Radiation with Participating Media

1. Radiationwith Participating Media Consider the general heat equation We know that we can write the flux in terms of advective, diffusive, andradiativecomponents heat flux due to radiation What the radiation heat flux? A balance of the emission and irradiation where κλ is the spectral absorption coefficient or the amount of energy absorbed over distance dxwith units of m-1 (absorptivity = emissivity) Integrate over entire solid angle which is a sphere in participating media

2. Emission Recall that we can relate the emission to blackbody emission with some factor where κλ is the spectral absorption coefficient or the amount of energy absorbed over distance dxwith units of m-1

3. Irradiation: Absorption Consider a beam starting at position x = 0 with intensity The reduction in intensity as it travels along x can be described by where κλ is the spectral absorption coefficient or the amount of energy absorbed over distance dxwith units of m-1 The solution of this first order ODE is Bier’s law radiation will decay over some length scale 1/κλ • Absorption • attenuates the intensity of the radiation beam by absorbing energy

4. Irradiation: Emission + Absorption There will also be emission along the beam’s path, and we can thus describe the change intensity based on emission (increase) and absorption Solution generates a balance of the two processes As our optical path goes to infinity, the intensity goes to the blackbody emission (perfect)

5. Irradiation: Scattering Consider a beam starting at position x = 0 with intensity The reduction in intensity can be described by Where σλ is the spectral scattering coefficient or the amount of radiation scattered over distance dxwith units of m-1 The solution of this first order ODE which is also Bier’s law radiation will decay over some length scale 1/σλ • Scattering • attenuates the intensity of the radiation beam by redirecting it

6. Irradiation: Extinction The optical thickness (dimensionless) is then a total path length equal to We can then rewrite Bier’s law as For very small optical thickness, there is virtually no attenuation. For large optical thickness, nearly all the radiation is attenuated • Extinction • combined effects of absorption and scattering

7. Irradiation: More Complete Scattering The phase function describes the probability of radiation being scattered into the direction corresponding to the angle between • Scattering • scattering can also increasethe beam intensity along the path x by scattering some radiation from another angle to be along x

8. Radiation Transfer Equation (RTE) Where the albedo is defined as the ratio of scattering to extinction Writing in terms of sources or radiation or source terms the RTE reduces to which has solution

9. Irradiation We now have an expression for the incident radiation on a control volume due to radiation emitted from some point x = 0 and all scattering, emission, and absorption along the path to the control volume.

10. Heat Equation What the radiation heat flux? A balance of the emission and irradiation Heat equation becomes an integro-differential equation