Burn Rate Models for Gun Propellants, CHEMIA I PIROTECHNIKA, Chemia i Pirotechnika
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142
Propellants, Explosives, Pyrotechnics
27
, 142 ± 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-
proaches have to be taken into account.In the case of solid
rocket propellants the modelling of the burning rate of
composites
(2,3)
and modified double base formulations
(4±6)
resulted also in new descriptions.
This paper reports some modifications to Vieille×s law to
account for the temperature dependence of the burning rate
or the combustion in porous propellants by a simplified
analysis of the heat flow in the solid phase.In addition,
results of applying more detailed reaction models, heat flow
and diffusion in the gaseous phase are briefly outlined.
In the past, Vieille×s law and minor modifications of it described
sufficiently the linear burning rate of gun propellants which
governs the design of charges by interior ballistic simulations.
Recent developments to increase the performance led to new gun
concepts and innovative propellants.These are the electrother-
mal-chemical gun
,
porous and foamed charges as well as
formulations exhibiting a temperature independent burning.
Vieille×s law cannot fully meet experimental results in these
cases.Approaches based on the heat flow equation in the solid
energetic material give simplified formulas to extend the validity.
These burning rate models have the ability to describe the
experimentally determined burning behavior at least in a sim-
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
gaseous reactions can be described.
2 Temperature Dependence of the Burning Rate
1 Introduction
The transition of the condensed phase to the gaseous
phase dominates the combustion of solid energetic materi-
als.This means that the cold solid is heated up to the
temperature of the burning surface caused by energy
transfer from the flame.It possibly undergoes phase
transitions e.g. to a liquid. The conversion to the gaseous
phase can occur by endothermic evaporation, exothermic
pyrolysis or heterogeneous reactions induced by some
unspecified energy sources (
Q
«
[
x
,
t
] from the flame) in the
gaseous phase.These effects can be included in the heat flow
equation, Eq.(3), whereas diffusion of species can be
neglected (the following outline is described in more detail
in Refs.6 ± 7).
The simulation of the interior ballistic behavior of gun
propellants requires as a major input the linear burning rate
and its dependence on pressure.It is obtained by closed
vessel experiments or gun firings
(1)
.The dependence of the
burning rate on pressure is usually described by Vieille×s law,
Eq.(1), which is then incorporated into these ballistic
simulations.The parameters of this law are derived by
analyzing experimental pressure records according to the
simplified relation of Eq.(2).
rp
a
p
n
1
c
p
T
t
2
T
Q
q
i
x
2
x
t
3
dp
dt
t
A
p
r
p
2
i
c
i
c
j
t
A
ij
e
j
RT
fc
i
4
Vieille×s law describes the burning rate for a broad variety
of solid propellant types with sufficient accuracy if minor
modifications are introduced.Recent developments of new
classes of propellants concern temperature insensitive
propellants, radiation absorbing formulations or compact
charges with porous structures.The burning behavior of
j
For an inert solid without phase transition energy the heat
flow equation can be solved by use of the Green×s function of
the heat flow equation
(8)
2
e
x
t
4
4
t
* Corresponding author; e-mail: ne@ict.fhg.de
G
U
x
x
t
t
5
t
t
¹ WILEY-VCH Verlag GmbH, 69469 Weinheim, Germany, 2002
0721-3113/02/2701-0142 $ 17.50+.50/0
c
i
E
i
Propellants, Explosives, Pyrotechnics
27
, 142 ± 149 (2002)
Burn Rate Models for Gun Propellants 143
which enables the construction of any solution for an inert
solid (
dc
i
/dt
(Note the sign convention: endotherm phase transition
0) for an arbitrary heat source
Q
«
[
x
,
t
].
A simplified formula for the ignition delay is derived
when calculating the time needed to increase the temper-
ature of the propellant to an unspecified pyrolysis temper-
ature
T
p
by the intensity of an external heat transfer
I
.
0).They
show that conductive and radiative heat transfer influences
the burning rate in the same way and that endothermic
phase transitions decrease and exothermic chemical reac-
tions increase it.Formulas like Eq.(11) were derived for
ablating surfaces when exposed to heat transfer
(9,10)
and
discussed by Glick
(11)
and Ewing and Osborn
(12)
for rocket
propellants to describe the dependence on initial temper-
ature.Crow and Grimshow proposed a similar law for gun
propellants
(13)
0 (Eq.(10), exothermal chemical reaction
q
T
0
t
2
I
t
c
p
6
t
ign
c
p
T
p
although the parameters were assigned
ap
7
different.
