Mono-propellants Octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine (HMX) and 1,3,5-
Trinitroperhydro-1,3,5-triazine (RDX) share multiple physical and chemical properties.
Experimental data shows that both propellants have similar burning rate exponents, flame
structures, and final flame temperatures at ambient temperatures [1, 2]. Data also shows
that the temperature sensitivity for RDX and HMX is relatively flat with a magnitude of
approximately 0.15%/K at higher pressures. However, at pressures lower than about 5 atm the
temperature sensitivity of HMX increases dramatically to a value of approximately 0.6%/K
at 1 atm[3]. In addition, the steady state surface temperature of HMX is approximately 10%
higher than that of RDX at low pressure even though the RDX gas phase flame thickness is
smaller than that of HMX. A simple explanation does not exist for these differences although
some explanations and models have been proposed [4’11].
Steady-state propellant models can be classified in three basic categories including models
based on global kinetics, semi-global models based on finite-rate kinetic mechanism in the gas
and/or condensed phases, and multi-phase models with detailed kinetic mechanisms. Given
increases in computational power over the last decade, recent studies of mono-propellant
combustion have focused on the latter of these three classifications [12]. Most of these detailed
kinetics models are able to accurately predict the burning rate pressure dependence for both
HMX and RDX[6’11]. In addition, these models calculate a near flat temperature sensitivity
for RDX within experimental accuracy, but only two of the models show an increase in the
temperature sensitivity of HMX at low pressures [10, 11]. However, the model by Kim et al.
predicts only a marginal increase in the low pressure temperature sensitivity to approximately
0.25%/K at 1 atm.
The model by Washburn et al. was successful in capturing the observed difference in the
temperature sensitivity of RDX and HMX at low pressure. This was accomplished by including
the effects of surface tension of the bubbles in the melt layer of the propellant. However,
addition of the surface tension and bubble models does increase the overall complexity of
the computation and it is not clear the proposed mechanism fully explains the observations
or whether or not simpler explanations are possible. This added complexity also makes it
difficult to isolate the underlying effects of the propellant properties (such as the condensed
phase heat release, density, activation energy, etc) on the temperature sensitivity and other
combustion phenomenon.
The objective of this work is to compare three simplified semi-global models developed by
Denison, Baum, and Williams (DBW)[13, 14]; Li, Williams, and Margolis (LWM) [15]; and
Ward, Son, and Brewster (WSB) [16]. All three of these models, when properly calibrated,
have previously been shown to predict the variance of burning rate with pressure at ambi-
ent temperature for HMX. However, only the Ward-Son-Brewster model has been evaluated
against more completely examining criteria such as the variation in flame structure, temper-
ature sensitivity, combustion instability, and burning rates at multiple initial temperatures.
Here a burning rate sensitivity study was first completed for each parameter to classify the
models based on the relevant physics. Next, computations from each model were compared to
experimental results for the variation in burning rate with initial temperature and pressure,
flame stand-off, temperature sensitivity, and combustion instability in order to determine
the accuracy of each model accuracy. Finally, the results from the models were used to help
explain the aforementioned temperature sensitivity phenomenon between RDX and HMX.
2. Model Overview
The following section presents an overview of the chosen combustion models for the present
study. All models assume a steady state deflagration with constant temperature boundaries
at x = ??1 as shown in Figure 1.
2.1 Denison-Baum-Williams
The basic form of the Denison-Baum-Williams model was originally developed by Denison and
Baum[13] and then derived by Williams using Activation Energy Asymptotics (AEA) [14].
Their analysis assumed that the condensed phase and gaseous phase energy were coupled
using an intermediate gaseious species such that [13],
R(c) ! R(g), R(g) +M ! P(g) +M. (1)
Page 2 of 16
URL: http://mc.manuscriptcentral.com/tctm E-mail: ctm@tandf.co.uk
Combustion Theory and Modelling
For Peer Review Only
February 9, 2015 Combustion Theory and Modelling paper
Combustion Theory and Modeling 3
They further assumed that the gaseous phase activation energy was large leading to an expres-
sion of the mass burning rate as a function of gaseous properties and the flame temperature
leading to,
m2 =
2gBgpnc2
pT4
f e’Eg/ ??RTf
E2
gQ2g
(2)
Tf =T0 +
Qc + Qg
cp
. (3)
This analysis is the classic high activation energy model for the deflagration of a homogeneous
solid propellant.
