- Email: [email protected]
Experimental study for thermal conductivity structure of lunar surface regolith: Effect of compressional stress
Icarus 221 (2012) 1180–1182
Contents lists available at SciVerse ScienceDirect
Icarus journal homepage: www.elsevier.com/locate/icarus
Note
Experimental study for thermal conductivity structure of lunar surface regolith: Effect of compressional stress Naoya Sakatani a,b,⇑, Kazunori Ogawa a, Yu-ichi Iijima a, Rie Honda c, Satoshi Tanaka a a
Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency, 3-l-l Yoshinodai, Chuo-ku, Sagamihara, Kanagawa 252-5210, Japan The Graduate University for Advanced Studies, Shonan Village, Hayama, Kanagawa 240-0193, Japan c Kochi University, 2-5-1 Akebono-cho, Kochi 780-8520, Japan b
a r t i c l e
i n f o
Article history: Received 12 January 2012 Revised 26 August 2012 Accepted 29 August 2012 Available online 13 September 2012
a b s t r a c t We measured the thermal conductivity of glass beads as a simple model material in order to investigate the compressional stress dependence of the thermal conductivity of powder materials. The measurements were conducted in a vertically elongated cylindrical sample container under vacuum conditions. Our results suggest that the compressional stress is one of the essential factors to understand the thermal conductivity structure in the regolith layer.
Ó 2012 Elsevier Inc. All rights reserved.
Keywords: Moon, Surface Regoliths Geophysics
1. Introduction Thermal conductivity of the surface regolith is an important parameter in understanding the interior thermal conditions of planets, asteroids, and satellites. For example, the crustal heat flow can be determined from subsurface thermal conductivity and the local temperature gradient. Measured values, and therefore, errors in measured thermal conductivity, directly affect the derived heat flow. The bulk (effective) thermal conductivity of powdered materials, such as lunar regolith, under vacuum, is orders of magnitude lower than that of solid rock. The regolith acts as a thermal insulating layer and controls short-term thermal evolution of small bodies (Akridge et al., 1998). The Heat Flow Experiments deployed during Apollo 15 and 17 missions serve as an important reference for understanding regolith thermal structure. Keihm and Langseth (1973) reported that the thermal conductivity of the upper few centimeters of the lunar surface is about 1:5 103 W=mK based on microwave brightness temperature analyses around the landing sites. On the other hand, Langseth et al. (1976) reported that the thermal conductivity at a few meters depth is about 102 W=mK as a result of in situ measurements in the Heat Flow Experiments. These results suggested that the thermal conductivity increases considerably within several tens of centimeters in depth. In the previous studies, the reason of this trend was deduced to be changes of bulk density (Fountain and West, 1970) and self-weighted stress (Horai, 1981) with depth. Laboratory measurements of Apollo samples showed that most lunar regolith has an effective conductivity from 5 104 to 4 103 W/mK for a bulk density from 1100 to 1950 kg/m3 and a temperature from 110 to 430 K (e.g. Cremers et al., 1970). Hence the lunar surface thermal conductivity up to 102 W/mK cannot be explained only by the bulk density variation. Horai (1981) measured the thermal conductivity of a basaltic regolith simulant and obtained 8:8 103 and 1:09 102 W/mK for the bulk density of 1700 and 1850 kg/m3, respectively. He suggested
⇑ Corresponding author at: Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency, 3-l-l Yoshinodai, Chuo-ku, Sagamihara, Kanagawa 252-5210, Japan. Fax: +81 42 759 8457. E-mail address: [email protected] (N. Sakatani). 0019-1035/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.icarus.2012.08.037
