Decagon Devices KD2 Pro - Single Needle Algorithm

KD2 Pro 7 KD2 PRO THEORY
Here, Ei is the exponential integral,
9
and b
o
, b
1
and b
2
are the con-
stants to be fit. T
o
is the temperature at the start of the measurement
and q is the heat input. The first equation applies for the first t
h
seconds, while the heat is on. The second equation applies when
the heat is off. Compute thermal conductivity from Equation 4 and
diffusivity from 5.
k =
1
b
1
(4)
D =
r
2
4b
2
(5)
You can find the conductivity and diffusivity by fitting equation 1
to the transformed data. The correct values of b
0
, b
1
and b
2
are the
ones which minimize the sum of squares of error between the equa-
tions and the measurements. Use the Marquardt (1963) non-linear
least squares procedure to find the correct values. This procedure
is susceptible to getting stuck in local minima and failing to find a
global minimum in some problems (the single needle problem is a
perfect example of a bad non-linear least squares problem) but the
dual needle problem typically works well. The KD2 Pro can find the
three model parameters quickly.
7.2 Single Needle Algorithm
Heat is applied to a single needle for a time, t
h
, and temperature is
monitored in that needle during heating and for an additional time
equal to t
h
after heating. Two needle sizes are used; One (the KS-
1) is 1.2 mm diameter and 6 cm long. The other (the TR-1) is 2.4
mm diameter and 10 cm long. The temperature during heating is
computed from equation 6.
T = m
0
+ m
2
t + m
3
ln t (6)
Where m
0
is the ambient temperature during heating (which could
include some offset for contact resistance and the heating element
being adjacent to the temperature sensor inside the needle), m
2
is
the rate of background temperature drift, and m
3
is the slope of a
line relating temperature rise to logarithm of temperature.
9
Abramowitz and Stegun, (1972).
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