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Attempt to study Marangoni flow of low-Pr-number fluids using a liquid bridge of silver
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Advances in Space Research 41 (2008) 2107–2111 www.elsevier.com/locate/asr
Attempt to study Marangoni flow of low-Pr-number fluids using a liquid bridge of silver Taketoshi Hibiya *, Keisuke Nagafuchi, Suguru Shiratori, Noriyoshi Yamane, Shumpei Ozawa Department of Aerospace Engineering, Tokyo Metropolitan University, 6-6, Asahigaoka, 191-0065 Hino, Japan Received 11 October 2006; accepted 6 April 2007
Abstract In order to experimentally investigate the Marangoni flow of low-Prandtl-number fluids in a liquid bridge geometry under the condition of small Marangoni numbers close to the critical Marangoni numbers Mac1 and Mac2, the formation of a liquid bridge of silver was attempted. The available temperature difference between the upper and lower rods to obtain a small Marangoni number, such as Ma = 50, was calculated for a 5 mm high liquid bridge for several molten metals. For molten silver, the possible temperature difference was estimated to be 16 K, whereas, for molten silicon, this was 0.38 K, which is unrealistic for the purposes of experiments. For silver, a free surface can be obtained in the wide range of oxygen partial pressures, whereas, for molten silicon, the available oxygen partial pressure range is very small; equilibrium oxygen partial pressure for SiO2 formation is as low as 1.1 · 1014 Pa. A liquid bridge of molten silver was successfully prepared and temperature oscillation was observed; the estimated Marangoni number was 160 and oscillation frequency was 0.26 Hz. 2007 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Marangoni flow; Low-Pr-number; Liquid bridge; Silver
1. Introduction Understanding and controlling Marangoni flow for lowPrandtl-number fluids (molten semiconductors, metals and alloys) are necessary to improve products from high temperature processes, such as crystal growth of semiconductors, casting of jet engine turbine blades, welding of automobile bodies and so on. When the high temperature melt has a free surface, Marangoni flow plays a significant role in the heat and mass transport processes and also affects the quality of the final products. The magnitude of the Marangoni flow is estimated using the Marangoni number, as follows, Ma ¼ ðjor=oT jDThÞ=ðlaÞ *
Corresponding author. Tel.: +81 42 585 8654. E-mail address: [email protected] (T. Hibiya).
ð1Þ
Here, r, l and a are the surface tension, the viscosity and the thermal diffusivity, respectively. DT is the temperature difference in the system, e.g., the temperature difference between the upper and lower ends of the liquid bridge sustained by solid rods in the liquid bridge geometry. h is the bridge height. Marangoni flow of low-Pr-number fluids has been studied experimentally for the case of large Marangoni numbers such as Ma = 10,000 (Nakamura et al., 1998), whereas it has been studied numerically in the small Marangoni number range, such as an order of 10 (Levenstam and Amberg, 1995; Wanschura et al., 1995; Imaishi et al., 2001). This is due to the fact that it is experimentally difficult to prepare the small Marangoni number conditions and that flow mode transition in the small Marangoni number region attracts attention of numerical people. It is also difficult to carry out the calculations at the large Marangoni number condition, because flow is almost turbulent and requires an advanced technique for calculation.
