Radiation

RadiationModel

Enumeration of available radiation transport models for pyrolysis simulations.

Values

• NO_RADIATION: No volumetric radiation absorption. External radiative boundary conditions (RadiativeBC, RadiativeFluxBC) still apply at surfaces. Use for transparent materials or when radiation effects are negligible.

• SURFACE_ABSORPTION: All incident radiation is absorbed in the surface cell. Computationally efficient O(1) with tridiagonal Jacobian. Appropriate for opaque materials where optical thickness τ >> 1 in the first cell. Component absorption coefficients (α) are ignored - a warning is issued if they are set.

• BEER_LAMBERT: In-depth volumetric absorption following Beer-Lambert law: I(z) = I₀ exp(-∫ α_eff dz). Radiation penetrates into the material based on component absorption coefficients. Creates non-local coupling in Jacobian (upper triangular pattern). Use for semi-transparent materials (polymers, foams).

• P1_QUASI_STEADY: P1 approximation with quasi-steady radiation field (∂G/∂t = 0). Solves for incident radiation intensity G including volumetric emission (4σαT⁴). Recommended when internal re-radiation is significant (T > 800K, optically thick). Uses Newton iteration to solve radiation field each timestep.

Physics

Beer-Lambert Model

Exponential attenuation through absorbing medium: I(z) = I₀ × exp(-∫ αeff(z') dz') q'''(z) = αeff(z) × I(z)

P1 Approximation

Solves for incident radiation intensity G [W/m²]: ∇·(Drad ∇G) = α G - 4σαT⁴ where Drad = 1/(3α) is the radiation diffusion coefficient.

Validity (Optical Thickness τ = αL)

• τ > 10: Optically thick - P1 excellent, surface absorption acceptable
• 1 < τ < 10: Intermediate - P1 good, Beer-Lambert recommended
• 0.1 < τ < 1: Optically thin - Beer-Lambert recommended
• τ < 0.1: Transparent - Use NO_RADIATION

For typical pyrolysis materials (τ ≈ 5-50), P1QUASISTEADY is ideal for high-temperature re-radiation, BEER_LAMBERT for semi-transparent polymers.

Example

# Surface absorption for opaque material
solution = solve(problem; radiation_model = SURFACE_ABSORPTION)

# In-depth absorption for semi-transparent PMMA
solution = solve(problem; radiation_model = BEER_LAMBERT)

# P1 for high-temperature char with re-radiation
solution = solve(problem; radiation_model = P1_QUASI_STEADY)

Performance Characteristics