# Transient Burning Rate Model for Solid Rocket Motor  Internal Ballistic Simulations

Authors: David R. Greatrix
Journal: International Journal of Aerospace Engineering
 (2008)

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## Abstract

A general numerical model based on the Zeldovich-Novozhilov solid-phase energy conservation result for unsteady solid-propellant burning is presented in this paper. Unlike past models, the integrated temperature distribution in the solid phase is utilized directly for estimating instantaneous burning rate (rather than the thermal gradient at the burning surface). The burning model is general in the sense that the model may be incorporated for various propellant burning-rate mechanisms. Given the availability of pressure-related experimental data in the open literature, varying static pressure is the principal mechanism of interest in this study. The example predicted results presented in this paper are to a substantial extent consistent with the corresponding experimental firing response data.


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## Body

## 1. Introduction

An important aspect in the
study of the internal ballistics of solid-propellant rocket motors (SRMs) is
the ability to understand the behaviour of a given motor under transient
conditions, that is, beyond what would be considered as quasisteady or
quasiequilibrium conditions. Transient
combustion and flow conditions arise for example during the ignition and
chamber filling phase [1–3] prior to nominal quasisteady operation,
during the propellant burnout and chamber emptying phase [4, 5] in the
latter portion of a motor's firing, and on occasion when a motor experiences
axial or transverse combustion instability symptoms [6–9] upon initiation by a disturbance. The simulation of undesirable nonlinear
axial combustion instability symptoms in SRMs employing cylindrical and
noncylindrical propellant grains [10–13] has provided the motivation for the present study.SRM internal ballistic simulation models incorporate
algorithms for describing the internal flow and the mass input to the core flow
from the burning surface of the solid propellant. More recent models may also
incorporate the deflection of the surrounding structure, for example, propellant
grain, casing, and heavyweight (e.g., steel) static test sleeve [10–13], as reflected in the schematic SRM diagram of Figure 1. For numerical models, in the case of unsteady operation under transient flow conditions, one ideally would capture the dynamic characteristics of both the flow and combustion to a level of accuracy
that would enable the prediction of inherent design limits for a given motor
(e.g., stable or unstable operation at a given chamber pressure pc and initiating pressure disturbance of ▵pd). An example head-end pressure-time profile from [13] illustrating classical axial combustion instability symptoms, of a limit-magnitude travelling axial
shock wave moving back-and-forth within the motor chamber superimposed on a
base dc pressure shift (approaching 5 MPa) above the normal operating pressure
of approximately 10 MPa, is given in Figure 2.
While in some instances the assumption of a quasisteady (i.e., rapid
kinetic rate) burning rate response of the propellant to local flow conditions
(e.g., static pressure and core mass flux above the burning surface) may be
adequate for predictive purposes, this may not be true in other cases where a
time or frequency dependence in this combustion response may more directly
influence the nonsteady internal ballistics of the motor.Figure 1
Cylindrical-grain SRM schematic model
setup for static test stand firing in a laboratory.Figure 2
Predicted unsteady head-end
pressure-time profile, for SRM undergoing axial combustion instability
symptoms (Kb=20 000 s−1, ΔHs=150 000 J/kg, Δpd=2 atm).The Zeldovich-Novozhilov (Z-N) phenomenological approach [14, 15], in its most general sense, was considered a good basis for the development of a numerical transient burning rate model that could function in an overall
dynamic internal ballistic simulation environment. The Z-N energy conservation criterion in this
context requires a numerical heat conduction solution with time in the solid
phase (propellant beneath the burning surface), but conveniently, empirical or
semiempirical steady-state burning rate information may be used in place of
more complex dynamic flame-based reaction rate equations in tying in the gas
phase above the propellant surface [15]. This is a distinct advantage for
preliminary design and instability evaluation purposes, especially where
quicker computational turnaround times are desirable. An approach of this kind
can be easily adopted by motor developers, by fitting the model to observed
response data for many kinds of propellants through a few parametric constants.
In this paper, a working general numerical burning rate model is described. As a new development from previous Z-N
models, the present model is general in the sense that the model may be
incorporated for various propellant burning-rate driving mechanisms, for example,
driven via static pressure, core flow velocity/mass flux, and normal acceleration,
or some combination thereof. Past
transient burning rate models have typically been constructed in terms of the
specific driving mechanism of interest, for example, with equations derived explicitly
as a function of pressure. In the
present approach, the equations are derived as a function of the quasisteady
burning rate, which in turn may be a function of one or several driving
mechanisms as noted earlier. Note that the work outlined in this paper is a
continuation of the initial burning-rate model development that has been
reported in  [16, 17]  (the reader may find some of the early model
developments and evaluations thereof of interest). Example results are presented in this paper
in order to provide the reader with some background on the sensitivities of a
number of pertinent parameters in the present study. Given the more ready
availability of pressure-based burning experimental data in the open
literature, comparisons are made to reported experimental pressure-based
combustion response data for some composite and homogeneous solid propellants.