that these relatively high conductivities were due to his vertically-long sample container, in which the sample was compressed more by self-weight. Therefore the self-weighted stress might be one of the essential parameters for the thermal conductivity structure of the lunar regolith layer. However, Horai measured the thermal conductivity only at one depth and its dependence has not been investigated experimentally. In this study we experimentally investigate and confirm the effect of self-weighted stress on the thermal conductivity of powder media under vacuum. 2. Experiments 2.1. Sample selection In this study, glass beads of 58 5; 98 8; 428 72, and 855 145 lm in diameters were used as a model material sample. The glass beads have the following advantages in the measurements. Spherical beads with narrow size distributions form a uniform packing structure, which results in a uniform porosity (between 37% and 39%). This cancels the effect of the bulk density variation. The effective thermal conductivity of glass beads is comparable to that of rock powders. 2.2. Thermal conductivity measurement method The thermal conductivity was measured by the so-called line heat source method (e.g. Carslaw and Jaeger, 1959). This method is based on the assumption of radial heat diffusion from an infinite thin and long line heat source in the medium. After a sufficiently long heating time, a linear relation between the temperature and the natural logarithm of time appears with slope s represented by q=4pk, where q is heat generation per unit length of the heat source and k is thermal conductivity of the media. The thermal conductivity is determined from the slope of the linear regime (Presley and Christensen, 1997). A vertically elongated cylinder (made of methylpentene resin) shown in Fig. 1 was used as a sample container. Four nichrome wires were arranged perpendicularly in the container at depths of 1, 5, 15, and 30 cm as the thermal conductivity measurement points. The wire diameter is 180 lm and the resistance per unit
1181
Note / Icarus 221 (2012) 1180–1182
pffiffiffiffiffi Andersson and Bäckström (1976) reported that the relation, 2R= at > 4, is required to ignore the boundary effect of the sample container, where R is the distance from the heater to the boundary, a is the thermal diffusivity, and t is the time. According to Jones (1988), the linear regime is found when satisfying 2 t 50b =a, where b is the distance between the heat source and a point where the temperature is measured. The measurement time was set to be 1000 s and the linear regime from 400 to 1000 s was used for the thermal conductivity estimation. These satisfy above equations assuming a thermal diffusivity of 108 m2 /s. The methodological measurement error was estimated according to the method proposed by Presley and Christensen (1997). The largest error source was the deviation from the straight lines in the linear regimes, and it was less than 8.2%. Other errors originated from heat loss through the heaters and thermocouples, variation in the resistance of the nichrome heaters with temperature. Finally, the total error in our experiments was estimated to be 10% in the thermal conductivity values (see Presley and Christensen, 1997 for detailed method). 2.3. Experimental procedure
Fig. 1. Experimental configuration. The switching system was used to switch channels of the thermocouples and reference temperature points of the thermocouples are mounted in it.
The bulk density of each sample was estimated from their volume and mass before the experiments. For the measurements, the sample container was evacuated for over 24 h in a less than 104 Pa environment by rotary and turbo molecular pumps. Then, a constant current 40 mA (corresponding to q ¼ 77 mW/m) was supplied to each heater in turn, and the temperature increase of the heaters was recorded for 1000 s. For all data the slopes s were determined by least square fitting to the linear regime. The thermal conductivity was determined from the equation k ¼ q=4p s. The measurements were conducted three times for each sample. A typical temperature curve obtained during the measurements is shown in Fig. 2. Initial temperature of the system was 20 °C. After 1000 s of the heating, the temperature of the heaters increased up to 40 °C. Therefore the measured values represent thermal conductivities at 20–40 °C.
3. Results
Fig. 2. Typical temperature curve during the thermal conductivity measurements. length is 48:12 X=m. For temperature measurements of the nichrome wires, junction points of alumel–chromel thermocouples (76 lm diameters) were bonded at the center of the nichrome wires with a-Cyanoacrylate adhesives, whose representative sizes were less than 500 lm. Since this adhesive’s thermal conductivity is typically 0.1 W/mK, two orders higher than that of the surrounding medium, it does not affect the thermal conductivity measurements.
