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Thermal conductivity measurement of liquid materials by a hot-disk method in short-duration microgravity environments
Materials Science and Engineering A276 (2000) 117 – 123 www.elsevier.com/locate/msea
Thermal conductivity measurement of liquid materials by a hot-disk method in short-duration microgravity environments Hideaki Nagai *, Fabrice Rossignol, Yoshinori Nakata, Takashi Tsurue, Masaaki Suzuki, Takeshi Okutani Hokkaido National Industrial Research Institute (HNIRI), 2 -17 -2 -1, Tsukisamu-higashi, Toyohira-ku, Sapporo 062 -8517, Japan Received 6 May 1999; received in revised form 25 June 1999
Abstract The thermal conductivities of silicone oils with various viscosities and mercury were measured by a hot-disk method in short-duration microgravity environments. The thermal conductivities of silicone oil with low viscosity were affected by the thermal convection on the ground, but the thermal convection was suppressed in microgravity. The thermal conductivities of highly viscous samples were not influenced by thermal convection. The thermal conductivity of mercury measured in microgravity was about 3% lower than that measured on the ground around room temperature. The thermal conductivity measurement conditions on the ground and in microgravity for which there was no influence from thermal convection could be estimated by using the Rayleigh number. © 2000 Elsevier Science S.A. All rights reserved. Keywords: Thermal conductivity; Hot-disk method; Microgravity; Silicone oil; Mercury
1. Introduction Thermal conductivity is an important property in many heat-transfer simulations. This value can be measured by the thermal response when heat or energy is applied to a specimen. However, it is difficult to measure the thermal conductivities of liquid materials (water, organic solvents, molten metals, etc.) precisely because heat is transferred by convection as well as conduction. In microgravity, however, it is possible to measure the thermal conductivities of liquid materials precisely because thermal convection is suppressed. The thermal conductivities of liquid materials are measured by (1) a steady heat flow method [1], (2) a stepwise heating method [2], (3) a transient hot-wire method [3] and (4) a laser flash method [4]. In microgravity experiments, the hot-wire method is more suitable because of its short time in measurement. Nakamura et al. [5,6] reported that the thermal conductivity of molten indium antimonide was measured by the hot-wire method in microgravity (using sounding * Corresponding author. Fax: +81-11-8578984. E-mail address: [email protected] (H. Nagai)
rockets (6 min microgravity) and drop shafts (10 s microgravity)). Recently, Gustafsson developed the hot-disk method for measuring thermal conductivity [7]. This method uses a transient plane source (TPS) element both as a heat source and temperature sensor, in the same way as a thin wire is used in the hot-wire method. TPS element is made of thin metal foil and its conducting pattern is a double spiral, which with some approximation resembles a hot disk. For the hot-wire method, a thin bare wire has been utilized, and only the thermal conductivity of insulating materials can be measured. In contrast, those of not only insulating materials but also electrical conducting materials can be measured by the hot-disk method because the both sides of TPS element was covered with a thin insulating layer. A reliable value could then be obtained by carefully calibrating the insulating layer effect [8]. In this study, the thermal conductivities of silicone oils with various viscosities were measured by the hot-disk method in microgravity during parabolic flight and that of mercury was measured in microgravity using a drop tower and drop shafts [9]. The effect of gravity on the experimental results on thermal conductivity is discussed in this study.
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the sample material (Eq. (2)). The thermal conductivity is calculated from the slope of the most straight line fit (Eq. (4)). Thermal conductivity and thermal diffusivity of the sample material are related as follows:
2. Experimental
2.1. Principle of the hot-disk method In hot-disk method experiments [7], the normal arrangement is to pass a constant current through a hot-disk sensor and simultaneously record its voltage changes. When analyzing the behavior of the hot-disk sensor during a transient recording, it is convenient to express the time dependent resistance (R(t)) with the following expression:
l= rCpk
R(t)= R0[1+aDT(t)]
2.2. Equipment setup and experimental procedure
(1)
where R0 is the resistance of the hot-disk sensor before the transient recording, a is the temperature coefficient of resistance, and DT(t) is the time-dependent temperature increase of the hot-disk sensor. In this equation, the temperature increase is expressed as a function of only one variable t, which is defined as: t = (t/u)1/2, u= d 2/k
(2)
where t is the time measured from the start of the transient heating, u is the characteristic time, d is the radius of the hot disk, and k is the thermal diffusivity of the sample material. For a hot-disk sensor, DT(t) is given by the following equation, from which the thermal conductivity and diffusivity can be obtained: DT(t)=P0(p 3/2dl)-1D(t)
(3)
where P0 is the total output power, l is the thermal conductivity of the sample material, and D(t) is the theoretical expression of the time-dependent temperature increase. We substitute Eq. (3) for Eq. (1), then obtain Eq. (4). R(t)=R0[1+aP0(p 3/2dl) − 1D(t)]
(4)
By plotting the measured resistivities R(t) vs. D(t), we can get a straight line according to Eq. (4) if the characteristic time (u) has the proper value. It is possible to find this value of u by varying it in an iterative procedure until the correlation coefficient between R(t) and D(t) reaches its maximum value. This particular value of u is used to obtain the thermal diffusivity of
Fig. 1. Schematic diagram of the hot-disk method.