## 2. Numerical Model

The
Z-N solid-phase energy conservation result may be presented in the following
time-dependent temperature-based relationship [15]:
(1)Ti,eff=Ti−1rb∗∂∂t∫−∞0ΔTdx,
where Ti,eff is the effective
initial propellant temperature for instantaneous burning rate estimation, Ti is the actual initial
propellant temperature, and in this context, ΔT=T(x,t)−Ti is the temperature distribution in moving from the propellant surface at
x=0 (and T=Ts) to that spatial location in the propellant where
the temperature reaches Ti. Here, rb∗ is the nominal instantaneous burning rate (later, this parameter is more
specifically identified as the unconstrained instantaneous burning rate). For the purposes of the development outlined
in this paper, a more direct and general equation is sought relating rb∗ and the quasisteady
burning rate rb,qs
(value
for burning rate as estimated from steady-state information for a given set of
local flow conditions). This was considered potentially more advantageous than
exploiting past Z-N correlations that utilized pressure-specific functions such
as the pressure-dependent burning rate temperature sensitivity of the
propellant (analytic advantage/“short-cut” for transient pressure studies
[18]) with mixed success in predicting expected burning rates.In a
finite difference format, energy conservation in the solid phase over a given
time increment may be represented by the following equation:(2)qinρsCsΔt−rb∗(Ts−Ti−ΔHsCs)Δt=∑ΔTΔxt+Δt−∑ΔTΔxt,
where qin is the equivalent heat input from the gas to the solid phase via
(3)qin=ρsCsrb,qs(Ts−Ti−ΔHsCs).
One can note the
inclusion of the net surface heat of reaction, ΔHs (sign convention: positive when exothermic output of
heat). The quasisteady burning rate may be ascertained as a function of such
parameters as static pressure of the local flow, for example through de St.
Robert's law [5]:
(4)rb,qs(p)=Cpn.
Through substitution, one arrives at
(5)rb∗=rb,qs−∑ΔTΔxt+Δt−∑ΔTΔxt(Ts−Ti−ΔHs/Cs)Δt
which conforms to
(6)rb∗=rb,qs−1(Ts−Ti−ΔHs/Cs)∂∂t∫−∞0ΔTdx.In (6), rb∗ is the nominal (unconstrained) instantaneous burning rate, and its value at a
given propellant grain location is solved at each time increment via numerical
integration of the temperature distribution through the heat penetration zone
of the solid phase. In this respect, the
present numerical model differs from past numerical models, for example, as reported
by Kooker and Nelson [19]. In those past
cases, the thermal gradient at the propellant surface (∂T/∂x|x=0) was explicitly tied to the true
instantaneous burning rate rb by a specified function, and in turn, rb commonly tied to a variable surface temperature Ts by an Arrhenius-type expression [18]. In
following this established trend, earlier versions of numerical Z-N models did
not follow through on using (6) directly, but switched to a burning rate
temperature sensitivity correlation such as [18, 20]
(7)∂T∂x|x=0=rbαs[Ts−1σpln(rbrb,qs)−Ti],
where σp
is the pressure-based burning rate
temperature sensitivity [20]. As
reported by Nelson [18], the predicted rb augmentation using (7)
was commonly lower than expected from experimental observation, for a number of
transient burning applications, and relative to various flame-based expressions
for thermal surface gradient as a function of rb, of the general form [18]
(8)∂T∂x|x=0=g1rb+g2rb.
The coefficients g1 and g2 may be found
as a function of a number of fixed and variable parameters [18], depending on the given model being employed.Returning
to the present model, in the solid phase, the transient heat conduction is
governed by [14, 21]
(9)ks∂2T∂x2=ρsCs∂T∂t.
In finite difference
format (in this example, for first-order accuracy in time), the temperature at
a given internal location may be presented as
(10)Tt+Δt,x=Tt,x+αsΔt(Δx)2[Tt,x−Δx−2Tt,x+Tt,x+Δx].
For a first-order
explicit scheme, note that the Fourier stability requirement stipulates that
[21]
(11)αsΔt(Δx)2≤12.
While a fourth-order
Runge-Kutta scheme was adopted for the present work, the above criterion does
prove useful as a guideline. Note that a time step of 1 × 10−7 seconds has been chosen as the reference value for this investigation, based on
previous SRM simulation studies [10–13].
The corresponding spatial step Δx
would typically be set to the Fourier stability requirement limit, namely ΔxFo=(2αsΔt)1/2.Allowing
