Range Breguet⚠ unverified

Understanding the Parameters

• True airspeed $V$ — flying faster covers more ground per unit time, so range scales linearly with $V$ at fixed everything else. But you cannot raise $V$ for free: drag rises, so $L/D$ eventually falls, and the best cruise is a compromise (see the Carson-speed note below).

• Thrust-specific fuel consumption $c_t$ — the engine's thirst: fuel burned per unit thrust per unit time. Lower is better, and it enters as $1/c_t$, so halving $c_t$ doubles range. Six decades of turbofan development — higher bypass ratios, higher turbine temperatures — have been a long war on $c_t$.

• Lift-to-drag ratio $L/D$ — the airframe's aerodynamic efficiency: newtons of lift per newton of drag (Lift-to-Drag Ratio). Range is maximised by cruising at the speed for maximum $L/D$ (the minimum-drag speed). A sailplane reaches $L/D \sim 50$; an airliner $\sim 17$; Concorde $\sim 7$.

• The weight ratio $W_i/W_f$ — the only term carrying the fuel fraction, and it enters as a logarithm. This is the equation's most consequential feature: because range depends on $\ln(W_i/W_f)$, adding fuel gives diminishing returns. Doubling the fuel fraction does not double range; to fly twice as far you must exponentially increase the fuel-to-empty ratio — the tyranny that makes ultra-long-haul, and in the extreme rocketry, so hard.

• Why they multiply — the four terms are independent because they live in different disciplines (propulsion, aerodynamics, structures, operations). A designer can improve any one without touching the others, and the product structure means a 10 % gain in each of the first three yields $1.1^3 \approx 33\%$ more range. This is why the Breguet equation is the organising principle of cruise-aircraft design.

Derivation (Approaching a Proof)

The core idea: fuel burned equals weight lost. As the engine burns fuel the aircraft gets lighter, and the whole derivation is bookkeeping on that fact.

The fuel weight burned per second for a jet is the TSFC times the thrust, $c_t\,T$. Since weight lost equals fuel burned,

$$\frac{dW}{dt} = -c_t\,T.$$

In steady, level cruise two equilibria hold: lift equals weight ($L = W$) and thrust equals drag ($T = D$). With $L = W$, the drag is $D = W/(L/D)$, so

$$\frac{dW}{dt} = -c_t\,\frac{W}{L/D}.$$

Convert time to distance: the aircraft covers $dR = V\,dt$, so $dt = dR/V$, giving

$$\frac{dW}{dR} = -\frac{c_t}{V}\,\frac{W}{L/D} \quad\Longrightarrow\quad dR = -\frac{V}{c_t}\,\frac{L}{D}\,\frac{dW}{W}.$$

Treating $V$, $c_t$, and $L/D$ as constant through the cruise (the classic assumption), integrate from the initial weight $W_i$ to the final weight $W_f$:

$$R = -\frac{V}{c_t}\,\frac{L}{D}\int_{W_i}^{W_f}\frac{dW}{W} = -\frac{V}{c_t}\,\frac{L}{D}\,\big[\ln W\big]_{W_i}^{W_f} = \frac{V}{c_t}\,\frac{L}{D}\,\ln\frac{W_i}{W_f}. \qquad\blacksquare$$

The logarithm is not an approximation — it is the exact integral of $dW/W$, the mathematical signature of "the aircraft you are propelling gets lighter as you go."

The propeller form. A propeller engine produces power, not thrust, so fuel flow is the power-specific fuel consumption $c_p$ times the shaft power $P = TV/\eta_p$ (propeller efficiency $\eta_p$). Repeating the derivation replaces $V/c_t$ with $\eta_p/c_p$:

$$R = \frac{\eta_p}{c_p}\,\frac{L}{D}\,\ln\frac{W_i}{W_f}.$$

Notably $V$ drops out — a propeller aircraft's range does not depend directly on speed, which is why efficient propeller cruise favours flying slowly at maximum $L/D$.

Dimensional check (jet form). $$\frac{V}{c_t}\,\frac{L}{D}\,\ln\frac{W_i}{W_f} = \frac{\text{m}/\text{s}}{1/\text{s}}\times(\text{–})\times(\text{–}) = \frac{\text{m}}{\text{s}}\times\text{s} = \text{m}. \checkmark$$ The weight ratio and $L/D$ are dimensionless; $V/c_t$ carries the metres.

History and Development

• The Breguet family. The equation is named for Louis Charles Breguet (1880–1955), the French aviation pioneer whose aircraft company later merged into Dassault and who founded the airline that became Air France. The range relation was developed in the 1920s as the theory of long-distance flight matured; it is sometimes attributed jointly to Breguet and the British aerodynamicist J. G. Coffin.

• Why it endured. The equation crystallised the insight that range is a product of separable efficiencies, giving designers a clear target in each discipline. It underpinned the transoceanic ambitions of the 1930s and became the sizing tool of the jet age.

• The logarithm's tyranny. The $\ln(W_i/W_f)$ term is the aeronautical cousin of the Tsiolkovsky rocket equation (1903), which has the identical logarithmic weight-ratio structure — both express the same brutal truth that a vehicle must carry the fuel it has not yet burned, linking aircraft range to orbital mechanics (see Escape Velocity).

• Modern refinements. Real cruise is flown as a slow climb (the "cruise-climb" or step-climb) to hold the aircraft near best $L/D$ as it lightens, approximating the constant-$C_L$ assumption of the derivation. Designers distinguish the max-range speed from the slightly faster Carson speed (about $1.32\times$ the min-drag speed), which trades a little fuel for a worthwhile gain in speed — "the least wasteful way of wasting fuel."

Related Concepts: Endurance, Specific Range, Lift-to-Drag Ratio, Glide Ratio, Drag Force, Lift Force, Thrust, Escape Velocity

Notes: Jet form; propeller form replaces $V/c_t$ with $\eta_p/c_p$ (and $V$ drops out). SFC is the thrust-specific fuel consumption $c_t$ (weight basis for the 1/s label to be correct). Range $\propto \ln$ of the weight ratio — diminishing returns on fuel (same structure as the rocket equation). Max range at the speed for maximum $L/D$. All registry defaults $1.0$ ⇒ $\ln(1)=0$, zero range until real values entered.