Eqs.(7) and (11) enable to analyze the effects of physical
and chemical parameters of solid propellants on ignition
delay and linear burning rate.As an example: the higher
pyrolysis temperature and the additional melting of HMX
or RDX which requires a latent heat
L
causes the strongly
increased ignition delays and the low burning rates of
nitramine propellants when compared to double base
propellants (
L
4
I
2
Eq.(7) indicates that the ignition delay depends on the
square of the pyrolysis temperature and on the inverse of the
square of the transferred energy intensity.Here, the second
term
a
(
p
) on the right hand side accounts for the initiation
and stabilization of gas phase reactions and depends there-
fore on pressure.It dominates only at high intensities
I
.The
high energy transfer induced by plasma ignition is the reason
for the short and reproducible ignition delay in this case.
Also laser ignition data can be described by Eq.(7).
Provided that a stable combustion has developed, Eq.(3)
can be transferred to the coordinate frame of the moving
flame front where chemical reactions are taken into account.
0) at low pressures.
In general the temperature sensitivity
is given by
d
ln[
r
]/
dT
.Using Eq.11 for the burning velocity this results
in:
dT
rT
1
L
c
p
i
12
T
q
i
c
p
c
p
T
c
p
T
x
2
T
t
r
Ignoring phase transition energies and chemical reactions,
Eq.(13) allows to obtain the unknown
Q
0
(
I
x
2
0) and
T
s
when fitting it to
r
-values measured at various
T
Q
x
t
q
i
c
i
t
r
q
i
c
i
8
(6,15)
.
x
i
Q
o
I
After cancelling the time dependent terms:
r
13
c
T
s
T
r
c
p
T
dT
dx
dx
2
Q
x
r
i
q
i
dc
i
dx
9
It was found that Eq.(13) represents the temperature
dependence of the linear burning rate of many solid
propellants very well.The fit parameter
T
s
is systematically
higher than the pyrolysis temperature obtained in thermal
analytical experiments (e.g. TG or DSC).
Q
0
turns out to
correspond to the pressure dependence of the burning rate
reproducing the pressure exponent
n
(6,14)
.Eq.(13) can
therefore be reformulated to modify Vieille×s law for taking
into account the radiative heat transfer and the dependence
on initial temperature:
This equation can be integrated.A phase transition is
described by a singular point in the specific heat
c
p
:
c
p
T
c
p
L
T
T
L
10
Heating by radiation is taken into account:
Q
x
b
I
e
bx
a
p
n
I
r
14
c
p
T
s
T
and using the boundary conditions:
T
[0]
T
s
T
[
]
T
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)).
The temperature dependence of the burning rate is
influenced by phase transitions and chemical reactions in
the condensed phase as obvious from Eq.(11). Investiga-
tions using the various phases of HMX confirmed the
influence of the latent heat of a structural phase transition
on the burning rate
(16)
dT
/
dx
Q
0
, (heat conduction from the flame) at
x
0
and
dT
/
dx
0at
x
c
i
[0]
1 and
c
i
[
]
0; (complete conversion)
r
Q
o
I
11
i
at low pressures, especially when
c
p
T
s
T
L
q
i
testing the phase
.In addition, the results of plasma pulse
L
dr
d
2
T
144 N.Eisenreich, T.Fischer, G.Langer, C.Kelzenberg, and V.Weiser
Propellants, Explosives, Pyrotechnics
27
, 142 ± 149 (2002)
phase to inert evaporation.In the mesa range the burning
rate
(6)
is given by a combination of Eqs.(11) and (15):
T
s
Z
e
E
R
dT
T
Q
o
I
r
T
0
17
c
p
T
s
T
L
Q
o
I
r
depending on
Q
0
I
exhibiting a minimum which is given
by Eq.(15).
4 Porous Propellants
Figure 1. Least squares fit of
Q
0
versus p curve;
Q
0
derived from
analysing JA2 closed vessel test results using Eq.(13).
The burning behavior of porous and foamed propellants
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
above those obtained by the linear burning of compact
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
and the permeability of the material
(17±19)
.
exposure of JA2 with intensities proportional to
I
can be
described by Eq.(11) as shown in Ref.(44).
3 Reactions in the Condensed Phase
If the chemical reactions are incomplete in the condensed
phase
p
k
d
v
hg
18
q
i
is unknown and a numerical solution has to be
found for the burning rate.