(5)
where r is the density of the sample, and Cp is the specific heat of the sample. The specific heat per unit volume (rCp) can be calculated by using the values of the thermal diffusivity and thermal conductivity of the sample material.
The thermal conductivities of liquid materials were measured by the hot-disk thermal constant analyser (Hot Disk Inc., Sweden). The schematic diagram of the thermal conductivity measurement setup is shown in Fig. 1. The hot-disk method utilizes a thin disk-shaped sensor (hot-disk sensor) as a heat source and a temperature sensor to measure the thermal conductivity. This hot-disk sensor was made of nickel foil in the form of a double spiral (10 mm thick, 3.2 mm radius, a=4.7× 10 − 3 K − 1 (293 K)) covered on both sides with an insulating layer of Kapton (25 mm thick). A constant electric power was supplied to the hot-disk sensor by the source meter to measure the sensor resistance. A computer was used to control the source meter, record the data, and analyze the thermal response during measurement. The hot-disk sensor was immersed vertically in the sample container made of polystyrene. Air bubbles on the container wall and hot-disk sensor were removed by evacuation. After removal of air bubbles, the container was sealed to eliminate free surfaces. The samples were commercial products of silicone oils (Polydimethylsiloxane (PDMS); viscosities were 4.6 (PDMS-5), 29 (PDMS-30), 291 (PDMS-300), and 29300×10 − 3 Nsm − 2 (PDMS-3000); Shin-Etsu Chemical Co., Ltd.) and mercury (99.5% pure; Wako Pure Chemical Industries, Ltd.). The microgravity experiments were performed using the 10 m drop tower of HNIRI (Fig. 2), the parabolic flight of Diamond Air Service (DAS) [10], the underground drop shaft of the Micro-Gravity Laboratory of Japan (MG-LAB) [11], and the underground drop shaft of Japan Microgravity Center (JAMIC) [6]. The HNIRI and JAMIC facility drop capsules have a double structure consisting of inner and outer capsules. The inner capsule, in which the apparatus was installed, was allowed to fall freely within the outer capsule. The MG-LAB drop capsule was allowed to fall freely within the vacuum of the underground shaft. In the parabolic flight experiment, the apparatus was fixed in the airplane. The microgravity qualities and times are shown in Table 1 (see Fig. 3).
H. Nagai et al. / Materials Science and Engineering A276 (2000) 117–123
Fig. 2. 10m drop tower facility (HNIRI). Table 1 Microgravity qualities and times of microgravity facilities Facility
Microgravity quality (g) Microgravity time (s)
HNIRI facility Parabolic flight MG-LAB facility JAMIC facility
10−3 10−2 10−4
1.2 (Fig. 3) 20 4.5
10−4
10
119
ductivities of silicone oils on the ground and in microgravity. Each thermal conductivity was calculated using data recorded for 1 s and the known specific heats and densities of silicone oils [12]. Data recorded from 0 to 0.2 s was not used to calculate the thermal conductivity because this data was strongly affected by the insulating layer of the hot-disk sensor. Measurements were repeated 8–10 times at the same output power. Error bars show the maximum and minimum measured values. In the hot-disk method, thermal conductivity, thermal diffusivity and specific heat per unit volume of sample can be measured in one experiment, if measurement time is of the order of the characteristic time (ideally between 0.5 and 1.0 times as long as characteristic time). The minimum necessary measurement times (tm; half of the characteristic time) were 57.5 s (PDMS5), 45.7 s (PDMS-30), 38.2 s (PDMS-300), and 38.2 s (PDMS-3000). When the measurement time is far from the above times, it is necessary to know the specific heat of the sample to get reliable results. As mentioned in Chapter 2.1, the sample’s thermal conductivity, thermal diffusivity, and specific heat per unit volume are related as Eq. (5). According to the hot-disk method theory, thermal conductivity and thermal diffusivity are calculated separately. If the specific heat per unit volume of the sample is known, thermal conductivity is expressed as a function of thermal diffusivity by using Eq. (5). This simplifies the calculation of the hot-disk method, and reliable values can be obtained in a short measurement time. In the low-viscous silicone oil (PDMS-5; similar to the viscosity of water [1×10 − 3 Nsm − 2]), the thermal conductivities measured on the ground were constant until 4 s, but seemed to increase after 6 s. In contrast, the thermal conductivity measured in micro-
Fig. 3. Microgravity quality and time by 10 m drop tower facility.
3. Results and discussion
3.1. Thermal conducti6ity of silicone oils The thermal responses of four silicone oils were measured on the ground at around 300 K and in microgravity during parabolic flight for 10 s at 0.05 W output power. Fig. 4 shows the apparent thermal con-
Fig. 4. Time depedence of apparent thermal conductivity of silicone oils with different viscosity on the ground and in microgravity. Measurement conditions: 0.05 W, 10 s. Initial temperature of sample, 300 91 K; Sample: , , PDMS-5 (as specific heat: 1760 J kg − 1 K − 1); , , PDMS-300 (as specific heat: 1460 J kg − 1K − 1). Each point was calculated by the recording data for 1 s.
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measurement because thermal convection was suppressed with their higher viscosity.
