Civil Engineering Reference
In-Depth Information
9.2.3 Rarefi ed photon and phonon regimes
for next-generation HPTIs
The Knudsen effect has shown that macroscopic laws of heat transport no
longer apply at nanoscale. Convective heat transport at this level is well
understood, but the background of solid conduction and radiative heat
transfer at this characteristic length is a rather unexplored fi eld for non-
metallic materials.
Whereas the gas molecules are the energy carriers for gaseous conduc-
tion, the energy carriers of solid conduction and radiation are respectively
photons and phonons. The behavior of phonons and photons is similar to
gas molecules in several ways: they are treated as classical particles beyond
a certain length scale, i.e., the coherence length for phonons and the wave-
length for photons, and their propagation is described by a Boltzmann
equation. As such, similar to strong reduction of the gaseous heat transfer
φ
cd,g
in the ballistic gas regime due to the so-called Knudsen effect, a reduc-
tion of the heat transfer by solid conduction
cd,s
and radiative heat transfer
φ
rd
can be denoted in the ballistic regimes of their energy carriers.
Radiative heat transfer is described by defi nition of the specifi c intensity
L
v
(
u, r
) in a frequency band [
v
,
v
φ
d
v
] depending on the direction
u
and
the considered point
r
. This intensity can be interpreted as the product of
the number of photons per unit volume
n
v
(
u, r
) with the energy per photon
h
v
and the speed of propagation
v
v
. The resulting radiative heat transfer
+
φ
rd
is expressed as:
1
4
∫∫
ø
rd
=
L
u
d
Ω
d
ω
with
L
=
n h
υ
v
[9.8]
υ
υ
υ
υ
π
in the solid angle d
. The transport of specifi c intensity
L
v
is described by
the radiative transfer equation (Chandrasekhar, 1960) as:
Ω
1
∂
∂
L
t
σ
υ
(
)
()
+
υ
∫
+⋅∇ =−
u
L
μσ
+
L
+
μ
n L
20
T
p L
d
Ω
′
[9.9]
υ
υ
υ
υ
υ
υ
υυ
v
4
π
υ
stating the radiative energy balance of a beam, with negative terms from
extinction by absorption and scattering, and positive terms from scattering
and thermal emission, and where
μ
υ
is the monochromatic absorption coef-
v
is the scattering coeffi cient,
n
is the real part of the refraction
index of the medium,
T
is the local temperature and
p
v
is the fraction of
the energy fl ux in direction
u
that is scattered in direction
u
fi cient,
σ
.
Analogous to radiative heat transfer, solid conduction can be described
by defi nition of the phonon radiative intensity
I
ω
(
u, r
), interpreted as the
product of the number of modes per unit volume d
′
with the number of
phonons per unit volume
n
ω
(
u, r
), with the energy per photon
Ω
¯
η
ω
and the
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