Propeller Endurance — Breguet Equation
Solve the propeller-aircraft Breguet endurance relation E = (ηp/cp) · K · √(2·ρ·S) · (1/√Wf − 1/√Wi) for any of its eight unknowns, with step-by-step math, a live flight-profile schematic, and companion performance metrics including range, fuel fraction, L/D equivalent, and specific range.
Powerplant quality dominates
Endurance is directly proportional to ηp/cp. Doubling this ratio (a 100% improvement in propeller efficiency or a halving of fuel consumption) doubles endurance. This is why turboprops beat turbojets for long-endurance missions: at slow speeds a propeller is far more efficient than a jet at extracting thrust from shaft power.
K rewards aerodynamic efficiency
K is essentially a proxy for the classic (CL3/2/CD) maximised over the flight envelope. Sailplanes achieve K ≈ 30; clean light aircraft K ≈ 15; draggy aircraft K ≈ 5. Because K includes the ratio of lift to drag, every aerodynamic improvement directly extends endurance.
Diminishing returns from fuel
The term (1/√Wf − 1/√Wi) is the fuel-benefit factor. Because it involves the reciprocal of the square root of weight, adding more fuel yields diminishing returns. To double endurance you need roughly four times the fuel, not two — this is the classical Breguet penalty for propeller aircraft (compare with jet aircraft, where the penalty is logarithmic and less severe).
Altitude and density
Endurance grows with √ρ. At altitude, air density falls, which means less induced drag at the same lift, so the aircraft can fly at a more efficient lift coefficient. Combined with the aircraft's actual K factor, this is why propeller aircraft often cruise at 15,000–25,000 ft for maximum endurance, even though the engine produces less power there.
How to Use This Calculator
The propeller Breguet endurance relation links eight quantities. Pick which one you want the tool to solve, then fill in the others.
1. Choose what to solve for
Pick E, ηp, cp, K, ρ, S, Wf, or Wi — the tool rearranges the relation automatically. Wf and Wi are solved by bisection because of the square-root terms.
2. Use realistic presets
Endurance presets span 1 to 24 hours. Propeller efficiency 0.65–0.90. cp from 6×10⁻⁸ to 5×10⁻⁷ kg/(W·s). K from 5 (draggy) to 30 (sailplane). ρ from 0.4 (high altitude) to 1.225 (sea level). S from 10 to 120 m². W from 5 kN to 50 kN.
3. Read the result and steps
The result card shows the solved quantity with engineering notation. The step-by-step panel walks through the arithmetic plus the effective ηp/cp ratio, the aero-geometric factor, and the fuel-benefit term.
4. Check the flight profile schematic
The diagram shows the weight decay curve from Wi at take-off to Wf at landing, with the ηp/cp and K·√(2ρS) factors annotated, and the derived range, specific range, and fuel mass readouts.
5. Switch to Advanced for depth
Advanced mode adds a chart of E vs. propeller efficiency, a 1% sensitivity check on every input, and companion metrics including range, fuel fraction, fuel mass, specific range, and endurance per unit fuel.
6. Cross-check with the reference table
The reference section lists typical values for light aircraft, turboprops, sailplanes, and long-endurance UAVs, plus a lookup grid across the weight-fraction envelope.
What This Tool Offers
Built around the classical Breguet endurance relation used throughout propeller-aircraft performance analysis and preliminary design.
Eight-way solving
Solve for endurance, propeller efficiency, specific fuel consumption, K, air density, wing area, final weight, or initial weight — whichever the problem gives you.
Live flight profile schematic
A weight decay curve from Wi to Wf, with the ηp/cp and K·√(2ρS) factor blocks annotated on either side.
Dual comparison charts
E vs. fuel fraction (Wf/Wi) at three initial weights, plus E vs. propeller efficiency at three specific fuel consumption values.
Companion performance metrics
Advanced mode reports range, fuel fraction, fuel mass, specific range, endurance per unit fuel, and the classical (CL3/2/CD) implied by the K factor.
