Civil Engineering Reference
In-Depth Information
k
s
ð
T
Þ¼
G
ð
q
Þ
k
0
ð
T
Þ;
ð
3
Þ
where k
0
the thermal conductivity of the backbone material and G the geometry
factor which depends on the density. The geometry factor G shows in many cases
weak nonlinear density dependence, i.e.
G
/
q
a
;
ð
4
Þ
with typical values of a = 1…2.
Energy within the gas phase is transported by the interaction of the gas mol-
ecules with itself and with the surrounding solid backbone via collisions. There-
fore, the conductive heat transfer in the gaseous phase depends on the ratio of the
mean free path l
g
of the gas molecules to the effective pore dimension D which is
characteristic for the porous structure of the insulation material. This ratio is also
known as Knudsen number Kn:
Kn
¼
l
g
D
ð
5
Þ
For Kn
1, i.e. the average pore size is significantly smaller than the mean
free path of the gas molecules, the gas molecules collide predominantly with the
solid backbone of the insulation material and the resulting thermal conductivity
contribution is proportional to the number of gas molecules, i.e. the gas pressure.
The contrary extreme is the case where the mean free path of the gas molecules is
much smaller than the average pore size, i.e. Kn
1. The gas molecules collide
predominantly with each other, which is the classical case of diffusive heat
transfer. The resulting thermal conductivity of the gaseous component equals the
thermal conductivity of the free gas, which is independent of the gas pressure for
ambient and moderate pressures. For Kn & 1, the gas molecules collide with both
the walls and each other. An analytic expression of the gas pressure dependence of
the effective thermal conductivity of a pore gas, k
g
, which considers all three
regimes, is provided by Kaganer (
1969
):
¼
1
þ
2
b
ð
T
Þ
Kn
ð
T
Þ
¼
P
k
g
;
0
ð
T
Þ
P
k
g
;
0
ð
T
Þ
k
g
p
g
;
T
ð
6
Þ
1
þ
p
1
=
2
ð
T
Þ
p
g
where P the porosity of the insulation material, b the coefficient, dependent on the
type of gas and temperature, p
1/2
the gas pressure at which the thermal conduc-
tivity is one-half of k
g,0
and k
g,0
the thermal conductivity of the non-convecting
free gas.
Figure
4
shows the ratio k
g
/k
g,0
as a function of the gas pressure for different
pore diameters D and P = 1.
The curves in the regime of molecular heat transfer increase linearly with the
gas pressure, which is hard to see in Fig.
4
, followed by the transition regime
where the gaseous thermal conductivity increases significantly until the saturation
regime is reached. In this case, the thermal conductivity is independent of the gas
pressure. The plot also shows that with decreasing Knudsen number, i.e. increasing
