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 ID: 217256
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Slide1
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)Slide2
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-1Slide3
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-1The solution of this first order ODE is Bier’s law
radiation will decay over some length scale 1/
κ
λSlide4
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)Slide5
Irradiation: Scattering
Scatteringattenuates 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-1The 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 Slide6
Irradiation: Extinction
Extinctioncombined effects of absorption and scatteringWe 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 attenuatedSlide7
Irradiation: More Complete Scattering
Scatteringscattering 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 Slide8
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
w
hich has
solution Slide9
IrradiationWe 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. Slide10
Heat EquationWhat the radiation heat flux? A balance of the emission and irradiation
Heat equation becomes an
integro
-differential equation