In the present investigation, various factors and trends, related to the usage of two or more sets of inert particles comprised of the same material (nominally aluminum) but at different diameters for the suppression of axial shock wave development, are numerically predicted for a composite-propellant cylindrical-grain solid rocket motor. The limit pressure wave magnitudes at a later reference time in a given pulsed firing simulation run are collected for a series of runs at different particle sizes and loading distributions and mapped onto corresponding attenuation trend charts. The inert particles’ presence in the central core flow is demonstrated to be an effective means of instability symptom suppression, in correlating with past experimental successes in the usage of particles. However, the predicted results of this study suggest that one needs to be careful when selecting more than one size of particle for a given motor application.

Over the last number of decades, a multitude of research efforts have been directed towards understanding the physical mechanisms, or at least the surrounding factors, behind the appearance of symptoms typically associated with nonlinear axial combustion instability in solid-propellant rocket motors (SRMs). The principal symptoms are the presence within the motor chamber of stronger finite-amplitude traveling axial pressure waves that may be shock fronted, commonly (although not always) accompanied by some degree of base chamber pressure rise (dc shift). Note that low-magnitude pressure waves due to vortex shedding from segmented/gapped components in the motor chamber are not included (here) in this more traditional category of nonlinear axial instability. Studies of nonlinear axial combustion instability have ranged from numerous experimental test firing series on the one hand [

The motivation for the experimental, analytical, and numerical studies noted above was and is of course to bring this better understanding to bear in more precisely suppressing, if not eliminating, these axial instability symptoms. For example, it has been long known that inert (nonreactive) or reactive particles in the internal core flow can help to suppress axial combustion instability symptoms [

An effective numerical model combines the effects of the unsteady one- or two-phase flow, the transient combustion process, and the structural dynamics of the surrounding propellant/casing structure. A case study reported by Blomshield [

In the present investigation, an updated numerical model incorporating the above attributes is used in the prediction of the unsteady instability-related behavior in a cylindrical-grain motor and allows for an evaluation of the corresponding effectiveness of using two or more sets of inert spherical particles (same material (nominally aluminum), differing diameters) in suppressing instability symptoms. While aluminum as a common solid propellant fuel addition is reactive (noninert) in practice (and its burning and other behavior at and away from the propellant surface in the central flow may have a significant influence on the given SRM’s combustion stability), the properties of aluminum are assumed for the inert particles in this study so as to allow for comparison to the results of future studies where the aluminum particles are modelled as reactive. In practice, one can note that inert particles composed of such materials as aluminum oxide (which forms from the combustion of aluminum and oxygen) or zirconium carbide do see usage for combustion stabilization purposes. The present study is a followon to the study reported in [

In the present paper, the focus for presented results will largely be on those cases where the transient burning response of the propellant is the primary mechanism for sustaining appreciable traveling pressure waves in the combustion chamber. A few additional results will illustrate the effect of normal acceleration (through radial vibration) as a complementary mechanism acting on the transient combustion process.

A simplified schematic diagram of the physical system of an SRM, that is placed on a static test stand, is provided in Figure

Schematic diagram of sleeved cylindrical-grain SRM, showing reference

One defines

When one has two or more sets of particles of differing sizes (comprised of the same material; thus, the same solid specific heat

The effect of particle mass loading

The equations of motion describing the nonsteady core flow within the SRM must be solved in conjunction with the local pyrolysis rate

Longitudinal acceleration ^{−7} s for the present study, given the motor solution node allocation in the axial direction from head end to nozzle exit plane), in sequence with additional equations for structural motion and propellant burning rate as described below.

Structural vibration can play a significant role in nonsteady SRM internal ballistic behavior, as evidenced by observed changes in combustion instability symptoms as allied to changes in the structure surrounding the internal flow (e.g., propellant grain configuration, wall thickness, and material properties) [

With respect to transient, frequency-dependent burning rate modeling, the Z-N (Zeldovich-Novozhilov) solid-phase energy conservation approach used in the present simulation program may be represented by the following time-dependent temperature-based relationship [

In (

The quasisteady burning rate

With respect to the burning surface temperature _{,}

The characteristics of the reference motor for this study are listed in Table ^{−1}) to be on the order of 1 kHz (a value within the range of what might be expected for this type of composite propellant at that base burning rate). This value for _{1L}

Reference motor characteristics.

Parameter | Value |
---|---|

Propellant grain length, | 52 cm |

Initial port diameter, | 3.6 cm |

Nozzle throat diameter, | 1.6 cm |

Grain/nozzle-conv. length ratio, | 16 : 1 |

Propellant specific heat, | 1500 J/kg-K |

Propellant density, | 1730 kg/m^{3} |

Propellant thermal conductivity, | 0.4 W/m-K |

Propellant thermal diffusivity, | 1.54 × 10^{-7 }m^{2}/s |

Propellant flame temperature, | 3000 K |

Propellant surface temperature, | 1000 K |

Propellant initial temperature, | 294 K |

Gas specific heat, | 1920 J/kg-K |

Specific gas constant, | 320 J/kg-K |

Gas thermal conductivity, | 0.2 W/m-K |

Gas absolute viscosity, | 8.07 × 10^{-5 }kg/m-s |

Gas specific heat ratio, | 1.2 |

De St. Robert exponent, | 0.35 |

De St. Robert coefficient, | 0.05 cm/s-(kPa)^{n} |

Particle solid density, | 2700 kg/m^{3} |

Particle specific heat, J/kg-K | 900 J/kg-K |

Particle mass fraction, | 0% |

Propellant elastic modulus, | 45 MPa |

Propellant Poisson’s ratio, | 0.497 |

Casing inner wall radius, | 3.24 cm |

Casing wall thickness, | 0.127 cm |

Casing material density, | 2700 kg/m^{3} |

Casing elastic modulus, | 80 GPa |

Casing material Poisson’s ratio, | 0.33 |

Sleeve wall thickness, | 0.47 cm |

Sleeve material density, | 7850 kg/m^{3} |

Sleeve elastic modulus, | 200 GPa |

Sleeve material Poisson’s ratio, | 0.30 |

Casing/prop. rad. damping ratio, | 0.35 |

Casing/prop. long. damping ratio, | 0.10 |

Frequency response of reference propellant (^{−1}, differing

An initial pulsed-firing simulation run was completed as a starting reference for this study, in which no particles are present or any other suppression technique being applied. In Figure _{1L}

