Perturbation analysis of the Heterogeneous Quasi 1-D model – a theoretical framework for predicting frequency response of AP–HTPB composite solid propellants

Abstract

In this paper, the Heterogeneous Quasi 1-D model for steady combustion of AP–HTPB propellants is extended to the unsteady regime. The extended model is used to calculate the pressure-coupled frequency response ($R_p$) of low-smoke (non-aluminised) multi-modal AP–HTPB propellants. The $R_p$ of a multi-modal propellant is expressed in terms of that of the individual binder-matrix coated AP particles constituting the statistical particle path. The weighting function, as expected from the serial burning approach, is the burn-time of particles. A closed-form expression is derived for the $R_p$ of the particles by perturbation analysis of the quasi 1-D burn rate model. In this equation, all except the two parameters that quantify the amplitude ($A_c$) and phase (${\phi }_c$) of fluctuating heat flux on the solid side of the interface, are shown to be from the steady-state model. This result establishes a strong connection between the steady and unsteady framework as compared to earlier models, where $R_p\to n$ (propellant pressure index) as $f\to 0$ was explicitly imposed. The model is used to predict $R_p$ for a few low-smoke compositions. Effects of AP particle size distribution, mean pressure and initial temperature are brought out. When expressed as $R_p/n$ vs $f_s$ (non-dimensional frequency based on conduction time scale), the peak response magnitude is of $\mathcal{O}\left(1\right)$ and occurs close to non-dimensional frequency ($f_s=f\alpha /{\overline{\dot{r}}}^2$) value of 1. While this conclusion is in line with the earlier results, it does not explain the ubiquitous nature of acoustic instability in tactical missile rockets, which requires the peak response to be at least an order of magnitude higher than n. Burn rate oscillations associated with the binder-melt effect caused by inhibitors is brought out as the most likely mechanism for the observed instabilities. Methods to extend the theory to include this effect is outlined.

Keywords:

• AP–HTPB composite propellants
• frequency response
• serial burning
• modelling
• pressure coupling

Acknowledgments

Extensive discussions with Prof. H S Mukunda (IISc, Bangalore, retired) were instrumental in development of the theory and we thank him for his contributions.

Notation

$A_c$	=	Heat flux amplification factor
$A_s$	=	pre-exponential constant, mm/s
B	=	Transfer number
$B_{\mathrm{e}\mathrm{f}\mathrm{f}}$	=	effective transfer number
$c_p$	=	specific heat capacity, J/kg K
D	=	Diffusion constant, μ m${}^2$/s
$d_0$	=	Diffusion distance, μ m
$d_{0,\mathrm{r}\mathrm{e}\mathrm{f}}$	=	Reference diffusion distance, μ m
$d_i$	=	Diameter of AP particle, i, $\text{μ}$ m
$E_s$	=	Solid phase activation energy, J/kg
$E_g$	=	Gas phase energy of activation, J/kg
f	=	Frequency, Hz
$f_{\mathrm{e}\mathrm{x}}$	=	Burn rate modifier mass fraction
$f_{\mathrm{e}\mathrm{x}}$	=	Extinct AP particle mass fraction
$f_{\mathrm{H}\mathrm{T}\mathrm{P}\mathrm{B}}$	=	HTPB mass fraction
$f_{\mathrm{p}\mathrm{m}}$	=	AP mass fraction below premixed cutoff diameter
$f_{ll}$	=	Fraction of the surface covered by binder matrix
$f_{ll,\mathrm{a}\mathrm{m}\mathrm{p}}$	=	Amplitude of fluctuation in $f_{ll}$
$f_{nll}$	=	Fraction of the surface not covered by binder matrix
$f_s$	=	Non-dimensional frequency
$g_f$	=	Geometric factor
$H_{\mathrm{A}\mathrm{P}}$	=	Enthalpy change for AP at surface due to phase change, kJ/kg
$H_{\mathrm{b}\mathrm{m}}$	=	Enthalpy change for binder-matrix at surface due to phase change, kJ/kg
$H_{\mathrm{H}\mathrm{T}\mathrm{P}\mathrm{B}}$	=	Enthalpy change for HTPB at surface due to phase change, kJ/kg
$H_s$	=	Net enthalpy change at surface due to phase change, KJ/kg
k	=	Thermal conductivity, W/m K
$k_g$	=	Thermal conductivity of gas phase, W/m K
$K_r$	=	Gas phase reaction rate, s/m${}^2$ atm
$K_{r,\mathrm{e}\mathrm{f}\mathrm{f}}$	=	Effective gas-phase reaction rate, s/m${}^2$ atm
$l_i$	=	Line average intersection of binder-matrix coated AP particle of diameter, $d_i$
n	=	pressure index
$n_i$	=	pressure index of binder-matrix coated AP particle of diameter, $d_i$
p	=	pressure, atm
${\dot{q}}_c$	=	Solid phase heat flux rate, W/m${}^2$ s
${\dot{q}}_g$	=	Gas phase heat flux rate, W/m${}^2$ s
R	=	Universal gas constant, J/mol-K
$\dot{r}$	=	Propellant burn rate, mm/s
${\dot{r}}_i$	=	Burn rate of binder-matrix coated AP particle of diameter, $d_i$, mm/s
$R_p$	=	Pressure coupled frequency response
$R_{p,i}$	=	Pressure coupled frequency response of AP particle
$T_0$	=	Initial temperature of solid propellant, K
$t_{\mathrm{b}\mathrm{m}}$	=	Thickness of binder matrix, $\text{μ}$ m
$T_{\mathrm{e}\mathrm{f}\mathrm{f}}$	=	Effective flame temperature, K
$T_f$	=	Adiabatic flame temperature, K
$T_{f,\mathrm{a}\mathrm{d}}$	=	Adiabatic flame temperature of homogenised binder-matrix coated AP particle, K
$T_s$	=	Surface Temperature, K
$V_i$	=	Volume fraction
$x^{\ast }$	=	Flame stand-off distance, μ m
$x_{\mathrm{e}\mathrm{f}\mathrm{f}}^{\ast }$	=	Effective flame stand off distance, μ m
α	=	Thermal diffusivity, m${}^2$/s
${ϵ}_T$	=	Amplitude of fluctuation in surface temperature
${\phi }_c$	=	Phase difference between $q_c$ and $T_s$, rad
${\phi }_p$	=	Phase difference between $\dot{r}$ and p, rad
φ	=	Homogenised binder-matrix equivalence ratio
${\rho }_{\mathrm{b}\mathrm{r}\mathrm{m}}$	=	Density of burn rate modifier, kg/m${}^3$
${\rho }_{\mathrm{H}\mathrm{T}\mathrm{P}\mathrm{B}}$	=	Density of HTPB, kg/m${}^3$
${\rho }_{\mathrm{A}\mathrm{P}}$	=	Density of AP, kg/m${}^3$
${\rho }_p$	=	Density of propellant, kg/m${}^3$
τ	=	Non-dimensional time
${\xi }^{\ast }$	=	Non-dimensional flame stand off distance
${\xi }_{\mathrm{e}\mathrm{f}\mathrm{f}}^{\ast }$	=	Effective non-dimensional flame stand off distance

Disclosure statement

No potential conflict of interest was reported by the author(s).