Cryogenic Propellant Boil-Off Calculator

Thermal Balance Model and Formulas

The model assumes a uniform average heat flux q (W/m²) through the tank boundary and a surface area A (m²). The total heat leak rate is

Total heat leak:

Q̇ = q × A   (watts)

Over a period of one hour (3600 s), the energy absorbed by the propellant is

Energy per hour:

Q = Q̇ × 3600   (joules)

If the latent heat of vaporization is L_v (kJ/kg), converted to J/kg as 1000 × L_v, the corresponding mass boiled off per hour ṁ (kg/h) is

Mass boil-off rate:

ṁ = (q × A × 3600) / (1000 × L_v)

Using the liquid density ρ (kg/m³), this is converted to a volumetric boil-off rate V̇ (m³/h):

Volumetric boil-off rate:

V̇ = ṁ / ρ

For a tank with liquid volume V (m³), the time to lose half of that volume, t₅₀, is

Storage half-life (50% loss time):

t₅₀ = (0.5 × V) / V̇

The calculator also provides a logistic-style risk score to express how quickly this 50% loss occurs relative to a one-day (24 h) reference.

Risk score formula (0–100%):

Risk = 100 × σ((24 − t₅₀) / 4)

where σ is the logistic sigmoid function. When t₅₀ is much shorter than 24 h, Risk approaches 100%; when t₅₀ is much longer than 24 h, Risk approaches 0%.

The key relationships can also be written in MathML form:

How to Use the Calculator

1. Tank volume (m³): Enter the liquid volume you expect to have in the tank. For small test tanks this might be 1–5 m³; large launch tanks can exceed 100 m³.
2. Surface area (m²): Use the external surface area of the tank. Spherical tanks minimize area per unit volume; cylindrical tanks with domed ends have higher area. If you do not know the exact value, a first-order geometric estimate is sufficient.
3. Heat leak rate (W/m²): This is the average heat flux through the insulation plus support penetrations. Typical orders of magnitude:
   • Well-insulated LH₂ tanks with MLI: ~0.05–0.3 W/m²
   • Moderate insulation or aged systems: ~0.3–1 W/m²
   • Poorly insulated tanks or warm piping: >1 W/m²
4. Latent heat of vaporization (kJ/kg): At typical storage conditions, LH₂ is around 446 kJ/kg and LOX around 213 kJ/kg. Other cryogens (e.g., LNG, liquid methane, liquid nitrogen) have different values; use data consistent with your operating temperature and pressure.
5. Liquid density (kg/m³): Representative densities near the normal boiling point are roughly 70 kg/m³ for LH₂ and 1,140 kg/m³ for LOX. Density changes with temperature and subcooling; choose a value appropriate for your condition.

After entering the inputs, run the calculation to obtain:

• Mass boil-off rate (kg/h)
• Volumetric boil-off rate (m³/h)
• Time to 50% volume loss (h)
• Qualitative risk score (0–100%) for rapid loss within a day

Interpreting the Risk Score

The risk score is a compact indicator of how severe boil-off is relative to a one-day storage window. It is not a safety classification, but a planning aid. Use the following guidance as a rule of thumb:

Worked Example

Consider a launch facility storing LH₂ in a well-insulated spherical tank.

1. Inputs:
   • Tank volume V = 10 m³
   • Surface area A = 50 m²
   • Heat leak rate q = 0.1 W/m²
   • Latent heat L_v = 446 kJ/kg
   • Density ρ = 70 kg/m³
2. Total heat leak: Q̇ = q × A = 0.1 × 50 = 5 W.
3. Energy absorbed per hour: Q = 5 × 3600 = 18,000 J.
4. Mass boiled off per hour: L_v = 446 kJ/kg = 446,000 J/kg, so ṁ = 18,000 / 446,000 ≈ 0.040 kg/h.
5. Volumetric boil-off rate: V̇ = 0.040 / 70 ≈ 5.7 × 10⁻⁴ m³/h.
6. Time to 50% loss: Half the tank volume is 5 m³, so t₅₀ = 5 / (5.7 × 10⁻⁴) ≈ 8,800 h (roughly 365 days).
7. Risk score: Because t₅₀ is much greater than 24 h, the logistic risk evaluates near 0%, indicating very slow boil-off relative to a one-day window.

In practice, this configuration would support long-duration storage with minimal mass loss, assuming the average heat leak assumptions are valid.

Comparison of Typical LH₂ vs LOX Properties

While the calculator accepts arbitrary inputs, the table below highlights approximate property ranges for two common propellants at conditions near their normal boiling points. Use data from your specific design or test for detailed work.

Assumptions and Limitations

This calculator is intended as a first-order engineering and planning tool, not as a substitute for detailed cryogenic system design or safety review. Key assumptions include:

• Steady, uniform heat flux: The heat leak rate (W/m²) is treated as constant over the entire tank surface and over time. Local hot spots, support penetrations, and transient conditions are not modeled.
• Constant fluid properties: Latent heat and density are held fixed. In reality they vary with temperature, pressure, and degree of subcooling.
• No pressure or vent control feedback: The model assumes boil-off is driven only by energy balance, without effects from tank pressurization, vent schedules, or pressure-control setpoints.
• No stratification or mixing effects: The liquid is assumed to be well mixed, with no vertical temperature gradients or layering.
• No active refrigeration: Cryocoolers, reliquefaction systems, and subcooling hardware are not represented; all incoming heat is assumed to produce boil-off.
• Single-tank, single-fluid model: Coupled tanks, complex plumbing networks, and multi-fluid configurations are outside the model scope.

Because of these simplifications, results should be interpreted as indicative trends rather than certified performance predictions. For mission- or safety-critical decisions, use detailed thermal models, ground-test data, and formal engineering analysis in addition to this calculator.

Practical Use Cases and Next Steps

In launch operations, you can use the tool to check how long a cryogenic stage can sit on the pad before boil-off becomes operationally significant, to quantify the benefit of adding or improving insulation, or to compare different propellants for long-coast upper stages. Test facilities can use it for rough sizing of storage tanks and vent systems, and for estimating consumable losses over multi-shift campaigns.

For more refined studies, consider coupling these boil-off estimates with structural supports heat-leak models, line chilldown calculations, or detailed CFD/thermal simulations. Internal documentation or related tools on insulation performance and tank sizing can also complement the insights from this calculator.