(a)
We found the average thermal conductivities of the glass beads under vacuum to be between 0.0021 and 0.0112 W/mK depending on the particle sizes and measurement depths (Fig. 3a). At each depth, higher thermal conductivity was found for larger glass beads, which was consistent with the results reported by Huetter et al. (2008). The smallest two samples had almost the same conductivity, whose difference would be hardly detectable due to the measurement error. The three measured values at each depth agreed within ±10%, except for the result of 855 145 lm beads at 15 cm depth. As shown in Fig. 3a, higher conductivities were found in the deeper layers for all samples. Possible explanations are an increase of the bulk density or the compressional stress with depth. The beads’ density variation from the surface to 30 cm depth was estimated to be less than 10%, as a result of a simple vibrating press experiment for the glass beads. Gusarov et al. (2003) predicted that the conductive contribution to the effective conductivity is proportional to the bulk density and the coordination number. Given that the coordination number depends linearly on the bulk density, the bulk density difference can provide approximately 20% variation at maximum in the conductive contribution. Therefore the observed large increase in the thermal conductivity is caused primarily by the self-weighted compressional stress. The contacts broadened by the compressional stress provide wide heat paths, through which heat can effectively flow between particles, and the conductive contribution increases.
(b)
Fig. 3. (a) Measurement results of glass beads’ thermal conductivity are summarized as a function of the measurement depth. The vertical error bars represent ranges in the three measurements. (b) Thermal conductivity of glass beads as a function of stress. Stress field in the sample container estimated from Janssen’s law is given in the inset (shaded area).
1182
Note / Icarus 221 (2012) 1180–1182
4. Discussion Stress distribution rzz of beads filled in a cylindrical container is expressed by following Janssen’s law (Masuda et al., 2006);
rzz ¼ qg
D 4K l z ; 1 exp D 4K l
ð1Þ
where q is the bulk density, g is the gravitational constant, D is the diameter of the cylinder, K is the ratio of horizontal to vertical stress, l is the friction coefficient between beads and the cylindrical wall, and z is depth. K and l are unknown parameters in this study. K is given by ð1 sin /Þ=ð1 þ sin /Þ, where / is the angle of internal friction. Castellanos et al. (2009) measured the angle of internal friction using glass beads and obtained 21–27 deg (corresponding to K = 0.38–0.47) depending on the normal stress. No values for l between glass and methylpenten resin were found in the literature, so here we assume l ¼ 0:3—0:5, which is the friction coefficient between glass beads and glass sheet (Ishibashi et al., 1994). According to the relationship between depth and stress, the experimental results in Fig. 3a can be plotted as a function of stress (Fig. 3b). The thermal conductivity variation with stress is interpreted as the change of the contact area, or the conductive contribution. Here we assume that the conductive contribution is proportional to the stress with an exponent of a constant x, and the effective thermal conductivity is expressed as k ¼ Arxzz þ krad , where k is the effective thermal conductivity, A is a constant, and krad represents the radiative contribution. The radiative conductivity was modeled by Schotte (1960) as follows;
krad ¼
1 ks
1u þ 4reuDp T 3 ; þ 4reD1 T 3
ð2Þ
p