2.2 Li-Williams-Margolis
More detailed high activation energy analyses have also been performed. For example, the Li-
Williams-Margolis model employs a two-phase liquid and gaseous interface which accounts for
the melting and subsequent vaporization of an energetic material [15]. A three step simplified
chemistry model consisting of a condensed reactant, gaseous reactant, and a gaseous product
was assumed,
R(c) ! P(g), R(c) $ R(g), R(g) ! P(g). (4)
The temperature and the mass fraction of gaseous reactant at the propellant surface is ap-
proximated as,
Ti =
cpTf ‘ YRiQg
cp
(5)
YRi = (1 + [(Wc/Wg) ‘ 1][1 ‘ YRi])
p0
p
e’Hv/ ??RTi (6)
where, Tf = 1320p0.008 (7)
Qc = cp(Tf ‘ T0) (8)
Qg = Qc + Qv (9)
(10)
Using the values for YRi and Ti, the ratio of product flux to total mass flux (G) from the
burning surface is calculated using
G =1 ‘ 0(1 ‘ YRi) ‘ YRi 0C
1 ‘ A 0
1 + B 0
(11)
where 0 is obtained from
0 =D
0(1 + B)(1 + B 0/2)
(1 + B 0)2 +
Qvln(1 + B 0)
Bcp(Ti ‘ T0)
‘
Qg[1 ‘ 0(1 ‘ YRi)]
cp[Ti ‘ T0]
‘
B 0[1 ‘ 0]
1 + B 0
2
.
(12)
Coefficients in the above equation are further defined as
A =
Qc(1 ‘ YRi) ‘ QvYRi
cp(Ti ‘ T0)
, (13)
B =
(Qc + Qv)CYRi
cp(Ti ‘ T0)
, (14)
C =
(Ti ‘ T0)QvWg
??R
T2
i Wc
[1 + (Wg/Wc ‘ 1)(1 ‘ YRi)], (15)
D =
2(g/cp)(cpTf /Qg)2(??RTf /Eg)2 ?? Agpnexp(‘Eg/(??RTf ))
(cTi/Qg)(??RTi/Ec)Accexp(‘Ec/(??RTi))
. (16)
Page 3 of 16
URL: http://mc.manuscriptcentral.com/tctm E-mail: ctm@tandf.co.uk
Combustion Theory and Modelling
For Peer Review Only
February 9, 2015 Combustion Theory and Modelling paper
4 L.B. Whitson Jr. and S.F. Son
Finally, the mass burning rate can then be calculated a function of G
m2 =
2gcpT4
f
??R
A??gpne’Eg/ ??RTf
G2QgEg
. (17)
2.3 Ward-Son-Brewster
Ward, Son, and Brewster utilized the same basic model as the Denison-Baum-Williams model.
However, the Ward-Son-Brewster model assumes a negligible gaseous phase activation en-
ergy [16]. The in-depth reaction term in the condensed phase energy equation was replaced
by the approximate solution for the zeroth order reaction assuming a high activation energy
yielding [17],
m2 =
Ac ??RT2
0 cce’Ec/ ??RTi
Ec[cp(Ti ‘ T0) ‘ Qc/2]
. (18)
The interface temperature is then determined by applying energy conservation to the con-
densed phase such that,
Ti = T0 +
Qc
cp
+
Qg
cp(mxgcp/g + 1)
(19)
Further assuming that the gaseous activation energy approaches zero yields,
xg =
2g
cp(
p
m2 + 4Dg ‘ m)
, (20)
Dg =
gBgp2W2
g
??R
2cp
. (21)
Equations 18-21 are a non-linear, algebraic equation set that can then be solved simultaneously
to determine the solid propellant deflagration characteristics.