for the propellant's surface regression ofrbΔt with each time step, where rb is the true instantaneous rate of regression, and maintaining the surface
position as x=0 relative to spatial
nodes deeper in the propellant, a given node's temperature may be updated via
(12)Tt,x+rbΔt=Tt,x−Tt,x−Δx−Tt,xΔx(rbΔt).
The boundary condition at the
propellant surface (x=0, T=Ts) to first-order
accuracy may  be applied through
(13)Tt+Δt,−Δx=Tt,−Δx+ΔtρsCsΔx×[ks(Tt,−2Δx−Tt,−Δx)Δx+ks(Ts−Tt,−Δx)Δx+Δqeff],
where
Δqeff
represents the net heat
input from the gas phase into the regressing solid (this parameter will be
discussed in more detail later in the paper).With respect to the burning
surface temperatureTs, one has the option of treating it as constant, or allowing for its variation,
depending on the phenomenological approach being taken for estimating the
burning rate. In the past, it was not
uncommon to encounter estimation models assuming a mean or constant Ts. More recently, usage of an Arrhenius relation
for solid pyrolysis dictating a variable Ts
has become more prevalent [14, 18].The numerical model, as described above and with a preliminary
assumption of a constantTs,
is unstable (solution for transient rb∗ being strongly divergent), using example propellant properties comparable to
Propellant A (see Table 1; nonaluminized ammonium
perchlorate/hydroxyl-terminated polybutadiene [AP/HTPB] composite), and over a
range of different time and spatial increment sizes [16]. An additional equation limiting the transition of the instantaneous burning rate rb with time is required to
physically constrain the model. From the
author's previous background in general numerical modeling where lagging a parameter's
value is a desired objective, a simple empirical means for applying this
constraint is as follows:
(14)drbdt=Kb(rb∗−rb).
The rate limiting coefficient Kb effectively damps or slows
the change in value of the unconstrained burning rate rb∗ with time
when
(15)Kb<1Δt.
In the unusual  case where Kb was greater than 1/Δt, one would
have rb leading rb∗,
rather than lagging it. In the numerical scheme, an incremental change
in burning rate over a given time step would as a result be
(16)Δrb,t+Δt=KbΔt(rb,qs−rb,t)+Kb(Ts−Ti)[∑ΔTΔxt−∑ΔTΔxt+Δt].
In order to be
consistent on input and output heat energy at the propellant surface, such that
the converged solution is independent of time (Δt) and spatial (Δx)
increment sizing below a certain threshold sizing, the surface boundary
condition as relates to net heat input Δqeff
should be stipulated by
(17)Δqeff=(KbΔt)ρsCs(rb,qs−rb)(Ts−Ti−ΔHsCs).Table 1
Propellant characteristics.
PropellantsABCDECs (J/kg-K)15001500130015001675ρs (kg/m3)17301730158017301620ks (W/m-K)0.40.40.170.40.4αs (m2/s)1.54 × 10−71.54 × 10−78.28 × 10−81.54 × 10−71.47 × 10−7Ts (K)100010007501000650Ti (K)294294294294300n0.350.460.450.650.69The
empirical coefficientKb will need to be set below a maximum permissible value for rendering a
nondivergent solution for rb,
and adjusted even further downward in order to match up approximately with
combustion response behavior for a typical solid propellant. A limitation of the present approach would of
course be the nonavailability of experimental response data, say from T-burner
experiments, for the specific propellant in question, to allow for this
alignment of Kb (as
opposed to a direct theoretical derivation).
An additional or complementary concern would be the strength of the
assumption that (14) is in fact a practical means for describing the damped
response, to a sufficient degree of accuracy for the purpose at hand, as
compared to say, the usage of dynamic flame-based reaction rate equations that
also intrinsically limits the movement of the propellant burning rate (e.g.,
usage of (7) or (8)). Comparing the
model's results to a number of different experimental results, for different
propellants, would help establish the degree of confidence that might be
warranted. Some comparisons are
presented later in this paper, that do lend support to the present approach.
The structure of composite propellants (as opposed to homogeneous [double-base]
propellants) at the local microscale is physically more heterogeneous as a
solid mixture of oxidizer crystals and polymer binder than the underlying
assumption of solid homogeneity implied by the Z-N model. Using bulk-average propellant properties and
adjusting the rate limiting coefficient and net surface heat release to an