where u is porosity, which is deduced using bulk density q and true density qs as u ¼ 1 q=qs ; ks is the thermal conductivity of the solid material, r is the Stefan– Boltzmann constant, e is the emissivity of the particle surface, Dp is the particle diameter, and T is temperature (Huetter et al., 2008). With the temperature of 30 °C (average temperature during the measurements) and median values of the particle sizes, we can obtain krad from 3:3 104 to 4:8 103 W/mK depending on the particle size. Using these values, k ¼ Arxzz þ krad was fitted to the data in Fig. 3b. The resultant values of x are 0:37 0:07; 0:39 0:05; 0:39 0:10, and 0:50 0:18 in order of increasing the particle size, which lie in the range between 1/3 and 3/5, as suggested by Halajian and Reichman (1969) and Pilbeam and Vaišnys (1973), respectively. Note that these values depend on the estimated stress distribution in the sample container. 5. Conclusion We investigated the stress dependence of the powder thermal conductivity, using glass beads as a simple model material. The greater the stress applied on the particles, the higher conductivity was obtained. This work revealed that the compressional stress is a very important factor for the thermal conductivity of powder materials. Even for the complex natural regolith, the self-weighted compressional stress must be considered for determining its thermal conductivity structure. Our results support the idea of Horai (1981), offering an explanation that selfweighted compressional stress is responsible for the increase in the thermal con-
ductivity between his experiments and those performed on Apollo regolith samples. We also suggest that not only the packing density but also the self-weighted stress is an important parameter for the thermal conductivity distribution of the regolith layer, especially in the deeper region. Moreover, this study indicates that regolith on a smaller body, with lower self-weighted stress due to its lower gravity, has a lower thermal conductivity. Acknowledgments The authors thank two reviewers for their constructive feedback of this paper. References Akridge, G., Benoit, P.H., Sears, D.W.G., 1998. Regolith and megaregolith formation of H-chondrites: Thermal constraints on the parent body. Icarus 132, 185–195. Andersson, P., Bäckström, G., 1976. Thermal conductivity of solids under pressure by the transient hot wire method. Rev. Sci. Instrum. 47 (2), 205–209. Carslaw, H.S., Jaeger, J.C., 1959. Conduction of Heat in Solids, second ed. Clarendon Press, Oxford. Castellanos, A., SoriaHoyo, C., Valverde, J.M., Quintanilla, M.A.S., 2009. Cohesion and internal friction of fine glass beads as affected by small in tensity vertical vibration. AIP Conf. Proc. 1145, 707–710. Cremers, C.J., Birkebak, R.C., Dawson, J.P., 1970. Thermal conductivity of fines from Apollo 11. Proc. Apollo 11 Lunar Sci. Conf. 3, 2045–2050. Fountain, J.A., West, E.A., 1970. Thermal conductivity of particulate basalt as a function of density in simulated lunar and martian environments. J. Geophys. Res. 75, 4063–4069. Gusarov, A.V., Laoui, T., Froyen, L., Titov, V.I., 2003. Contact thermal conductivity of a powder bed in selective laser sintering. Int. J. Heat Mass Transfer 43, 1103– 1109. Halajian, J.D., Reichman, J., 1969. Correlation of mechanical and thermal properties of the lunar surface. Icarus 10, 179–196. Horai, K., 1981. The effect of interstitial gaseous pressure on the thermal conductivity of a simulated Apollo 12 lunar soil sample. Phys. Earth Planet. Inter. 27, 60–71. Huetter, E.S., Koemle, N.I., Kargl, G., Kaufmann, E., 2008. Determination of the effective thermal conductivity of granular materials under varying pressure conditions. J. Geophys. Res. 113, E12004. Ishibashi, I., Perry, C., Agarwal, T.K., 1994. Experimental determinations of contact friction for spherical glass particles. Solid Fond. 34 (4), 79–84. Jones, B.W., 1988. Thermal conductivity probe: Development of method and application to a coarse granular medium. J. Phys. E: Sci. Instrum. 21, 832–839. Keihm, S.J., Langseth, M.G., 1973. Surface brightness temperatures at Apollo 17 heat flow site: Thermal conductivity of the upper 15 cm of regolith. Proc. 4th Lunar Sci. Conf. 3, 2503–2513. Langseth, M.G., Keihm, S.J., Peters, K., 1976. Revised lunar heat-flow values. Proc. Lunar Sci. Conf. 7, 3143–3171. Masuda, H., Higashitani, K., Yoshida, H., 2006. Powder Technology Handbook, third ed. CRC Press, New York. Presley, M.A., Christensen, P.R., 1997. Thermal conductivity measurements of particulate materials: 2. Results. J. Geophys. Res. 102 (E3), 6551–6566. Pilbeam, C.C., Vaišnys, J.R., 1973. Contact thermal conductivity in lunar aggregates. J. Geophys. Res. 78, 5233–5236. Schotte, W., 1960. Thermal conductivity of packed beds. AIChE J. 6, 63–67.