3. Model Parameters
The Li-Williams-Margolis model parameters for HMX and RDX were taken from the original
paper [15]. The Ward-Son-Brewster and Denison-Baum-Williams model parameters for HMX
were taken from [16]. However, the Ward-Son-Brewster and Denison-Baum-Williams RDX
model parameters were determined by using measured property values [1] and calibrating the
condensed phase Arrhenius rate constant, gas phase conductivity, gas phase heat release, and
gas phase frequency factor to match the burning rate, surface temperature, flame temperature,
and flame stand-off at a baseline 20 atm and 298K similar to how the HMX model parameters
were originally determined [16]. The following section details the determination of the Ward-
Son-Brewster model parameters for RDX. Final parameters for all models are shown in Table
5.
3.1 Calibration Equations
While values for the condensed phase specific heat, conductivity, activation energy, and density
are well understood and accepted in literature. However, multiple values are suggested for
the condensed phase heat release of RDX. These reported values vary from approximately
-100 kJ/kg to 500 kJ/kg [1]. Once a value for the condensed phase heat release is assumed, the
gas phase heat release can be calculated as
Qg = cp(Tf ‘ T0) ‘ Qc. (22)
The final flame temperature, Tf , can be obtained from either experimental measurements or
from equilibrium calculations [18]. Eq (19) can then be rewritten to estimate the effective gas
phase conductivity as a function of the experimentally determined flame stand-off such that
g = xg
mc2
p(Ti ‘ Tf )
Qg + mcp
. (23)
Page 4 of 16
URL: http://mc.manuscriptcentral.com/tctm E-mail: ctm@tandf.co.uk
Combustion Theory and Modelling
For Peer Review Only
February 9, 2015 Combustion Theory and Modelling paper
Combustion Theory and Modeling 5
The calculated gas phase conductivity is then compared with reported values to ensure that
is falls within the experimental uncertainty. The condensed phase Arrhenius rate coefficient
can be determined using the experimental interface temperature and burn rate,
Ac =
m2Ec(cp(Ti ‘ T0) ‘ Qc/2)
??R
T2
i ce’Ec/ ??RTi
, (24)
which is a rearrangement of Eq (18). Finally, the gas phase frequency factor can be estimated
by combining Eq (20) and (21) and matching the experimental burn rate yielding
Bg =
((2g/(cpxg) + m)2 ‘ m2)??R2cp
4gp2M2 . (25)
Once the parameters are obtained, they are held constant throughout further calculations.
3.2 RDX Parameter Comparison
Using Eq (22) through (25), calibration parameters were determined for multiple values of
the condensed phase heat release for RDX (-100, 100, 200, 300, 400, 500 kJ/kg). Each set
of parameters was evaluated on how well it matched the experimental burning rate, surface
temperature, and flame stand-off at various initial temperatures and pressures.
3.2.1 RDX Burning Rate
The calculated burning rate for RDX was compared to experimental data at multiple tem-
peratures and pressures [2, 19]. Figure 2 is a plot of the calculated burning rate versus the
experimental measurements. Subfigures for the 373K and 423K initial temperatures clearly
show that the three largest values of Qc (300, 400, 500 kJ/kg) overestimate the burning rate
at lower pressures and therefore cannot be acceptable values. This implies that RDX has a
lower condensed phase heat release than that of HMX which is reported as 400 kJ/kg.
Further evaluation of the burning rates was completed by calculating the error of each model
compared to the experimental measurements. The calculation error for each experimental data
point was then normalized by the burning rate at that condition yielding a l2-norm function
of the form
||”||2 =
sPn
j=1([rb,j ‘ rb,m(pj , T0,j )]/rb,j )2
n
. (26)
Table 2 is a comparison of the burning rate errors for each of the Qc values. Again, the
values in this table support the conclusion that the larger Qc values do not represent the
experimental data well for RDX.
3.2.2 RDX Temperature Sensitivity
Temperature sensitivity is defined as
p =
1
rb
@rb
@T0
p
. (27)
Experimental values for temperature sensitivity were taken from studies by Atwood et al. [3]
and model values were determined numerically.