3.2. Thermal conducti6ity of mercury
Fig. 5. Time dependence of apparent thermal conductivity of mercury on the ground and in microgravity. Measurement conditions: 1.0 W, 5 s. Initial temperature of sample, 291 K. Each point was calcultaed by the recording data for 0.5 s (as specific heat: 140 J kg − 1 K − 1).
Fig. 6. Effect of output power on the measurement of thermal conductivity of mercury. Condition: Initial temperature of sample, 295 K; Measurement time, 1.2 s. Table 2 Thermal conductivity and specific heat of mercury at 293 K Thermal conductivity (Wm−1K−1)
Specific heat (J kg−1K−1)
Literature
8.45 8.03 8.69a 7.96a 8.61a – 7.91b
139.7 139
[13] [14] [15] [16] [17] [18] [19]
– – 139.2 –
a
This value was extrapolated from the thermal conductivity of mercury on higher temperature. b This value was measured at 300 K.
gravity was constant during the 10-s measurement. The increase of thermal conductivity on the ground was thought to be caused by thermal convection. In the silicone oils with viscosities exceeding 29× 10 − 3 Nsm − 2, thermal conductivities measured on the ground and in microgravity were similar and constant during
The thermal responses of mercury at 291 K were measured on the ground and in microgravity using the MG-LAB facility in a 5-s at 1.0 W output power. Fig. 5 shows the apparent thermal conductivities of mercury on the ground and in microgravity as a function of measurement time. Each thermal conductivity was calculated by recording data for 0.5 s on the assumption that the specific heat of mercury was 140 J kg − 1 K − 1 and that its density was 1.3546×104 kg m − 3 [13] because tm of mercury was about 1.2 s. The thermal conductivities measured in microgravity were constant during measurement. The thermal conductivities measured on the ground were similar to those in microgravity up to 1 s, but apparently increased after 1.5 s. The thermal conductivity on the ground could not be calculated from the recording data after 3.5 s since it was beyond the theory of the hot-disk method. It was found that heat flow from the hot-disk sensor caused thermal convection on the ground, and that this convection was suppressed in microgravity. The thermal conductivities on the ground and in microgravity were then studied in detail using a 10 m drop tower (1.2 s microgravity) by recording data for 1.2 s. Fig. 6 shows the effect of output power on the measurement of the thermal conductivity of mercury. Measurements were repeated several times at the same output power. Error bars show maximum and minimum measured values. The thermal conductivities at 295 K were constant in this range of output power. Thermal conductivity measured on the ground was 8.10 Wm − 1 K − 1, and that in microgravity was 7.85 Wm − 1 K − 1. Table 2 shows the thermal conductivities and specific heats of mercury reported in literature. Our result on the ground was in the range of the values shown in Table 2. The thermal conductivity measured in microgravity was 0.2 Wm − 1 K − 1 (3%) lower than that on the ground. It was reported that the standard deviation of thermal conductivity measured by the hotdisk method was 2% [20]. This revealed that the thermal conductivity of mercury measured on the ground included the effect of thermal convection introduced by the heat of the hot-disk sensor. The specific heat of mercury could also be measured by the hot-disk method under this measurement condition. The specific heats of mercury on the ground and in microgravity were similar (140 J kg − 1 K − 1), assuming that the density of mercury was 1.3546× 104 kg m − 3, in good agreement with the values shown in Table 2. Fig. 7 shows the effect of temperature on the thermal conductivity of mercury. The thermal conductivities of
H. Nagai et al. / Materials Science and Engineering A276 (2000) 117–123
mercury on the ground and in microgravity increased with the initial temperature of the sample. The temperature coefficient of the thermal conductivity of mercury was reported to be 0.013 – 0.025 Wm − 1 K − 2 [13–17]. Our results (0.025 Wm − 1 K − 2) were in the range of those values. The thermal conductivities in microgravity were about 0.2 Wm − 1 K − 1 lower than those on the ground around room temperature. The specific heats on the ground and in microgravity were the same for the following two reasons; First, specific heat accuracy of the hot-disk method was 9 10% [20]. Second, the temperature coefficient of the specific heat of mercury was very small (0.03 J kg − 1 K − 2) [13]. Data shown in Fig. 7 were obtained in microgravity using the MG-LAB and JAMIC facilities. The data recorded in the first 2.4 s of these experiments were used for calculating the thermal conductivity because the characteristic time of mercury for thermal conductivity measurement was
Fig. 7. Effect of temperature on the measurement of thermal conductivity of mercury. Condition: Output power of measurement; 1.0 W; measurement time, 1.2 s (HNIRI), 2.4 s (MG-ALB and JAMIC).
121
about 2.4 s. The thermal conductivities measured at the MG-LAB and JAMIC facilities were in good agreement with those measured at the HNIRI facility, except one set of experimental data.