0273-1177/$34.00 2007 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2007.04.106
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T. Hibiya et al. / Advances in Space Research 41 (2008) 2107–2111
The target of the current research on the Marangoni flow of low-Pr-number fluids in a liquid bridge structure is to understand the transition from an axisymmetric to a 3Dstationary flow and the transition from a 3D-axisymmetric flow to an oscillatory flow. For these transitions, the critical Marangoni numbers Mac1 and Mac2 are indices, above which the transitions from an axisymmetric to a 3D-stationary and from a 3D-stationary to an oscillatory flow take place, respectively. The Mac1 is estimated to be 30 and the Mac2 about 100. However, for a half-zone liquid bridge of low-Pr-number fluids, it is difficult to observe these transitions experimentally. This is due to the fact that it is difficult to produce a small temperature difference between the upper and lower rods for a liquid bridge with a specific height. As an exception to this, Matsumoto et al. observed successfully Mac1 experimentally for a molten tin bridge (Matsumoto et al., 2004). In order to overcome this problem experimentally, Cro¨ll et al. (1991) proposed an idea of partial confinement of the liquid bridge of molten silicon. They coated a liquid bridge with an SiO2 film with a narrow circular window, where the driving force is limited within this open area, although total height of the liquid bridge is larger than this window. Using this technique, the Mac2 for molten silicon was experimentally determined to be in the range of 100–200. The partial confinement technique has been studied more in detail for a half-zone liquid bridge by Shiratori et al. (2007) and it was found that the flow structure within the partially confined half-zone bridge is different from that of the normal half-zone structure depending on the position of the open window. The surface of low-Pr-number fluids is easily oxidized. This means that the Marangoni flow is suppressed by formation of a thin oxide layer (Hibiya et al., 2003). Furthermore, for some molten transient metals and alloys, inversion of the temperature coefficient of surface tension is observed; the sign for the temperature coefficient of surface tension changes from negative to positive with the increase in adsorption of a surfactant such as oxygen or sulfur (Takiuchi et al., 1991). When these materials are melted, inversion of the Marangoni flow direction is expected to be observed; i.e. Marangoni flow from the cold side to the
hot side. This effect is dependent on the solubility of oxygen or sulfur in melts. Molten silver is one of candidate materials for use in study of the Marangoni flow of molten metals because its thermophysical properties assure a liquid bridge with a small Marangoni number and a free surface in a wide range of oxygen partial pressures in the ambient atmosphere. In the present study, use of molten silver is proposed to simplify the experimental condition, so that the liquid bridge of low-Pr-number fluids with small Marangoni number can be prepared. 2. Considerations concerning the half-zone liquid bridge of low-Pr-number fluids with small Ma-number Let us now discuss the conditions that apply to the preparation of a liquid bridge of low-Pr-number fluids with small Marangoni numbers. For example, for a 5-mm high half-zone liquid bridge of molten silicon, which represents a model of a floating zone crystal growth, the required temperature difference between the upper and lower edges is estimate to be 0.38 K, when the Marangoni number is required to be at Ma = 50 based on the reported thermophysical properties, as shown in Table 1. Table 1 also shows the estimated temperature difference for Sn and Ag melts, i.e. 4.5 K and 16 K, respectively. This suggests that an increase in temperature difference of 1 K between the upper and lower edges gives rise to a Marangoni number of Ma = 130 K for molten silicon, whereas Marangoni number increases by 3.1 for molten silver. The calculated temperature difference of 0.38 K for Si melt between the upper and lower edges is not realistic, whereas it is 4.5 K and 16 K for molten tin and silver, respectively, and is available experimentally. According to chemical thermodynamics, molten silver cannot be oxidized easily, but can absorb huge amounts of oxygen. For example, the equilibrium oxygen partial pressure for formation of Ag2O at the silver melting point of 1233 K is almost 0.1 MPa (Knacke et al., 1991), which is higher than the oxygen partial pressure of air, whereas equilibrium oxygen partial pressure for SiO2 formation is
Table 1 The thermophysical properties of molten silver, tin and silicona
Melting temperature Tm (K) Isobaric mass heat capacity Cp (J/kg K) Density q (kg/m3) Temperature coefficient of surface tension or/oT (mN/m · K) Viscosity l (Pa · s) Kinematic viscosity m (m2/s) Thermal conductivity k (W/m K) Thermal diffusivity a (m2/s) Pr-number Pr = m/a () DT (K) for the liquid bridge 5-mm high, 10-mm in diameter with Ma = 50
Ag
Sn
Si
1234 283 9.35 · 103 0.16 3.88 · 103 4.15 · 107 175 6.60 · 105 6.3 · 103 16
505 250 7.00 · 103 0.07 1.85 · 103 2.64 · 107 30 1.7 · 105 1.6 · 102 4.5
1687 1.00 Kobatake et al. (2006) 2.56 · 103 (Sato et al., 2000) 0.36 Mukai et al. (2000) 0.56 · 103 Sato et al. (2003) 2.19 · 107 62 Kobatake et al. (2007) 2.42 · 105 9.0 · 103 0.38
a See ‘‘Smithells Metals Research Book Seventh Edition 1992’’ edited by Brandes and Brook for thermophysical properties of molten Ag and Sn. For molten Si, latest data are cited; see each reference. Kinematic viscosity and thermal diffusivity are defined as follows: l = q · m and k = Cp · q · a.
