Lift and Buoyancy

ABSTRACT

The fundamental operating principle of Luna is its helium-based lift system. This page details the physics of buoyancy, the weight budget of the system, and the critical volume-to-lift calculations that determine the balloon’s size and payload capacity.

The Physics of Lift

Luna relies on Archimedes’ principle: the upward buoyant force is equal to the weight of the air displaced by the balloon. For Luna to fly, this force must exceed the total weight of the system (helium, envelope, frame, electronics, and tether).

Lift Equation

The net lift force ($F_{net}$) is given by:

$F_{net}=(V\cdot {\rho }_{air})-(V\cdot {\rho }_{He})-M_{system}$

Where:

• $V$ = Volume of the balloon ($m^3$)
• ${\rho }_{air}$ = Density of air ($\approx 1.225kg/m^3$ at sea level)
• ${\rho }_{He}$ = Density of helium ($\approx 0.179kg/m^3$)
• $M_{system}$ = Mass of all solid components (envelope, frame, electronics, cable)

Gross Lift Capacity

The gross lift (before subtracting system weight) generated by helium is approximately 1 kg per cubic meter.

Volume ($m^3$)	Gross Lift (kg)
3	~3.1
5	~5.2
7	~7.3
10	~10.5

NOTE

Helium density varies with pressure and temperature. Our calculations assume standard atmospheric conditions. Internal pressure is maintained slightly above ambient (superpressure) to maintain shape, which slightly increases helium density and reduces lift per unit volume, but this is negligible for our initial sizing.

Weight Budget Breakdown

To ensure flight and stability, every gram counts. The total system mass must be kept significantly below the gross lift to provide “free lift” for wind stability and tension on the tether.

1. The Envelope (Cover)

The Mylar envelope is the heaviest single component.

• Material: Mylar (PET) film, 125 $\mu m$ thickness
• Density: $\approx 1390kg/m^3$
• Surface Area: For a sphere, $A=4\pi r^2$
• Mass: $M_{env}=A\cdot \text{thickness}\cdot \text{density}$

For a 5 $m^3$ balloon ($r\approx 1.06m$):

• Surface Area $\approx 14.1m^2$
• Mass $\approx 2.45kg$ (This is high; we may need thinner material or optimized gores)

Optimization: Using 75 $\mu m$ or 50 $\mu m$ film for parts of the envelope can significantly reduce weight.

2. The Tether (Cable)

The power and data cable adds weight that increases with height.

• Length: 10 - 50 meters
• Linear Density: Target $<20g/m$
• Total Mass: 0.2 - 1.0 kg

3. Internal Payload (Electronics & Frame)

This is the fixed payload that must be lifted.

• LED Modules: ~100g
• Cooling Fans: ~50g
• Frame: ~150g
• Sensors & Wiring: ~100g
• Total Payload: < 500g

Lift vs. Volume Analysis

We model the net lift capacity as a function of balloon radius to find the “sweet spot” where lift is sufficient without creating excessive drag.

Critical Thresholds

1. Minimum Flight Volume: The volume where Net Lift > 0.
2. Operational Volume: The volume providing enough excess lift (tension) to keep the tether taut in wind.

From our Python simulations:

• A 2.5m radius is too large (excessive drag).
• A 1.0m radius (~4.2 $m^3$) is marginal for lifting a robust cable.
• Optimal Range: 1.2m - 1.5m radius (7 - 14 $m^3$).

Helium Properties

We use helium not just for lift, but for its thermodynamic properties.

Property	Helium	Air	Advantage
Density ($kg/m^3$)	0.179	1.225	Provides Lift
Thermal Conductivity ($W/(m\cdot K)$)	0.151	0.025	6x Better Cooling
Specific Heat ($J/(kg\cdot K)$)	5193	1005	High Heat Capacity

TIP

The high thermal conductivity of helium is a critical design feature. It allows us to cool the high-power LEDs much more efficiently than would be possible in air, enabling the use of smaller, lighter heatsinks (or no heatsinks at all, relying on convection). See Thermal Management for details.

Pressure Management

Luna operates as a superpressure balloon (constant volume) to maintain its spherical shape against wind pressure.

• Internal Pressure: Slightly higher than ambient ($\Delta P>0$).
• Regulation: Passive. The envelope is non-extensible.
• Safety: Pressure sensors monitor for leaks or over-pressure due to solar heating.

Permeation

Helium atoms are small and can diffuse through envelope materials.

• Mylar Permeability: Very low compared to latex or PVC.
• Loss Rate: Estimated < 5% volume loss over 1000 hours of operation.
• Impact: Negligible for typical event/construction durations (hours to days).

Design Decisions

Based on this analysis, the Luna Type One specification is:

• Target Volume: 7.5 $m^3$ ($r\approx 1.21m$)
• Gross Lift: ~8.0 kg
• Target System Mass: < 4.0 kg
• Net Lift (Tension): ~4.0 kg (provides stability)

This configuration offers a robust safety margin for payload variations and ensures stability in moderate winds.