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地上での液体金属拡散実験で対流を抑制する Ra 数の条件
Rayleigh numbers for suppressing convection in groundbased diffusion measurements of liquid metals
山中亜里紗 1, 登林兼丸 1, 小林由央 1, 椎木政人 1,2, 鈴木進補 1
Arisa YAMANAKA1, Kanemaru NOBORIBAYASHI1, Yoshihiro KOBAYASHI1,
Masato SHIINOKI1,2 , and Shinsuke SUZUKI1
1 早稲田大学, Waseda Univercity
2 ドイツ航空宇宙センター, Deutsches Zentrum für Luft- und Raumfahrt

1. Introduction
Diffusion coefficients in liquid alloys are industrially important for understanding the diffusion phenomena.
Natural convection should be suppressed to measure diffusion coefficients accurately. A stable density
layering which is the density increases monotonically in the direction of gravity is used to suppress naturalconvection in ground-based measurements 1, 2). However, in some cases of impurity diffusion experiments witha small concentration difference, the conditions for stable density layering are not clear because of the density
difference caused by a slight temperature distribution in the sample. The objective of this study is to identify
the conditions for suppressing natural convection in impurity diffusion measurement expressed as Rayleighnumber (Ra). Computational Fluid Dynamics (CFD) was performed to evaluate the Ra correctly undercomplex density and temperature conditions.
2. Experimental and analytical methods
Diffusion experiments were conducted using the shear cell technique at 973 K. The diffusion couples were
Al-AlSn (labeled Cap. A, B, C, and D) and prepared with four different initial concentrations of Sn (c0 = 0.05,0.46, 1.55, 2.88 at.%, respectively). The main procedure of the experiment was the same as that in reference 3).
The fluid analysis was performed using the COMSOL Multiphysics🄬 version 5.6 with the same dimensions
as a capillary of the shear cell device. Simulation conditions were pure Al (c0 = 0 at.%) and the temperature
gradient was set according to the temperature measurement as shown in Fig. 1 (ii). To visualize the convection
behavior during the diffusion experiment, Fick's law equations, Navier-Stokes equations, and energy balance
equations were solved.
3. Results
Experimental results using the shear cell method have already been published in reference 3). Figure 1 (iv)shows the CFD results of the velocity field. The maximum absolute velocity at the bottom (x1~x10) of the gravity
direction was 1.93× 10-14 ms-1 and the maximum absolute velocity at the top (x11~x20) was 8.22× 10-14 ms-1.
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Therefore, a stable density layering was observed and convection was suppressed at the bottom. On the otherhand, the density layering was not stable at the top, and convection occurred.

Figure 1. Conditions and results of simulation reproduced a shear cell experiment(Simulation conditions were pure Al (c0 = 0 at.%)). (i) Schematic diagram of acapillary in a shear cell device, (ii) Temperature distribution, (iii) Density
distribution in the initial state, and (iv) Velocity field of pure Al in the capillary.
4. Discussion
The calculation method of the maximum value of the density gradient 𝐺𝑚𝑎𝑥 is demonstrated in Eq. (1) 2)
and Fig. 2, which shows the results of Cap. B as an example.
𝜌𝑖+1 − 𝜌𝑖−1
𝐺𝑚𝑎𝑥 = 𝑚𝑎𝑥 []2∆𝑥Rayleigh number was calculated by Eq. (2) 4,5),𝑅𝑎 =𝑔𝑟 4𝐺𝜂𝐷 𝑚𝑎𝑥(1)(2)

where 𝑟 is a capillary radius (7.5×10-4 m), 𝑔 is gravitational acceleration (9.8 m/s2), 𝜂 is the viscosity of pureAl (1.29×10-3 Pa∙s) 6), and 𝐷 is the diffusion coefficient of Sn in liquid Al which is measured by shear cell
technique (6.7×10-9 m2/s). The Ra at the beginning and the end of the experiment for each capillary is shownin Fig. 3. For small initial concentrations (Cap. A and B), Ra increased with time. It is suggested that convection
due to a slight temperature gradient, which generated in pure Al, occurred initially in the initial states andwas not suppressed. On the other hand, for large initial concentrations (Cap. C and D), Ra decreased with time.
Therefore, accurate experiments can be performed under experimental conditions where Ra decreases during
diffusion experiments.

Figure 2. Density distribution of Cap. B (The initial concentration of Sn c0 = 0.46 at.%) andthe method to calculate the density gradient.
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Figure 3. Relationship between Rayleigh number (Ra) and the diffusion coefficient of Sn
in liquid Al (D). The number next to the plot is the initial concentration of Sn (c0).
5. Conclusion
The natural convection can be suppressed when the Rayleigh number (Ra) decreases during the diffusion
experiment.
Acknowledgment
This study was supported by financial support by Kimura Foundry Co. Ltd, Grant-in-Aid for Scientific Research(C)Grant Number JP23K0364, Grant-in-Aid for JSPS Research Grant Number JP20J14950, and Grant-in-Aid for JST SPRING,
Grant Number JPMJSP2128.

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© 2023 by the authors. Submitted for possible open access publication under theterms and conditions of the Creative Commons Attribution (CC BY) license(http://creativecommons.org/li censes/by/4.0/).

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