3.3. Effect of gra6ity on the measurement of thermal conducti6ity To measure thermal conductivity precisely, it is important to suppress thermal convection while measuring the thermal response. Krishnamurti [21] reported that several distinct transitions occurred in a horizontal convection layer of fluid at a certain Rayleigh number (Ra) and for a given Prandtl number (Pr), as summarized in Fig. 8. The Rayleigh number and Prandtl number are defined as follows: Ra=(gb/kn)DT·a 3
(6)
Pr= n/k
(7)
where g is the acceleration of gravity; b, k and n are the thermal expansion coefficient, thermal diffusivity and kinematic viscosity of the sample; DT is the temperature difference between the bottom and top of the layer; and a is the layer depth. In our studies, DT is the temperature difference between the hot-disk sensor surface and the sample, and a is the probing depth (i.e. the distance from any part of the hot-disk sensor to the nearest outside boundary of the sample). Table 3 shows the data for calculating Ra and Pr. Circles in Fig. 8 represent the calculated results of silicone oils and mercury on the ground; dots represent those results in microgravity. On the ground, thermal convection was suppressed in every silicone oil until 6.5 s, and only PDMS-5 was affected by thermal convection after 6.5 s. Thermal convection occurred in mercury on the ground. In microgravity, there was no convective flow in any sample. These results were in good agreement with Krishnamurti’s result. Therefore, we could estimate the thermal conductivity measurement conditions for which there is no influence of thermal convection on the ground or in microgravity by using Fig. 8. The lowest viscosities of silicone oil on the ground were 190× 10 − 3 Nsm − 2, and in microgravity (10 − 2 g), 1.8× 10 − 3 Nsm − 2. The necessary gravity level for measuring the thermal conductivity of mercury was less than 10 − 2 g, in good agreement with Nakamura’s result [10].
4. Conclusions
Fig. 8. The regime diagram to the sample flow on the measurement of thermal conductivity.
In this study, the thermal conductivities of silicone oils and mercury were measured by the hot-disk method in short-duration microgravity. The major findings are described below.
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Sample
Silicone oil
Mercury
a
Measurement condition of Gravity level g thermal conductivity (g)
PDMS-5 PDMS-30 PDMS-300 PDMS-30000 PDMS-5 PDMS-30 PDMS-300 PDMS-30000
Output power (W)
Time (s)
0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 1.0 1.0
6.5 10 10 10 10 10 10 10 1.2 1.2
These data were taken from references [12], [14] and [18].
1 1 1 1 10−2 10−2 10−2 10−2 1 10−3
Thermal expansion coefficienta b (K−1)
Thermal diffusitity k (m2 s−1)
Kinematic viscositya n (m2 s−1)
DT (K)
Probing depth a (m)
1.1×10−3 9.9×10−4 9.5×10−4 9.4×10−4 1.1×10−3 9.9×10−4 9.5×10−4 9.4×10−4 1.8×10−4 1.8×10−4
9.1×10−8 1.1×10−7 1.3×10−7 1.3×10−7 8.9×10−8 1.1×10−7 1.3×10−7 1.3×10−7 4.1×10−6 3.9×10−6
5.0×10−6 3.0×10−5 3.0×10−4 3.0×10−2 5.0×10−6 3.0×10−5 3.0×10−4 3.0×10−2 1.2×10−7 1.2×10−7
3.7 4.0 3.9 3.9 4.0 4.0 3.9 3.9 4.5 4.5
1.6×10−3 2.1×10−3 2.3×10−3 2.3×10−3 1.9×10−3 2.1×10−3 2.3×10−3 2.3×10−3 4.5×10−3 4.4×10−3
H. Nagai et al. / Materials Science and Engineering A276 (2000) 117–123
Table 3 The data for calculating Ra and Pr on the ground and in microgravity
H. Nagai et al. / Materials Science and Engineering A276 (2000) 117–123
1. The thermal conductivities of silicone oil with low viscosity were affected by thermal convection on the ground, but thermal convection was suppressed in microgravity. Thermal convection did not influence measurements of high-viscosity samples. 2. The thermal conductivity of mercury in microgravity was about 3% lower than that on the ground around room temperature. 3. The thermal conductivity measurement conditions for which there is no influence of thermal convection on the ground or in microgravity could be estimated by using the Rayleigh number.
Acknowledgements This work was partly supported by the National Space Development Agency of Japan and the Hokkaido Microgravity Society. The authors wish to thank Professor O. Odawara of Tokyo Institute of Technology for his valuable comments.
References [1] S. Fischer, E. Obermeier, High Temp. High Pressures 17 (1985) 699. [2] Y. Kato, K. Furukawa, N. Arai, K. Kobayashi, High Temp High Pressures 15 (1983) 191.
