🔥 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

Radiation

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⚡ Solves 22

💾 Downloads 235 📦 Fortran Code 5.4 KB

📅 Released Jun 2026

⏱️ Latency < 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 = 31.89 kW/m²

Transparent Limit: 109.21 kW/m² | Attenuation: 70.8 %

τ₀ = 4.75

Optical Thickness (τ₀) 4.75 Optically Thick

Rosseland Conductivity (k_rad) 1.95 W/(m·K) Equivalent Diffusion

Scattering Albedo (ω) 0.632 Extinction β = 95.00 m⁻¹

Rosseland Diffusion Flux 35.19 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                 : Porous Ceramic Surface Radiant Gas Burner
Medium Slab Thickness (L)  : 0.050 m
Wall Temperatures (T1 x T2): 1300.0 K (1026.8 C) x 400.0 K (126.9 C)
Optical Coefficients       : a = 35.00 m-1, sigma_s = 60.00 m-1 (Extinction beta = 95.00 m-1)
Wall Emissivities (e1 x e2): 0.88 x 0.75
-----------------------------------------------------------------
OPTICAL THICKNESS (tau_0)  : 4.7500
Scattering Albedo (omega)  : 0.6316
P1 RADIATIVE HEAT FLUX (qr): 31.895 kW/m2
Transparent Limit Flux     : 109.206 kW/m2
Rosseland Diffusion Flux   : 35.190 kW/m2
Rosseland Conductivity     : 1.955 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