Brake Energy Sizing

Wheel Brake Energy Sizing (vehicle-design/sizing/brake-energy-sizing)

Use when the task is sizing the wheel brake heat sink of an aircraft at the conceptual level: converting the kinetic energy the brakes must absorb at the rejected takeoff (RTO) decision speed and at the landing touchdown speed into a per-brake energy, a required carbon heat sink mass, and a temperature rise check of the selected heat sink. This leaf implements the standard brake energy sizing model in pure Python, stdlib only (scripts/brake_energy_sizing_logic.py). It pairs with vehicle-design/sizing/landing-gear-sizing for the gear context and vehicle-design/sizing/tire-sizing for the tire context; the flight-test-operations/performance/accelerate-stop-distance leaf covers the rejected takeoff distance demonstration of a flight test, while this leaf sizes the brakes from the kinetic energy of the same stop. The carbon specific heat, the reverse-thrust credit and the design deceleration are program inputs with documented typical defaults.

Domain quick reference

• RTO energy at the rejected takeoff: E_rto = 0.5 * m_to * V1^2, with m_to the takeoff mass (normally MTOW) and V1 the decision speed.
• Landing-stop energy: E_land = 0.5 * m_lw * v_td^2, with m_lw the landing mass (normally MLW) and v_td the touchdown speed.
• Per-brake share with reverse-thrust credit: E_b = E_total * (1 - r_rev) / n_b, over n_b braked wheels; the conservative default credit r_rev is 0.
• Required heat sink mass per brake: m_hs = E_b / (c_p * delta_t_allow), with c_p the heat sink specific heat (carbon default 1200 J/(kg K)) and delta_t_allow the allowable temperature rise.
• Actual temperature rise of the selected heat sink: delta_t = E_b / (m_hs * c_p); the margin is delta_t_allow - delta_t.
• Governing case: the stop with the larger per-brake energy sizes the heat sink; for a normal transport the RTO stop at V1 governs.
• Braking distance at the design deceleration: s = v^2 / (2 * a) with a = decel_g * G0 and G0 = 9.80665 m/s^2.
• Certification context: FAR-25 and CS-25 treat the rejected takeoff condition as the brake energy design case for transport category airplanes; the brakes must absorb that energy within the heat sink temperature limits. Summary context only, standards not reproduced.
• SI units throughout: J, kg, K, m/s.

Workflow

1. Fix the program inputs: MTOW and V1 for the RTO stop, MLW and touchdown speed for the landing stop, the number of braked wheels, the heat sink specific heat, the allowable temperature rise, the available heat sink mass per brake and the design deceleration.
2. Compute the stop energies: rto_energy_J(mtow_kg, v1_m_s) and landing_energy_J(mlw_kg, touchdown_speed_m_s).
3. Divide each stop energy over the braked wheels, applying the reverse-thrust credit, with per_brake_energy_J(total_energy_J, n_braked_wheels, reverse_credit).
4. Identify the governing case: the larger per-brake energy (rto or landing) drives the sizing.
5. Size the heat sink: required_heat_sink_mass_kg(energy_per_brake_J, cp, delta_t_K) for the governing per-brake energy.
6. Check the selected heat sink: temperature_rise_K(energy_per_brake_J, mass_kg, cp) with the available mass per brake, and form the margin delta_t_allowable - actual rise.
7. Estimate the braking distance at V1 with braking_distance_m(v_m_s, decel_g) at the design deceleration.
8. Run analyze(inputs) on the full input dict for the complete report: both energies, both per-brake values, the governing case, the required mass, the actual rise, the margin, the braking distance and the verdict.
9. Read the verdict: brake-energy-pass when the margin is non-negative and the required mass fits the available mass, else brake-energy-fail. Confirm with the contract test.

Worked example

Regional transport: MTOW 70000 kg, V1 = 70 m/s; MLW 58000 kg, touchdown speed 65 m/s; 4 braked wheels; carbon cp 1200 J/(kg K); allowable rise 300 K; available heat sink 130 kg per brake; deceleration 0.35 g; reverse-thrust credit 0.

