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.

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 [

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 [

Cylindrical-grain SRM schematic model setup for static test stand firing in a laboratory.

Predicted unsteady head-end
pressure-time profile, for SRM undergoing axial combustion instability
symptoms (^{−1},

The Zeldovich-Novozhilov (Z-N) phenomenological approach [

The
Z-N solid-phase energy conservation result may be presented in the following
time-dependent temperature-based relationship [

In a
finite difference format, energy conservation in the solid phase over a given
time increment may be represented by the following equation:

In (

Returning
to the present model, in the solid phase, the transient heat conduction is
governed by [^{−7} seconds has been chosen as the reference value for this investigation, based on
previous SRM simulation studies [

Allowing
for the propellant's surface regression of

With respect to the burning
surface temperature

The numerical model, as described above and with a preliminary
assumption of a constant

Propellant characteristics.

Propellants | |||||
---|---|---|---|---|---|

A | B | C | D | E | |

1500 | 1500 | 1300 | 1500 | 1675 | |

^{3}) | 1730 | 1730 | 1580 | 1730 | 1620 |

0.4 | 0.4 | 0.17 | 0.4 | 0.4 | |

^{2}/s) | 1.54 × 10^{−7} | 1.54 × 10^{−7} | 8.28 × 10^{−8} | 1.54 × 10^{−7} | 1.47 × 10^{−7} |

1000 | 1000 | 750 | 1000 | 650 | |

294 | 294 | 294 | 294 | 300 | |

0.35 | 0.46 | 0.45 | 0.65 | 0.69 |

The
empirical coefficient

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 (

As part of the model development studies, a variable propellant surface
temperature ^{6} J/kg) for differing steady burning rates (Figure

Steady-state temperature distribution in regressing solid propellant.

Analogous to constraining the burning rate to some degree via ^{−1}, the
true surface temperature

As reported in [^{−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

Frequency response of Propellant A, ^{−1},

Examining propellants with characteristics as provided in Table

Frequency response of Propellant A,

Referring to Figure

Frequency response of AP/HTPB Propellant B, theory
(^{−1},

Experimental combustion response data for an AP/PBAN
(polybutadiene acrylic-acid/acrylonitrile binder) composite propellant
designated as A-13 in the literature [

Frequency response of AP/PBAN Propellant C, theory
(^{−1},
^{−1},

Frequency response of AP/PBAN Propellant C, theory
(^{−1},
^{−1},

In [^{−4} in^{2}/s or 1.9355 × 10^{−7}m^{2}/s. This reference value differs substantially
with the assumed actual value for the A-13 propellant (C in Table ^{−8} m^{2}/s).
While it was not indicated explicitly in [

Frequency response of AP/PBAN
Propellant C, theory (^{−1},

In [^{6} 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 [

Frequency response of AP/HTPB
Propellant D, theory (^{−1},

In [^{−1}. In
order to provide a better comparison between theory and experiment, the value
for

Frequency response of double-base
Propellant E, theory (^{−1},

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 for

Arrhenius coefficient, m/s

De St. Robert coefficient, m/s-Pa^{n}

Specific heat (solid phase), J/kg-K

Activation energy, J/kg

Frequency, Hz

Resonant frequency, Hz

Net surface heat of reaction, J/kg

Burn rate limiting coefficient, s^{−1}

Surface temperature
damping coefficient, s^{−1}

Thermal conductivity (solid phase), W/m-K

Limit magnitude (cyclic input)

Mass flow, kg/s

Exponent (de St. Robert's law)

Local gas static pressure, Pa

Equivalent heat input, W/m^{2}

Net heat input, W/m^{2}

Universal gas constant, J/kg-K

Pressure-based response function

Instantaneous burning rate, m/s

Reference burning rate, m/s

Quasisteady burning rate, m/s

Unconstrained burning rate, m/s

Initial temperature (solid phase), K

Burning surface temperature, K

Time increment, s

Spatial increment, m

Fourier limit spatial increment, m

Thermal diffusivity (solid phase),
m^{2}/s

Density (solid phase), kg/m^{3}

Dimensionless frequency