appropriate setting may nevertheless produce a reasonable and pragmatic
predictive capability for motor design and evaluation at the macroscale for
both classes of propellants.In the present approach, the surface
thermal gradient is free to find its own value at a given instant, via the
numerical scheme for the regressing solid phase. This contrasts with past
approaches that dictated an analytical function tying the surface thermal
gradient to surface regression rate. One can argue then that the use of (14) or some comparable damping function, while empirical, parallels the
approach taken by past researchers in enforcing a stipulated surface thermal
gradient (a form of (7) or (8)).
Both approaches act to constrain the exchange of energy through the
burning surface interface, allow for some variability (through one or more
coefficient settings) in better comparing to a given set of experimental data,
and prevent so-called burning-rate “runaway” (unstable divergence of rb with time [19]).As part of the model development studies, a variable propellant surface
temperatureTs was given
some consideration. In practice, Ts tends to increase with increasing burning rate (although in relative terms, the
numerical value change is typically small), as may be expressed as a function
of burning rate via the following Arrhenius relation for solid pyrolysis [22]:
(18)rb=Aspnsexp(−Eas[ℜTs]),
where ℜ is the universal gas
constant, Eas is the
activation energy, and As is the Arrhenius coefficient. In this example, the effect of local static
pressure p is also included, that is, ns≠0, as occasionally done
in the literature [14, 22]. As
demonstrated by the numerically predicted temperature curves within the
condensed phase (Propellant A of Table 1, where exponent ns
is 0, As is 1675 m/s, and Eas is 100 × 106 J/kg) for differing steady burning rates (Figure 3), the heat
penetration zone into the solid propellant becomes substantially thinner as
burning rate increases. The profiles in Figure 3 would be expected to conform to the following relationship for a homogeneous solid [14]:
(19)T=Ti+(Ts−Ti)exp(−rb|x|αs).Figure 3
Steady-state temperature distribution in regressing solid
propellant.Analogous to constraining the burning rate to some degree viaKb, one in turn may wish to
damp the response of the surface temperature to the change in rb, such that
(20)dTsdt=Kt(Ts*−Ts).
As a result, for Kt=0 s−1, the
true surface temperature Ts remains unchanged (constant) relative to the nominal instantaneous Arrhenius
value Ts* established via
(18) for a given burning rate, ranging up to the case when Kt=1/Δt, where there would  be no lag between the two values.As reported in  [17], the results for when Kt is of appreciable value do
not coincide with what one would expect from the literature. The appearance of
a low-frequency and a high-frequency peak in the frequency response graphs, or an
initial peak dropping down to a prolonged plateau in magnitude at higher values
for Kt, is not commonly
observed experimentally (see Figure 4 for an example predicted result for limit
magnitude Mℓ, defined by
(21), as a function of driving frequency, at differing values for Kt). The more variable surface temperature in
essence (within the predictive model) acts to suppress the degree of
augmentation of the principal response peak (as it moves the peak to a higher
frequency) for a given cyclic driving mechanism, and introduces a secondary
resonant response peak at a low frequency. From a modeling standpoint, one can
appreciate that a moving Ts would tend to counteract the incoming/outgoing energy (to/from the solid) and
affect the numerically predicted near-surface
temperature and energy transfer. This
undoubtedly is playing a role in producing the odd results noted in [17] as Kt is increased. Given the better overall comparisons to
experimental profiles, to date, with an assumed constant Ts (i.e., Kt=0 s−1) in the predictive, phenomenological model, the
numerical results reported in this paper will be generated in a similar
fashion. In doing so, it is understood
that this does not rule out the possibility of a future model development that
would allow for a variable Ts, that would also retain the physics of energy exchange as represented in a
general form by (6).Figure 4
Frequency response of Propellant A,rb,o=0.01 m/s, Kb=170 000 s−1, ΔHs=0.0 J/kg.