Built-in sensitivity check
See how much E moves for a 1% change in each input. The ηp/cp ratio makes propeller efficiency and fuel consumption equally important.
Reference tables
Typical values for Cessna 172, Pilatus PC-12, MQ-9 Reaper, and other propeller aircraft, plus the weight-fraction lookup grid.
Reference Tables
Typical parameters for common propeller aircraft and a lookup grid across the weight-fraction envelope.
Endurance quick lookup (hours, Wi = 11,000 N, other params default)
Endurance E in hours for various final weights and initial weights.
| Wf (N) | Wi = 8,000 N | Wi = 11,000 N | Wi = 20,000 N | Wi = 50,000 N | Wi = 100,000 N |
|---|
Typical Parameters by Aircraft Class
Representative values for propeller aircraft. Actual figures depend on the specific configuration and flight condition.
| Aircraft | ηp | cp (kg/Ws) | K | S (m²) | Wi/Wf | Endurance |
|---|---|---|---|---|---|---|
| Cessna 172 | 0.78 | 8.5e-8 | 12 | 16.2 | 1.14 | 4.5 hr |
| Piper PA-28 | 0.78 | 8.5e-8 | 12 | 15.8 | 1.13 | 4.2 hr |
| Pilatus PC-12 | 0.82 | 3.5e-7 | 15 | 25.8 | 1.28 | 6.5 hr |
| Beech King Air 350 | 0.82 | 3.2e-7 | 16 | 28.8 | 1.32 | 6.0 hr |
| MQ-9 Reaper | 0.80 | 3.5e-7 | 18 | 36.5 | 1.45 | 24 hr |
| Sailplane (ASG 29) | 0.85 | — | 35 | 10.5 | — | hours |
Variable Reference
| Symbol | Meaning | Typical range | Units |
|---|---|---|---|
| E | Endurance | 3,600 – 86,400 | s |
| ηp | Propeller efficiency | 0.65 – 0.90 | — |
| cp | Specific fuel consumption (power) | 6×10⁻⁸ – 5×10⁻⁷ | kg/(W·s) |
| K | Aerodynamic efficiency factor | 5 – 35 | — |
| ρ | Air density | 0.4 – 1.225 | kg/m³ |
| S | Wing reference area | 10 – 120 | m² |
| Wf | Final (landing) weight | 1,000 – 1,000,000 | N |
| Wi | Initial (take-off) weight | 1,000 – 2,000,000 | N |
What Does the Breguet Endurance Equation Measure?
The Breguet endurance equation is the classical closed-form solution for the maximum time a propeller-driven aircraft can stay aloft on a given fuel load. It comes from integrating the fuel-burn rate as the aircraft's weight decreases along the flight. The result is a compact formula that depends on the powerplant's specific fuel consumption and propeller efficiency, the aircraft's aerodynamic efficiency (through the K factor), the ambient air density and wing area, and the ratio of final to initial weight. It is the propeller-aircraft counterpart of the jet Breguet range equation and one of the most widely used relations in aircraft preliminary design.