T. Hibiya et al. / Advances in Space Research 41 (2008) 2107–2111
as low as 1.1 · 1014 Pa for molten silicon. Also, a surface of molten tin is easily oxidized. This suggests that the surface of molten silver is always bare at this experimental condition and study of the Marangoni flow can be carried out without having to take into consideration of an oxide layer in the atmospheric condition. Furthermore, temperature coefficient can be changed from the negative to the positive value with an increase in oxygen partial pressure in an ambient atmosphere (Bernard and Lupis, 1971). This is an advantage of use of silver, compared with the use of molten tin and silicon. 3. Experimental setup A silver rod 5-mm high and 10-mm in diameter was melted and a liquid bridge was successfully prepared between the upper and lower alumina ceramic rods heated by a PBN ceramic heater. Fig. 1a shows a molten silver bridge. As shown in Fig. 1b, platinum plates were inserted between molten silver and alumina ceramic rods, so that
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wettability was assured between rods and molten silver. The entire experimental process was carried out at 6N– Ar atmosphere of 105 Pa. Prior to the melting, the surface of 4N–silver rod was cleaned using molten NaHCO3, so that naturally formed sulfide and or oxide at the surface was removed. The temperature difference between the upper and lower rods was controlled by adjusting input power of the ceramic heaters for the upper and lower alumina rods and was monitored using R-type thermocouples installed through the upper and lower carbon rods. The error in the temperature difference measurement was estimated to be 10%, because the thermocouples did not directly touch the silver bridge, as shown in Fig. 1b. The temperature oscillation was measured at a 2.5-mm diameter spot of the molten silver surface using a two-colorpyrometer of CHINO IR-CAQ2CN though a quartz window of the chamber. The shape of the liquid bridges was observed using a CCD camera. 4. Results and discussion – Temperature oscillation analysis Fig. 2a shows the observed temperature oscillation of the molten silver bridge with an aspect ratio (height/radius)
Fig. 1. (a) A molten silver bridge sustained between the upper and lower rods of alumina ceramics. (b) A sketch of a molten silver bridge. Between molten silver and alumina ceramic rods platinum plates were inserted, so that wettability was assured between rods and molten silver.
Fig. 2. (a) Temperature oscillation due to instability of the Marangoni flow of molten silver. Temperature oscillation was observed using a twocolor pyrometer for the liquid bridge with an aspect ratio of 1.0; the temperature difference between the upper and lower ends was 52 K and the estimated Marangoni number was 160. (b) Spectrum analysis of the Marangoni flow instability for the liquid bridge of molten silver with an aspect ratio of 1.0 and the estimated Marangoni number of 160.
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silver). A Marangoni number of 160 means that flow is in a condition close to the second critical Marangoni number Mac2. Imaishi et al. (2001) reported that the Marangoni flow of the Pr = 0.01 liquid with a half-zone configuration of As = 1.0 showed m = 2 + 1 twisting mode oscillation with non-dimensional frequency of xc = 76.5 just above the second critical Marangoni number of Mac2 = 66.5, where xc is defined as xc = 2pfca2/m, and a and m are the radius of the liquid bridge and the kinematic viscosity, respectively. This corresponds to f = 0.20 Hz and agrees with the experimentally observed frequency of 0.26 Hz, when kinetic viscosity of m = 4.15 · 107 m2/s for molten silver is substituted into the above relationship. 5. Conclusion A molten silver bridge was successfully prepared to experimentally observe the Marangoni flow near the critical Marangoni number, such as Mac1 and Mac2. By employing molten silver, one will be able to measure the Mac1 of the transition form 3D axisymmetric to 3D nonaxisymmetric flow. Acknowledgement Fig. 3. (a) Temperature oscillation due to instability of the Marangoni flow of molten silicon, detected by an R-type thermocouple. The aspect ratio was 1.0; the temperature difference between the upper and lower ends at 52 K; the estimated Marangoni number was 6500. (b) Spectrum analysis of the Marangoni flow instability for the liquid bridge of molten silicon with an aspect ratio of 1.0 and a Marangoni number of 6500.
