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Abstract
An unsteady three-dimensional numerical model has been formulated, coded, and solved to study ignition and flame development over a composite solid fuel sample upon heating by a localised radiant beam in a buoyant atmosphere. The model consists of an unsteady gas phase and an unsteady solid phase. The gas phase formulation consists of full Navier-Stokes equations for the conservation of mass, momentum, energy, and species. A one-step, second-order overall Arrhenius reaction is adopted. Gas radiation is included by solving the radiation transfer equation. For the solid phase formulation, the energy (heat conduction) equation is employed to solve the transient solid temperature. A first-order in-depth solid pyrolysis relation between the solid fuel density and the local solid temperature is assumed. Numerical simulations provide time-and-space resolved details of the ignition transient and flame development and the existence of two types of ignition modes: one with reaction kernel initiated on the surface and the other with ignition kernel initiated in the gas phase. Other primary outputs of the computation are the minimum ignition energy (Joule) for the solid as a function of the external heating rate (Watt). Both the critical heat input for ignition and the optimal ignition energy are identified. Other parameters that were varied over the simulations include: sample thickness, ignition heat source spatial shape factor, and gravity level.
Acknowledgement
The help from Michael Johnston and Xiao-Yang Zhao is appreciated. This research is supported by a grant from NASA with Dr. Gary Ruff as the technical monitor.
Nomenclature
| As | = | Solid pyrolysis pre-exponential factor ( |
| Bg | = | Gas phase pre-exponential factor |
| Bo | = | Boltzmann number |
| Cp | = | Gas phase specific heat |
| Cp,F | = | Fuel vapour specific heat |
| Cs | = | Solid phase specific heat |
| Da | = | Damköhler number |
| Dj | = | Diffusion coefficient of species j |
| E | = | Gas phase activation energy ( |
| Es | = | Solid phase activation energy ( |
| fj | = | Stoichiometric mass ratio of species j to fuel vapour |
| g | = | Gravity level ( |
| ge | = | Earth gravity (= 981 cm/s2) |
| Hmax | = | Heat flux peak value of the ignition source ( |
| k | = | Gas thermal conductivity |
| kratio | = | = k*s/k* |
| ks | = | Solid thermal conductivity |
| L | = | Latent heat of the solid: L = XL + (Cp,F−Cs)(Ts−TL), |
| Lej | = | Lewis number of species j |
| LR | = | Reference length (LR = α*/UR) |
| = | Fuel vapour mass flux (or the pyrolysis rate) ( | |
| P | = | Pressure ( |
| Pr | = | Prandtl number |
| = | Convective heat flux on the solid surface from the gas phase | |
| = | External heat flux on the solid surface | |
| = | Heating rate of the ignition source ( | |
| = | Net heat flux on the solid surface ( | |
| qr | = | Flame radiation heat flux vector |
| = | Net radiative heat loss on the solid surface | |
| qtotal | = | Total heat input to the solid (J) ( |
| Q | = | Energy released in gas-phase combustion ( |
| r | = | Distance to the ignition heating centre on the solid surface |
| R | = | Heating radius of the ignition source |
| Re | = | Reynolds number |
| Ru | = | Universal gas constant (= 8.31 J/gmol/K) |
| SF | = | External heat source shape factor: SF = Hmax/R2 |
| t | = | Time |
| tg,ref | = | Characteristic time of the gas phase |
| theating | = | Heating time |
| theat-up | = | Solid heat-up time |
| treact | = | Gas-phase reaction time |
| ts,ref | = | Characteristic time of the solid phase |
| T | = | Gas-phase temperature |
| Tg,max | = | Maximum gas temperature |
| TL | = | Solid latent heat reference temperature (= 750 K) |
| Tp | = | Pyrolysis temperature |
| Ts | = | Solid-phase temperature |
| u | = | Flow velocity vector |
| UR | = | Reference velocity: UR = [g(ρ∞−ρflame)α*/ρ*]1/3 |
| x | = | x-coordinate |
| xmin | = | Lower bound of the x domain (xmin = −5 cm) |
| xmax | = | Upper bound of the x domain (xmax = 50 cm) |
| Xj | = | Mole fraction of species j |
| XL | = | Solid latent heat at TL |
| y | = | y-coordinate |
| ymax | = | Upper bound of the y domain (ymax = 20 cm) |
| ys | = | y-coordinate in the solid phase |
| Yj | = | Mass fraction of species j |
| z | = | z-coordinate |
| zcentre | = | z location at the sample centre line (zcentre = 20 cm) |
| zsample | = | z location of the sample/sample-holder boundary (zsample = 17.5 cm) |
| zholder | = | z location of the sample-holder/air boundary (zholder = 16 cm) |
| Γ | = | = ρ*s∝s*/ρ*∝* |
| ΔHR | = | Combustion heat release ( |
| ΔHR0 | = | Heat of combustion at reference temperature T∞ ( |
| Δi | = | Grid size in i direction ( |
| Δt | = | Time step size |
| α | = | Gas thermal diffusivity |
| αs | = | Solid thermal diffusivity |
| μ | = | Gas viscosity |
| ρ | = | Gas density |
| ρflame | = | Gas density at adiabatic flame temperature (= 2500 K) |
| ρs | = | Solid density |
| ρs,F0 | = | Virgin solid combustible density |
| σ | = | Stefan–Boltzmann constant (= 5.67 × 10−12 W/cm2/K4) |
| τ | = | Solid thickness |
| τp | = | Heat penetration thickness |
| ωF | = | Fuel vapour reaction rate ( |
| ωF,max | = | Maximum fuel vapour reaction rate |
| ωj | = | Sink or source term of species j in the species equations (ωj = −fj ωF) |
Subscripts
| c | = | Ignition source centre |
| F | = | Fuel |
| flame | = | Adiabatic flame |
| holder | = | Sample holder |
| i | = | In i direction |
| I | = | Inert |
| j | = | Species j |
| tip | = | At the pyrolysis tip |
| w | = | On the solid surface |
| ∞ | = | Ambient environment |
Superscripts
| * | = | Evaluated at reference temperatures (1250 K for gas phase, 300 K for solid phase) |
| − | = | Over-bar denotes the non-dimensional quantity |