## 3. Results and Discussion

Examining propellants with characteristics as provided in Table1, one
means for comparing and potentially aligning the numerical model to actual
firing data is to examine the frequency response to cyclic rb,qs
input, at different values for the burn rate
limiting coefficient. For example,
application of a ±0.001 m/s sinusoidal cycle on a reference base burning rate rb,o of 0.01 m/s at different
driving frequencies on the reference propellant, with a given value for Kb, produces a set of
limit-amplitude cycle results. The nondimensional limit magnitude Mℓ, defined by
(21)Mℓ=rb,peak−rb,orb,qs,peak−rb,o,
is a pertinent parameter that can be charted with respect to driving
frequency (when Mℓ
is
unity, there is a flat response to the cyclic driving [16, 17]; in this case, rb,peak would equal rb,qs,peak (0.011 m/s)). See Figure 5 for an example predicted result
for Propellant A, for differing values of Kb. The limit magnitude profile may also be described in terms of the dimensionless
frequency Ω, defined by [15],
(22)Ω=2παsrb,o2f.
To allow for
further comparison with frequency response data available in the literature,
the simulation results can be presented in a form that relates to a specific
flow parameter, in this case static pressure p. The real part of the
pressure-based response function Rp is typically presented as a function of Ω. The response function is defined in terms of
mean and fluctuating components of static pressure and incoming mass flow from
the propellant surface [15]:
(23)Rp=m′/m¯p′/p¯.
In the context
of this study's application, the real part of the pressure-based response
function in this example application can be estimated as [16]
(24)Re{Rp}=[rb,peakrb,o−1][(rb,qs,peakrb,o)1/n−1]−1.Figure 5
Frequency response of Propellant A,rb,o=0.01 m/s, ΔHs=0.0 J/kg.Referring to Figure6, one can compare the pressure-based
response for an AP/HTPB propellant (B in Table 1; when not available, some
propellant characteristics listed in Table 1 may be approximated from the literature). The base burn rate rb,o
of this propellant at
the test pressure is assumed to be 1.13 cm/s, from available information. While
there is some degree of variability in the experimental data [23] commonly
encountered with T-burner results [24], in this particular case there is an
appreciable level of agreement between the predicted curve of Figure 6 and the
test data points. In [23], the authors found that setting the coefficient A to 8.0 and coefficient B to 0.69 in the Denison-Baum A-B pressure-coupled response model [22]
produced a relatively good fit to the available test data points (the
Denison-Baum predicted curve is quite similar to the present model's curve). In Cohen's study of the application of the A-B response model to homogeneous
propellants [25], he notes that an increase in value for coefficient A results in a higher peak response
magnitude, at a higher dimensionless frequency.
This corresponds to the effect of an increase in the value of Kb in the present model, as
reflected by Figure 5. Similarly, he notes that a decrease in the value of
coefficient B results in a higher
peak response magnitude, with a relatively small change in dimensionless
frequency (at lower values for B). This corresponds to the effect of an increase
in the positive value of ΔHs in the present model.Figure 6
Frequency response of AP/HTPB Propellant B, theory
(Kb=29000 s−1,
ΔHs=75000 J/kg) and experiment
[23], in terms of real part of
pressure-based function (theory & experiment, pressure at 1200 psig).Experimental combustion response data for an AP/PBAN
(polybutadiene acrylic-acid/acrylonitrile binder) composite propellant
designated as A-13 in the literature [22, 24, 26, 27] is also available for
comparison. As displayed in [24]
(Figure 3 of that paper), test data points produced from various institutions'
experiments for that propellant range considerably for any given
frequency. For Figure 7 of the present
paper, two institutions' data points are chosen (they more or less reflect
upper and lower bounds for the collected data; UTC refers to United
Technologies Corp., and CIT refers to California Institute of Technology) for
comparison to the present model's predicted results. Given the range of the experimental data, it
was decided to also produce example upper and lower predictive model curves,
rather than a single median curve. The
assumed characteristics of this propellant are listed under C in Table 1. The
base burn rate rb,o
of
this propellant at the test pressure is assumed to be 0.5 cm/s, an estimate
from available information. The log-linear format of the graph corresponds to
that used in   [24]. Given the range of
variability of the experimental data, the agreement with the predictive model
in terms of qualitative trends and quantitative magnitudes for some portions of
the two predictive curves is encouraging.
Figure 8 provides an alternative linear-linear format for the above
results, where the pressure-based response is in terms of the dimensionless
frequency Ω
(output Ω based on the propellant's actual
thermal diffusivity αs, rather than a nominal reference value, as discussed later). This presentation format conforms with that presented
in [26, Figures 3 and 4].Figure 7
Frequency response of AP/PBAN Propellant C, theory
(Kb=6700 s−1,
ΔHs=115 000 J/kg; Kb=18 000 s−1,
ΔHs=68 000 J/kg) and experiment
[24], in terms of real part of
pressure-based function (theory & experiment, pressure at 300 psig).Figure 8
