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Numerical studies on erosive burning in cylindrical solid propellant grain

Abstract

This paper addresses erosive burning of a cylindrical composite propellant grain. Equations governing the steady axisymmetric, chemically reacting boundary layer are solved numerically. The turbulence is described by the two equation (k-ɛ) model and Spalding’s eddy break up model is employed for the gas phase reaction rate. The governing equations are transformed and solved in the normalized stream function coordinate system. The results indicate that the dominant reaction zone lies within 20% of the boundary layer thickness close to the wall. The sharp gradient of the temperature profile near the wall is found responsible for bringing the maximum heat release zone near the surface and hence enhancement in the burning rate. The model reproduces the experimental observation that erosive burning commences only above a threshold value of axial velocity.

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Abbreviations

a :

pre exponent in strand-burning rate law, (0.245 ×  10−2 (m/s)/(Mpa)n)

A :

cross-sectional flow area

A s :

Arrhenius frequency factor in propellant surface decomposition, (5.65 m/s)

A + :

damping constant in van Driest’s hypothesis (26)

C 1C 4,C μ, C ω :

constants in turbulence models (C 1 =  1,  C 2 =  1.3,  C 3 =  1.57, C 4 =  2,  C ω =  0.18,  C μ =  0.09)

C p :

\({\sum\limits_k {Y_{k}}}{C_{pk}}\) average heat capacity of reacting gases, (1.254 kJ/kg K)

C pk :

heat capacity of kth species (kJ/kg K)

C s :

heat capacity of solid propellant, (1.59 kJ/kg-K)

d Ap :

average diameter of ammonium per-chlorate particles

D :

port diameter of rocket motor (m)

D f :

diffusion coefficient in Fick’s law (m2/s)

E as :

activation energy in propellant surface decomposition, (62.7 kJ/kmole)

Δh o f,k :

heat of formation of kth species, (233.662 kJ/kg for fuel, −3937.56 kJ/kg for oxidizer, −4753.914 kJ/kg for products)

k :

von Karman’s constant (0.41)

K :

\(\overline{{u_{i}^{\prime} u_{i}^{\prime}}}/2,\) turbulent kinetic energy (m2/s2)

ℓ:

mixing length (m)

n :

exponent in strand-burning rate law (0.41)

P :

pressure (Pa)

Pr:

C p  μ/λ, Prandtl number based upon molecular properties of fluid

Pr t :

Prandtl number for turbulent flow (0.9)

r :

coordinate in radial direction (m)

r b :

total burning rate of a solid propellant (m/s)

r bo :

ap n strand burning rate of a solid propellant (mm/s)

R:

port radius of rocket motor (m)

R h :

roughness height (m)

R u :

universal gas constant (J/kmol k)

Sc :

\(\mu /\bar{\rho}D_{i},\) Schmidt number based upon molecular properties of fluid

Sc t :

Schmidt number for turbulent flow

T :

temperature (K)

T ci :

initial centerline temperature (K)

T o :

reference temperature, 298.14 K

Q s,ref :

surface heat release due to pyrolysis at reference temperature (J/kg)

T p :

propellant temperature (K)

T pi :

propellant initial temperature, (298 K)

T ps :

propellant surface temperature, (800 K)

T oi :

initial stagnation temperature (K)

\(\bar{T}_{ps}\) :

reference surface temperature of propellant (K)

u :

gas velocity in x-direction (m/s)

U :

axial velocity outside boundary layer (m/s)

U ci :

initial centerline velocity (m/s)

u * :

\({\sqrt \frac{\tau_{w}}{{\rho_{\infty}}}},\) friction velocity (m/s)

v :

gas velocity in y-direction (m/s)

W :

\({\left({{\sum\limits_k {Y_{k}}}/W} \right)}^{{- 1}} \) average molecular weight of gases (kg/kmol)

W k :

molecular weight of kth species, kg/kmol (30 kg/kmol for fuel, 27.9 kg/kmol for oxidizer, 20.4 kg/kmol for products)

x :

coordinate in axial direction (m)

y :

coordinate normal to propellant surface (m)

Y k :

mass fraction of k-th species

Y FS :

mass fraction of fuel in a composite solid propellant (0.25)

Y OS :

mass fraction of oxidizer in a composite solid propellant (0.75)

\(\overline{{{\left({} \right)}}}\) :

time-averaged quantity

\((^{\tilde{}})\) :

Favre averaged quantity

\({\left({} \right)}^{\prime}\) :

fluctuating quantity

δ:

boundary-layer thickness (m)

ɛ:

\(\mu \overline{{u^{\prime}_{{ij}} u^{\prime}_{{ij}}}}/\overline{\rho},\) turbulent dissipation (m2/s3)

γ:

constant-pressure to constant-volume specific heat ratio (1.26)

λ:

thermal conductivity of gas (kJ/m s K)

λ s :

thermal conductivity of solid propellant (kJ/m s K)

μ:

gas viscosity (kg/m s)

μ eff :

μ + μ t , effective viscosity (kg/m s)

μ t :

turbulent viscosity (kg/m s)

(μ/Pr ) eff :

μ/Pr + μ t /Pr t (kg/m s)

(μ/Sc ) eff :

μ/Sc + μ t /Sc t (kg/m s)

ν k :

number of moles of kth species (1 kmol for fuel, 3.23 kmol for oxidizer, 5.9 kmol for products)

ρ:

gas density (kg/m3)

ρ s :

solid propellant density (1600 kg/m3)

τ:

\(\mu_{{eff}} \partial \overline{u}/\partial y,\) local shear stress (N/m2)

ω k :

rate of production of species k due to chemical reactions (kg/m3 s)

b :

bulk or averaged variable

c :

centerline condition

k :

species index representing fuel gas [F], oxidizer gas [O], and product gas [P].

∞:

free stream condition

w :

wall (propellant surface) condition

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Srinivasan, K., Narayanan, S. & Sharma, O.P. Numerical studies on erosive burning in cylindrical solid propellant grain. Heat Mass Transfer 44, 579–585 (2008). https://doi.org/10.1007/s00231-007-0280-5

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  • DOI: https://doi.org/10.1007/s00231-007-0280-5

Keywords

  • Burning Rate
  • Turbulent Boundary Layer
  • Ammonium Perchlorate
  • Core Flow
  • Composite Propellant