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Analytical model validation for melting probe performance using applied computational fluid dynamics

Michael Ullman a, b , Michael Durka a, b , Kevin Glunt a, b , and Matthew Barry, PhD a, b a Applied Computational Fluid Dynamics Lab, b Department of Mechanical Engineering and Materials Science, University of Pittsburgh, PA, USA

Michael Ullman

Michael Ullman graduated from the University of Pittsburgh in December 2019 with a Bachelor’s degree in Mechanical Engineering and a minor in Physics. He plans to continue his education in the fall by pursuing a Ph.D. in Aerospace Engineering, focusing on computational fluid dynamics and fluid modeling.

Matthew Barry, PhD

Dr. Barry’s research focuses on multi-physics modeling of energy systems. This ranges from terrestrial thermal-fluid-electric coupled modeling of waste-heat recovery systems to thermalelectric-mechanical coupled modeling of space power-generation systems, and includes phase-change modeling for extraterrestrial probe design and evaluation.

Significance Statement

NASA Jet Propulsion Laboratory is developing a melting probe to access Europa’s subterranean oceans in search of extraterrestrial life. This research advances the mathematical formulation of the melting process by identifying and quantifying discrepancies between models, ultimately providing insights into how the probe shall be designed to maximize its performance.

Category: Computational research Keywords: Melting probe, Europa Clipper, Europa Lander, model validation

Abstract

Observations of water vapor plumes ejected from the waterice surface of Jupiter’s moon Europa have prompted scientists to hypothesize that liquid water oceans lie beneath the surface, making Europa a primary focus in the search for extraterrestrial life. NASA Jet Propulsion Laboratory (JPL) is developing plans for its Europa Lander mission, during which a probe will melt through the surface ice sheets to access Europa’s subterranean oceans. An analytical model is being developed by JPL and the University of Pittsburgh to compute the probe’s melting performance envelope, but this model requires corroboration from numerical models. In this project, the boundary conditions of the analytical model were implemented into ANSYS-CFX finite-volume models to evaluate the analytical model’s validity.

All of the numerical models exhibited heat flux distributions qualitatively similar to the analytical model on the front and side of the ice cavities, but the cavity shapes differed from what was desired. The greatest discrepancies occurred at the front corner of the probe, suggesting that radial dissipation of axial heat flux must be considered at this location. The results provide insights into the applicability of the analytical model and how the desired melt profile may be achieved.

1. Introduction

Because terrestrial life originated and thrives in Earth’s oceans, biologists hypothesize that the presence of water may be essential for life to emerge. To test this theory, astrobiologists look to examine extraterrestrial environments where water and organic compounds are plentiful. The discovery of life in these environments would help to elucidate how life developed on Earth and answer one of the most profound questions in science—are we alone in the universe?

About 400 million miles from Earth, the smallest of the Galilean moons, Europa, orbits its home planet of Jupiter. This moon is notable for its thin, oxygen-rich atmosphere and fractured water-ice surface, which is splotched with red-brown hues believed to result from salt and sulfur compounds discolored by radiation [1]. Recent analyses of data from NASA’s Galileo orbiter suggest that the spacecraft flew through a plume of water vapor when passing close to Europa in 1997 [2]. Water vapor ejections have also been observed by the Hubble Space Telescope, with periodicity consistent with predicted variations in Europa’s tidal forces [3]. Scientists have hypothesized that these plumes originate within a water ocean beneath Europa’s icy surface, making it a primary focus in the search for extraterrestrial life.

Because of this promise, NASA is developing plans for its Europa Lander mission, which will consist of a probe landing on and penetrating the moon’s surface to explore its subterranean oceans. The proposed method for penetrating the ice is a combination of drilling and melting—the latter of which will be caused by nuclear heat generation within the probe. Engineers at NASA Jet Propulsion Laboratory (JPL) and researchers at the University of Pittsburgh are developing a system of nondimensional equations—hereafter referred to as the JPL analytical model—to

