Rotorcraft Range Endurance

Rotorcraft Range and Endurance (flight-mechanics/performance/rotorcraft-range-endurance)

Use when you must close the fuel budget of a rotorcraft into hover endurance and cruise range and endurance. This leaf turns fuel into time and distance: the hover power from the weight, the rotor disk area and the figure of merit; the exact weight-decay integral that turns the fuel mass into hover time; and, at a chosen cruise speed, the range and endurance over the same fuel load from an input power-required curve scaled with the average weight. It is the fuel closure member of the rotorcraft performance pair: it pairs with flight-mechanics/performance/rotorcraft-hover-performance (the rotor power physics: induced velocity, ideal power and figure of merit at a single weight, no fuel budget) and with flight-mechanics/performance/ rotorcraft-forward-flight-performance (the power-required curve and its best-speed search, which this leaf consumes as its cruise power input). The cruise power-required curve is an input here, not recomputed.

Domain quick reference

All quantities are SI. The fuel burn rate follows the weight decay dW/dt = -g0 * c * P, with g0 = 9.80665 m/s^2, c the specific fuel consumption in kg/(s W), and P the power required at the current weight.

• Disk area: A = PI * R^2, R the rotor radius in m.
• Hover power at a weight W (induced-dominated rotor, figure of merit FM as the hover efficiency input): P = k_h * W^1.5 with k_h = 1 / (FM * sqrt(2 * rho * A)), FM in (0, 1], default rho = 1.225 kg/m^3 at sea level.
• Weight at fuel burnout: W1 = W0 - g0 * m_f, with W0 the takeoff weight and m_f the fuel mass in kg; the fuel load must leave W1 > 0.
• Hover endurance (exact integral of the weight decay): t = (2 / (g0 * c * k_h)) * (1/sqrt(W1) - 1/sqrt(W0)), seconds.
• Fuel flow at a weight: mdot = c * P(W), kg/s.
• Specific range at a cruise point: SR = V / (g0 * c * P), metres of range per kg of fuel.
• Best range speed over the curve: V maximizing SR; best endurance speed: V minimizing P. Deterministic scans over the (V, P) pairs.
• Cruise fuel closure with the average-weight power scaling: P_avg = P_ref * (W_avg / W_ref)^1.5 with W_avg = (W0 + W1) / 2, then R = V * (W0 - W1) / (g0 * c * P_avg) and E = (W0 - W1) / (g0 * c * P_avg). The W^1.5 power scaling reflects the induced-dominated rotor power; the average-weight approximation follows the house average-ROC convention of the climb-performance leaves.
• Default specific fuel consumption c = 1.0e-7 kg/(s W) (about 0.36 kg/kWh); pass the engine value when known.
• FAR 29 frames the rotorcraft certification context; the relations above are standard engineering methodology, summary-only.

Workflow

1. Fix the operating point: takeoff weight W0 in N, fuel load m_f in kg, rotor radius R in m, density rho at the operating altitude, rotor figure of merit FM and specific fuel consumption c.
2. Get the rotor disk area with disk_area(radius).
3. Compute the hover power constant with hover_power_constant(radius, rho, figure_of_merit) and the hover power at the takeoff weight with hover_power(weight_n, radius, rho, figure_of_merit).
4. Close the hover endurance fuel budget with hover_endurance( weight_initial_n, fuel_mass_kg, radius, rho, figure_of_merit, c_specific): the weight-decay power integration into hover time.
5. Get the fuel flow at the operating weights with fuel_flow(weight_n, radius, rho, figure_of_merit, c_specific).
6. For cruise, take the power-required curve at the reference weight from the forward-flight sibling. Evaluate the specific range at a candidate speed with specific_range(v_ms, power_w), and pick the best range speed with best_range_speed(power_curve) and the best endurance speed with best_endurance_speed(power_curve) over the (V, P) pairs.
7. Close the cruise fuel budget at the chosen speed with cruise_range(v_ms, weight_initial_n, fuel_mass_kg, power_at_ref_w, weight_ref_n, c_specific) and cruise_endurance(v_ms, ...), which apply the average-weight power scaling to the reference power.
8. Confirm the deterministic checks with the contract test scripts/test_rotorcraft_range_endurance.py.

Worked example

Six-tonne class helicopter, W0 = 60000 N, fuel 1500 kg, rotor radius 8 m, sea level (rho = 1.225), FM 0.75, c = 1.0e-7 kg/(s W). Running the module functions:

• disk_area(8.0) = 201.062 m^2.
• hover_power_constant(8.0) = 0.0600746 (SI).
• hover_power(60000, 8.0) = 882912 W (about 883 kW).
• fuel_flow(60000, 8.0) = 0.0882912 kg/s.
• W1 = 60000 - 9.80665 * 1500 = 45290.0 N; hover_endurance(60000, 1500, 8.0) = 20927.3 s = 5.81 h.
• Power-required curve at the reference weight W_ref = 60000 N, (V m/s, P W): (40, 620000), (50, 560000), (60, 540000), (70, 555000), (80, 600000). best_range_speed = 80 m/s, best_endurance_speed = 60 m/s.
• specific_range(60, 540000) = 113.302 m per kg of fuel.
• cruise_range(80, 60000, 1500, 600000, 60000) = 2433442 m = 2433 km.
• cruise_endurance(60, 60000, 1500, 540000, 60000) = 33797.8 s = 9.39 h.

