e - Propulsion and Combustion Laboratory

Combined Heat Transfer – Formulations‣ Assumptions• The gas flow is incompressible and has constant properties.• Viscous work is negligible.• The gray gas absorbs, emits, and scatters radiation isotropically.• The wall is isothermal, and diffusely emits and reflects radiation.• The inflow is fully developed, and its temperature is uniform* *‣ Governing Equations ∇⋅ V = 0 1 ( V * ⋅∇ * ) V * = −∇ * p * + ∇*2 V*2 Re* * 1 *2 1 τ0 ⎛ 4 1 ⎞( V ⋅∇ ) Θ= ∇ Θ− ( 1 −ω0) ⎜Θ − GdΩ⎟RePr RePr N4Ω= 4π⎝∫⎠* * * *where x = x L, y = y L, u = u u0, v = v u0, τ0= β0L, ω0 = σsβ0T p − p* ref ρ0uLC0p0µ0Θ= , p = ,Re = ,Pr=2Twρ0u0 µ0k0k0β0IN = , G=3 44σTw σTw‣ Dimensionless RTE 1 dG( r , s ) 1 −ω4 ω =− Grs ( , ) + 0 Θ ( r) + 0 Grs ( , ′) Φ( ssd ′,) Ω′τ dsπ 4π ∫Ω=′ 4π0y(0,1)INLET(0,0)SYMMETRY PLANEy (x)=1.0(Re/3,1)HOT WALL (Re/3, y (Re/3))ly (x)=[tanh(2-30x/Re)-tanh(2)]/2‣ Source Term Linearization*2 2* ⎛ dS ⎞ * ⎡ 4 τ0 *3 ⎤ ⎡ 1 τ0⎛ *4 1 ⎞⎤S = S + ⎜ * ( Θ−Θ ) = − ( 1−ω0) Θ Θ + ( 1−ω0)⎜3Θ + GdΩ ⎟ = S Θ + SdΘ ⎟ ⎢RePr N⎥ ⎢RePr N 4Ω= 4π⎥⎝∫⎝ ⎠ ⎣ ⎦ ⎣ ⎠⎦T T P P P P P P CPl33 Propulsion and Combustion Lab.xy (x)luEXIT