determine the melting descent speed of the probe as a function of its internal heat generation and the characteristics of the ice environment. This model is comparatively simple to compute and allows for rapid trade-space studies by minimizing the number of system variables. Because of this utility, the model will eventually be implemented in a Monte Carlo simulation to ascertain the performance envelope of the probe while accounting for uncertainties associated with the ice environment. Many simplifications and assumptions are employed in the analytical formulation, so it requires validation from numerical modeling and experimental testing. While experimental data exists for terrestrial ice probes, steady-state melting probe performance in cryogenic ice has not been extensively studied. Thus, numerical modeling must play a critical role in the validation of the analytical model. The purpose of this project is to use finite-volume models within ANSYS-CFX to provide a framework for validating the analytical model and evaluating its shortcomings. These numerical models facilitate a more comprehensive understanding of the melting process, thereby providing insight into how the probe’s design can be optimized.

2. Methods 2.1 Assumptions and Conditions

The foundation for the JPL analytical model was drawn from the work of Haldor W.C. Aamot [4]. This formulation models a cylindrical cavity in the ice with a constant radius and length, and calculates the heat required to produce a given descent speed in a given ice environment. The heat terms considered in this project are the axial heat beneath the front end of the probe, which maintains the prescribed descent speed, and the radial heat along the side of the probe, which maintains the desired water jacket thickness and prevents the probe from getting stuck on refrozen ice. The front end heat has two components—conduction and melting. The conduction term is the heat required to warm the ice beneath the probe from its far-field temperature to its melting temperature, while the melting term is the heat required to complete the phase change from solid to liquid. Both of these terms, as well as the side heat, depend upon the properties of the ice—namely, its density, specific heat, thermal conductivity, and latent heat of fusion. While future versions of the analytical model will include temperature-dependent ice properties, the model was only equipped to utilize temperature-independent properties at the time of this project. For the sake of comparison, the numerical models presented utilize temperature-independent water and ice properties, taken to be tabulated values at the melting temperature of ice—273.15 K. These values, listed in Table 1, were provided by project collaborators at JPL.

98 Undergraduate Research at the Swanson School of Engineering Material

Water Ice Density (kg/m 3 )

999.9 916.2 Specific Heat (J/kg-K)

4217 2100 Thermal Conductivity (W/m-K)

0.569 2.373 Reference Specific Enthalpy (J/kg) 0 -334,000

Table 1: Temperature-independent material properties, taken to be tabulated values at 273.15 K.

Apart from the melting heat at the front end, the analytical model only considers heat conducted into the ice surrounding the probe—convection in the water jacket which develops around the probe is omitted from the front and side heat terms. The model assumes an infinitesimal water jacket thickness along the front of the probe, such that it is in contact with the phase change region. To maintain negligibly small viscous frictional forces on the probe’s walls and allow debris to flow around the probe, a 6 mm gap is desired between the side of the probe and the ice cavity. Thus, because the probe design is a cylinder of radius 11.5 cm and length 2.1 m, the analytical model prescribes a 12.1 cm radius for the ice cavity. Another assumption of the model is that the melting process is at steady state. In this project, steady state was defined as the point at which the cavity profiles no longer changed with time in transient CFX simulations.

All of the models created for this project used an advecting ice scheme. Like a car in a wind tunnel, the probe was held stationary, while ice was forced through the domain at the prescribed descent speed of 37.5 cm/hr and far-field ice temperature of 160 K. This allowed for greater computational speed, as a moving mesh model—i.e., using stationary ice and a moving probe—would require the domain to be remeshed after each time step. 2.2 Ice-Only Model

The first numerical model developed for the analytical comparison was an ice-only model, which used a cavity geometry with a 12.1 cm radius and only solved the energy balance equation within the CFX solver. The front and side walls of the cavity were prescribed to be the melting temperature of the ice—273.15 K—while the rear wall was prescribed to be adiabatic—i.e., perfectly insulated. This model served as the most direct analog for the analytical model, as the ice cavity profile was explicitly imposed, and only the heat conducted into the surrounding ice was modeled. In this case, the analytical model would be corroborated if the conductive heat fluxes at the front and side of the ice cavity matched the values and profiles from the analytical calculations. 2.3 Water-Ice Model (Conduction-Only)