A limiting situation is encountered if the energy transfer
from the gaseous phase can be neglected.Then, the temper-
ature gradient is small in the zone where the chemical
reactions take place and the highest temperatures are
reached.The burning rate does not depend on the heat
release of the reaction but on chemical kinetics
(6,11,12)
.For a
0
th
order reaction the Zeldovich formula can be derived
(details also for n
th
order reactions see Ref.(6)) if
dT
/
dx
0
The gases generate hot spots in the propellant pores which
evolve to (quasi) spherical burning zones.This leads to
increased burning surfaces and in consequence to a higher
mass conversion rate.This porous burning occurs when the
stand-off distance of the flame which depends on pressure
(see e.g. Refs. (15, 20, 21)) is lower than the pore radius. The
total burning surface of the propellant consists of the sum of
all expanding (by the burning rate) pore surfaces within the
penetration depth where a flame could develop.The mass
conversion or the pressure is given by:
is assumed:
2
Z
T
s
e
RT
dT
dp
dt
dm
dt
4
i
j
k
x
2
ijk
t
r
19
r
T
L
c
p
15
c
p
T
s
T
When the pores unite, small propellant residues still burn
resulting in a degressive burning behavior at the end of the
conversion.The behavior of the pores is illustrated by Fig.2.
A surface increase can be up to a factor of 20.
The implementation of these effects to interior ballistic
calculations at Fraunhofer ICT is obtained by two different
models: A phenomenological model using the concept of
cellular automates and a hot spot model using a simplified
numerical solution of the heat flow equation.
The phenomenological model
(22)
enables to apply the
linear burning rate to the enlarging pores of burning
energetic materials.Two-dimensional form functions are
obtained by a formal procedure.In addition, on the basis of
the Noble-Abel-equation the adiabatic pressure rise in a
closed vessel is simulated.Closed vessel tests with non-
porous and porous propellants were used to modify the
model parameters.
The dependence on
T
of the integral in Eq.(15) can
usually be ignored because the Arrhenius term is 0
compared to it at
T
s
.The temperature sensitivity is now
given by:
1
2
T
s
L
c
p
16
Such a behavior was first found for modified double base
rocket propellants at low pressures in the super rate burning
regime.Some modified double base propellants even exhibit
a mesa effect which means that the burning rate decreases
with increasing pressure for some pressure range.Within this
pressure interval the temperature sensitivity is low due to a
change of the mechanisms from reactions in the condensed
Propellants, Explosives, Pyrotechnics
27
, 142 ± 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
flames can stabilise in their initial size.An increase of the burning
surface of a factor 8 is obtained here.
Explicit consideration of the internal structure of the
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 cm
3
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/cm
3
.
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
the same behavior is found.
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
(25±28)
.
The hot spots are approximated by a sum of Gaussian
curves; the temperature development is given by
Figure 3. Influence of the pore size (D
diameter) on the
pressure rise.
2
e
x
x
i
j
k
IJK
Q
ijk
4
t
t
i
j
k
T
hs
t
3
2
t
ijk
t
20
4
t
t
ijk
ijk
1
Figure 4. Influence of the loading density on the linear burning
rate.
A propagating hot gas flow with speed
v
hs
which could
depend on the conversion initiates hot spots at time
t
n, j,k
possibly including a response time
t
R
(
0, here):
x
146 N.Eisenreich, T.Fischer, G.Langer, C.Kelzenberg, and V.Weiser
Propellants, Explosives, Pyrotechnics
27
, 142 ± 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
t
njk
t
n
1
jk
x
n
1
jk
v
hS
t
R
21
solve the 3-D heat flow equation and examples using
nitrocellulose decomposition kinetics is described in
Refs.25 ± 28. The model could also qualitatively describe
the burning behavior of transparent JA2 after plasma
fragmentation
(44)
.
The propagation of a reaction front in a porous medium
with statistically distributed pores is shown in Figure 6,
where the decomposition kinetics of nitrocellulose is used.
The conversion rate continuously increases with rising
penetration depth of the reaction front when the initiation
of the pores already occurs before the flame front arrives.
(see Fig.7a). As an alternative, below a conversion (pres-
In addition, it is possible to initiate the hot spots depend-
ing on an achieved propellant conversion or after contact
with the flame front.Chemical reactions of Arrhenius type
lead to nonlinear behavior of the heat flow equation which
can no longer be solved analytically.An initial temperature
distribution is converted to an instantaneous heat source
that would provide this temperature distribution.Chemical
reactions take place which, in addition, contribute to
instantaneous heat sources.The procedure to numerically
x
njk
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142
Propellants, Explosives, Pyrotechnics
27
, 142 ± 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-
proaches have to be taken into account.In the case of solid
rocket propellants the modelling of the burning rate of
composites
(2,3)
and modified double base formulations
(4±6)
resulted also in new descriptions.
