45
VARIOS
Using only the first term of the series expansion is only valid at long heating times, so
early time data are left out of the analysis.
can, in fact, be shown to give correct
results after long enough times, but the times are very long, especially for low-conductivity
materials.
shows that conductivity is proportional to the inverse of the slope
when temperature is plotted versus {ln [t]} (
). At long times the temperature hardly
changes, so noise in the measurements can strongly affect the measurement. Part of the
problem with shorter measurement times is that the neglected terms in the exponential
integral expansion are functions of diffusivity, so sample diffusivity affects the conductivity
estimates. A bigger problem, though, is that an ideal line heat source has no heat capacity
while the real probe has significant heat capacity. Another issue is the often significant
thermal contact resistance between the probe and the medium in which it is placed.
To investigate these effects, the Knight et al. (2012) model was used to simulate sensor
data for a wide range of conductivities, diffusivities, and contact resistances. After fitting
to these data, it was determined that the biggest problem is in the time scale. By
changing the equation to
ΔT =
(
q
_
4
πk
)
ln
(
t + t
0
)
+ C
Equation 3
where
t
0
is a time offset, all of the data fit well. Effects of contact resistance and diffusivity
are eliminated or significantly reduced. The values of
k
,
t
0
, and
C
are determined by nonlinear
least square fitting. This procedure was used on samples with known conductivity, such as
glycerin, ultrasound gel, ceramic, on dry soil, and wet soil. The 1-min readings on all of these
samples were more accurate than 10-min readings using
.