of As = 1.0 and an estimated Marangoni number of 160, where the temperature difference between the upper and lower edges of the liquid bridge was DT = 52 K. The Marangoni number of 160 is almost twice that of the Mac2 for low-Pr-number fluids. Note that the Marangoni number was lowered by more than one order of magnitude compared with that of 6500 for the molten silicon of the 10mm diameter liquid bridge with the aspect ratio of 1.0 and temperature difference between the upper and lower edges of 80 K. Fig. 2b shows the result of the fast Fourier transformation analysis of temperature oscillation, suggesting that the flow is oscillatory. Fig. 3a and 3b show the temperature oscillation and power spectrum density for the molten silicon bridge with an aspect ratio of 1.0 and the Marangoni number of 6500. For the silicon case, the temperature oscillation was measured with an R-type thermocouple in the course of research for time evolution of flow instability (Yamane et al., 2005). It is clear that the Marangoni flow was stabilized from almost turbulence (Ma = 6500) to oscillation (Ma = 160) for the liquid bridge with almost the same aspect ratio (As = 1) at temperature differences between the upper and lower edges of the liquid bridge (DT = 52 K for molten silicon and 80 K for molten
This work was supported by the Grant-in-Aid of the Japan Society of the Promotion of Science (JSPS) No.16360377 and Kato foundation. References Bernard, B., Lupis, C.H.P. The surface tension of liquid silver alloys: Part II. Ag-O alloys. Metall. Trans. 2, 2991–2998, 1971. Brandes, E.A., Brook, G.B. Smithells Metals Research Book, Seventh ed Butterworth-Heinemann, Oxford, UK, 1992. Cro¨ll, A., Mu¨ller-Sebert, W., Benz, K.W., Nitsche, R. Natural and thermocapillary convection in partially confined silicon melt zones. Microgravity Sci. Technol. 3, 204–214, 1991. Hibiya, T., Asai, Y., Sumiji, M., Kojima, T. Effect of oxygen partial pressure on silicon single crystal growth by floating zone technique: surface oxidation and Marangoni flow. Crystal Res. Technol. 38, 619– 626, 2003. Imaishi, N., Yasuhiro, S., Akiyama, Y., Yoda, S. Numerical simulation of oscillatory Marangoni flow in half-zone liquid bridge of low Prandtl number fluid. J. Crystal Growth 230, 164–171, 2001. Knacke, O., Kubaschewski, O., Hesselmann, K. Thermochemical Properties of Inorganic Substances, second ed Springer-Verlag, Verlagn Stahleisen, 1991. Kobatake, H., Fukuyama, H., Minato, I., Tsukada, T., Awaji, S. Noncontact AC calorimetry of liquid silicon with suppressing convections in a static magnetic field, Proceedings of the 16th Symposium on Thermophysical Properties, Boulder, CO, 30 July-4 2006, p. 625. Kobatake, H., Fukuyama, H., Minato, I., Tsukada, T., Awaji, S. Noncontact measurement of thermal conductivity of liquid silicon in a static magnetic field. Appl. Phys. Lett. 90, 094102, 3 pages, 2007. Levenstam, M., Amberg, G. Hydrodynamical instabilities of thermocapillary flow in a half-zone. J. Fluid Mech. 297, 357–372, 1995. Matsumoto, S., Hayashida, H., Yoda, S., Komiya, A., Natsui, H., Imaishi, N. Transition phenomena on Marangoni convection in low Pr number liquid bridge. Thermal Sci. Eng. 12, 21–22, 2004.
