1. AME 60634 Int. Heat Trans. D. B. Go Radiation with Participating Media Consider the general heat equation We know that we can write the flux in terms of advective, diffusive, and radiative components heat flux due to radiation What the radiation heat flux? A balance of the emission and irradiation Integrate over entire solid angle which is a sphere in participating media where κ λ is the spectral absorption coefficient or the amount of energy absorbed over distance dx with units of m -1 (absorptivity = emissivity)

2. AME 60634 Int. Heat Trans. D. B. Go Emission Recall that we can relate the emission to blackbody emission with some factor where κ λ is the spectral absorption coefficient or the amount of energy absorbed over distance dx with units of m -1

3. AME 60634 Int. Heat Trans. D. B. Go Irradiation: Absorption Absorption –attenuates the intensity of the radiation beam by absorbing energy Consider a beam starting at position x = 0 with intensity The reduction in intensity as it travels along x can be described by where κ λ is the spectral absorption coefficient or the amount of energy absorbed over distance dx with units of m -1 The solution of this first order ODE is Bier’s law radiation will decay over some length scale 1/ κ λ

4. AME 60634 Int. Heat Trans. D. B. Go 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. AME 60634 Int. Heat Trans. D. B. Go Irradiation: Scattering Scattering –attenuates the intensity of the radiation beam by redirecting it The reduction in intensity can be described by Where σ λ is the spectral scattering coefficient or the amount of radiation scattered over distance dx with 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/ σ λ Consider a beam starting at position x = 0 with intensity

6. AME 60634 Int. Heat Trans. D. B. Go Irradiation: Extinction Extinction –combined effects of absorption and scattering We can then rewrite Bier’s law as The optical thickness (dimensionless) is then a total path length equal to For very small optical thickness, there is virtually no attenuation. For large optical thickness, nearly all the radiation is attenuated

7. AME 60634 Int. Heat Trans. D. B. Go Irradiation: More Complete Scattering Scattering –scattering can also increase the beam intensity along the path x by scattering some radiation from another angle to be along x The phase function describes the probability of radiation being scattered into the direction corresponding to the angle between

8. AME 60634 Int. Heat Trans. D. B. Go 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. AME 60634 Int. Heat Trans. D. B. Go 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. AME 60634 Int. Heat Trans. D. B. Go Heat Equation What the radiation heat flux? A balance of the emission and irradiation Heat equation becomes an integro-differential equation