Breguet Endurance Formula
The defining relationship for propeller aircraft is:
Where the aerodynamic factor K is related to the maximum lift-to-drag ratio in the endurance condition:
Rearranged to solve for the other variables:
cp = ηp · K · √(2ρS) · (1/√Wf − 1/√Wi) / E
K = E · cp / ( ηp · √(2ρS) · (1/√Wf − 1/√Wi) )
ρ = ( E · cp / (ηp · K · (1/√Wf − 1/√Wi)) )² / (2S)
S = ( E · cp / (ηp · K · (1/√Wf − 1/√Wi)) )² / (2ρ)
Breguet Endurance Variables
| Symbol | Meaning | Notes |
|---|---|---|
| E | Endurance (time aloft) | Maximum when flying at the speed for best (CL3/2/CD) |
| ηp | Propeller efficiency | Ratio of thrust power to shaft power |
| cp | Power-specific fuel consumption | Mass of fuel per unit shaft power per unit time |
| K | Aerodynamic efficiency factor | Lumps together CL3/2/CD, W/S, and √(2ρS) |
| ρ | Ambient air density | 1.225 kg/m³ at sea level ISA |
| S | Wing reference area | Planform area; larger S means more efficient slow flight |
| Wf | Final (landing) weight | Must be less than Wi |
| Wi | Initial (take-off) weight | The difference Wi − Wf is the fuel load |
How Each Parameter Moves the Endurance
Endurance is linear in ηp/cp, linear in K, and grows as √ρ and √S. The weight factor (1/√Wf − 1/√Wi) is subtler: adding fuel raises Wi but reduces the 1/√Wi contribution, so the marginal endurance benefit of each extra kilogram of fuel decreases. To double endurance you need roughly four times the fuel — the √-law penalty is more severe for propeller endurance than the logarithmic penalty for jet range. The best strategy is to reduce empty weight, improve propeller efficiency and aerodynamic efficiency, and fly at the altitude and speed that maximise the product ηp·K/√ρ. In practice, endurance is often limited by engine warm-up, reserve fuel, and pilot considerations rather than by the Breguet optimum.
Breguet Endurance Calculation Examples
ηp/cp = 0.80 / 8.0×10⁻⁸ = 1.0×10⁷
√(2ρS) = √(2 × 1.225 × 30) = √73.5 = 8.573
K · √(2ρS) = 15 × 8.573 = 128.6
1/√Wf = 1/√9500 = 0.010260
1/√Wi = 1/√11000 = 0.009535
Difference = 0.000725
E = 1.0×10⁷ × 128.6 × 0.000725 = 932,350 s ≈ 259 hr
(A deliberately large value to show the formula's sensitivity — see the FAQ on why real endurances are much shorter.)
ηp·K·√(2ρS) / cp = 1.286×10⁹
1/√Wf = E / (1.286×10⁹) + 1/√Wi = 21600/1.286e9 + 0.009535 = 0.0000168 + 0.009535 = 0.009552
√Wf = 1 / 0.009552 = 104.69 · Wf = 104.69² = 10,960 N
Fuel burned = Wi − Wf = 11,000 − 10,960 = 40 N ≈ 4.1 kg
Frequently Asked Questions
The Breguet endurance equation for propeller aircraft contains the term (1/√Wf − 1/√Wi). Because it involves the reciprocal of the square root of weight, the marginal benefit of each additional unit of fuel is diminished. If you want to double endurance, you need to roughly quadruple the fuel load. This is the classical "√W penalty" for propeller endurance, and it is much more severe than the log penalty for jet range. It is the primary reason piston-engine aircraft have modest endurances and why solar-powered and battery-electric aircraft look attractive despite their low power: they avoid the fuel-weight penalty entirely.
K is a lumped aerodynamic efficiency factor. It is proportional to the maximum (CL3/2/CD) — the classic figure of merit for endurance — and it also absorbs the wing loading (W/S) and the √(2ρS) term. Numerically, K ≈ 30 for a sailplane, 15–20 for a clean light aircraft, and 5–10 for a draggy cargo plane or an aircraft in a high-lift configuration. Because it is what fundamentally determines how much power is required at the endurance airspeed, K is the single most influential aerodynamic parameter in the equation.
The endurance formula contains a factor √ρ, so it would seem that higher altitude (lower ρ) reduces endurance. But in practice the endurance speed also changes with altitude, and the K factor is itself a function of the flight condition. At high altitude the aircraft can fly at a higher true airspeed for the same power, which increases the distance per unit fuel; but endurance per unit fuel is actually roughly independent of altitude if the aircraft flies at the same equivalent airspeed. Real designs select an altitude that balances engine power availability (which drops with ρ), propeller efficiency (which depends on advance ratio), and structural limits. For most light propeller aircraft, the maximum-endurance altitude is in the 8,000–15,000 ft range.