T. Hibiya et al. / Advances in Space Research 41 (2008) 2107–2111 Mukai, K., Yuan, Z., Nogi, K., Hibiya, T. Effect of oxygen partial pressure on the surface tension of molten silicon and its temperature coefficient. ISIJ Int. 40, S148–S152, 2000. Nakamura, S., Hibiya, T., Kakimoto, K., Imaishi, N., Nishizawa, S., Hirata, A., Mukai, K., Yoda, S., Morita, T.J. Temperature Fluctuations of the Marangoni Flow in a Liquid Bridge of Molten Silicon under Microgravity on Board the TR-IA Rocket. J. Crystal Growth 186, 402–412, 1998. Sato, Y., Nishizuka, T., Hara, K., Yamamura, T., Waseda, Y. Density measurement of molten Silicon by a pycnometric method. Int. J. Thermophysics 21, 1463–1471, 2000. Sato, Y., Kameda, Y., Nagasawa, T., Sakamoto, T., Moriguchi, S., Yamamura, T., Waseda, Y. Viscosity of molten silicon and the factors affecting measurement. J. Crystal Growth 249, 404–415, 2003.
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Shiratori, S., Kuhlmann, H.C., Hibiya, T. Linear stability of thermocapillary flow in partially confined half-zones. Physics Fluids 19, 044103, 19 pages, 2007. Takiuchi, N., Taniguchi, T., Tanaka, Y., Shinozaki, S., Mukai, K. Effects of oxygen and temperature on the surface tension of liquid iron an its wettability of alumina. J. Japan Inst. Metals 55, 180–185, in Japanese, 1991. Wanschura, M., Shevtsova, V.M., Kuhlmann, H.C., Rath, H.J. Convective instability mechanisms in thermocapillary liquid bridges. Phys. Fluids 7, 912–925, 1995. Yamane, N., Nagafuchi, K., Shiratori, S., Okubo, H., Sato, N., Hibiya, T. Experimental study on time evolution of Marangoni flow instability in molten silicon bridge. J. Materials Sci. 40, 2221– 2225, 2005.
Advances in Space Research 41 (2008) 2107–2111 www.elsevier.com/locate/asr
Attempt to study Marangoni flow of low-Pr-number fluids using a liquid bridge of silver Taketoshi Hibiya *, Keisuke Nagafuchi, Suguru Shiratori, Noriyoshi Yamane, Shumpei Ozawa Department of Aerospace Engineering, Tokyo Metropolitan University, 6-6, Asahigaoka, 191-0065 Hino, Japan Received 11 October 2006; accepted 6 April 2007
Abstract In order to experimentally investigate the Marangoni flow of low-Prandtl-number fluids in a liquid bridge geometry under the condition of small Marangoni numbers close to the critical Marangoni numbers Mac1 and Mac2, the formation of a liquid bridge of silver was attempted. The available temperature difference between the upper and lower rods to obtain a small Marangoni number, such as Ma = 50, was calculated for a 5 mm high liquid bridge for several molten metals. For molten silver, the possible temperature difference was estimated to be 16 K, whereas, for molten silicon, this was 0.38 K, which is unrealistic for the purposes of experiments. For silver, a free surface can be obtained in the wide range of oxygen partial pressures, whereas, for molten silicon, the available oxygen partial pressure range is very small; equilibrium oxygen partial pressure for SiO2 formation is as low as 1.1 · 1014 Pa. A liquid bridge of molten silver was successfully prepared and temperature oscillation was observed; the estimated Marangoni number was 160 and oscillation frequency was 0.26 Hz. 2007 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Marangoni flow; Low-Pr-number; Liquid bridge; Silver
1. Introduction Understanding and controlling Marangoni flow for lowPrandtl-number fluids (molten semiconductors, metals and alloys) are necessary to improve products from high temperature processes, such as crystal growth of semiconductors, casting of jet engine turbine blades, welding of automobile bodies and so on. When the high temperature melt has a free surface, Marangoni flow plays a significant role in the heat and mass transport processes and also affects the quality of the final products. The magnitude of the Marangoni flow is estimated using the Marangoni number, as follows, Ma ¼ ðjor=oT jDThÞ=ðlaÞ *
Corresponding author. Tel.: +81 42 585 8654. E-mail address: [email protected] (T. Hibiya).
