KD2 calculates its values for thermal conductivity, resistivity, and diffusivity by monitoring the dissipation of heat from a line heat source given a known voltage.   The equation for radial heat conduction in a homogeneous, isotropic medium is given by   (1)   where T is temperature (°C), t is time (s), k is thermal diffusivity (m2 s-1), and r is radial distance (m). When a long, electrically heated probe is introduced into a medium, the rise in temperature from an initial temperature, T0, at some distance, r, from the probe is   (2)   where q is the heat produced per unit length per unit time (W m-1) and Ei is the exponential integral function   (3)   with a = r2/4kt and g is Euler's constant (0.5772…). When t is large, the higher order terms can be ignored, so combining Eqs. (2) and (3) yields (4)   where lh is the thermal conductivity of the medium (W m-1C-1). It is apparent from the relationship between thermal conductivity and DT = T-T0, shown in Eq. (4), that DT and ln(t) are linearly related with a slope m = (q/4plh). Linearly regressing DT on ln(t) yields a slope that, after rearranging, gives the thermal conductivity as (5)   where q is known from the power supplied to the heater. The diffusivity can also be obtained from Eq. (4). The intersection of the regression line with the t axis (DT = 0) gives (6).   From the calculated t0(from the intercept of DT vs. ln(t)) and finite r, Eq. (6) gives the diffusivity. Because the higher order terms of Eq. (3) have been neglected, Eq. (4) is not exact. However, if the slope and intercept are computed only for DT and ln(t) values, where t is large enough to ignore the higher order terms, Eq. (5) and (6) give correct values for lh and k. To verify these relationships, realistic values of lh and k were supplied to Eq. (2), varying both lh and volumetric heat capacity (rcp), and the resulting slope and intercept tabulated for t ranging from 1 to 30 s. Plots of slope vs. theoretical lh and ln(intercept) vs. ln(theoretical k) show an exact linear relationship (Fig. 1 and 2, respectively) with low cross-covariance. Figure 1: calculated lh vs. theoretical lh   　 　   Figure 2: calculated k vs. theoretical k   The experimental analysis differs from the theoretical shown in Eq. (2) in that the heater and sensor have their own conductivity and diffusivity, which, in general, differ from those of the medium being measured. We have shown experimentally that the relationships in Eq. (5) and (6) still allow calculations of lh and k, but empirical factors must be introduced to correct for heater thermal properties.   Assumptions: The thermal conductivity measurement assumes several things: the long heat source can be treated as a infinitely long heat source, the medium is both homogeneous and isotropic, and a uniform initial temperature, T0. Although these assumptions are not true in the strict sense, they are adequate for accurate thermal properties measurements.   Further Readings: Bristow, K.L., White, R.D., Kluitenberg, G.J., 1994 Comparison of Single and Dual Probes for Measuring Soil Thermal Properties with Transient Heating. Australian Journal of Soil Research 32, 447-464. Bruijn, P.J, van Haneghem, I.A., Schenk, J. 1983 An Improved Nonsteady-State Probe Method for Measurements in Granular Materials. Part 1: Theory. High Temperatures - High Pressures 15, 359-366 Shiozawa, S., Campbell, G.S., 1990. Soil Thermal Conductivity. Remote Sensing Rev. 5, 301-310. van Haneghem, I.A., Schenk, J., Boshoven, H.P.A., 1983. An Improved Nonsteady-State Probe method for Measurements in Granular Materials. Part II: Experimental Results. High Temperatures - High Pressures 15, 367-374.