This paper reports some modifications to Vieille×s law to
account for the temperature dependence of the burning rate
or the combustion in porous propellants by a simplified
analysis of the heat flow in the solid phase.In addition,
results of applying more detailed reaction models, heat flow
and diffusion in the gaseous phase are briefly outlined.
In the past, Vieille×s law and minor modifications of it described
sufficiently the linear burning rate of gun propellants which
governs the design of charges by interior ballistic simulations.
Recent developments to increase the performance led to new gun
concepts and innovative propellants.These are the electrother-
mal-chemical gun
,
porous and foamed charges as well as
formulations exhibiting a temperature independent burning.
Vieille×s law cannot fully meet experimental results in these
cases.Approaches based on the heat flow equation in the solid
energetic material give simplified formulas to extend the validity.
These burning rate models have the ability to describe the
experimentally determined burning behavior at least in a sim-
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
gaseous reactions can be described.
2 Temperature Dependence of the Burning Rate
1 Introduction
The transition of the condensed phase to the gaseous
phase dominates the combustion of solid energetic materi-
als.This means that the cold solid is heated up to the
temperature of the burning surface caused by energy
transfer from the flame.It possibly undergoes phase
transitions e.g. to a liquid. The conversion to the gaseous
phase can occur by endothermic evaporation, exothermic
pyrolysis or heterogeneous reactions induced by some
unspecified energy sources (
Q
«
[
x
,
t
] from the flame) in the
gaseous phase.These effects can be included in the heat flow
equation, Eq.(3), whereas diffusion of species can be
neglected (the following outline is described in more detail
in Refs.6 ± 7).
The simulation of the interior ballistic behavior of gun
propellants requires as a major input the linear burning rate
and its dependence on pressure.It is obtained by closed
vessel experiments or gun firings
(1)
.The dependence of the
burning rate on pressure is usually described by Vieille×s law,
Eq.(1), which is then incorporated into these ballistic
simulations.The parameters of this law are derived by
analyzing experimental pressure records according to the
simplified relation of Eq.(2).
rp
a
p
n
1
c
p
T
t
2
T
Q
q
i
x
2
x
t
3
dp
dt
t
A
p
r
p
2
i
c
i
c
j
t
A
ij
e
j
RT
fc
i
4
Vieille×s law describes the burning rate for a broad variety
of solid propellant types with sufficient accuracy if minor
modifications are introduced.Recent developments of new
classes of propellants concern temperature insensitive
propellants, radiation absorbing formulations or compact
charges with porous structures.The burning behavior of
j
For an inert solid without phase transition energy the heat
flow equation can be solved by use of the Green×s function of
the heat flow equation
(8)
2
e
x
t
4
4
t
* Corresponding author; e-mail: ne@ict.fhg.de
G
U
x
x
t
t
5
t
t
¹ WILEY-VCH Verlag GmbH, 69469 Weinheim, Germany, 2002
0721-3113/02/2701-0142 $ 17.50+.50/0
c
i
E
i
Propellants, Explosives, Pyrotechnics
27
, 142 ± 149 (2002)
Burn Rate Models for Gun Propellants 143
which enables the construction of any solution for an inert
solid (
dc
i
/dt
(Note the sign convention: endotherm phase transition
0) for an arbitrary heat source
Q
«
[
x
,
t
].
A simplified formula for the ignition delay is derived
when calculating the time needed to increase the temper-
ature of the propellant to an unspecified pyrolysis temper-
ature
T
p
by the intensity of an external heat transfer
I
.
0).They
show that conductive and radiative heat transfer influences
the burning rate in the same way and that endothermic
phase transitions decrease and exothermic chemical reac-
tions increase it.Formulas like Eq.(11) were derived for
ablating surfaces when exposed to heat transfer
(9,10)
and
discussed by Glick
(11)
and Ewing and Osborn
(12)
for rocket
propellants to describe the dependence on initial temper-
ature.Crow and Grimshow proposed a similar law for gun
propellants
(13)
0 (Eq.(10), exothermal chemical reaction
q
T
0
t
2
I
t
c
p
6
t
ign
c
p
T
p
although the parameters were assigned
ap
7
different.
