High performance thermal insulation materials for buildings - Nanotechnology in Eco-efficient Construction - page 190

gas molecules. The resulting conductivity drop is described by the so-called
Knudsen effect (Kennard, 1938). The heat transfer in rarefi ed gas regimes
is generally expressed as derived from parallel surfaces by normalizing the
heat transfer

, * denoting the apparent macro-
scopic heat transfer of the gas in the collisional regime. The resulting ratio
φ cd,g / ø cd g

φ cd,g by the heat transfer ø cd g

*

) describ-
ing the heat transfer by collision between the gas molecule and the pore
wall, depending on the thermal accommodation coeffi cient

depends only on the Knudsen number and a term β ( α , γ

,

α and heat

capacity ratio γ , expressed as:

1

*

ø = ø [9.6]

cd g , cd g ,

1 +⋅ β Kn

and as shown graphically in Fig. 9.2, whereas the is generally simplifi ed

as constant. The required Knudsen number can be determined based on the
kinetic theory for an ideal gas as:

β

l 1

2

1 p

k B

Kn == with n = [9.7]

Λ Λ

n σ

d 2 with d the particle
diameter, k B the Boltzmann constant, T the average gas temperature, and
p the total gas pressure.

where σ is the collisional cross-sectional area equal to π

λ (W/(mK))

0.028

0.024

0.020

￿ ￿ ￿ ￿ ￿ ￿

Pore diameter 10 mm
1 mm

0.016

0.1 mm

0.012

0.01 mm

0.001 mm

0.008

Fumed
silica/aerogel

0.004

0.000
0.0001

0.001 0.01 0.1

p gas (mbar)

1 10 100 1000

9.2 The thermal conductivity of air as a function of the air pressure
and the average pore diameter of the medium.