
Transient initial temperature change in methane hydrate at 9.6°C. The contact parameter, H, is calculated from a curve fit to the first ~0.5 s of data (Click for larger image).


Given the timedependent sample temperature change used to calculate thermal conductivity, λ:
ΔT = A·ln(t) + B, (1)
the thermal diffusivity, κ, is calculated from the slope, A, and intercept, B:
(3)
where γ is Euler's constant, 0.5772, and H is a parameter describing the thermal contact between the probe and sample [Blackwell, 1954].
For fluids, contact between sample and probe is often assumed to be ideal, H is infinite [Glatzmaier and Ramirez, 1985; Nagasaka and Nagashima, 1981], and both conductivity and diffusivity can be determined from the steadystate dependence of ΔT on time. For solid materials, ideal thermal contact between probe and sample cannot be assumed, and H must be determined from:
(4)
where Z_{1} and Z_{2} are fit parameters for the initial, transient dependence of ΔT on time [Blackwell, 1954]:
ΔT = Z_{1}·t  Z_{1}· Z_{2}·t^{2} + Z_{1}· Z_{2}· Z_{3}·t^{2.5}. (5)
The transient temperature change occurs for times t«r_{p}^{2}/κ. This time can be on the order of one second or less, meaning the data acquisition rate must be fast enough to collect enough points for a meaningful fit for calculating H. Our minimum data acquisition rate is 18.2 readings per second, compared to the ASTM standard of 0.2 readings/second. Nonetheless, the determination of κ is made based on a nonlinear fit to perhaps a dozen points, rather than the linear fit to hundreds of points used in the determination of λ. As a result, the scatter in repeat measurements is much higher for κ than for λ, and the scatter worsens as κ increases. For hydrates, the scatter can be as low as ±3%, but for ice, with κ an order of magnitude higher than that of hydrate (~10^{6} m^{2}/s vs. ~10^{7} m^{2}/s), the scatter can be ±15% or more.
In standard needle probe measurements, t = 0 is generally taken to be the moment current is applied to the heater wire. Eq. 5, however, assumes t = 0 when the probe first begins heating the sample. Since the initial transient temperature change lasts only ~2 seconds in hydrate, the finite time delay between applying current to the needle probe's heater wire and the time at which the probe begins heating the sample can not be ignored [Hammerschmidt, 2005]. This delay depends on the needle probe dimensions and construction, and must be determined as part of a probe calibration.