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[3] Y. Nagasaka, A. Nagashima, Rev. Sci. Instrum. 52 (2) (1981) 229. [4] H. Ohta, G. Ogura, Y. Waseda, Rev. Sci. Instrum. 61 (10) (1990) 2645. [5] S. Nakamura, T. Hibiya, F. Yamamoto, T. Yokota, Int. J. Thermophys. 12 (5) (1991) 783. [6] S. Nakamura, T. Hibiya, Microgravity Sci. Technol. 6 (2) (1993) 119. [7] S.E. Gustafsson, Rev. Sci. Instrum. 62 (3) (1991) 797. [8] S.E. Gustafsson, E. Karawacki, M.A. Chohan, J. Phys. D: Appl. Phys. 19 (1986) 727. [9] H. Nagai, F. Rossignol, Y. Nakata, T. Tsurue, M. Suzuki, O. Odawara, T. Okutani, Proceedings of Joint 1st Pan-Pacific Basin Workshop and 4th Japan-China Workshop on Microgravity, Sciences, Tokyo, Japan, July 8 – 11, J. Jpn. Soc. Microgravity Appl. 15 Suppl. 2 (1998) 358. [10] S. Nakamura, T. Hibiya, F. Yamamoto, Microgravity Sci. Technol. 5 (3) (1992) 156. [11] S. Asano, S. Ito, T. Tokuo, J. Jpn. Soc. Microgravity Appl. 12 (3) (1993) 168. [12] These data come from the technical data catalog of silicone oils KF96 prepared by Shin-Etsu Chemical Co., Ltd. [13] Nihon Netsubussei Gakkai, Thermophysical Properties Handbook, Yokendo, Tokyo 1990, p.101 – 104. [14] E.A. Brandes, G.B. Brook, Smithells Metals Reference Book 7th, Butterworth, Heinemann, London, 1992, Chapter 14, pp. 1–11. [15] W.C. Hall, Phys. Rev. 53 (1938) 1004. [16] M.J. Duggin, Phys. Lett. 27A (5) (1968) 257. [17] C.T. Ewing, R.E. Seebold, J.A. Grand, R.R. Miller, J. Phys. Chem. 59 (1955) 524. [18] G.J.F. Holman, C.A. ten Seldam, J. Phys. Chem. Ref. Data 23 (5) (1994) 807. [19] S. Nakamura, T. Hibiya, Rev. Sci. Instrum. 59 (12) (1988) 2600. [20] T. Log, S.E. Gustafsson, Fire Materials 19 (1995) 43. [21] R. Krishnamurti, J. Fluid Mech. 60 (2) (1973) 285.
Thermal conductivity measurement of liquid materials by a hot-disk method in short-duration microgravity environments Hideaki Nagai *, Fabrice Rossignol, Yoshinori Nakata, Takashi Tsurue, Masaaki Suzuki, Takeshi Okutani Hokkaido National Industrial Research Institute (HNIRI), 2 -17 -2 -1, Tsukisamu-higashi, Toyohira-ku, Sapporo 062 -8517, Japan Received 6 May 1999; received in revised form 25 June 1999
Abstract The thermal conductivities of silicone oils with various viscosities and mercury were measured by a hot-disk method in short-duration microgravity environments. The thermal conductivities of silicone oil with low viscosity were affected by the thermal convection on the ground, but the thermal convection was suppressed in microgravity. The thermal conductivities of highly viscous samples were not influenced by thermal convection. The thermal conductivity of mercury measured in microgravity was about 3% lower than that measured on the ground around room temperature. The thermal conductivity measurement conditions on the ground and in microgravity for which there was no influence from thermal convection could be estimated by using the Rayleigh number. © 2000 Elsevier Science S.A. All rights reserved. Keywords: Thermal conductivity; Hot-disk method; Microgravity; Silicone oil; Mercury
1. Introduction Thermal conductivity is an important property in many heat-transfer simulations. This value can be measured by the thermal response when heat or energy is applied to a specimen. However, it is difficult to measure the thermal conductivities of liquid materials (water, organic solvents, molten metals, etc.) precisely because heat is transferred by convection as well as conduction. In microgravity, however, it is possible to measure the thermal conductivities of liquid materials precisely because thermal convection is suppressed. The thermal conductivities of liquid materials are measured by (1) a steady heat flow method [1], (2) a stepwise heating method [2], (3) a transient hot-wire method [3] and (4) a laser flash method [4]. In microgravity experiments, the hot-wire method is more suitable because of its short time in measurement. Nakamura et al. [5,6] reported that the thermal conductivity of molten indium antimonide was measured by the hot-wire method in microgravity (using sounding * Corresponding author. Fax: +81-11-8578984. E-mail address: [email protected] (H. Nagai)
rockets (6 min microgravity) and drop shafts (10 s microgravity)). Recently, Gustafsson developed the hot-disk method for measuring thermal conductivity [7]. This method uses a transient plane source (TPS) element both as a heat source and temperature sensor, in the same way as a thin wire is used in the hot-wire method. TPS element is made of thin metal foil and its conducting pattern is a double spiral, which with some approximation resembles a hot disk. For the hot-wire method, a thin bare wire has been utilized, and only the thermal conductivity of insulating materials can be measured. In contrast, those of not only insulating materials but also electrical conducting materials can be measured by the hot-disk method because the both sides of TPS element was covered with a thin insulating layer. A reliable value could then be obtained by carefully calibrating the insulating layer effect [8]. In this study, the thermal conductivities of silicone oils with various viscosities were measured by the hot-disk method in microgravity during parabolic flight and that of mercury was measured in microgravity using a drop tower and drop shafts [9]. The effect of gravity on the experimental results on thermal conductivity is discussed in this study.