Contents lists available at SciVerse ScienceDirect
Icarus journal homepage: www.elsevier.com/locate/icarus
Note
Experimental study for thermal conductivity structure of lunar surface regolith: Effect of compressional stress Naoya Sakatani a,b,⇑, Kazunori Ogawa a, Yu-ichi Iijima a, Rie Honda c, Satoshi Tanaka a a
Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency, 3-l-l Yoshinodai, Chuo-ku, Sagamihara, Kanagawa 252-5210, Japan The Graduate University for Advanced Studies, Shonan Village, Hayama, Kanagawa 240-0193, Japan c Kochi University, 2-5-1 Akebono-cho, Kochi 780-8520, Japan b
a r t i c l e
i n f o
Article history: Received 12 January 2012 Revised 26 August 2012 Accepted 29 August 2012 Available online 13 September 2012
a b s t r a c t We measured the thermal conductivity of glass beads as a simple model material in order to investigate the compressional stress dependence of the thermal conductivity of powder materials. The measurements were conducted in a vertically elongated cylindrical sample container under vacuum conditions. Our results suggest that the compressional stress is one of the essential factors to understand the thermal conductivity structure in the regolith layer.
Ó 2012 Elsevier Inc. All rights reserved.
Keywords: Moon, Surface Regoliths Geophysics
1. Introduction Thermal conductivity of the surface regolith is an important parameter in understanding the interior thermal conditions of planets, asteroids, and satellites. For example, the crustal heat flow can be determined from subsurface thermal conductivity and the local temperature gradient. Measured values, and therefore, errors in measured thermal conductivity, directly affect the derived heat flow. The bulk (effective) thermal conductivity of powdered materials, such as lunar regolith, under vacuum, is orders of magnitude lower than that of solid rock. The regolith acts as a thermal insulating layer and controls short-term thermal evolution of small bodies (Akridge et al., 1998). The Heat Flow Experiments deployed during Apollo 15 and 17 missions serve as an important reference for understanding regolith thermal structure. Keihm and Langseth (1973) reported that the thermal conductivity of the upper few centimeters of the lunar surface is about 1:5 103 W=mK based on microwave brightness temperature analyses around the landing sites. On the other hand, Langseth et al. (1976) reported that the thermal conductivity at a few meters depth is about 102 W=mK as a result of in situ measurements in the Heat Flow Experiments. These results suggested that the thermal conductivity increases considerably within several tens of centimeters in depth. In the previous studies, the reason of this trend was deduced to be changes of bulk density (Fountain and West, 1970) and self-weighted stress (Horai, 1981) with depth. Laboratory measurements of Apollo samples showed that most lunar regolith has an effective conductivity from 5 104 to 4 103 W/mK for a bulk density from 1100 to 1950 kg/m3 and a temperature from 110 to 430 K (e.g. Cremers et al., 1970). Hence the lunar surface thermal conductivity up to 102 W/mK cannot be explained only by the bulk density variation. Horai (1981) measured the thermal conductivity of a basaltic regolith simulant and obtained 8:8 103 and 1:09 102 W/mK for the bulk density of 1700 and 1850 kg/m3, respectively. He suggested
⇑ Corresponding author at: Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency, 3-l-l Yoshinodai, Chuo-ku, Sagamihara, Kanagawa 252-5210, Japan. Fax: +81 42 759 8457. E-mail address: [email protected] (N. Sakatani). 0019-1035/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.icarus.2012.08.037