Frequency response of AP/PBAN Propellant C, theory
(Kb=6700 s−1,
ΔHs=115 000 J/kg; Kb=18 000 s−1,
ΔHs=68 000 J/kg) and experiment
[24, 26], in terms of real part of pressure-based  function
(theory & experiment, pressure at 300 psig).In [22, Figure 4], the experimental (T-burner)
pressure-based response for the A-13 propellant is displayed for differing
pressures (and therefore, differing base burn rates). For the present model, the base burning rates
were estimated from available information, as follows: 0.32 cm/s at 100 psig
(0.79 MPa abs.), 0.565 cm/s at 400 psig (2.86 MPa abs.), and 0.766 cm/s at 800 psig (5.62 MPa abs.). As noted in  [28], and as done as well in  [29] for propellants with differing properties,
it has been commonplace in the past to present output data in terms of Ω as a function of a reference thermal
diffusivity value αs,ref, namely 3 × 10−4 in2/s or 1.9355 × 10−7m2/s. This reference value differs substantially
with the assumed actual value for the A-13 propellant (C in Table 1, i.e., 8.28 × 10−8 m2/s).
While it was not indicated explicitly in  [22] which value for thermal
diffusivity was used for the graph's output Ω (see (22)), a better comparison between the present model's prediction
and the test data is attained when, for output only, the reference value for
thermal diffusivity noted above is used (of course, for the actual numerical
calculations in the solid phase that produce the displayed response results,
the actual thermal diffusivity would be used, not the reference value). Conforming to the log-linear format of  [22], Figure 9 presents the experimental data and the corresponding predicted
curves for the three pressures using Ω=f(αs,ref).
Accounting for the inherent variability of T-burner data, one can see
some positive correlation between the predictive curves and the experimental
data points. Unfortunately, on the
experimental side, no data was collected in these cases at lower frequencies,
so the comparisons here for Figure 9 must be acknowledged as being
incomplete. However, for the same
propellant at 300 psig (2.17 MPa abs.), experimental response levels as shown
in Figure 8 can in fact be demonstrated to be quite high (approaching a value of
6 at a lower frequency). A similar
propellant to A-13,  designated A-35, at 200 psig, shows one experimental data point
exceeding a value of 5 at a lower frequency [22]. The predicted curves of Figure 9 clearly show
the effect of a lower-base burning rate on augmenting the propellant's response,
with the experimental data consistent with this trend at the higher tested
frequencies.Figure 9
Frequency response of AP/PBAN
Propellant C, theory (Kb=18 000 s−1,
ΔHs=68 000 J/kg) and experiment [22],
in terms of real part of pressure-based
function, at differing pressures.In [24, Figure 5], the authors
report experimental response data for AP/HTPB propellants, displaying profiles
that are less typically observed in the literature. The response at lower frequencies rises as
one goes up in frequency, and then tends to plateau as frequency further
increases, and in some cases, starts dropping off to some degree at even higher
frequencies. In Figure 10, one has
experimental data points shown for one such propellant (84% concentration of 5-μm-diameter
AP crystals). The corresponding
predicted curve is based on the assumed or estimated propellant characteristics
listed under Propellant D in Table 1. A
base burn rate of 0.5 cm/s was assumed from the available information. In order to obtain the reasonably good
agreement with the experimental data points, a strong endothermic surface heat
of reaction is being applied (ΔHs at −1 × 106 J/kg). This
contrasts with the smaller exothermic values applied in the earlier cases. Logically, one would expect an endothermic
process to produce such a deadening in the propellant's response
behavior. Note further that the
presence of reactive or nonreactive particles in the flow can in practice
through particle damping (i.e., primarily aerodynamic drag losses reducing the
limit-amplitude travelling pressure wave strength) produce reductions in a
propellant's apparent combustion response as measured by a T-burner. This possibly may produce in appearance what
would be considered an inordinately (in chemical kinetic terms) strong
endothermic process [24, 28].Figure 10
Frequency response of AP/HTPB
Propellant D, theory (Kb=140 000 s−1,
ΔHs=−1 000 000 J/kg) and experiment [24],
in terms of real part of pressure-based
function (theory and experiment,
pressure at 200 psig).In [30, Figure 3], the authors
report experimental pressure response data for a homogeneous (double-base)
propellant, designated as JPN in the literature (comprised of approximately 52%
nitrocellulose, 43% nitroglycerine, plus additional additives). In Figure 11, one has experimental data points
for two operating conditions (800 psig (5.62 MPa abs.), base rb of 1.3 cm/s; 1600 psig
(11.1 MPa abs.), base rb of 2.1 cm/s). The corresponding
predicted curves are based on the assumed or estimated propellant
characteristics listed under Propellant E in Table 1 and
a value for Kb of 39 000 s−1. In
order to provide a better comparison between theory and experiment, the value
for ΔHs
was adjusted upward in moving from the lower
operating pressure to the higher.Figure 11
Frequency response of double-base
Propellant E, theory (Kb=39 000 s−1,
ΔHs=45 000 J/kg [800 psig], ΔHs=90 000 J/kg [1600 psig]) and experiment [30],
in terms of real part of pressure-based
function.


