🔥 Radiation in Participating Media (P1 Model)

Evaluate radiative heat flux in absorbing, emitting, and scattering semitransparent gray media using P1 differential spherical harmonics and Rosseland diffusion conductivity.

⚡ Fortran 90 Engine Double Precision (IEEE 754) ✓ ISO / ASME Validated

📊 Solver Telemetry ● ACTIVE

👁️ Consultations 27

⚡ Calculs faits 22

💾 Téléchargements 140 📦 Code Fortran 5.4 KB

📅 Mise en service Jun 2026

⏱️ Latence < 1 ms

🔥 Participating Semitransparent Medium & Photon Scattering Field

Real-time visual simulation of photon emission, volumetric gas absorption & isotropic scattering attenuation

📝 Configuration & Presets

🥃 Glass Melting Furnace 🏭 Coal Boiler Flue Gas 🚀 Rocket Soot Plume 🧱 Porous Radiant Burner

📐 Geometry & Wall Temperatures

Medium Thickness L [m]:

Hot Wall Temperature T₁ [K]:

Cold Wall Temperature T₂ [K]:

🧪 Optical & Radiative Properties

Absorption Coeff a [1/m]:

Scattering Coeff σ_s [1/m]:

Hot Wall Emissivity ε₁:

Cold Wall Emissivity ε₂:

P1 Spherical Harmonics Formulation:
• Extinction Coeff: β = a + σ_s | Optical Thickness: τ₀ = β · L
• P1 Heat Flux: q_r = σ (T₁⁴ − T₂⁴) / [ (1/ε₁ − ½) + (1/ε₂ − ½) + ¾ τ₀ ]
• Rosseland Diffusion: k_rad = 16 σ T_mean³ / (3 β) [W/(m·K)]
• Scattering Albedo: ω = σ_s / β

📊 Radiative Flux Results

📊 Output Summary
💾 Fortran Source

P1 Radiative Heat Flux (q_r)

q_r = 234.45 kW/m²

Transparent Limit: 1,141.03 kW/m² | Attenuation: 79.5 %

τ₀ = 6.00

Optical Thickness (τ₀) 6.00 Optically Thick

Rosseland Conductivity (k_rad) 49.22 W/(m·K) Equivalent Diffusion

Scattering Albedo (ω) 0.333 Extinction β = 12.00 m⁻¹

Rosseland Diffusion Flux 187.04 kW/m² Thick medium limit

📈 P1 Heat Flux q_r (kW/m²) vs Optical Thickness τ₀

📉 Radiative Heat Flux vs Hot Wall Temperature T₁ (K)

=================================================================
 THERMOFLUIDCALC — PARTICIPATING MEDIA RADIATION (P1 MODEL) REPORT
=================================================================
Case Title                 : Hydrocarbon Rocket Exhaust Plume Soot Radiation
Medium Slab Thickness (L)  : 0.500 m
Wall Temperatures (T1 x T2): 2200.0 K (1926.8 C) x 300.0 K (26.9 C)
Optical Coefficients       : a = 8.00 m-1, sigma_s = 4.00 m-1 (Extinction beta = 12.00 m-1)
Wall Emissivities (e1 x e2): 0.95 x 0.90
-----------------------------------------------------------------
OPTICAL THICKNESS (tau_0)  : 6.0000
Scattering Albedo (omega)  : 0.3333
P1 RADIATIVE HEAT FLUX (qr): 234.449 kW/m2
Transparent Limit Flux     : 1,141.025 kW/m2
Rosseland Diffusion Flux   : 187.044 kW/m2
Rosseland Conductivity     : 49.222 W/(m.K)
=================================================================

📘 Calculation Methodology & P1 Spherical Harmonics Standards

P1 Differential Approximation

The P1 method expands the directional radiative intensity into spherical harmonics, converting the complex integro-differential RTE into an elliptic Helmholtz equation:

                    ∇²G − 3aβ G = −12aβ σ T⁴
                

Rosseland Diffusion Analogy

In optically thick media ($\tau_0 \ge 3$), radiation acts like pure non-linear heat conduction with equivalent radiative conductivity $k_{rad} = \frac{16\sigma T^3}{3\beta}$.

Key Engineering Assumptions

1D planar participating medium slab.
Gray gas with wavelength-independent absorption and isotropic scattering.
Opaque diffuse gray wall boundaries ($\epsilon_1, \epsilon_2$).

ThermoFluidCalc Engineering Report