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Exercise 600.4 — Hydrogen tank thermodynamics

Pressure build-up and boil-off in a sealed LH₂ tank

🧪 Script
h2_tank.py

Reference

This exercise implements the models described in Section 6.4.5 of the course notes
(Hydrogen aircraft design: the storage challenge).

Students are expected to read the reference section carefully before working on this exercise. Equations and assumptions are not re-derived here.

How to run

From the script folder (chapters/600_hydrogen_combustion/scripts):

python h2_tank.py

Learning objectives

By completing this exercise, you will learn to:

  • Interpret the thermodynamic evolution of a sealed cryogenic hydrogen tank
  • Distinguish between non-equilibrium (isothermal) and two-phase equilibrium modeling approaches
  • Understand how heat leaks translate into pressure rise and boil-off
  • Identify numerical and physical consistency requirements in two-phase simulations
  • Connect time-domain results to the hydrogen phase diagram

Physical problem (context)

A rigid, non-vented cryogenic tank contains liquid hydrogen near its normal boiling point. A small but continuous heat leak from the environment causes:

  • partial vaporization of the liquid,
  • growth of the vapor mass and volume,
  • increase in tank pressure.

Two modeling approaches are considered in the reference material:

  1. Approach 1 — Constant liquid temperature, fixed ullage, ideal-gas vapor
  2. Approach 2 — Two-phase equilibrium at saturation, with evolving temperature and vapor volume

The script focuses on illustrating these mechanisms numerically, not on detailed tank design.

What the script does

Depending on the selected model option, the script:

  • applies a prescribed heat leak \( Q \),
  • evolves the tank state in time,

  • computes and plots quantities such as:

    • pressure \( P(t) \),
    • temperature \( T(t) \),
    • vapor mass \( M_{\mathrm{vap}}(t) \),
    • liquid mass \( M_{\mathrm{liq}}(t) \).

For the two-phase model, saturation properties are enforced consistently with the phase diagram, as described in the reference notes.

Guided questions

1) Physical interpretation of pressure rise

  • Why can very small heat inputs (order of 1–10 W) lead to large pressure increases over a few hours?

  • Which physical mechanism dominates the pressure rise:

    • sensible heating,
    • or latent heat associated with vaporization?

Relate your answer explicitly to the energy balance discussed in the reference.

2) Comparison of modeling approaches

Using the results from the script:

  • How does the pressure evolution differ between:

    • the constant-temperature / ideal-gas approach,
    • the two-phase equilibrium approach?
    • In which sense can Approach 1 be interpreted as a bounding or limiting case?

Discuss under which physical conditions each approach might be more appropriate.

3) Role of saturation and the phase diagram

  • In the two-phase model, why is the condition
    \(P(t) = p_{\mathrm{sat}}(T(t))\) enforced at all times?
  • How does this constraint shape the trajectory of the solution in the \( P\text{–}T \) plane?

Use the hydrogen phase diagram from the reference notes to support your explanation.

4) Initial-condition consistency (numerical robustness)

The reference emphasizes the importance of choosing a consistent initial vapor mass.

  • What happens if the initial vapor mass is not consistent with:

    • total mass,
    • tank volume,
    • saturation properties at \( T(0) \)?
    • Which unphysical behaviors may appear in the numerical solution?

Explain why these issues are not merely numerical bugs, but indicators of an inconsistent physical state.

5) Time scales and operational relevance

  • Over what time scale does pressure reach potentially critical values?

  • How does this compare with:

    • typical ground operation times,
    • turnaround times,
    • cruise durations?

What does this imply for tank venting, pressure regulation, or mission planning?

Student tasks

Task 1 — Baseline reproduction

Run the script using the baseline parameters provided.

Produce plots of:

  • \( P(t) \),
  • \( T(t) \),
  • \( M_{\mathrm{vap}}(t) \),
  • \( M_{\mathrm{liq}}(t) \).

Briefly describe the qualitative behavior of each variable.

Task 2 — Heat-leak sensitivity

Increase and decrease the heat leak \( Q \) by at least a factor of 5.

For each case:

  • compare pressure rise rates,
  • identify whether temperature or phase change dominates.

Discuss whether the response is linear or nonlinear.

Task 3 — Initial-condition robustness test

Deliberately modify the initial vapor mass so that it is inconsistent with saturation at \( T(0) \).

Observe:

  • early-time behavior of pressure and temperature,
  • any numerical instabilities or nonphysical values.

Explain why enforcing consistency at \( t = 0 \) is essential in two-phase modeling.

Task 4 — Engineering interpretation

In ~10–12 lines, answer:

  • Why is hydrogen storage primarily a thermal management problem, not an energy problem?
  • Why does tank pressurization remain a concern even with excellent insulation?
  • Which design or operational strategies could mitigate these issues?

Limitations (important)

This exercise uses a lumped-parameter, rigid-tank model. It does not account for:

  • structural stresses and tank mechanics,
  • active venting or pressure regulation,
  • sloshing or stratification,
  • coupling with aircraft mission phases,
  • transient non-equilibrium phase-change kinetics.

The goal is physical insight, not certification-level modeling.

Key takeaway

Even in an idealized setting, cryogenic hydrogen storage exhibits strong coupling between heat transfer, phase change, and pressure rise.

This coupling — not combustion — is one of the dominant constraints on hydrogen-powered aircraft.