Verification

• Confirm disk_area(8.0) = 201.062 m^2 (within 0.01) and that a zero or negative radius raises ValueError.
• Confirm hover_power_constant(8.0) = 0.0600746 (within 1e-5) and that non-positive density or a figure of merit outside (0, 1] raises ValueError.
• Confirm hover_power(60000, 8.0) = 882912 W (within 10), with ValueErrors at zero weight, at FM 0 and at FM 1.5.
• Confirm fuel_flow(60000, 8.0) = 0.0882912 kg/s (within 1e-5) and the identity fuel_flow = c * hover_power at any weight.
• Confirm hover_endurance(60000, 1500, 8.0) = 20927.3 s (within 1.0), returns exactly 0.0 s with zero fuel, grows with fuel mass and with figure of merit, and raises ValueError for fuel that zeroes the weight and for negative fuel.
• Confirm specific_range(60, 540000) = 113.302 m/kg (within 0.01), ValueErrors at zero speed and zero power, and monotone growth with speed at a fixed power.
• Confirm best_range_speed and best_endurance_speed over the worked curve return 80.0 and 60.0 m/s, reject an empty curve and any non-positive pair, and match the argmax of specific range and the argmin of power computed independently.
• Confirm cruise_range(80, 60000, 1500, 600000, 60000) = 2433442 m (within 100) and cruise_endurance(60, 60000, 1500, 540000, 60000) = 33797.8 s (within 10). At a fixed cruise speed the reference power sits in the denominator, so a lower reference power closes a longer range (the fuel burns more slowly); across the worked curve the lower-power lower-speed point closes a shorter range than the best-range point, and the weight-scaling (W_avg/W_ref)^1.5 is a strict reduction below the reference weight, so the scaled range is longer than the no-scaling estimate.
• Confirm determinism: repeated runs return identical floats (no RNG, stdlib only).

Pitfalls

• Feeding the hover-power figure of merit into a from-geometry rotor build: this leaf takes FM (and the disk area) as inputs and computes power and fuel closure only; the rotor power physics and geometry terms live in the hover-performance and forward-flight siblings.
• Treating the cruise power as constant over the whole fuel burn: induced-dominated rotor power decays with weight, so the closure must scale the reference power with (W_avg/W_ref)^1.5; using the takeoff power under-predicts the cruise range.
• Burning the fuel load below positive weight: the weight at burnout W1 = W0 - g0 * m_f must stay above zero, and the module raises ValueError when the fuel mass violates that.
• Confusing the two best-speed picks: best range speed maximizes the specific range (speed per fuel flow), best endurance speed minimizes the power; on the worked curve they sit at 80 m/s and 60 m/s.
• Mixing units: weight must be in N (mass in kg times g0), not in kgf, and the specific fuel consumption is in kg/(s W), not kg/(kW h); pass c = 1.0e-7 default only when the engine value is unknown.

Related leaves

• flight-mechanics/performance/rotorcraft-hover-performance: the rotor power physics at a single weight (induced velocity, ideal power, profile power, figure of merit) that this leaf consumes as its hover power input.
• flight-mechanics/performance/rotorcraft-forward-flight-performance: the power-required curve and its best-speed search over the physical curve, which this leaf takes as the cruise power input; the two leaves together gate a rotorcraft mission performance check.
• flight-mechanics/performance/breguet-range and flight-mechanics/performance/breguet-endurance: the fixed-wing cruise fuel-closure leaves with a fixed lift-to-drag factor; this leaf owns the rotorcraft fuel closure with average-weight power scaling instead.
• flight-mechanics/performance/rotorcraft-vertical-climb-performance: the rotorcraft vertical flight closure that consumes the same hover power law in climb checks.

Behavior contract (gate 3)

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

python3 skills/flight-mechanics/performance/rotorcraft-range-endurance/scripts/test_rotorcraft_range_endurance.py

The test covers the worked example against the spec anchors (disk area 201.062 m^2, hover power constant 0.0600746, hover power 882912 W, fuel flow 0.0882912 kg/s, hover endurance 20927.3 s, cruise range 2433442 m, cruise endurance 33797.8 s, specific range 113.302 m/kg), the zero-fuel and burnout bounds, the weight-scaling identities of the cruise closure, the best-speed argmax and argmin identities over the power-required curve, the fuel-flow equals c times hover power identity, run-to-run determinism, absence of random or external imports, and ValueError rejection of every non-physical input in the verification list.

Compliance

• Standards referenced, not reproduced: FAR 29 (rotorcraft airworthiness, certification context only). The fuel-closure relations above are standard engineering methodology, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.