that these relatively high conductivities were due to his vertically-long sample container, in which the sample was compressed more by self-weight. Therefore the self-weighted stress might be one of the essential parameters for the thermal conductivity structure of the lunar regolith layer. However, Horai measured the thermal conductivity only at one depth and its dependence has not been investigated experimentally. In this study we experimentally investigate and confirm the effect of self-weighted stress on the thermal conductivity of powder media under vacuum. 2. Experiments 2.1. Sample selection In this study, glass beads of 58 5; 98 8; 428 72, and 855 145 lm in diameters were used as a model material sample. The glass beads have the following advantages in the measurements. Spherical beads with narrow size distributions form a uniform packing structure, which results in a uniform porosity (between 37% and 39%). This cancels the effect of the bulk density variation. The effective thermal conductivity of glass beads is comparable to that of rock powders. 2.2. Thermal conductivity measurement method The thermal conductivity was measured by the so-called line heat source method (e.g. Carslaw and Jaeger, 1959). This method is based on the assumption of radial heat diffusion from an infinite thin and long line heat source in the medium. After a sufficiently long heating time, a linear relation between the temperature and the natural logarithm of time appears with slope s represented by q=4pk, where q is heat generation per unit length of the heat source and k is thermal conductivity of the media. The thermal conductivity is determined from the slope of the linear regime (Presley and Christensen, 1997). A vertically elongated cylinder (made of methylpentene resin) shown in Fig. 1 was used as a sample container. Four nichrome wires were arranged perpendicularly in the container at depths of 1, 5, 15, and 30 cm as the thermal conductivity measurement points. The wire diameter is 180 lm and the resistance per unit
1181
Note / Icarus 221 (2012) 1180–1182
pffiffiffiffiffi Andersson and Bäckström (1976) reported that the relation, 2R= at > 4, is required to ignore the boundary effect of the sample container, where R is the distance from the heater to the boundary, a is the thermal diffusivity, and t is the time. According to Jones (1988), the linear regime is found when satisfying 2 t 50b =a, where b is the distance between the heat source and a point where the temperature is measured. The measurement time was set to be 1000 s and the linear regime from 400 to 1000 s was used for the thermal conductivity estimation. These satisfy above equations assuming a thermal diffusivity of 108 m2 /s. The methodological measurement error was estimated according to the method proposed by Presley and Christensen (1997). The largest error source was the deviation from the straight lines in the linear regimes, and it was less than 8.2%. Other errors originated from heat loss through the heaters and thermocouples, variation in the resistance of the nichrome heaters with temperature. Finally, the total error in our experiments was estimated to be 10% in the thermal conductivity values (see Presley and Christensen, 1997 for detailed method). 2.3. Experimental procedure
Fig. 1. Experimental configuration. The switching system was used to switch channels of the thermocouples and reference temperature points of the thermocouples are mounted in it.
The bulk density of each sample was estimated from their volume and mass before the experiments. For the measurements, the sample container was evacuated for over 24 h in a less than 104 Pa environment by rotary and turbo molecular pumps. Then, a constant current 40 mA (corresponding to q ¼ 77 mW/m) was supplied to each heater in turn, and the temperature increase of the heaters was recorded for 1000 s. For all data the slopes s were determined by least square fitting to the linear regime. The thermal conductivity was determined from the equation k ¼ q=4p s. The measurements were conducted three times for each sample. A typical temperature curve obtained during the measurements is shown in Fig. 2. Initial temperature of the system was 20 °C. After 1000 s of the heating, the temperature of the heaters increased up to 40 °C. Therefore the measured values represent thermal conductivities at 20–40 °C.
3. Results
Fig. 2. Typical temperature curve during the thermal conductivity measurements. length is 48:12 X=m. For temperature measurements of the nichrome wires, junction points of alumel–chromel thermocouples (76 lm diameters) were bonded at the center of the nichrome wires with a-Cyanoacrylate adhesives, whose representative sizes were less than 500 lm. Since this adhesive’s thermal conductivity is typically 0.1 W/mK, two orders higher than that of the surrounding medium, it does not affect the thermal conductivity measurements.