ð1Þ
Here, r, l and a are the surface tension, the viscosity and the thermal diffusivity, respectively. DT is the temperature difference in the system, e.g., the temperature difference between the upper and lower ends of the liquid bridge sustained by solid rods in the liquid bridge geometry. h is the bridge height. Marangoni flow of low-Pr-number fluids has been studied experimentally for the case of large Marangoni numbers such as Ma = 10,000 (Nakamura et al., 1998), whereas it has been studied numerically in the small Marangoni number range, such as an order of 10 (Levenstam and Amberg, 1995; Wanschura et al., 1995; Imaishi et al., 2001). This is due to the fact that it is experimentally difficult to prepare the small Marangoni number conditions and that flow mode transition in the small Marangoni number region attracts attention of numerical people. It is also difficult to carry out the calculations at the large Marangoni number condition, because flow is almost turbulent and requires an advanced technique for calculation.
0273-1177/$34.00 2007 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2007.04.106
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T. Hibiya et al. / Advances in Space Research 41 (2008) 2107–2111
The target of the current research on the Marangoni flow of low-Pr-number fluids in a liquid bridge structure is to understand the transition from an axisymmetric to a 3Dstationary flow and the transition from a 3D-axisymmetric flow to an oscillatory flow. For these transitions, the critical Marangoni numbers Mac1 and Mac2 are indices, above which the transitions from an axisymmetric to a 3D-stationary and from a 3D-stationary to an oscillatory flow take place, respectively. The Mac1 is estimated to be 30 and the Mac2 about 100. However, for a half-zone liquid bridge of low-Pr-number fluids, it is difficult to observe these transitions experimentally. This is due to the fact that it is difficult to produce a small temperature difference between the upper and lower rods for a liquid bridge with a specific height. As an exception to this, Matsumoto et al. observed successfully Mac1 experimentally for a molten tin bridge (Matsumoto et al., 2004). In order to overcome this problem experimentally, Cro¨ll et al. (1991) proposed an idea of partial confinement of the liquid bridge of molten silicon. They coated a liquid bridge with an SiO2 film with a narrow circular window, where the driving force is limited within this open area, although total height of the liquid bridge is larger than this window. Using this technique, the Mac2 for molten silicon was experimentally determined to be in the range of 100–200. The partial confinement technique has been studied more in detail for a half-zone liquid bridge by Shiratori et al. (2007) and it was found that the flow structure within the partially confined half-zone bridge is different from that of the normal half-zone structure depending on the position of the open window. The surface of low-Pr-number fluids is easily oxidized. This means that the Marangoni flow is suppressed by formation of a thin oxide layer (Hibiya et al., 2003). Furthermore, for some molten transient metals and alloys, inversion of the temperature coefficient of surface tension is observed; the sign for the temperature coefficient of surface tension changes from negative to positive with the increase in adsorption of a surfactant such as oxygen or sulfur (Takiuchi et al., 1991). When these materials are melted, inversion of the Marangoni flow direction is expected to be observed; i.e. Marangoni flow from the cold side to the
hot side. This effect is dependent on the solubility of oxygen or sulfur in melts. Molten silver is one of candidate materials for use in study of the Marangoni flow of molten metals because its thermophysical properties assure a liquid bridge with a small Marangoni number and a free surface in a wide range of oxygen partial pressures in the ambient atmosphere. In the present study, use of molten silver is proposed to simplify the experimental condition, so that the liquid bridge of low-Pr-number fluids with small Marangoni number can be prepared. 