AME 60634 Int. Heat Trans.

D. B. Go

Radiation with Participating MediaConsider the general heat equation

We know that we can write the flux in terms of advective, diffusive, and radiative components

heat flux due to radiationWhat the radiation heat flux? A balance of the emission and irradiation

Integrate over entire solid angle which is a sphere in participating media

where κλ is the spectral absorption coefficient or the amount of energy absorbed over distance dx with units of m-1 (absorptivity = emissivity)

AME 60634 Int. Heat Trans.

D. B. Go

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 dx with units of m-1

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 κλ 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/κλ

AME 60634 Int. Heat Trans.

D. B. Go

Irradiation: Emission + AbsorptionThere 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)

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 σλ 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/σλ

Consider a beam starting at position x = 0 with intensity

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

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

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

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.

AME 60634 Int. Heat Trans.

D. B. Go

Heat EquationWhat the radiation heat flux? A balance of the emission and irradiation

Heat equation becomes an integro-differential equation