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the sample material (Eq. (2)). The thermal conductivity is calculated from the slope of the most straight line fit (Eq. (4)). Thermal conductivity and thermal diffusivity of the sample material are related as follows:
2. Experimental
2.1. Principle of the hot-disk method In hot-disk method experiments [7], the normal arrangement is to pass a constant current through a hot-disk sensor and simultaneously record its voltage changes. When analyzing the behavior of the hot-disk sensor during a transient recording, it is convenient to express the time dependent resistance (R(t)) with the following expression:
l= rCpk
R(t)= R0[1+aDT(t)]
2.2. Equipment setup and experimental procedure
(1)
where R0 is the resistance of the hot-disk sensor before the transient recording, a is the temperature coefficient of resistance, and DT(t) is the time-dependent temperature increase of the hot-disk sensor. In this equation, the temperature increase is expressed as a function of only one variable t, which is defined as: t = (t/u)1/2, u= d 2/k
(2)
where t is the time measured from the start of the transient heating, u is the characteristic time, d is the radius of the hot disk, and k is the thermal diffusivity of the sample material. For a hot-disk sensor, DT(t) is given by the following equation, from which the thermal conductivity and diffusivity can be obtained: DT(t)=P0(p 3/2dl)-1D(t)
(3)
where P0 is the total output power, l is the thermal conductivity of the sample material, and D(t) is the theoretical expression of the time-dependent temperature increase. We substitute Eq. (3) for Eq. (1), then obtain Eq. (4). R(t)=R0[1+aP0(p 3/2dl) − 1D(t)]
(4)
By plotting the measured resistivities R(t) vs. D(t), we can get a straight line according to Eq. (4) if the characteristic time (u) has the proper value. It is possible to find this value of u by varying it in an iterative procedure until the correlation coefficient between R(t) and D(t) reaches its maximum value. This particular value of u is used to obtain the thermal diffusivity of
Fig. 1. Schematic diagram of the hot-disk method.
(5)
where r is the density of the sample, and Cp is the specific heat of the sample. The specific heat per unit volume (rCp) can be calculated by using the values of the thermal diffusivity and thermal conductivity of the sample material.
The thermal conductivities of liquid materials were measured by the hot-disk thermal constant analyser (Hot Disk Inc., Sweden). The schematic diagram of the thermal conductivity measurement setup is shown in Fig. 1. The hot-disk method utilizes a thin disk-shaped sensor (hot-disk sensor) as a heat source and a temperature sensor to measure the thermal conductivity. This hot-disk sensor was made of nickel foil in the form of a double spiral (10 mm thick, 3.2 mm radius, a=4.7× 10 − 3 K − 1 (293 K)) covered on both sides with an insulating layer of Kapton (25 mm thick). A constant electric power was supplied to the hot-disk sensor by the source meter to measure the sensor resistance. A computer was used to control the source meter, record the data, and analyze the thermal response during measurement. The hot-disk sensor was immersed vertically in the sample container made of polystyrene. Air bubbles on the container wall and hot-disk sensor were removed by evacuation. After removal of air bubbles, the container was sealed to eliminate free surfaces. The samples were commercial products of silicone oils (Polydimethylsiloxane (PDMS); viscosities were 4.6 (PDMS-5), 29 (PDMS-30), 291 (PDMS-300), and 29300×10 − 3 Nsm − 2 (PDMS-3000); Shin-Etsu Chemical Co., Ltd.) and mercury (99.5% pure; Wako Pure Chemical Industries, Ltd.). The microgravity experiments were performed using the 10 m drop tower of HNIRI (Fig. 2), the parabolic flight of Diamond Air Service (DAS) [10], the underground drop shaft of the Micro-Gravity Laboratory of Japan (MG-LAB) [11], and the underground drop shaft of Japan Microgravity Center (JAMIC) [6]. The HNIRI and JAMIC facility drop capsules have a double structure consisting of inner and outer capsules. The inner capsule, in which the apparatus was installed, was allowed to fall freely within the outer capsule. The MG-LAB drop capsule was allowed to fall freely within the vacuum of the underground shaft. In the parabolic flight experiment, the apparatus was fixed in the airplane. The microgravity qualities and times are shown in Table 1 (see Fig. 3).
H. Nagai et al. / Materials Science and Engineering A276 (2000) 117–123
Fig. 2. 10m drop tower facility (HNIRI). Table 1 Microgravity qualities and times of microgravity facilities Facility
Microgravity quality (g) Microgravity time (s)
HNIRI facility Parabolic flight MG-LAB facility JAMIC facility
10−3 10−2 10−4
1.2 (Fig. 3) 20 4.5
10−4
10
119
ductivities of silicone oils on the ground and in microgravity. Each thermal conductivity was calculated using data recorded for 1 s and the known specific heats and densities of silicone oils [12]. Data recorded from 0 to 0.2 s was not used to calculate the thermal conductivity because this data was strongly affected by the insulating layer of the hot-disk sensor. Measurements were repeated 8–10 times at the same output power. Error bars show the maximum and minimum measured values. In the hot-disk method, thermal conductivity, thermal diffusivity and specific heat per unit volume of sample can be measured in one experiment, if measurement time is of the order of the characteristic time (ideally between 0.5 and 1.0 times as long as characteristic time). The minimum necessary measurement times (tm; half of the characteristic time) were 57.5 s (PDMS5), 45.7 s (PDMS-30), 38.2 s (PDMS-300), and 38.2 s (PDMS-3000). When the measurement time is far from the above times, it is necessary to know the specific heat of the sample to get reliable results. As mentioned in Chapter 2.1, the sample’s thermal conductivity, thermal diffusivity, and specific heat per unit volume are related as Eq. (5). According to the hot-disk method theory, thermal conductivity and thermal diffusivity are calculated separately. If the specific heat per unit volume of the sample is known, thermal conductivity is expressed as a function of thermal diffusivity by using Eq. (5). This simplifies the calculation of the hot-disk method, and reliable values can be obtained in a short measurement time. In the low-viscous silicone oil (PDMS-5; similar to the viscosity of water [1×10 − 3 Nsm − 2]), the thermal conductivities measured on the ground were constant until 4 s, but seemed to increase after 6 s. In contrast, the thermal conductivity measured in micro-
Fig. 3. Microgravity quality and time by 10 m drop tower facility.