Figure 3 is a plot of the temperature sensitivity for RDX from the various Qc values
compared with the experimental measurements. The results from this plot in conjunction
with the burning rate plots indicate that the condensed phase reaction for RDX may be
endothermic since the most accurate value for the condensed phase heat release is approxi-
mately -100 kJ/kg for the WSB model. It is shown below that this difference from HMX has
significant implications.
4. Model Evaluation
In the following section, each of the models is compared to experimental data taken from
multiple sources for HMX[2, 4, 20’22] and RDX[2, 19].
Page 5 of 16
URL: http://mc.manuscriptcentral.com/tctm E-mail: ctm@tandf.co.uk
Combustion Theory and Modelling
For Peer Review Only
February 9, 2015 Combustion Theory and Modelling paper
6 L.B. Whitson Jr. and S.F. Son
4.1 Model Sensitivity
A model sensitivity study was performed in order to identify which model parameters had
the largest effect on the propellant burning rate. The burning rate sensitivity is defined as
m =
m
@m
@
(28)
where is the parameter being evaluated. Figure 4 is a plot of the burn rate sensitivity for
each of the models studied.
Figure 4(a) clearly shows that controlling parameters for the Denison-Baum-Williams
model are the specific heat, gas phase activation energy, and gas phase heat release. In ad-
dition, the condensed phase heat release has a relatively small effect on the burning rate
while no other condensed phase property plays a role. This results from the gas phase high
activation energy assumption made in the original model derivation.
The Li-Williams-Margolis model shows a similar overall sensitivity towards the gas phase
properties as shown in Figure 4(b). However, the inclusion of the bubble layer results in the
condensed phase activation energy playing an equally important role in the calculation. Also
of interest is the relative lack of sensitivity to the specific heat and conductivity.
Figure 4(c) indicates that the condensed phase activation energy and specific heat play the
most significant role in the burning rate calculation for the Ward-Son-Brewster model which
is a consequence of the zero activation energy assumption for the gas phase. However, unlike
the Denison-Baum-Williams and Li-Williams-Margolis models, theWard-Son-Brewster model
shows an overall more balanced sensitivity with respect to other parameters. For example, the
highest sensitivity parameters for both the Denison-Baum-Williams and Li-Williams-Margolis
models show a sensitivity more than three times higher than the next closest parameters while
the Ward-Son-Brewster models shows only a two fold difference.
If one compares any of these models to only burning rate as a function of pressure for one
initial temperature, very little can be learned. However, comparing to all available data yields
contrasts between the models as a consequence of these sensitivities.
4.2 Burning Rate
A comparison of the burning rate calculated by each model to experimental data was com-
pleted for both HMX and RDX. Plots of burning rate versus pressure at varying initial
temperature allows for a qualitative evaluation of the burning rate models while calculation
of the l2-norm of the error gives a quantitative comparison.
Graphs of burning rate versus pressure for varying initial temperature and the overall l2-
norm error for HMX are shown in Figure 5 and Table 3(a) respectively. The l2-norm of the
burning rate error shows that while at lower temperature the Denison-Baum-Williams model
performs the best, the Ward-Son-Brewster model is the most accurate across all experimen-
tal conditions. Of interest also is the shape of the burning rate curves. At the lower initial
temperatures, the burning rate curve has a nearly constant slope meaning that the propellant
burning rate can be accurately modeled using the standard power law (r pn) functional
form. However, Figure 5(h) shows that at higher initial temperatures the burning rate expo-
nent decreases from 0.85 at pressures above 10 atm to very small values ( 0.14) at pressures
below 1 atm. The reason for the decrease in pressure exponent at low pressure and high initial
temperature is the interaction between the temperature independent gas phase flame (negli-
gible gas activation energy) and the high activation energy surface reaction. Specifically, at
lower pressures the flame has moved far enough away from the propellant surface that it pro-
vides much less heat feedback and therefore plays a more minor role. The Ward-Son-Brewster
model is the only model in the present study that also captures this observed trend.
