires the reduction of the numerous version zone with strongly increased mass conversions.
reaction steps in the elementary reaction schemes to some Recent progress of fundamental research concerning
representative steps of simplified mechanisms. In principle, elementary reactions in the gas phase, their availability in
the reduced mechanism should reproduce the dominating computer codes and in fluid dynamics enables the develop-
features (conversion, time and temperature scales) of the ment of burning rate models based upon physical and
original one. As an example, the reduction for the liquid chemical parameters.
148 N. Eisenreich, T. Fischer, G. Langer, C. Kelzenberg, and V. Weiser Propellants, Explosives, Pyrotechnics 27, 142 ą 149 (2002)
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Nomenclature
T1 initial temperature
T temperature Q(x , t) heat flux from an external source
E activation energy Q0 heat flux at burning surface
A pre-exponential factor qi heat from conversion of species ci
r linear burning rate xi, j,k radii of burning hot spots in porous propellants
P temperature sensitivity l heat conductivity
p pressure r density
a factor cp specific heat
n pressure exponent ci concentration of species i
I radiation intensity vhs velocity of hot gases in a porous solid
b absorbance kd drag coefficient of the porous solid to gas
L heat of phase transition penetration
TL temperature of phase transition
Tp temperature of solid phase pyrolysis (Received April 4, 2002; Ms 2002/019)