Development of a Small-Sized Line Heat Source Apparatus for the Thermal Conductivity Measurement of Extraterrestrial Soils

Abstract

The thermal conductivity of planetary soils, or regolith, is essential for understanding the present global thermal state of bodies. The thermal conductivity of lunar soils is important with respect to lunar crustal heat flow. Although in situ measurements were performed by the Apollo 15 and 17 missions, laboratory measurements of the returned lunar samples have not reproduced the estimated subsurface values. Since the amount of extraterrestrial soil samples is limited, a small apparatus is needed to measure their thermal conductivity. In this study, we developed an apparatus enabling the measurement of the thermal conductivity of a small amount of soil (< 10 g) via the line heat source method as a function of compressional pressure under vacuum conditions. The thermal conductivity of glass beads and lunar regolith simulant derived by the new apparatus is higher than that obtained from the larger line heat source, and then, reliable apparatus. To evaluate the experimental results, we performed numerical simulation of the temperature evolution during the line heat source measurement, and found that the thermal conductivity derived from the simulation data is higher than the input thermal conductivity. This is consistent with the experimental results and is caused by the heat loss through a line heat source with a limited length. The difference depends on the contact conductance between the sample and the line heat source, and the calibration factors for each sample are determined.

Appendices

Appendix 1: Model Verification

To test the validity of our numerical model of the line heat source measurements, we performed additional simulations with some specific conditions.

1.1 1.1. Effect of Heating Wire Length

First, the dependence of the length of the nichrome wire L on the estimated thermal conductivity is investigated. The length of the wire is varied from 10 mm to infinitely long with $k_{true}$ = 0.004–0.012 W$\cdot$m$^{-1}$ K$^{-1}$. Figure 7 shows the results assuming the contact conductance parameter c = 10 000 m$^{-1}$. We confirmed that the simulation with the infinitely long heating wire gives true thermal conductivity as expected from the principle of the line heat source measurements. If $L < 30$ mm, then $k_{measure}$ deviates from $k_{true}$ due to the heat loss through the heating wire.

Fig. 7

Dependence of $k_{measure} - k_{true}$ on the length of the heating wire L. The contact conductance parameter c is fixed at 10,000 $m^{-1}$, and three cases of $k_{true}$ = 0.004, 0.008, and 0.012 W m$^{-1}$ K$^{-1}$ are shown. Our small apparatus has $L = 15$ mm

1.2 1.2. Effect of Boundary Conditions

The sensitivity of the boundary distance R and the boundary conditions (isothermal or adiabatic) is also investigated (Fig. 8). As a result, we found that $k_{measure}$ does not significantly depend on R or the boundary condition. This means that a higher $k_{measure}$ than $k_{true}$ is not contributed from the boundary effect, but only from the heat loss through the heating wire.

Fig. 8

Effect of the boundary distance and boundary conditions. On the horizontal axis, 5 and 10 are boundary distances in mm, and “iso” and “ad” indicate isothermal and adiabatic boundary conditions, respectively

1.3 1.3. Effect of Heating Wire Diameter

Our simulation does not involve the effect of the thermocouple. Heat loss through the thermocouple lines (alumel and chromel wires of 50 ${\upmu }$m) will also affect the thermal conductivity estimate in addition to the nichrome wire. To approximately evaluate this effect, we carried out the numerical simulation with changing the diameter of the nichrome wire. Because the thermal conductivity of alumel and chromel is roughly comparable with that of the nichrome, taking into account the heat loss via the thermocouple is equivalent to doubling the cross-sectional area of the nichrome wire. In other words, it corresponds to the case for 70 ${{\upmu }}$m diameter. Figure 9 shows the thermal conductivity estimate in terms of the wire diameter. As seen in this graph, the wire diameter significantly affect the offset from the true thermal conductivity. If the wire diameter is doubled, the offset is also doubled. Therefore, the effect of the thermocouples as paths of the heat loss is not negligible, and the appropriate contact conductance parameter c reproducing the experimental results would change. At the same time, we find that thinner wires are very useful in reducing the offset. However, the offset related to the thermocouple may be overestimated, since the thermocouples themselves do not generate heat unlike the nichrome wire and the temperature gradient is expected to be smaller than that of the nichrome wire. For a more realistic simulation, our two-dimensional model would need to be extended to three dimensions.

Fig. 9

Effect of the heating wire diameter. The contact conductance parameter c is fixed at 10,000 $m^{-1}$, and three cases of $k_{true}$ = 0.004, 0.008, and 0.012 W·m$^{-1}$·K$^{-1}$ are shown. The nichrome wire diameter we used for the small line heat source is 50 ${{\upmu }}$m

Appendix 2: Numerical Simulation for the Large Apparatus

In Sect. 3.2, we compared the measurement results of the new small apparatus and the large apparatus by Sakatani et al. [16, 17]. The latter should be more reliable. Here, we evaluate the reliability of the large apparatus based on the numerical simulation of the line heat source measurements.

The geometric setting of the simulation is basically the same as the small one, shown in Fig. 6a, but the dimensions are optimized for the large apparatus. The large apparatus has the nichrome wire with the length and diameter of $L = 100$ mm and 180 ${\upmu }$m, respectively. The radial boundary distance is $R = 20$ mm. The contact conductance parameter c is fixed at 10,000 m$^{-1}$. We used a linear fitting regime between 400 and 1000 s to derive the thermal conductivity. Fig. 10 shows the numerical result. In the range of the true thermal conductivity $k_{true}$ from 0.002 to 0.012 W·m$^{-1}$·K$^{-1}$, which covers the collected thermal conductivity data by Sakatani et al. [16, 17], the relative difference of the measured conductivity $k_{measure}$ from the true conductivity is less than 1.2%. Therefore, it is reasonable to treat the large apparatus data as true thermal conductivity.

Fig. 10

Result of numerical simulation of large apparatus experiments. The vertical axis shows the relative difference (in percentage) of the estimated thermal conductivity from the true conductivity