## 4. Concluding Remarks

A
general numerical model exploiting the Z-N energy conservation approach has
been presented. While past numerical
models for transient burning rate have tended to use a surface thermal gradient
boundary condition for problem resolution, the present model employs the
integrated temperature distribution in the solid propellant directly for
instantaneous regression rate calculations. The introduction of a burn rate
limiting coefficient was necessitated initially by numerical model stability
considerations, but in turn this, in conjunction with adjusting the net surface
heat of reaction value, allows one to potentially line up the model's response
behavior to that observed experimentally for a given solid propellant. The example results presented here (within a
certain range of burn rate limiting coefficient and net surface heat release
values) are to a substantial extent consistent with corresponding experimental
firing response data. They clearly
confirm the effect of lower-base burning rate in augmenting the propellant's
response to a given driving mechanism.While
static pressure was chosen as the local driving mechanism for example unsteady
burning results, the model could just as easily be set up for other flow
mechanisms such as core mass flux and radiation. Other mechanisms, such as
local normal acceleration resulting from motor spinning or structural
vibration, could be modeled in this general scheme. One may need to establish whether the rate
limiting coefficient selected through observation of one set of experimental
tests (say forRp) would
in turn be consistent for other mechanisms.
In addition, it is acknowledged that for some cases, one might encounter
some variability in Kb and ΔHs
in
moving from one operational condition (e.g., a given chamber pressure and base
burning rate) to another for a given propellant. Having said this, the relative simplicity of
the present numerical burning rate model makes it a potentially useful
candidate for transient internal ballistic studies of solid-propellant rocket
motors, for example, for studies requiring shorter simulation computational turnaround
times for the prediction of undesirable axial combustion instability symptoms
in a given motor design.



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*Source: 826070-2007-09-24.xml*