The sensor is a double spiral etched from nickel foil and sandwiched between thin insulating films, usually polyimide. A constant current heats it, and its resistance, which rises with temperature, is recorded through a bridge. The technique grew out of the transient hot-strip method (Gustafsson et al., 1979). In the ideal model the spiral is replaced by a set of concentric ring sources in an infinite homogeneous medium, and the solution of the conduction equation (Carslaw and Jaeger, 1959) gives the mean temperature rise of the sensor as ΔT(τ) = P0 D(τ) / (π3/2 r k). Here P0 is the heating power, r the sensor radius, k the thermal conductivity, D(τ) a tabulated dimensionless function and τ = (αt)1/2/r the dimensionless time formed with the thermal diffusivity α (He, 2005).

Because τ contains the unknown diffusivity, the evaluation is iterative. The software searches for the diffusivity, and for a small time correction, that make the recorded temperature rise a straight line against D(τ); the conductivity then follows from the slope. The earliest points are discarded, because they are dominated by the insulating film and the contact between sensor and specimen, and the recording must end before the heat reaches any outer boundary. Bohac et al. (2000) examined how the choice of time window affects the sensitivity of the fit to each parameter and showed that both properties are well determined only when the window spans a suitable range of dimensionless time.

Variants of the basic experiment exist for slabs, thin films and anisotropic materials (Gustavsson et al., 1994; Log and Gustafsson, 1995). Several of them require the volumetric heat capacity as an input, which adds a further measured quantity to the model.

The GUM expresses a result as a function of input quantities, assigns each input a standard uncertainty from statistical analysis of repeated observations (Type A) or from other knowledge (Type B), and combines them through first-order sensitivity coefficients (JCGM, 2008a; Taylor and Kuyatt, 1994). The approach assumes that the model is nearly linear over the range of the uncertainties. Supplement 1 replaces the linear combination with numerical propagation of full probability distributions by Monte Carlo sampling and is recommended when linearity or normality is doubtful (JCGM, 2008b). A fitted, iterative model such as that of the transient plane source is a natural candidate for the second route. The engineering literature on experimental uncertainty offers the same ideas in different vocabulary (Moffat, 1988).

The statement begins by saying what is being measured: for example, the thermal conductivity of a named material at a stated mean temperature, in a stated direction if the material is anisotropic, at a stated moisture condition if it is porous. The evaluation model is then written down in full, including the sensor geometry, the form of D(τ), the fitted parameters and the criterion by which the time window is chosen. Writing it out exposes which quantities are inputs with their own uncertainty and which are outputs of the fit.

Every quantity that enters the model, or that could make the experiment depart from it, is listed. Table 1 gives a starting list. Some inputs are instrumental and the same for every test with a given sensor. Others belong to the specimen and its mounting and change from test to test.

Each input receives a probability distribution. Where repeated observations exist, such as the noise of the temperature signal, a normal distribution with the observed standard deviation is appropriate. Where only limits are known, such as a tolerance on radius, the GUM recommends a rectangular distribution between the limits. Calibration certificates supply a standard uncertainty directly. The reasons for each assignment are recorded, because a reader can only judge the final figure by judging these choices. Correlations should be noted: the temperature coefficient and the power both depend on the same resistance measurement.

Before any propagation, each input is perturbed by a small amount and the evaluation is rerun on the recorded transient to find the change in conductivity. The resulting sensitivity coefficients show which inputs matter, and they can differ sharply from one material to another. They also reveal the condition of the fit. Parameter estimation theory shows that two parameters cannot be separated if their sensitivity curves are proportional over the time window (Beck and Arnold, 1977), and a near-proportionality between the sensitivities to conductivity and diffusivity is a warning that the window is poorly chosen. Where inputs interact strongly, a variance-based global analysis is more informative than one-at-a-time perturbation (Saltelli et al., 2008).

Values of all inputs are drawn from their distributions, the full evaluation, including the iteration and the fit, is repeated for each draw, and the resulting conductivities are collected. Supplement 1 gives guidance on the number of trials and on extracting a coverage interval from the output distribution (JCGM, 2008b). Noise in the transient is included by resampling the residuals of the fit or by adding synthetic noise of the observed variance. The linear GUM combination, using the coefficients from Step 4, should be computed alongside. Agreement between the two gives confidence in both, and disagreement shows that the model is too nonlinear for the simpler method.

Propagation only covers uncertainty within the model. Whether the model fits the experiment is a separate question, and three checks address it. The probing depth, conventionally taken as 2(αt)1/2 at the end of the window, must be smaller than the distance from the sensor edge to the nearest specimen boundary. The residuals of the fit should scatter randomly about zero; a systematic curve indicates boundary effects, contact resistance or drift. And the whole procedure should be run on a reference material of similar conductivity. If a check fails, the set-up is changed and the measurement repeated; a larger uncertainty is not a substitute.

The third check matters most for insulating materials. Zheng et al. (2020) showed that the standard evaluation can overestimate the conductivity of low-conductivity specimens because the sensor and its insulating film store and conduct a non-negligible share of the heat, which the ideal model ignores. A bias of this kind has to be corrected, or included as a separate component with its own justification.

The result is reported as a value with an expanded uncertainty and a stated coverage probability, together with the conditions of Step 1, the sensor, the power, the measurement time and the window used. The uncertainty budget, giving each input, its distribution, its sensitivity and its contribution, should accompany the result or be available on request, as laboratories working to recognised quality systems already do. Reporting the fitted diffusivity with its own uncertainty costs nothing extra and lets a reader test the result for consistency against an independent heat capacity.