7 KD2 PRO THEORY KD2 Pro
Equation 7 represents the model during cooling.
T = m
1
+ m
2
t + m
3
ln
t
t − t
h
(7)
The thermal conductivity is computed from Equation 8.
k =
q
4πm
3
(8)
Since these equations are long-time approximations to the exponen-
tial integral equations (Equation 1), we use only the final
2
3
of the
data collected (ignore early-time data) during heating and cooling.
This approach has several advantages. One is that effects of contact
resistance appear mainly in these earlytime data, so by analyzing
only the later time data the measurement better represents the ther-
mal conductivity of the sample of interest. Another advantage is that
equations 5 and 6 can be solved by linear least squares, giving a solid
and definite result. The same data, subjected to a non-linear least
squares analysis, can give a wide range of results depending on the
starting point of the iteration because the single needle problem is
susceptible to getting stuck in local minima. The linear least squares
computation is also very fast.
7.3 The Error (Err) Value
When heat at a constant rate, q is applied to the KD2 Pro needles,
the temperature response of the sensor over time can be described
by the equation 9.
∆T = −
q
4πk
Ei
−r
2
4Dt
(9)
where k is the thermal conductivity of the medium in which the
needle is buried, D is the thermal diffusivity of the medium, r is
the distance between the heater and the sensor where temperature
is measured, and Ei is the exponential integral. The KD2 Pro uses
this equation to model temperature rise in the dual needle sensor,
and uses just the first term of the exponential integral expansion (the
logarithm of time) for the single probe sensors. Values of k and D are
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