4.3 Surface Temperature
Modeling of the propellant surface temperature is an integral part of both the Li-Williams-
Margolis and Ward-Son-Brewster deflagration models. For comparison, the surface tempera-
ture for the Denison-Baum-Williams model can be calculated from the mass burning rate by
using Eq. (2) and Eq. (18).
In Figure 7, the calculated surface temperature is compared to the experimental measure-
ments of Zenin et al. [1]. All three models match experimental data for HMX equally well.
However, the Denison-Baum-Williams model and the Li-Williams-Margolis model tend to
over-predict and under-predict the RDX surface temperature respectively, but the data un-
certainty is quite high so clear conclusions are not possible. More accurate surface temperature
measurements are needed, but this is very challenging.
Page 6 of 16
URL: http://mc.manuscriptcentral.com/tctm E-mail: ctm@tandf.co.uk
Combustion Theory and Modelling
For Peer Review Only
February 9, 2015 Combustion Theory and Modelling paper
Combustion Theory and Modeling 7
4.4 Flame Stand-Off
The gas phase flame thickness or stand-off distance has been reported for HMX and RDX by
Parr et al [23, 24] using the location of the maximum CN concentration. Calculation of the
gas phase thickness, xg, is an integral part of the burning rate calculation for the Ward-Son-
Brewster model. The gas phase thickness for the Denison-Baum-Williams and Li-Williams-
Margolis models are calculated as a post-process step using
xg = g
mcp ln
Qg
cp(Ts ‘ T0) ‘ Qc
’1
(29)
and
xg = ln
hgG
hgG ‘ hc + 1
g
cpm
(30)
respectively. For the Li-Williams-Margolis model, the variables hg and hc represent the non-
dimensionalized gas and condensed phase heat release, hj = Qj/[cp(Ti ‘ T0)].
In Figure 8 the flame stand-off calculated by each model is shown. Two flame stand-off
distances (or thicknesses) are reported for the Ward-Son-Brewster model. The 0.63 Tf value is
equivalent to the value of xg and the 0.99 Tf value is the 99% flame temperature distance. As
noted by Parr et al., the flame stand-off for RDX is lower than that for HMX[23]. A similar
trend is seen for all models in that the flame stand-off distance for RDX is always smaller than
that for HMX. In addition, as noted by Ward et al., the Ward-Son-Brewster model calculates
a flame stand-off that is consistent with experiments while the Denison-Baum-Williams model
calculates a stand-off that is an order of magnitude too low[16]. Of most interest though is the
significantly smaller flame stand-off calculated by the Li-Williams-Margolis model compared
to either of the other two models.
4.5 Gas Phase Temperature Profile
Figure 9 is a plot of the gas phase temperature profile as calculated by theWard-Son-Brewster
and Denison-Baum-Williams models compared to the experimental temperature profile from
Zenin [1]. The Li-Williams-Margolis model does not provide for a straight forward determi-
nation of the gas phase temperature profile and has therefore not been considered. This plot
shows that the Ward-Son-Brewster model accurately captures the concave down gas phase
temperature profile that is observed experimental measurements for both HMX and RDX
as well as the correct flame thickness scale as discussed above. The Denison-Baum-Williams
model however, yields a concave up temperature profile in the gas phase. It is expected that
the Li-Williams-Margolis model would show a similar concave up profile due to the assumed
high activation energy for the gas phase.
4.6 Temperature Sensitivity
A comparison of the experimental and model temperature sensitivity at 298K for pressures
up to 11MPa is presented in Figure 10. The experimental values for HMX show a relatively
constant sensitivity near 0.1%/K at pressures above 5MPa but show a sharp increase in
sensitivity as the pressure decreases. Both the Li-Williams-Margolis and Ward-Son-Brewster
models capture the trend of increasing sensitivity at low pressure, however the Denison-
Baum-Williams model does not. There are fewer experimental data points for RDX, and the
profile is less distinct, but the results seem to show the same trend. Again, an increase in the
burning rate sensitivity at low pressures is captured only by the Li-Williams-Margolis and
Ward-Son-Brewster models.