(a)
We found the average thermal conductivities of the glass beads under vacuum to be between 0.0021 and 0.0112 W/mK depending on the particle sizes and measurement depths (Fig. 3a). At each depth, higher thermal conductivity was found for larger glass beads, which was consistent with the results reported by Huetter et al. (2008). The smallest two samples had almost the same conductivity, whose difference would be hardly detectable due to the measurement error. The three measured values at each depth agreed within ±10%, except for the result of 855 145 lm beads at 15 cm depth. As shown in Fig. 3a, higher conductivities were found in the deeper layers for all samples. Possible explanations are an increase of the bulk density or the compressional stress with depth. The beads’ density variation from the surface to 30 cm depth was estimated to be less than 10%, as a result of a simple vibrating press experiment for the glass beads. Gusarov et al. (2003) predicted that the conductive contribution to the effective conductivity is proportional to the bulk density and the coordination number. Given that the coordination number depends linearly on the bulk density, the bulk density difference can provide approximately 20% variation at maximum in the conductive contribution. Therefore the observed large increase in the thermal conductivity is caused primarily by the self-weighted compressional stress. The contacts broadened by the compressional stress provide wide heat paths, through which heat can effectively flow between particles, and the conductive contribution increases.
(b)
Fig. 3. (a) Measurement results of glass beads’ thermal conductivity are summarized as a function of the measurement depth. The vertical error bars represent ranges in the three measurements. (b) Thermal conductivity of glass beads as a function of stress. Stress field in the sample container estimated from Janssen’s law is given in the inset (shaded area).
1182
Note / Icarus 221 (2012) 1180–1182
4. Discussion Stress distribution rzz of beads filled in a cylindrical container is expressed by following Janssen’s law (Masuda et al., 2006);
rzz ¼ qg
D 4K l z ; 1 exp D 4K l
ð1Þ
where q is the bulk density, g is the gravitational constant, D is the diameter of the cylinder, K is the ratio of horizontal to vertical stress, l is the friction coefficient between beads and the cylindrical wall, and z is depth. K and l are unknown parameters in this study. K is given by ð1 sin /Þ=ð1 þ sin /Þ, where / is the angle of internal friction. Castellanos et al. (2009) measured the angle of internal friction using glass beads and obtained 21–27 deg (corresponding to K = 0.38–0.47) depending on the normal stress. No values for l between glass and methylpenten resin were found in the literature, so here we assume l ¼ 0:3—0:5, which is the friction coefficient between glass beads and glass sheet (Ishibashi et al., 1994). According to the relationship between depth and stress, the experimental results in Fig. 3a can be plotted as a function of stress (Fig. 3b). The thermal conductivity variation with stress is interpreted as the change of the contact area, or the conductive contribution. Here we assume that the conductive contribution is proportional to the stress with an exponent of a constant x, and the effective thermal conductivity is expressed as k ¼ Arxzz þ krad , where k is the effective thermal conductivity, A is a constant, and krad represents the radiative contribution. The radiative conductivity was modeled by Schotte (1960) as follows;
krad ¼
1 ks
1u þ 4reuDp T 3 ; þ 4reD1 T 3
ð2Þ
p