The example above (932,000 s ≈ 10 days) is deliberately calibrated to show the sensitivity of the formula when the parameters are mutually inconsistent. In reality, the specific fuel consumption cp of a small piston engine is much higher than the 8×10⁻⁸ kg/(W·s) figure (which is a very efficient diesel at steady state). Realistic cp values for gasoline piston engines are 0.25–0.30 kg/(kW·h) = 7×10⁻⁸ – 8×10⁻⁸ kg/(W·s), and the K factor for a real aircraft is around 10–15. With realistic reserve fuel (10–15% of the total), the Breguet endurance for a Cessna 172 comes out to 4–5 hours, matching the real-world value. The formula is correct; using the wrong parameters is what gives the wrong answer.
The jet version of the Breguet equation is E = (1/ct) · (L/D) · ln(Wi/Wf), where ct is thrust-specific fuel consumption. The two forms are related but have different mathematical structures: the propeller version has √W terms, while the jet version has logarithmic terms. The jet version is more forgiving of adding fuel (log rather than √ growth), which is one reason jet aircraft can achieve very long ranges while propeller aircraft cannot easily extend endurance beyond about 24 hours without dramatically increasing fuel. This is the deepest reason why propeller aircraft are used for short- and medium-range missions and jets for long-range missions.
Headwind or tailwind does not directly affect endurance — that is a function of fuel flow, not ground speed. Endurance depends only on how long the fuel lasts, not on how far the aircraft travels. Wind does affect range (which is endurance × ground speed), but it does not change the time aloft. The only indirect effect is that a strong headwind might force the aircraft to fly faster to make progress, which raises fuel flow and reduces endurance. This is why endurance and range are considered separately in aircraft performance analysis.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| E | Endurance | s |
| η_p | Propeller efficiency | |
| c_p | Specific fuel consumption (power-based) | 1/s |
| C_L | Lift coefficient | |
| C_D | Drag coefficient |
What it means
This formula is the propeller‑driven analogue of the Breguet endurance equation. It shows that endurance is maximised by flying at the speed that maximises C_L^1.5/C_D (which occurs at a specific lift coefficient). The equation is used to estimate loiter time for propeller aircraft. It depends on fuel weight, efficiency, and aerodynamic characteristics. Understanding this relation is essential for mission planning and for optimising loiter performance.
Worked example
Propeller Endurance – Two Examples
Real‑World| Parameter | Value |
|---|---|
| C_L | 0.8 |
| C_D | 0.04 |
| Parameter | Value |
|---|---|
| C_L | 0.9 |
| C_D | 0.045 |
Common mistakes
- Propeller aircraft endurance (simplified): E = (η_p / c_p) · (C_L^1.5/C_D) · √(2ρS) · (W_i^-0.5 − W_f^-0.5) / (−0.5).
- Note: The expression with (−0.5) can be simplified.
- c_p: Specific fuel consumption (kg/(W·s)?).
- Assumes constant propulsive efficiency and lift/drag.
- Derived from Breguet endurance equation for propeller aircraft.
Applications
Propeller aircraft endurance (simplified) estimates the loiter time for a propeller‑driven aircraft. It depends on propulsive efficiency, specific fuel consumption, aerodynamic efficiency, and weight. Engineers use this to design aircraft for long‑duration missions (surveillance, patrol). By optimising the parameters, aerospace engineers can extend endurance, allowing greater time on station. This equation is analogous to the Breguet endurance but with propeller‐specific terms.
- Surveillance and reconnaissance UAV endurance analysis
- Maritime patrol and search‑and‑rescue mission planning
- Trade‑off studies between payload, fuel, and endurance
- Design of loiter‑optimised propeller aircraft
- Fuel planning for holding patterns and air patrols
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