4.7 Stability Analysis
The Zel’dovich-Novozhilov (ZN) stability condition states that for a stable burning propellent
either k < 1 or, if k > 1, r > (k ‘ 1)2/(k + 1) where [25]
k = (Ti ‘ T0)p (31)
r =
@Ti
@T0
p
(32)
Using this criterion, the stability for HMX and RDX can be calculated using both the mea-
sured and calculated values. Experimental values for the surface temperature come from Zenin
Page 7 of 16
URL: http://mc.manuscriptcentral.com/tctm E-mail: ctm@tandf.co.uk
Combustion Theory and Modelling
For Peer Review Only
February 9, 2015 Combustion Theory and Modelling paper
8 REFERENCES
et al. [1] and values for the temperature sensitivity come from Atwood et al. [3] and Parr et
al. [22] Table 4 compares the experimental and computational values for the burning stability
of HMX and RDX.
The experimental values of k for HMX show that as the pressure decreases, the burning
becomes marginally stable with k > 1. This is consistent with the observed behavior of HMX
burning erratically near atmospheric pressures. Both the Li-Williams-Margolis and Ward-
Son-Brewster calculations are qualitatively correct such that at lower pressures the value of
the k increases. However, when the computed k is combined with the r stability criterion,
the Ward-Son-Brewster calculations agree with the experimental observation that at lower
pressures the burning becomes unstable while the Li-Williams-Margolis calculations show that
the burning remains stable for all pressures considered.
The experimental and computational values for RDX show that as pressure increases, the
value of the k also increases. This would indicate that as the pressure decreases, the propellant
would be inherently stable. However, experimental observations have shown that, like HMX,
the combustion of RDX becomes unstable at atmospheric pressure. Both the Li-Williams-
Margolis and the Ward-Son-Brewster calculations show a trend that near atmospheric pres-
sures the k increases dramatically with decreasing pressure. Application of the r criteria to the
Ward-Son-Brewster model shows that for sub-atmospheric pressures, the burning of RDX is
unstable. Neither of the high activation energy model calculations yield this same instability.
5. Conclusions
This work has presented an objective analytical comparison of three simplified energetic ma-
terial combustion models. Also, the WSB model has been successfully applied to RDX. This
comparison encompassed not only the determination of the material burning rate, but also
other physical quantities that are important to the combustion problem including the ma-
terial surface temperature, flame stand-off distance, gas phase temperature profile, and the
burning rate temperature sensitivity. This illuminates the difference between models, as well
as provides insight into the differences between RDX and HMX.
Each model was evaluated against experimental measurements of the relevant physical
quantities. The comparison of burning rate data to the models shows that each model has
ranges of data where it performs the best; however, the Ward-Son-Brewster model shows
the best overall comparison to the experimental data. With regards to propellant surface
temperature, all of the models fall within experimental uncertainty for pressures up to 10MPa.
Comparisons of flame stand-off distance show that both the Denison-Baum-Williams and
Li-Williams-Margolis model drastically under-predict experimental measurements while the
Ward-Son-Brewster model captures both the shape and thickness of the curve. Finally, the
Ward-Son-Brewster and Li-Williams-Margolis models calculate a similar trend with regards
to the temperature sensitivity of HMX such that the temperature sensitivity increases at lower
pressures. The Denison-Baum-Williams model, however, predicts that pressure has no effect
on temperature sensitivity for either propellant. The results presented in Section 3.2 indicate
that the condensed phase heat release for RDX is much lower than that of HMX and that
the condensed phase RDX reaction may be endothermic versus exothermic. An endothermic
condensed phase heat release can lead to an overall lower surface temperature due to the
reaction acting as a sink to the overall heat feedback from the gas phase reactions.
The analysis has shown that overall the Ward-Son-Brewster model is able to not only
predict the effect of initial temperature and pressure on the burning rate of an energetic
material, but is also able to accurately capture many of the important physical phenomena.
While both the Denison-Baum-Williams and the Li-Williams-Margolis models perform fairly
well at predicting the effect of initial pressure and temperature on the propellant burning
rate, neither is able to accurately determine the flame stand-off or temper