The second model was a water-ice conduction model, which used the probe radius of 11.5 cm and only solved the energy equations within CFX. According to the analytical model, which calculates a heat flux profile conducted into the side of the ice cavity, the requisite heat flux on the side of the probe can be found by scaling the side cavity flux by the ratio of the cavity and probe radii. Thus, the analytical side cavity heat flux profile was scaled by this factor—(12.1 cm)/(11.5 cm) = 1.052—and applied to the

side wall of the probe. On the front end of the probe, the sum of the analytically-calculated conduction and melting heat fluxes was applied. On the rear wall, an adiabatic condition was applied. In this scenario, the analytical model would be supported if there were a miniscule cavity thickness at the front end, a 6-mm-thick cavity along the side wall, and heat fluxes at the cavity interface which match the analytically-calculated values and profiles. 2.4 Water-Ice Model (With Convection)

In order to capture the pertinent physics that the analytical model omits, the second numerical model was modified to include the effects of convection within the water annulus by enabling the fluid momentum solver within CFX. In previous simulations, it was found that placing fillets on the corners of the probe allowed for more substantial water flow from beneath the probe, facilitating complete ice melting in the front corner region and preventing

undesirable water pooling. Consequently, a 1 cm fillet was added to the front and rear corners of the probe in this model. The purpose of this model was to examine the effects of implementing convection on the shape of the ice cavity and the conductive heat fluxes at the cavity interface. This model illustrates whether the previously omitted physical effects play a negligibly small role in the analysis, thereby providing insight into whether the analytical model can be applied in a more realistic context.

3. Results

Once each of the models reached steady-state, the results were analyzed and compared to the values calculated by the analytical model. Figure 1 shows the conductive heat flux and cavity thickness profiles along the length of the probe, while Figure 2 shows the profiles along the front end. The integrated heat values and mean cavity thicknesses are given in Table 2.

Figure 1: Left (a): Heat fluxes into side of ice cavity along probe height, Right (b): Thickness of ice cavity along side of probe.

Figure 2: Left (a): Heat fluxes into front of ice cavity along probe radius, Right (b): Thickness of ice cavity along front end of probe.

Model

Aamot (Analytical) Ice-Only Water-Ice: ConductionOnly Water-Ice: With Convection Side Wall: Ice Conduction (W) Mean Ice Cavity Thickness (mm)

4540.7

4521.6 6.000

4452.3 2.841

4908.3 5.811 Front End: Ice Conduction (W) Mean Ice Cavity Thickness (mm)

1044.1

1121.5 0

860.5 0.178

771.5 0.534

Table 2: Comparisons between analytical and numerical models: heat and cavity thicknesses.

4. Discussion

Figure 1a shows that all three models exhibit radial heat flux profiles qualitatively similar to the analytical model along the side of the ice cavity. This is especially true for the ice-only and water-ice conduction profiles, which are nearly indistinguishable from the analytical profile for nearly the entire probe length. However, near the front end, the radial heat fluxes at the ice interface are much smaller than the analytical values, which asymptotically approach infinity. Notably, the radial heat flux profile for the water-ice model with convection initially matches the profiles from the other models, but begins to increasingly deviate at a quarter of the probe’s length. Figure 1b shows that this increase in radial conduction facilitated by convection within the water annulus drastically increases the side cavity thickness. Indeed, the mean side cavity thickness in this model, 5.811 mm, is nearly the desired 6 mm value. The smaller side cavity thickness in the water-ice conduction model illustrates that the scaled analytical radial conduction profile does not yield an ice cavity with the desired 12.1 cm radius—an erroneous assumption of the analytical model. Also, neither of the water-ice models yield a vertical side cavity, which is necessary if the numerical models are to serve as analogous validation points for the analytical formulation.