• E_rto = 0.5 * 70000 * 70^2 = 171,500,000 J (171.5 MJ).
• E_land = 0.5 * 58000 * 65^2 = 122,525,000 J (122.5 MJ).
• Per-brake RTO energy = 171.5e6 / 4 = 42,875,000 J (42.875 MJ); per-brake landing energy = 122.525e6 / 4 = 30.63 MJ, so the RTO stop governs.
• Required heat sink mass = 42.875e6 / (1200 * 300) = 119.10 kg.
• Actual rise with 130 kg = 42.875e6 / (130 * 1200) = 274.84 K, margin 300 - 274.84 = 25.16 K; required mass 119.10 <= 130 kg, verdict brake-energy-pass.
• With a 100 kg heat sink the rise is 357.29 K, above the 300 K allowable, verdict brake-energy-fail.
• Braking distance at V1 = 70^2 / (2 * 0.35 * 9.80665) = 713.8 m.

Pitfalls

• Sizing the heat sink on the wrong stop: the governing case is the stop with the larger per-brake energy (the RTO at V1 for a normal transport - 42.875 MJ per brake against 30.63 MJ landing in the worked example), not whichever stop is easier to check.
• Forgetting the energy scales with the square of the speed: both stop energies are 0.5 * m * v^2, so a small V1 or touchdown speed error dominates the energy and hence the heat sink mass.
• Crediting reverse thrust that is not there: the per-brake share applies (1 - r_rev) with a conservative default credit of 0; assuming thrust reversal during sizing without the credit change under-sizes the heat sink.
• Checking the temperature rise at the required mass instead of the available one: the margin is allowable rise minus the actual rise of the SELECTED heat sink (274.84 K with 130 kg, 357.29 K with 100 kg in the worked example), so the verdict flips with the available mass.
• Mixing up specific heat units: m_hs = E_b / (c_p * delta_t) with carbon c_p near 1200 J/(kg K); using a kJ-based specific heat shifts the required mass by 1000.
• Sizing from energy while ignoring the distance case: this leaf sizes the brake heat sink from kinetic energy; the rejected takeoff DISTANCE demonstration belongs to the flight-test accelerate-stop-distance leaf.

Verification

• Confirm rto_energy_J(70000, 70) returns 171.5e6 J and landing_energy_J(58000, 65) returns 122.525e6 J.
• Confirm required_heat_sink_mass_kg(42.875e6, 1200, 300) returns 119.10 kg and temperature_rise_K(42.875e6, 130, 1200) returns 274.84 K with margin 25.16 K.
• Confirm braking_distance_m(70, 0.35) returns 713.8 m.
• Confirm analyze on the example dict reports governing_case "rto" and verdict "brake-energy-pass"; with heat_sink_mass_available_kg 100 the verdict flips to "brake-energy-fail".
• Confirm the round trip: the required mass computed for the allowable rise gives exactly that rise back when re-checked.
• Confirm every non-positive mass, speed, wheel count, specific heat, allowable rise and deceleration, and every reverse credit outside 0..1, raises ValueError.
• Run the contract test offline: python3 scripts/test_brake_energy_sizing.py (35 tests, deterministic).

Related leaves

• vehicle-design/sizing/landing-gear-sizing: the landing gear and shock strut context around the wheels.
• vehicle-design/sizing/tire-sizing: tire load and footprint context for the same landing gear.
• flight-test-operations/performance/accelerate-stop-distance: the rejected takeoff distance demonstration whose stop this leaf sizes the brakes for.
• vehicle-design/sizing/weight-estimation: the MTOW and MLW inputs for the stop energies.

Behavior contract (gate 3)

Run the deterministic contract test (stdlib unittest, offline):

python3 scripts/test_brake_energy_sizing.py

The test covers the regional transport sizing contract (RTO energy 171.5 MJ, landing energy 122.5 MJ, per-brake RTO energy 42.875 MJ, required mass 119.10 kg, rise 274.84 K with 130 kg and margin 25.16 K, rise 357.29 K with 100 kg, braking distance 713.8 m), the quadratic speed scaling of both stop energies and of the braking distance, the per-brake split with a 20% reverse-thrust credit and the zero-credit default, the inverse scaling of required mass with the allowable rise, the rto and landing governing cases, the pass and fail verdicts, the round-trip identity between required mass and temperature rise, and ValueError rejection of non-positive mass, speed, wheel count, specific heat, allowable rise and deceleration and out-of-range reverse credit.

Compliance

• Standards referenced, not reproduced: FAR-25 (14 CFR Part 25 Airworthiness Standards for Transport Category Airplanes) and CS-25 (Certification Specifications for Large Aeroplanes) frame the transport category brake energy absorption context; the relations above are standard engineering methodology, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.