Examining the Influence of Turbulence on Viscosity Measurements of Molten Germanium under Reduced Gravity

doi:10.21203/rs.3.rs-1593954/v1

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Examining the Influence of Turbulence on Viscosity Measurements of Molten Germanium under Reduced Gravity

https://doi.org/10.21203/rs.3.rs-1593954/v1

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published 24 Nov, 2022

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The thermophysical properties of liquid germanium were recently measured both in parabolic flight experiments and on the ISS in the ISS-EML facility. The viscosity measurements differed between the reduced gravity experiments and the literature values. Since the oscillating drop method has been widely used in EML, further exploration into this phenomenon was of interest. Models of the magnetohydrodynamic flow indicated that turbulence was present during the measurement in the ISS-EML facility which accounts for the observed difference.

viscosity

oscillating drop

turbulence

germanium

computational fluid dynamics

magneto-hydrodynamics

micro-gravity processing

Recent experiments on the International Space Station (ISS) in the Electromagnetic Levitation (EML) facility and in parabolic flight experiments have taken measurements on the density, thermal expansion, viscosity, and surface tension of molten germanium^1,2. The results of these viscosity measurements are shown in Fig. 1, where it can be seen that the viscosity measurements taken during the parabolic flight experiments on pure germanium are approximately an order of magnitude larger than the measurement taken in the ISS-EML facility^2,3. The viscosity measurement was taken at 1310°C and observed to be 2.9 mPa·s ³ in the ISS-EML. however, ground based measurements by Gruner⁴, using an oscillating cup viscometer, indicate a viscosity of 0.367 mPa·s, an order of magnitude lower. Gruner’s oscillating cup measurements were fit to the following Arrhenius relationship for pure germanium in which η_∞ = 0.206 mPa·s and E_η = 7.60 kJ/mol:

$$\eta \left(T\right)= {\eta }_{\infty }*\text{exp}\left(\frac{{E}_{\eta }}{RT}\right)$$

In the microgravity experiments during parabolic flight¹ and in the ISS-EML^2,3, the oscillating drop method was used to measure the surface tension and the viscosity of the melt over a range of processing temperatures in the facility described by Lohöfer⁵. The oscillating drop method utilizes the electromagnetic force field to excite surface oscillations in the sample. The properties of the melt are inferred from the response of the oscillations according to the relationships calculated by Rayleigh⁶ and Lamb⁷. The frequency of the oscillations is determined by the surface tension and the damping coefficient is determined by the viscosity. Lamb’s equation relating the damping coefficient to the viscosity of the melt assumes that there is no flow other than the flow driven by the surface oscillations and that that flow is laminar. While it has been assumed that laminar flow driven by the EML forces can be superimposed over the flow driven by the perturbations without affecting the surface oscillations^8,9, turbulent eddies greatly accelerate the damping. During turbulent flow, the momentum of the surface oscillations is redistributed by the turbulent eddies and damping is dominated by the turbulent dissipation rather than by the inherent viscosity of the liquid. As a result, it is important to calculate the Reynolds number describing the flow within the drop^10,11.

However, it is difficult to observe the behavior and velocity of the flow during EML experiments directly. In the liquid state, germanium is a metallic conductor. Like other molten metals, germanium is opaque preventing optical access to the internal flow. While surface particles may be present in EML experiments, these particles are swept into the stagnation lines of the flow and do not provide quantitative insight into the flow behavior. Instead, magnetohydrodynamic models are used to relate the experimental conditions and properties of the melt with the resulting internal flow of the sample.

Since the flow of the drop cannot be directly observed during most EML experiments, models are used to assess and quantify the flow behavior. The flow is driven by the EML force field which is calculated using the coil geometry, sample geometry, conductivity of the melt, and the applied current to the EML system. Further details on the magnetic model are described by Hyers et al.¹² and Bracker et al.¹³ The flow is modeled using computational fluid dynamics (CFD) in conjunction with the magnetic model. The work presented here uses ANSYS Fluent to calculate the flow present during the experiment.

The CFD model for the microgravity EML experiments is defined by the following boundary conditions: The free surface of the drop cannot be crossed by the flow and is free of traction. Second, the sample is represented by a 2-dimensional axisymmetric mesh. At the axis of symmetry, the derivatives must be zero.

The model has been validated against a physical experimental case in which the sample, a copper-cobalt alloy, formed a two-phase liquid¹⁴. This case provided a rare opportunity to use particle imaging velocimetry to directly quantify the flow on the surface of the drop. This work by J. Lee et al. found that the model agreed within the experimental accuracy, better than 7% error¹⁴

The flow was analyzed using both laminar and turbulent flow models. The laminar model directly solves the discussed Navier-stokes equations in ANSYS Fluent. Prior work has found that the laminar-turbulent transition occurs near Reynolds number 600, with flow above this observed to be turbulent while flow described by lower Reynolds numbers is seen to be laminar¹⁵. Microgravity EML experiments can display both laminar and turbulent flows. In EML the Reynolds numbers of the turbulent flow are relatively low, when compared to traditional turbulence studies. Despite the low Reynolds numbers, turbulent EML flow maintain the key characteristics of turbulence: chaos, mixing and vorticity. Turbulent flow in EML is best described by the RNG K-ε turbulence model¹⁶ which includes additional transport equations to the Navier-Stokes equations to account for turbulent kinetic energy and dissipation, and account for low Reynolds number effects ^17,18.

