High performance thermal insulation materials for buildings - Nanotechnology in Eco-efficient Construction - page 192

group velocity v ω . The resulting (solid) conductive heat transfer φ cd,s is

expressed as:

1
4

∫∫

()

ø cd s = I u d Ω d ω with I = D ω n ηω v [9.10]

, ω ω ω ω

in the solid angle d Ω . Also the transport of phonon radiative intensity I ω

is described by a phonon radiative transfer equation (Srivastava, 1990) as:

1 ∂

∂

I
t

[ ]

ω

+⋅∇ =− u I μ I − I

0

[9.11]

ω ω ω ω

v

ω

stating the radiative phonon energy balance, with creation and destruction
of phonons during collision, and where u ω is the reciprocal of the mean free
phonon path Λ ω equal to v ω τ ω .

Both the radiative transfer equation and the phonon radiative transfer
equation denote that also radiative and conductive heat transfer has a bal-
listic regime with a characteristic length l ext equal to respectively ( μ υ + σ υ ) −1

μ v −1 . The characteristic length of radiation and solid conduction is,
however, lower than the typical values for gaseous conduction and can be
found in the range of 1-0.01 nm. This means that an equally strong reduc-
tion on

and

φ rd can be achieved as depicted by the Knudsen effect.
Solid layers or structures with a thickness below this l ext show a lower k due
to ballistic phonon and photon transport. Here, the phonon radiative trans-
fer equation is reduced to boundary scattering, whereas the radiative
transfer equation for the ballistic regime can be obtained by inserting the
decomposition of the specifi c intensity L v as L v,bal ·

φ cd,s and

δ ( u − u 0 ) into the RTE,

resulting in a ballistic component equal to

1
v

∂
∂

L
t

υ

( )

+⋅∇ =− u L μσ + L [9.12]

υ υ υ υ

υ

denoting a strongly reduced φ rd and resulting k -value. The reduction of both

the solid conductive and radiative heat transfer component in

￿ ￿ ￿ ￿ ￿ ￿

can be used
in two ways. Firstly, it could be used to reduced the effective thermal con-
ductivity k of current state-of-the-art thermal insulators to values below
0.014 W/(mK) at ambient conditions. Secondly, it could be employed to
develop more robust high performance thermal insulators with equal
thermal properties to current state-of-the-art materials but without their
drawbacks by reducing the importance of the gaseous conductive compo-
nent

φ

φ cd,g into the overal low heat transfer.

9.3 State-of-the-art insulators

As traditional thermal isolators are distinguished by how they trap a gaseous
material, current high performance thermal insulators are distinguished by