Predicted head-end pressure-time profile, reference motor (^{−1},

One can refer to Figure

Predicted head-end pressure wave profile, reference motor (^{−1},

In considering an example of two particle sets being used (5 and 10

Predicted head-end pressure wave profile, reference motor (^{−1},

Predicted head-end pressure wave profile, reference motor (^{−1},

Nondimensional attenuation as function of particle diameter and loading of a single particle set, reference motor, acceleration nullified.

Let’s consider the case when vibration-induced acceleration is active as a mechanism working in conjunction with the transient response of the burning solid propellant. Referring to Figure

Predicted head-end pressure-time profile, reference motor (^{−1},

Predicted head-end pressure wave profile, reference motor (^{−1}, _{p3}

Nondimensional attenuation as function of particle diameter and loading of a single particle set, reference motor, acceleration active.

One adjustment in particle size produces a significant change in the result of Figure

Predicted head-end pressure wave profile, reference motor (^{−1},

Along the lines of Figures

Nondimensional attenuation as function of particle diameter and loading distribution

Nondimensional attenuation as function of particle diameter and loading distribution

Nondimensional attenuation as function of particle diameter and loading distribution

An alternative format for illustrating trends associated with using two sets of particles is provided in Figures

Nondimensional attenuation as function of particle diameter

Nondimensional attenuation as function of particle diameter

Nondimensional attenuation as function of particle diameter

A numerical evaluation of the use of two or three sets of different-sized nonburning particles within the flow as a means for suppressing axial pressure wave development has been completed for a reference cylindrical-grain composite-propellant motor, in cases where the transient burning response of the propellant is the primary mechanism for driving the instability symptoms, and in cases where both the transient burning response in conjunction with vibration-induced acceleration is playing a role. The ability of the particles to suppress axial wave development is evident, at relatively low loading percentages, results that are consistent with experimental experience. This is clearly reflected by the respective attenuation maps. If this or a comparable numerical model proves to be suitably accurate, such maps could prove a useful tool for motor designers evaluating their own motor configurations for instability behavior.

The adverse effect of loading a second (or a third) particle set that has a size that is less effective in suppressing pressure wave motion is also made evident in the present results. This study also gives some indication of what might be expected with reactive particles, where due to particle size reduction with time (under burning) the suppression effectiveness may not be quite what one would have ideally expected. In a similar vein, the present results are an indicator of the potential adverse effects of the agglomeration of particles in producing particle sizes bigger than what one would ideally find desirable for suppressing wave activity. These and other issues, some of them potentially quite complex, remain to be explored in regards to the use and modelling of reactive particles for suppression of combustion instability in SRMs.

local core cross-sectional area, m^{2}

gas sound speed, m/s

longitudinal (or lateral) acceleration, m/s^{2}

normal acceleration, m/s^{2}

nonequilibrium sound speed of two-phase mixture

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

particle specific heat, J/kg-K

gas specific heat, J/kg-K

specific heat, solid phase, J/kg-K

drag of gas on a particle from

local core hydraulic diameter, m

mean particle diameter for

local total specific energy of gas in core flow, J/kg

local total specific energy,

frequency, Hz, or Darcy-Weisbach friction factor

accelerative mass flux, kg/m^{2}-s

convective heat transfer coefficient, W/m^{2}-K

net surface heat of reaction, J/kg

burn rate limiting coefficient, s^{−1}

gas thermal conductivity, W/m-K

thermal conductivity, solid phase, W/m-K

magnitude of attenuation

limit magnitude, cyclic input

mean mass of a particle from

number of particles from the

total number of particle sets

exponent, de St. Robert’s law

local gas static pressure, Pa

initial pulse disturbance step pressure, Pa

heat transfer from gas to a particle from

specific gas constant, J/kg-K

instantaneous burning rate, m/s

reference burning rate, m/s

quasisteady burning rate, m/s

unconstrained burning rate, m/s

base burning rate, m/s

flame temperature, gas phase, K

initial temperature, solid phase, K

temperature of particle from

burning surface temperature, K

time increment, s

core axial gas velocity, m/s

core axial particle velocity for particle from

nominal flamefront velocity, m/s

elemental volume, m^{3}

distance from head end, m

spatial increment in axial direction, m

radial distance from burning surface, m

spatial increment in radial direction, solid phase, m

Fourier limit spatial increment, m

gas phase void fraction

total particle mass fraction of overall core flow

particle mass fraction of

thermal diffusivity, solid phase, m^{2}/s

particle-gas mass flux ratio

gas ratio of specific heats

reference combustion zone thickness, m

resultant combustion zone thickness, m

vibration-based wall dilatation term (^{−1}

absolute gas viscosity, kg/m-s

gas density, kg/m^{3}

total density of particles in core flow, kg/m^{3}

density, ^{3}

solid density of particle, kg/m^{3}

solid density of propellant, kg/m^{3}

reference solid density of propellant (no particle loading), kg/m^{3}

acceleration orientation angle, rad

longitudinal/lateral-acceleration-based displacement orientation angle, rad.