The flow was modeled using conditions present in the ISS-EML experiment at 1310°C when the property measurements were taken. The magnetic model used 1.52x10⁶ S/m for the conductivity of molten germanium as measured by Skinner at 1250K with negligible changes as a function of temperature over the range of interest¹⁹. The EML force field is defined by a control voltage of 7.72 V for the positioner and a control voltage of 0.00 V for the heater circuit which was ON at the time of interest. At this temperature the density was calculated to be 5308 kg/m³ using Iida and Gutherie’s density fit²⁰ and the viscosity was calculated to be 0.367 mPa·s using Gruner’s viscosity fit⁴. The resulting flow velocity in the drop was then calculated to be 0.147 m/s, corresponding to a Reynolds number of approximately 17,000, which indicates clearly turbulent flow. The flow vector field and turbulent viscosity are plotted in Fig. 2.

Under these conditions, the assumptions of Lamb’s equation are not satisfied. The model shows that the damping reported by Luo et al^2,3. from the ISS-EML experiments was largely due to turbulent dissipation. The higher viscosity values reported by Luo et al.¹ on the parabolic flight experiments correspond to even higher flow velocities than the models presented here. These higher flow velocities explain the even faster damping observed for the parabolic flight experiments.

The flow within the drop explored over a wider range of conditions to find whether or not laminar flow was accessible for any combination of parameters. The flow was modeled using the EML force fields used during the experiment over the range of cooling. The slowest flow occurred at recalescence, immediately before solidification. Recalescence, in this sample, occurred when the sample was at 885°C. At this time, the control voltages on the EML field were 3.90 V positioner and 0.00 V heater, with the heater circuit ON. The density at 885°C was calculated to be 5516 kg/m³ using Iida’s density fit²⁰ and the viscosity was calculated to be 0.454 mPa·s at 885°C using Gruner’s viscosity fit⁴. The resulting flow velocity in the drop was then calculated to be 0.0744 m/s which corresponds to a Reynolds number of 7240, which still is more than an order of magnitude larger than the expected laminar- turbulent transition at Reynolds number 600¹⁵.

The turbulent flow in the sample was further validated through a video of the experiment on the ISS in which oxide rafts on the surface of the sample can be seen to move chaotically throughout the cycle up to recalescence. The chaotic motions indicate that the flow was, in fact, turbulent. The turbulent flow at the minimum flow conditions indicates that it is not possible to achieve laminar flow in EML for a sample of this size with such low viscosity.

Measurement of the viscosity of molten germanium using oscillating drop in microgravity EML on the ISS² and in parabolic flights¹ reported values much higher than those obtained using an oscillating cup viscosimeter⁴. Models of fluid flow in the EML samples reveal that the reported difference in viscosity was caused by turbulent flow in the levitated samples. This turbulence is not always observed in EML, but only for specific combinations of sample size, material, and operating parameters. Further calculations show that for germanium samples of this size, the turbulence persists for all achievable experimental conditions. It is recommended that the flow effects are characterized using projected experimental parameters with the properties of the melt during the planning phase to ensure that the experimental conditions satisfy the requirements of the measurements during the experiment.

Data Availability

The data generated and analyzed during the current study are available from the corresponding author upon reasonable request.

Acknowledgements

The authors acknowledge collaborate support by team members from the Microgravity User Support Center (MUSC) through access to the ISS-EML facility which is a joint undertaking of the European Space Agency (ESA) and the German Aerospace Administration (DLR).

Support for this project was provided to the USTIP project through NASA Grants NNX16AB40G and 80NSSC21K0103. This work was completed in collaboration with the ESA research project SEMITHERM (AO-2000-068) which is financially supported by DLR Bonn via project 50WM1750.

The sample material preparation was done by N. Abrosimov et al. in Leibniz Institute for Crystal Growth in Berlin.

Author Contributions

The models presented in this work were created and evaluated in ANSYS Fluent by G.P. Bracker. Manuscript preparation was done by G.P. Bracker and R.W. Hyers. Many fruitful discussions and suggestions to the manuscript were provided by Y. Luo, B. Damaschke, and K. Samwer regarding the experiments and results of the experiments which were modeled in this study.

Competing Interests

The Authors declare no Competing Financial or Non-Financial Interests.

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Examining the Influence of Turbulence on Viscosity Measurements of Molten Germanium under Reduced Gravity

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