Thermal conductivity measurement

Two classes of methods exist to measure the thermal conductivity of a sample: steady-state and non-steady-state methods.

Steady-state methods

(In Geology/Geophysics)The most common method for consolidated rock samples is the Divided Bar. There are various modifications to these devices depending on the temperatures and pressures needed as well as sample sizes.A sample of unknown conductivity is placed between two samples of known conductivity (usually brass plates). The setup is usually vertical with the hot brass plate at the top, the sample in between then the cold brass plate at the bottom. Heat is supplied at the top and made to move downwards to stop any convection within the sample.Measurements are taken after the sample has attained equilibrium (same heat over entire sample), this usually takes about 10 minutes.

Transient methods

Non-steady-state methods to measure the thermal conductivity do not require the signal to obtain a constant value. Instead, the signal is studied as a function of time. The advantage of these methods are that they can in general be performed more quickly, since there is no need to wait for a steady-state situation. The disadvantage is that the mathematical analysis of the data is in general more difficult.

Transient line source method

The physical model behind this method is the infinite line source with constant power per unit length. The temperature profile $T(t,r)$ at a distance $r$ at time $t$ is as follows

:$T(t,r)\; =\; frac\{Q\}\{4\; pi\; k\}\; mathrm\{Ei\}\; left(\; frac\{r^2\}\{4\; a\; t\}\; ight)$

where: $Q$ is the power per unit length, in [W·m^-1] : $k$ is the thermal conductivity of the sample, in [W·m^-1·K^-1] : $mathrm\{Ei\}(x)$ is the exponential integral, a transcendent mathematical function: $r$ is the radial distance to the line source: $a$ is the thermal diffusivity, in [m²·s^-1] : $t$ is the amount of time that has passed since heating has started, in [s]

When performing an experiment, one measures the temperature at a point at fixed distance, and follows that temperature in time. For large times, the exponential integral can be approximated by making use of the following relation

:$mathrm\{Ei\}(x)\; =\; -\; gamma\; -\; ln\; (x)\; +\; O(x)$

where: $gamma\; approx\; 0.577$ is the Euler gamma constant

This leads to the following expression

:$T(t,r)\; =\; frac\{Q\}\{4\; pi\; k\}\; left\{\; -gamma\; -ln\; left(\; frac\{r^2\}\{4\; a\; t\}\; ight)\; +\; ln\; (t)\; ight\}$