Eqs.(7) and (11) enable to analyze the effects of physical
and chemical parameters of solid propellants on ignition
delay and linear burning rate.As an example: the higher
pyrolysis temperature and the additional melting of HMX
or RDX which requires a latent heat
L
causes the strongly
increased ignition delays and the low burning rates of
nitramine propellants when compared to double base
propellants (
L
4
I
2
Eq.(7) indicates that the ignition delay depends on the
square of the pyrolysis temperature and on the inverse of the
square of the transferred energy intensity.Here, the second
term
a
(
p
) on the right hand side accounts for the initiation
and stabilization of gas phase reactions and depends there-
fore on pressure.It dominates only at high intensities
I
.The
high energy transfer induced by plasma ignition is the reason
for the short and reproducible ignition delay in this case.
Also laser ignition data can be described by Eq.(7).
Provided that a stable combustion has developed, Eq.(3)
can be transferred to the coordinate frame of the moving
flame front where chemical reactions are taken into account.
0) at low pressures.
In general the temperature sensitivity
is given by
d
ln[
r
]/
dT
.Using Eq.11 for the burning velocity this results
in:
dT
rT
1
L
c
p
i
12
T
q
i
c
p
c
p
T
c
p
T
x
2
T
t
r
Ignoring phase transition energies and chemical reactions,
Eq.(13) allows to obtain the unknown
Q
0
(
I
x
2
0) and
T
s
when fitting it to
r
-values measured at various
T
Q
x
t
q
i
c
i
t
r
q
i
c
i
8
(6,15)
.
x
i
Q
o
I
After cancelling the time dependent terms:
r
13
c
T
s
T
r
c
p
T
dT
dx
dx
2
Q
x
r
i
q
i
dc
i
dx
9
It was found that Eq.(13) represents the temperature
dependence of the linear burning rate of many solid
propellants very well.The fit parameter
T
s
is systematically
higher than the pyrolysis temperature obtained in thermal
analytical experiments (e.g. TG or DSC).
Q
0
turns out to
correspond to the pressure dependence of the burning rate
reproducing the pressure exponent
n
(6,14)
.Eq.(13) can
therefore be reformulated to modify Vieille×s law for taking
into account the radiative heat transfer and the dependence
on initial temperature:
This equation can be integrated.A phase transition is
described by a singular point in the specific heat
c
p
:
c
p
T
c
p
L
T
T
L
10
Heating by radiation is taken into account:
Q
x
b
I
e
bx
a
p
n
I
r
14
c
p
T
s
T
and using the boundary conditions:
T
[0]
T
s
T
[
]
T
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)).
The temperature dependence of the burning rate is
influenced by phase transitions and chemical reactions in
the condensed phase as obvious from Eq.(11). Investiga-
tions using the various phases of HMX confirmed the
influence of the latent heat of a structural phase transition
on the burning rate
(16)
dT
/
dx
Q
0
, (heat conduction from the flame) at
x
0
and
dT
/
dx
0at
x
c
i
[0]
1 and
c
i
[
]
0; (complete conversion)
r
Q
o
I
11
i
at low pressures, especially when
c
p
T
s
T
L
q
i
testing the phase
.In addition, the results of plasma pulse
L
dr
d
2
T
144 N.Eisenreich, T.Fischer, G.Langer, C.Kelzenberg, and V.Weiser
Propellants, Explosives, Pyrotechnics
27
, 142 ± 149 (2002)
phase to inert evaporation.In the mesa range the burning
rate
(6)
is given by a combination of Eqs.(11) and (15):
T
s
Z
e
E
R
dT
T
Q
o
I
r
T
0
17
c
p
T
s
T
L
Q
o
I
r
depending on
Q
0
I
exhibiting a minimum which is given
by Eq.(15).
4 Porous Propellants
Figure 1. Least squares fit of
Q
0
versus p curve;
Q
0
derived from
analysing JA2 closed vessel test results using Eq.(13).
The burning behavior of porous and foamed propellants
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
above those obtained by the linear burning of compact
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
and the permeability of the material
(17±19)
.
exposure of JA2 with intensities proportional to
I
can be
described by Eq.(11) as shown in Ref.(44).
3 Reactions in the Condensed Phase
If the chemical reactions are incomplete in the condensed
phase
p
k
d
v
hg
18
q
i
is unknown and a numerical solution has to be
found for the burning rate.