3. Results and discussion
3.1. Thermal conducti6ity of silicone oils The thermal responses of four silicone oils were measured on the ground at around 300 K and in microgravity during parabolic flight for 10 s at 0.05 W output power. Fig. 4 shows the apparent thermal con-
Fig. 4. Time depedence of apparent thermal conductivity of silicone oils with different viscosity on the ground and in microgravity. Measurement conditions: 0.05 W, 10 s. Initial temperature of sample, 300 91 K; Sample: , , PDMS-5 (as specific heat: 1760 J kg − 1 K − 1); , , PDMS-300 (as specific heat: 1460 J kg − 1K − 1). Each point was calculated by the recording data for 1 s.
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measurement because thermal convection was suppressed with their higher viscosity.
3.2. Thermal conducti6ity of mercury
Fig. 5. Time dependence of apparent thermal conductivity of mercury on the ground and in microgravity. Measurement conditions: 1.0 W, 5 s. Initial temperature of sample, 291 K. Each point was calcultaed by the recording data for 0.5 s (as specific heat: 140 J kg − 1 K − 1).
Fig. 6. Effect of output power on the measurement of thermal conductivity of mercury. Condition: Initial temperature of sample, 295 K; Measurement time, 1.2 s. Table 2 Thermal conductivity and specific heat of mercury at 293 K Thermal conductivity (Wm−1K−1)
Specific heat (J kg−1K−1)
Literature
8.45 8.03 8.69a 7.96a 8.61a – 7.91b
139.7 139
[13] [14] [15] [16] [17] [18] [19]
– – 139.2 –
a
This value was extrapolated from the thermal conductivity of mercury on higher temperature. b This value was measured at 300 K.
gravity was constant during the 10-s measurement. The increase of thermal conductivity on the ground was thought to be caused by thermal convection. In the silicone oils with viscosities exceeding 29× 10 − 3 Nsm − 2, thermal conductivities measured on the ground and in microgravity were similar and constant during
The thermal responses of mercury at 291 K were measured on the ground and in microgravity using the MG-LAB facility in a 5-s at 1.0 W output power. Fig. 5 shows the apparent thermal conductivities of mercury on the ground and in microgravity as a function of measurement time. Each thermal conductivity was calculated by recording data for 0.5 s on the assumption that the specific heat of mercury was 140 J kg − 1 K − 1 and that its density was 1.3546×104 kg m − 3 [13] because tm of mercury was about 1.2 s. The thermal conductivities measured in microgravity were constant during measurement. The thermal conductivities measured on the ground were similar to those in microgravity up to 1 s, but apparently increased after 1.5 s. The thermal conductivity on the ground could not be calculated from the recording data after 3.5 s since it was beyond the theory of the hot-disk method. It was found that heat flow from the hot-disk sensor caused thermal convection on the ground, and that this convection was suppressed in microgravity. The thermal conductivities on the ground and in microgravity were then studied in detail using a 10 m drop tower (1.2 s microgravity) by recording data for 1.2 s. Fig. 6 shows the effect of output power on the measurement of the thermal conductivity of mercury. Measurements were repeated several times at the same output power. Error bars show maximum and minimum measured values. The thermal conductivities at 295 K were constant in this range of output power. Thermal conductivity measured on the ground was 8.10 Wm − 1 K − 1, and that in microgravity was 7.85 Wm − 1 K − 1. Table 2 shows the thermal conductivities and specific heats of mercury reported in literature. Our result on the ground was in the range of the values shown in Table 2. The thermal conductivity measured in microgravity was 0.2 Wm − 1 K − 1 (3%) lower than that on the ground. It was reported that the standard deviation of thermal conductivity measured by the hotdisk method was 2% [20]. This revealed that the thermal conductivity of mercury measured on the ground included the effect of thermal convection introduced by the heat of the hot-disk sensor. The specific heat of mercury could also be measured by the hot-disk method under this measurement condition. The specific heats of mercury on the ground and in microgravity were similar (140 J kg − 1 K − 1), assuming that the density of mercury was 1.3546× 104 kg m − 3, in good agreement with the values shown in Table 2. Fig. 7 shows the effect of temperature on the thermal conductivity of mercury. The thermal conductivities of
H. Nagai et al. / Materials Science and Engineering A276 (2000) 117–123
mercury on the ground and in microgravity increased with the initial temperature of the sample. The temperature coefficient of the thermal conductivity of mercury was reported to be 0.013 – 0.025 Wm − 1 K − 2 [13–17]. Our results (0.025 Wm − 1 K − 2) were in the range of those values. The thermal conductivities in microgravity were about 0.2 Wm − 1 K − 1 lower than those on the ground around room temperature. The specific heats on the ground and in microgravity were the same for the following two reasons; First, specific heat accuracy of the hot-disk method was 9 10% [20]. Second, the temperature coefficient of the specific heat of mercury was very small (0.03 J kg − 1 K − 2) [13]. Data shown in Fig. 7 were obtained in microgravity using the MG-LAB and JAMIC facilities. The data recorded in the first 2.4 s of these experiments were used for calculating the thermal conductivity because the characteristic time of mercury for thermal conductivity measurement was
Fig. 7. Effect of temperature on the measurement of thermal conductivity of mercury. Condition: Output power of measurement; 1.0 W; measurement time, 1.2 s (HNIRI), 2.4 s (MG-ALB and JAMIC).