where u is porosity, which is deduced using bulk density q and true density qs as u ¼ 1 q=qs ; ks is the thermal conductivity of the solid material, r is the Stefan– Boltzmann constant, e is the emissivity of the particle surface, Dp is the particle diameter, and T is temperature (Huetter et al., 2008). With the temperature of 30 °C (average temperature during the measurements) and median values of the particle sizes, we can obtain krad from 3:3 104 to 4:8 103 W/mK depending on the particle size. Using these values, k ¼ Arxzz þ krad was fitted to the data in Fig. 3b. The resultant values of x are 0:37 0:07; 0:39 0:05; 0:39 0:10, and 0:50 0:18 in order of increasing the particle size, which lie in the range between 1/3 and 3/5, as suggested by Halajian and Reichman (1969) and Pilbeam and Vaišnys (1973), respectively. Note that these values depend on the estimated stress distribution in the sample container. 5. Conclusion We investigated the stress dependence of the powder thermal conductivity, using glass beads as a simple model material. The greater the stress applied on the particles, the higher conductivity was obtained. This work revealed that the compressional stress is a very important factor for the thermal conductivity of powder materials. Even for the complex natural regolith, the self-weighted compressional stress must be considered for determining its thermal conductivity structure. Our results support the idea of Horai (1981), offering an explanation that selfweighted compressional stress is responsible for the increase in the thermal con-
ductivity between his experiments and those performed on Apollo regolith samples. We also suggest that not only the packing density but also the self-weighted stress is an important parameter for the thermal conductivity distribution of the regolith layer, especially in the deeper region. Moreover, this study indicates that regolith on a smaller body, with lower self-weighted stress due to its lower gravity, has a lower thermal conductivity. Acknowledgments The authors thank two reviewers for their constructive feedback of this paper. References Akridge, G., Benoit, P.H., Sears, D.W.G., 1998. Regolith and megaregolith formation of H-chondrites: Thermal constraints on the parent body. Icarus 132, 185–195. Andersson, P., Bäckström, G., 1976. Thermal conductivity of solids under pressure by the transient hot wire method. Rev. Sci. Instrum. 47 (2), 205–209. Carslaw, H.S., Jaeger, J.C., 1959. Conduction of Heat in Solids, second ed. Clarendon Press, Oxford. Castellanos, A., SoriaHoyo, C., Valverde, J.M., Quintanilla, M.A.S., 2009. Cohesion and internal friction of fine glass beads as affected by small in tensity vertical vibration. AIP Conf. Proc. 1145, 707–710. Cremers, C.J., Birkebak, R.C., Dawson, J.P., 1970. Thermal conductivity of fines from Apollo 11. Proc. Apollo 11 Lunar Sci. Conf. 3, 2045–2050. Fountain, J.A., West, E.A., 1970. Thermal conductivity of particulate basalt as a function of density in simulated lunar and martian environments. J. Geophys. Res. 75, 4063–4069. Gusarov, A.V., Laoui, T., Froyen, L., Titov, V.I., 2003. Contact thermal conductivity of a powder bed in selective laser sintering. Int. J. Heat Mass Transfer 43, 1103– 1109. Halajian, J.D., Reichman, J., 1969. Correlation of mechanical and thermal properties of the lunar surface. Icarus 10, 179–196. Horai, K., 1981. The effect of interstitial gaseous pressure on the thermal conductivity of a simulated Apollo 12 lunar soil sample. Phys. Earth Planet. Inter. 27, 60–71. Huetter, E.S., Koemle, N.I., Kargl, G., Kaufmann, E., 2008. Determination of the effective thermal conductivity of granular materials under varying pressure conditions. J. Geophys. Res. 113, E12004. Ishibashi, I., Perry, C., Agarwal, T.K., 1994. Experimental determinations of contact friction for spherical glass particles. Solid Fond. 34 (4), 79–84. Jones, B.W., 1988. Thermal conductivity probe: Development of method and application to a coarse granular medium. J. Phys. E: Sci. Instrum. 21, 832–839. Keihm, S.J., Langseth, M.G., 1973. Surface brightness temperatures at Apollo 17 heat flow site: Thermal conductivity of the upper 15 cm of regolith. Proc. 4th Lunar Sci. Conf. 3, 2503–2513. Langseth, M.G., Keihm, S.J., Peters, K., 1976. Revised lunar heat-flow values. Proc. Lunar Sci. Conf. 7, 3143–3171. Masuda, H., Higashitani, K., Yoshida, H., 2006. Powder Technology Handbook, third ed. CRC Press, New York. Presley, M.A., Christensen, P.R., 1997. Thermal conductivity measurements of particulate materials: 2. Results. J. Geophys. Res. 102 (E3), 6551–6566. Pilbeam, C.C., Vaišnys, J.R., 1973. Contact thermal conductivity in lunar aggregates. J. Geophys. Res. 78, 5233–5236. Schotte, W., 1960. Thermal conductivity of packed beds. AIChE J. 6, 63–67.