Figure 2a shows that the axial heat flux profiles along the front end of the cavity in the ice-only and water-ice conduction models closely agree with the analytical model for the first 10 cm of the cavity radius, but diverge at the corner of the cavity. This is likely due to radial dissipation of the front end heat flux, which is most pronounced at this location. In the ice-only model, the ice beneath the cavity must be warmed to 273.15 K to satisfy the prescribed front end boundary condition. When the heat used to warm the ice beneath the cavity radially dissipates into the far-field, more heat must be conducted through the cavity interface to maintain the prescribed temperature. In the case of the water-ice model conduction, the spike in heat flux at the corner of the cavity corresponds to incomplete melting at the probe’s front end, causing the conductive heat flux to approach the prescribed front wall heat flux. As shown in Figure 2b, this model yielded a cavity thickness of 0.178 mm at the front end of the probe, corresponding to the approximate thickness of the phase change region in the numerical models. This suggests that when convection is omitted, the analytical assumption of an infinitesimal water jacket thickness beneath the probe is correct, in a thermodynamic sense. However, as in the side cavity, the addition of convective heat transfer thickens the front cavity considerably. The flow within the water annulus carries heat away from the front cavity, corresponding to the smaller axial conductive heat fluxes illustrated in Figure 2a. These axial heat fluxes vanish at the front corner of the cavity because of the curvature of the cavity in this region, which itself is caused by the filleted corner of the probe in this model. As in previous simulations, the filleted corner allowed the ice in this region to full melt, thereby preventing water from pooling beneath the probe.

As shown in Table 2, the ice-only model has the best agreement with the analytical model on both of the front and side of the ice cavity. On the front end, the numerical value is 7.4% higher than the analytical value, due to the spike in axial heat flux at the corner. On the side, the numerical value is 0.4% lower than the analytical value, due to the analytical model’s unrealistically large radial heat flux at the front of the cavity. In the water-ice conduction model, there is a similarly attributed discrepancy in heat at the side of the ice cavity. However, the front end heat falls below the analytical value because of the smaller ice cavity radius in this model, which corresponds to a smaller area over which the heat fluxes are integrated. As expected from Figures 1a and 2a, the largest heat discrepancies occur in the water-ice model with convection. Here, warm water carried out of the front cavity and into the side leads to a smaller axial conduction along the front end and a larger radial conduction along the side wall.

5. Conclusion

The numerical models implemented in this project indicate that the JPL analytical model is not an exhaustive description of the probe’s melt-driven descent. When convection in the water annulus is not considered, the analytical model successfully models conductive heat transfer along the majority of the front and side of the ice cavity. However, large discrepancies in both conductive heat terms occur at the front corner of the cavity. This suggests that the radial dissipation of the axial heat fluxes at this location must be considered in the analytical formulation. As demonstrated in the water-ice conduction model, this radial dissipation leads to an ice cavity radius which is smaller than desired near the front corner of the probe. When convection is considered, the heat fluxes at the cavity interface deviate from the analytical model, but the average width of the side cavity expands to nearly the desired value of 6 mm. Unlike in the analytical model, though, this side cavity is not vertical—a necessity for the model comparison.

New probe boundary conditions are being investigated with the goal of creating a vertical 6-mm-thick side cavity within a regime which considers convection. Achieving this will create a robust numerical analog for the analytical model, which takes the pertinent physical phenomena into account and can then be used to identify correction factors for the analytical formulation. This revised analytical model will then be used in a Monte Carlo

simulation, which will determine the probe’s melting performance envelope for a variety of possible Europan ice environments. To this end, this project provides key insights into understanding where the analytical model can be improved and how the discrepancies with the desired melt cavity profile may be mitigated.

6. Acknowledgements

I would like to thank Dr. Barry, the Swanson School of Engineering, and the Office of the Provost for sponsoring my SSOE Summer Research Internship. I would also like to thank Dr. Miles Smith and the team at NASA-JPL for their contributions to this project. Computational resources were provided by the University of Pittsburgh Center for Research Computing.

7. References

[1] Europa, NASA Science Solar System Exploration (2019). https://solarsystem.nasa.gov/moons/jupiter-moons/europa/indepth/.

[2] J. Xianzhe et al, Evidence of a plume on Europa from Galileo magnetic and plasma wave signatures. Nature Astronomy 2 (2018) 59–464.

[3] L. Roth et al, Transient water vapor at Europa’s south pole, Science 343 (2014) 171-174.

[4] H.W.C. Aamot; Cold Regions Research & Engineering Laboratory, Heat transfer and performance analysis of a thermal probe for glaciers, Ft. Belvoir Defense Technical Information Center, 1967.

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