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about 2.4 s. The thermal conductivities measured at the MG-LAB and JAMIC facilities were in good agreement with those measured at the HNIRI facility, except one set of experimental data.
3.3. Effect of gra6ity on the measurement of thermal conducti6ity To measure thermal conductivity precisely, it is important to suppress thermal convection while measuring the thermal response. Krishnamurti [21] reported that several distinct transitions occurred in a horizontal convection layer of fluid at a certain Rayleigh number (Ra) and for a given Prandtl number (Pr), as summarized in Fig. 8. The Rayleigh number and Prandtl number are defined as follows: Ra=(gb/kn)DT·a 3
(6)
Pr= n/k
(7)
where g is the acceleration of gravity; b, k and n are the thermal expansion coefficient, thermal diffusivity and kinematic viscosity of the sample; DT is the temperature difference between the bottom and top of the layer; and a is the layer depth. In our studies, DT is the temperature difference between the hot-disk sensor surface and the sample, and a is the probing depth (i.e. the distance from any part of the hot-disk sensor to the nearest outside boundary of the sample). Table 3 shows the data for calculating Ra and Pr. Circles in Fig. 8 represent the calculated results of silicone oils and mercury on the ground; dots represent those results in microgravity. On the ground, thermal convection was suppressed in every silicone oil until 6.5 s, and only PDMS-5 was affected by thermal convection after 6.5 s. Thermal convection occurred in mercury on the ground. In microgravity, there was no convective flow in any sample. These results were in good agreement with Krishnamurti’s result. Therefore, we could estimate the thermal conductivity measurement conditions for which there is no influence of thermal convection on the ground or in microgravity by using Fig. 8. The lowest viscosities of silicone oil on the ground were 190× 10 − 3 Nsm − 2, and in microgravity (10 − 2 g), 1.8× 10 − 3 Nsm − 2. The necessary gravity level for measuring the thermal conductivity of mercury was less than 10 − 2 g, in good agreement with Nakamura’s result [10].
4. Conclusions
Fig. 8. The regime diagram to the sample flow on the measurement of thermal conductivity.
In this study, the thermal conductivities of silicone oils and mercury were measured by the hot-disk method in short-duration microgravity. The major findings are described below.
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Sample
Silicone oil
Mercury
a
Measurement condition of Gravity level g thermal conductivity (g)
PDMS-5 PDMS-30 PDMS-300 PDMS-30000 PDMS-5 PDMS-30 PDMS-300 PDMS-30000
Output power (W)
Time (s)
0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 1.0 1.0
6.5 10 10 10 10 10 10 10 1.2 1.2
These data were taken from references [12], [14] and [18].
1 1 1 1 10−2 10−2 10−2 10−2 1 10−3
Thermal expansion coefficienta b (K−1)
Thermal diffusitity k (m2 s−1)
Kinematic viscositya n (m2 s−1)
DT (K)
Probing depth a (m)
1.1×10−3 9.9×10−4 9.5×10−4 9.4×10−4 1.1×10−3 9.9×10−4 9.5×10−4 9.4×10−4 1.8×10−4 1.8×10−4
9.1×10−8 1.1×10−7 1.3×10−7 1.3×10−7 8.9×10−8 1.1×10−7 1.3×10−7 1.3×10−7 4.1×10−6 3.9×10−6
5.0×10−6 3.0×10−5 3.0×10−4 3.0×10−2 5.0×10−6 3.0×10−5 3.0×10−4 3.0×10−2 1.2×10−7 1.2×10−7
3.7 4.0 3.9 3.9 4.0 4.0 3.9 3.9 4.5 4.5
1.6×10−3 2.1×10−3 2.3×10−3 2.3×10−3 1.9×10−3 2.1×10−3 2.3×10−3 2.3×10−3 4.5×10−3 4.4×10−3
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Table 3 The data for calculating Ra and Pr on the ground and in microgravity
H. Nagai et al. / Materials Science and Engineering A276 (2000) 117–123
1. The thermal conductivities of silicone oil with low viscosity were affected by thermal convection on the ground, but thermal convection was suppressed in microgravity. Thermal convection did not influence measurements of high-viscosity samples. 2. The thermal conductivity of mercury in microgravity was about 3% lower than that on the ground around room temperature. 3. The thermal conductivity measurement conditions for which there is no influence of thermal convection on the ground or in microgravity could be estimated by using the Rayleigh number.
Acknowledgements This work was partly supported by the National Space Development Agency of Japan and the Hokkaido Microgravity Society. The authors wish to thank Professor O. Odawara of Tokyo Institute of Technology for his valuable comments.
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