A limiting situation is encountered if the energy transfer
from the gaseous phase can be neglected.Then, the temper-
ature gradient is small in the zone where the chemical
reactions take place and the highest temperatures are
reached.The burning rate does not depend on the heat
release of the reaction but on chemical kinetics
(6,11,12)
.For a
0
th
order reaction the Zeldovich formula can be derived
(details also for n
th
order reactions see Ref.(6)) if
dT
/
dx
0
The gases generate hot spots in the propellant pores which
evolve to (quasi) spherical burning zones.This leads to
increased burning surfaces and in consequence to a higher
mass conversion rate.This porous burning occurs when the
stand-off distance of the flame which depends on pressure
(see e.g. Refs. (15, 20, 21)) is lower than the pore radius. The
total burning surface of the propellant consists of the sum of
all expanding (by the burning rate) pore surfaces within the
penetration depth where a flame could develop.The mass
conversion or the pressure is given by:
is assumed:
2
Z
T
s
e
RT
dT
dp
dt
dm
dt
4
i
j
k
x
2
ijk
t
r
19
r
T
L
c
p
15
c
p
T
s
T
When the pores unite, small propellant residues still burn
resulting in a degressive burning behavior at the end of the
conversion.The behavior of the pores is illustrated by Fig.2.
A surface increase can be up to a factor of 20.
The implementation of these effects to interior ballistic
calculations at Fraunhofer ICT is obtained by two different
models: A phenomenological model using the concept of
cellular automates and a hot spot model using a simplified
numerical solution of the heat flow equation.
The phenomenological model
(22)
enables to apply the
linear burning rate to the enlarging pores of burning
energetic materials.Two-dimensional form functions are
obtained by a formal procedure.In addition, on the basis of
the Noble-Abel-equation the adiabatic pressure rise in a
closed vessel is simulated.Closed vessel tests with non-
porous and porous propellants were used to modify the
model parameters.
The dependence on
T
of the integral in Eq.(15) can
usually be ignored because the Arrhenius term is 0
compared to it at
T
s
.The temperature sensitivity is now
given by:
1
2
T
s
L
c
p
16
Such a behavior was first found for modified double base
rocket propellants at low pressures in the super rate burning
regime.Some modified double base propellants even exhibit
a mesa effect which means that the burning rate decreases
with increasing pressure for some pressure range.Within this
pressure interval the temperature sensitivity is low due to a
change of the mechanisms from reactions in the condensed
Propellants, Explosives, Pyrotechnics
27
, 142 ± 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
flames can stabilise in their initial size.An increase of the burning
surface of a factor 8 is obtained here.
Explicit consideration of the internal structure of the
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 cm
3
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/cm
3
.
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
the same behavior is found.
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
(25±28)
.
The hot spots are approximated by a sum of Gaussian
curves; the temperature development is given by
Figure 3. Influence of the pore size (D
diameter) on the
pressure rise.
2
e
x
x
i
j
k
IJK
Q
ijk
4
t
t
i
j
k
T
hs
t
3
2
t
ijk
t
20
4
t
t
ijk
ijk
1
Figure 4. Influence of the loading density on the linear burning
rate.
A propagating hot gas flow with speed
v
hs
which could
depend on the conversion initiates hot spots at time
t
n, j,k
possibly including a response time
t
R
(
0, here):
x
146 N.Eisenreich, T.Fischer, G.Langer, C.Kelzenberg, and V.Weiser
Propellants, Explosives, Pyrotechnics
27
, 142 ± 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
t
njk
t
n
1
jk
x
n
1
jk
v
hS
t
R
21
solve the 3-D heat flow equation and examples using
nitrocellulose decomposition kinetics is described in
Refs.25 ± 28. The model could also qualitatively describe
the burning behavior of transparent JA2 after plasma
fragmentation
(44)
.
The propagation of a reaction front in a porous medium
with statistically distributed pores is shown in Figure 6,
where the decomposition kinetics of nitrocellulose is used.
The conversion rate continuously increases with rising
penetration depth of the reaction front when the initiation
of the pores already occurs before the flame front arrives.
(see Fig.7a). As an alternative, below a conversion (pres-
In addition, it is possible to initiate the hot spots depend-
ing on an achieved propellant conversion or after contact
with the flame front.Chemical reactions of Arrhenius type
lead to nonlinear behavior of the heat flow equation which
can no longer be solved analytically.An initial temperature
distribution is converted to an instantaneous heat source
that would provide this temperature distribution.Chemical
reactions take place which, in addition, contribute to
instantaneous heat sources.The procedure to numerically
x
njk
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