Bubble Positioning, Stability, and Coolant Flow in BS-Fusion
To evaluate whether a small DT bubble can be positioned accurately and remain stable within a reactor, here is an illustrative scenario using a 1 mg deuterium-tritium bubble (50:50 mix, 100% burn yield = 330 MJ) immersed in molten Yb:FliBe at 750 K:
Parameters Used
- Fluid: Yb:FliBe
- ρ = 2000 kg/m³
- μ = 3 Pa·s
- σ = 0.19 N/m
- Cp = 2414 J/kg·K
- Boiling point = 1703 K
- Bubble:
- Mass = 1 mg
- γ = 1.4
- Yield = 330 MJ
- Acoustics:
- Square wave: 50 Hz, ±5 atm
- Triangular wave: 50 Hz, 40 m/s² (≈4 G)
Bubble Radii and Velocities
Using the ideal gas law (PV = NkT) for pressures of 10, 15, and 20 atm:
- Bubble radii: 0.53, 0.58, and 0.66 cm
- Terminal velocities (gravity only, G = 9.81 m/s²): 4.0, 4.9, and 6.4 cm/s
- Terminal velocities (acoustic drive, G = 40 m/s²): 16.5, 20.0, and 26.2 cm/s
- Average acoustic velocities (due to oscillation lag): 15.0, 17.3, and 20.5 cm/s
It should be possible to produce net directional motion (ratcheting) at ~5.5 cm/s by alternating between high/low pressures and fast/slow phases.
Dissolution Estimates
Under 1-minute durations, gas loss to dissolution is negligible:
- Estimated loss: 0.000012%–0.00372% of gas content
Pressure Effects on Bubble Properties
- Resonance frequencies (Hz): 1104, 1548, 2070
- Internal gas temperatures (K): 668, 750, 814
Figures of Merit for Stability (Various Accelerations)
| Metric | 10 atm | 15 atm | 20 atm | Criterion |
|---|---|---|---|---|
| Weber (1G) | 0.58 | 0.29 | 0.18 | Stable (We < 10) |
| Galilei (1G) | 1.13 | 0.92 | 0.80 | Stable (Ga < 25) |
| Eötvös (1G) | 4.56 | 3.48 | 2.87 | Stable (Eo < 20) |
| Weber (4G) | 5.88 | 3.65 | 2.51 | Slight deformation possible |
| Galilei (4G) | 2.29 | 1.87 | 1.62 | Stable |
| Eötvös (4G) | 18.61 | 14.20 | 11.72 | Stable |
Coolant Exchange Calculations (Per Shot)
- Fusion energy: 330 MJ
- Temp rise: 750 → 1100 K
- Coolant volume per shot: \frac{3.3 \times 10^8\ \text{J}}{2414\ \text{J/kg·K} \times 350\ \text{K} \times 2000\ \text{kg/m³}}\approx 0.195\ \text{m³/shot}
- Flow through 0.5 m diameter inlet:
- Cross-sectional area = π × (0.25 m)² ≈ 0.196 m²
- Distance per shot = 0.195 m³ / 0.196 m² ≈ 1.0 m
- Flow speed for 60 shots/hour = 1 m / 60 s = 1.7 cm/s
- Displacement at center of sphere (r = 2.5 m):
- Cross-section = π × (2.5 m)² ≈ 19.6 m²
- Displacement per shot = 0.195 m³ / 19.6 m² ≈ 1.0 cm
Bubble Transport and Centering
- Radius of sphere: 2.5 m
- Bubble rise (natural @15 atm): ~4.9 cm/s
- Acoustic transport speed: ~5.5 cm/s
- Time to center (if needed): < 1 min (without pumping)
Even if all pumps are off and acoustic waves are disabled, the bubble will rise by gravity in under a minute. However, acoustic driving offers faster, directional control.
Lowering ambient pressure increases bubble buoyancy, offering another way to reduce positioning time.
These bubbles operate in a zone of near-total stagnation at the center of the sphere. Far from being battered by turbulence, they can be positioned and held precisely.
Conclusion
This example isn’t optimized for peak output. Higher yields and faster repetition rates are easily achievable with more massive targets. Further analysis, including methods of real-time bubble tracking and positioning, can follow once the current modeling is agreed upon.