2. Considerations concerning the half-zone liquid bridge of low-Pr-number fluids with small Ma-number Let us now discuss the conditions that apply to the preparation of a liquid bridge of low-Pr-number fluids with small Marangoni numbers. For example, for a 5-mm high half-zone liquid bridge of molten silicon, which represents a model of a floating zone crystal growth, the required temperature difference between the upper and lower edges is estimate to be 0.38 K, when the Marangoni number is required to be at Ma = 50 based on the reported thermophysical properties, as shown in Table 1. Table 1 also shows the estimated temperature difference for Sn and Ag melts, i.e. 4.5 K and 16 K, respectively. This suggests that an increase in temperature difference of 1 K between the upper and lower edges gives rise to a Marangoni number of Ma = 130 K for molten silicon, whereas Marangoni number increases by 3.1 for molten silver. The calculated temperature difference of 0.38 K for Si melt between the upper and lower edges is not realistic, whereas it is 4.5 K and 16 K for molten tin and silver, respectively, and is available experimentally. According to chemical thermodynamics, molten silver cannot be oxidized easily, but can absorb huge amounts of oxygen. For example, the equilibrium oxygen partial pressure for formation of Ag2O at the silver melting point of 1233 K is almost 0.1 MPa (Knacke et al., 1991), which is higher than the oxygen partial pressure of air, whereas equilibrium oxygen partial pressure for SiO2 formation is
Table 1 The thermophysical properties of molten silver, tin and silicona
Melting temperature Tm (K) Isobaric mass heat capacity Cp (J/kg K) Density q (kg/m3) Temperature coefficient of surface tension or/oT (mN/m · K) Viscosity l (Pa · s) Kinematic viscosity m (m2/s) Thermal conductivity k (W/m K) Thermal diffusivity a (m2/s) Pr-number Pr = m/a () DT (K) for the liquid bridge 5-mm high, 10-mm in diameter with Ma = 50
Ag
Sn
Si
1234 283 9.35 · 103 0.16 3.88 · 103 4.15 · 107 175 6.60 · 105 6.3 · 103 16
505 250 7.00 · 103 0.07 1.85 · 103 2.64 · 107 30 1.7 · 105 1.6 · 102 4.5
1687 1.00 Kobatake et al. (2006) 2.56 · 103 (Sato et al., 2000) 0.36 Mukai et al. (2000) 0.56 · 103 Sato et al. (2003) 2.19 · 107 62 Kobatake et al. (2007) 2.42 · 105 9.0 · 103 0.38
a See ‘‘Smithells Metals Research Book Seventh Edition 1992’’ edited by Brandes and Brook for thermophysical properties of molten Ag and Sn. For molten Si, latest data are cited; see each reference. Kinematic viscosity and thermal diffusivity are defined as follows: l = q · m and k = Cp · q · a.
T. Hibiya et al. / Advances in Space Research 41 (2008) 2107–2111
as low as 1.1 · 1014 Pa for molten silicon. Also, a surface of molten tin is easily oxidized. This suggests that the surface of molten silver is always bare at this experimental condition and study of the Marangoni flow can be carried out without having to take into consideration of an oxide layer in the atmospheric condition. Furthermore, temperature coefficient can be changed from the negative to the positive value with an increase in oxygen partial pressure in an ambient atmosphere (Bernard and Lupis, 1971). This is an advantage of use of silver, compared with the use of molten tin and silicon. 3. Experimental setup A silver rod 5-mm high and 10-mm in diameter was melted and a liquid bridge was successfully prepared between the upper and lower alumina ceramic rods heated by a PBN ceramic heater. Fig. 1a shows a molten silver bridge. As shown in Fig. 1b, platinum plates were inserted between molten silver and alumina ceramic rods, so that
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wettability was assured between rods and molten silver. The entire experimental process was carried out at 6N– Ar atmosphere of 105 Pa. Prior to the melting, the surface of 4N–silver rod was cleaned using molten NaHCO3, so that naturally formed sulfide and or oxide at the surface was removed. The temperature difference between the upper and lower rods was controlled by adjusting input power of the ceramic heaters for the upper and lower alumina rods and was monitored using R-type thermocouples installed through the upper and lower carbon rods. The error in the temperature difference measurement was estimated to be 10%, because the thermocouples did not directly touch the silver bridge, as shown in Fig. 1b. The temperature oscillation was measured at a 2.5-mm diameter spot of the molten silver surface using a two-colorpyrometer of CHINO IR-CAQ2CN though a quartz window of the chamber. The shape of the liquid bridges was observed using a CCD camera. 4. Results and discussion – Temperature oscillation analysis Fig. 2a shows the observed temperature oscillation of the molten silver bridge with an aspect ratio (height/radius)
Fig. 1. (a) A molten silver bridge sustained between the upper and lower rods of alumina ceramics. (b) A sketch of a molten silver bridge. Between molten silver and alumina ceramic rods platinum plates were inserted, so that wettability was assured between rods and molten silver.
