# A Miniaturized 3D Heat Flux Sensor to Characterize Heat Transfer in Regolith of Planets and Small Bodies

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## Abstract

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## 1. Introduction

## 2. Sensor Description and Operation

## 3. Analytical Solution of Heat Conduction in the Regolith

#### 3.1. Properties of the Analytical Solution

#### 3.2. Effect of Contact Resistance and Covering Film

- for ${r}_{0}<r<{r}_{1}$: the film (or thermal contact resistance) with a thermal conductivity, ${k}_{1}$, and
- for $r>{r}_{1}$: the thermal conductivity of the regolith, ${k}_{r}$.

## 4. Experimental Results

- First, the average power injected in the sphere as a function of pitch angle is 48–64 mW, which is within the range of the expected values predicted by the solution: $4\pi {k}_{r}\delta T{r}_{0}$ = 37–88 mW for ${k}_{r}$ = 0.1–0.2 W/(mK) and $\delta $T = 6–6.5 ${}^{\circ}$C.It must be noted that the average power depends on the final height of the sensor with respect to the heater plate, within the regolith. Since the temperature of the sphere is kept constant at 35.5 ${}^{\circ}$C, depending on the resulting effective height after a pitch rotation, the temperature of the immediate regolith changes. We can observe this effect in Figure 11: in the first 3 intervals on the left, the effective height of the sphere was corrected after the corresponding changes in the angle, while that correction was dismissed in the rest of the intervals, to prevent slow temperature re-distributions in the regolith. As a consequence, in the second half of the experiment, for higher angles the effective height increases, which means that the temperature in the immediate regolith of the sensor decreases, and therefore more power is needed in the sectors.
- Second, the differences between the sectors grow as the heat flux increases. This effect is more clearly seen in Figure 12, where the difference between the power injected in the sectors and their average, $\Delta {P}_{i}$, is shown for all angles and both heat fluxes. The sector differences, $\Delta {P}_{i}$ increase for increasing flux.In order to see if Figure 12 matches the behaviour predicted by the theory, we may compare those with the results in Figure 7 for pitch angles between 0${}^{\circ}$ and 12.5${}^{\circ}$. The expected tendency is that the sectors in the upper position of the sphere (at 0${}^{\circ}$) have the same power (red and magenta lines), as well as the sectors at the lower position (blue and cyan lines) at 0${}^{\circ}$. For increasing angles, the sector powers separate (one increasing and the other decreasing). The main difference is that the magenta and red curves do not cross at 0${}^{\circ}$ pitch. However, if we apply the theoretical expressions for the experiment data, allowing an angle misalignment in the sectors of 11${}^{\circ}$, we obtain the theoretical solution seen in Figure 13. As it can be seen, the expected tendency, allowing for this adjustment, is very close to the experimental result. Other possible causes of this behaviour may be that the thin film wrapping the sphere is thicker near the north pole of the sphere. This may have generated a non-ideal behaviour for the upper sectors.

#### Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Photograph of the 4-sector spherical sensor (10 mm diameter), before being wrapped and inserted into the regolith simulant.

**Figure 2.**Schematic showing the placement of the Pt resistors associated with the four spherical sectors and cores.

**Figure 3.**Heat flux on the complete domain for ${k}_{r}=0.1$ W/(mK), $\delta T=0$ K and ${\dot{Q}}_{0}=1$ W/m${}^{2}$. As it may be seen the $\dot{Q}(r\to \infty )\to {\dot{Q}}_{0}\widehat{x}$. The bottom of the sphere is being heated whereas its upper part loses heat to the surrounding regolith. In this case, a heat flux in the $\widehat{x}$ direction has been considered.

**Figure 4.**(

**Left**) Heat flux on the surface of the sphere for ${k}_{r}=0.003$ W/(mK), $\delta T=10$ K, and ${\dot{Q}}_{0}$ = 1 W/m${}^{2}$. (

**Right**) Integral of the heat fluxes on each of the sectors for the same conditions. In this case, a heat flux in the $\widehat{x}$ direction has been considered.

**Figure 5.**(

**Left**) Heat flux on the surface of the sphere for ${k}_{r}=0.03$ W/(mK), $\delta T=10$ K, and ${\dot{Q}}_{0}$ = 1 W/m${}^{2}$. (

**Right**) Integral of the heat fluxes on each of the sectors for the same conditions. In this case, a heat flux in the $\widehat{x}$ direction has been considered.

**Figure 7.**(

**Left**) Heat flux on the surface of the sphere for ${k}_{r}=0.003$ W/(mK), $\delta T=10$ K and ${\dot{Q}}_{0}$ = 1 W/m${}^{2}$. (

**Right**) Integral of the heat fluxes on each of the sectors for pitch rotations, for the same conditions. In this case, a heat flux in the $\widehat{z}$ direction has been considered. This will be the configuration tested in the experimental results.

**Figure 8.**Amplitude of the heat flux variation on the surface, $3{k}_{1}{B}_{1}$, as a function of the conductivity of the regolith, for different values of the conductivity of the covering film, ${k}_{1}$. The film thickness is 100 $\mathsf{\mu}$m and ${\dot{Q}}_{0}=1$ W/m${}^{2}$.

**Figure 9.**Schematic of the experimental setup. The complete structure has been placed in a climate chamber. The temperature of the bottom plate allows setting different heat fluxes in the regolith. The inner walls containing the regolith simulant are made of polystyrene.

**Figure 11.**Sector powers in a 170 h experiment in which heat flux magnitude and angle are changed as a function of time. The plotted signals are the result of concatenating 30 min time intervals in which a steady state has been achieved. Note that height readjustment after angle change was made only after the first 3 angles during the first part of the experiment.

**Figure 12.**Sector power differences in the experiment of Figure 11.

**Figure 13.**Theoretical prediction of the results in Figure 12, in case a misalignment of 11${}^{\circ}$ in the sector placement is allowed.

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**MDPI and ACS Style**

Domínguez-Pumar, M.; Rodríguez-Manfredi, J.-A.; Jiménez, V.; Bermejo, S.; Pons-Nin, J. A Miniaturized 3D Heat Flux Sensor to Characterize Heat Transfer in Regolith of Planets and Small Bodies. *Sensors* **2020**, *20*, 4135.
https://doi.org/10.3390/s20154135

**AMA Style**

Domínguez-Pumar M, Rodríguez-Manfredi J-A, Jiménez V, Bermejo S, Pons-Nin J. A Miniaturized 3D Heat Flux Sensor to Characterize Heat Transfer in Regolith of Planets and Small Bodies. *Sensors*. 2020; 20(15):4135.
https://doi.org/10.3390/s20154135

**Chicago/Turabian Style**

Domínguez-Pumar, Manuel, Jose-Antonio Rodríguez-Manfredi, Vicente Jiménez, Sandra Bermejo, and Joan Pons-Nin. 2020. "A Miniaturized 3D Heat Flux Sensor to Characterize Heat Transfer in Regolith of Planets and Small Bodies" *Sensors* 20, no. 15: 4135.
https://doi.org/10.3390/s20154135