Fig. 2. (a) Temperature oscillation due to instability of the Marangoni flow of molten silver. Temperature oscillation was observed using a twocolor pyrometer for the liquid bridge with an aspect ratio of 1.0; the temperature difference between the upper and lower ends was 52 K and the estimated Marangoni number was 160. (b) Spectrum analysis of the Marangoni flow instability for the liquid bridge of molten silver with an aspect ratio of 1.0 and the estimated Marangoni number of 160.
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silver). A Marangoni number of 160 means that flow is in a condition close to the second critical Marangoni number Mac2. Imaishi et al. (2001) reported that the Marangoni flow of the Pr = 0.01 liquid with a half-zone configuration of As = 1.0 showed m = 2 + 1 twisting mode oscillation with non-dimensional frequency of xc = 76.5 just above the second critical Marangoni number of Mac2 = 66.5, where xc is defined as xc = 2pfca2/m, and a and m are the radius of the liquid bridge and the kinematic viscosity, respectively. This corresponds to f = 0.20 Hz and agrees with the experimentally observed frequency of 0.26 Hz, when kinetic viscosity of m = 4.15 · 107 m2/s for molten silver is substituted into the above relationship. 5. Conclusion A molten silver bridge was successfully prepared to experimentally observe the Marangoni flow near the critical Marangoni number, such as Mac1 and Mac2. By employing molten silver, one will be able to measure the Mac1 of the transition form 3D axisymmetric to 3D nonaxisymmetric flow. Acknowledgement Fig. 3. (a) Temperature oscillation due to instability of the Marangoni flow of molten silicon, detected by an R-type thermocouple. The aspect ratio was 1.0; the temperature difference between the upper and lower ends at 52 K; the estimated Marangoni number was 6500. (b) Spectrum analysis of the Marangoni flow instability for the liquid bridge of molten silicon with an aspect ratio of 1.0 and a Marangoni number of 6500.
of As = 1.0 and an estimated Marangoni number of 160, where the temperature difference between the upper and lower edges of the liquid bridge was DT = 52 K. The Marangoni number of 160 is almost twice that of the Mac2 for low-Pr-number fluids. Note that the Marangoni number was lowered by more than one order of magnitude compared with that of 6500 for the molten silicon of the 10mm diameter liquid bridge with the aspect ratio of 1.0 and temperature difference between the upper and lower edges of 80 K. Fig. 2b shows the result of the fast Fourier transformation analysis of temperature oscillation, suggesting that the flow is oscillatory. Fig. 3a and 3b show the temperature oscillation and power spectrum density for the molten silicon bridge with an aspect ratio of 1.0 and the Marangoni number of 6500. For the silicon case, the temperature oscillation was measured with an R-type thermocouple in the course of research for time evolution of flow instability (Yamane et al., 2005). It is clear that the Marangoni flow was stabilized from almost turbulence (Ma = 6500) to oscillation (Ma = 160) for the liquid bridge with almost the same aspect ratio (As = 1) at temperature differences between the upper and lower edges of the liquid bridge (DT = 52 K for molten silicon and 80 K for molten
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