Range and endurance of jet aircraft

Why it matters

Jet transports earn their money on range: the distance they can carry a payload decides which city pairs are possible without a fuel stop. Jet engines burn fuel in proportion to thrust rather than power, so the best speeds and altitudes for range and endurance differ from those of propeller aircraft — which is why airliners cruise high and fast, and hold at their minimum-drag speed.

Key ideas

Thrust specific fuel consumption (TSFC). A jet engine is rated by c_t, the weight of fuel burnt per unit thrust per unit time. On a weight basis its unit is N/(N·s) = 1/s; engine data often quote it per hour, e.g. 0.6 h⁻¹ (equivalently 0.6 kg/(kgf·h) or 0.6 lb/(lbf·h)), which is 0.6/3600 = 1.667 × 10⁻⁴ s⁻¹. If c_t is given in kg/(N·h), multiply by g and divide by 3600 to get 1/s. The weight-loss rate is then dW/dt = −c_t·T = −c_t·D (level flight, T = D).

Endurance. dt = −dW/(c_t·D) = −(1/c_t)·(L/D)·dW/W. With c_t and L/D constant, E = (1/c_t)·(L/D)·ln(W₀/W₁).

• Maximum endurance is at (L/D)_max, the minimum-drag condition (minimum fuel flow, because fuel flow ∝ thrust = drag).
• Altitude enters only through c_t, which falls somewhat with height in the troposphere; to first order, jet endurance does not depend on altitude.

Range. dR = V·dt = −(V/c_t)·(L/D)·dW/W. The speed now appears explicitly, so the result depends on how the flight is flown. Programme 1 — constant C_L and constant V (cruise climb). As weight falls, the aircraft drifts up so that W/ρ stays constant. Then R = (V/c_t)·(L/D)·ln(W₀/W₁) (the classic Breguet range equation for jets). The product V·(L/D) is the jet's "range factor"; airlines cruise near the Mach number that maximises M·(L/D). Programme 2 — constant C_L at constant altitude. Speed must fall as fuel burns, V = √(2W/(ρSC_L)). Integrating, R = (2/c_t)·√(2/(ρS))·(C_L^(1/2)/C_D)·(√W₀ − √W₁).

• Maximum range is at maximum C_L^(1/2)/C_D: C_L = √(C_D0/(3K)), induced drag = C_D0/3, C_D = (4/3)·C_D0.
• That speed is the Carson speed, 3^(1/4)·V_md ≈ 1.32·V_md, and there L/D = 0.866·(L/D)_max.
• Range ∝ 1/√ρ: flying higher increases range, until compressibility (drag rise near the drag-divergence Mach number) or thrust available limits it.

Why jets fly high and fast. At fixed C_L, higher altitude means higher TAS for the same drag, so distance per kilogram of fuel rises as 1/√σ. A propeller aircraft, whose fuel flow scales with power (∝ DV), gains nothing from this; a jet gains directly.

Assumptions. Constant c_t; parabolic polar with no Mach effects; steady level flight (or cruise climb); no allowance for climb, descent, reserves or wind. In real flight planning, c_t varies with throttle, Mach and altitude, and the cruise is split into segments ("step climbs").

Wind. Wind changes ground range (ground speed = TAS ± wind) but not endurance, which depends only on fuel flow.

Formulas

dW/dt = −c_t·T

E = (1/c_t)·(L/D)·ln(W₀/W₁) (jet endurance), maximum at (L/D)_max

R = (V/c_t)·(L/D)·ln(W₀/W₁) (constant V and C_L — cruise climb)

R = (2/c_t)·√(2/(ρS))·(C_L^(1/2)/C_D)·(√W₀ − √W₁) (constant altitude and C_L)

(C_L^(1/2)/C_D)_max = (3/4)·(1/(3·K·C_D0³))^(1/4) at C_L = √(C_D0/(3K))

V_(best range) = 3^(1/4)·V_md

Symbols: c_t thrust specific fuel consumption on a weight basis (1/s); T thrust, D drag, W weight (N); W₀ initial, W₁ final weight (N); E endurance (s); R range (m); V true airspeed (m/s); ρ density (kg/m³); S wing area (m²); C_L, C_D lift and drag coefficients; C_D0, K drag-polar constants; V_md minimum-drag speed (m/s).

Worked examples

Example 1 (standard): cruise-climb range. Given: V = 230 m/s, c_t = 0.6 h⁻¹, L/D = 16 held constant, fuel burnt = 30 % of initial weight (W₁/W₀ = 0.7).

1. Convert: c_t = 0.6/3600 = 1.6667 × 10⁻⁴ s⁻¹.
2. V/c_t = 230/1.6667 × 10⁻⁴ = 1.380 × 10⁶ m.
3. ln(W₀/W₁) = ln(1/0.7) = 0.35667.
4. R = (V/c_t)·(L/D)·ln(W₀/W₁) = 1.380 × 10⁶ × 16 × 0.35667 = 7.875 × 10⁶ m = 7875 km.

Example 2 (GATE level): best range at constant altitude, and endurance. Given: W₀ = 600 kN, W₁ = 450 kN, S = 150 m², C_D = 0.018 + 0.045·C_L², c_t = 0.6 h⁻¹, cruise at 10 km where ρ = 0.4127 kg/m³ (given) and a = 299.5 m/s.

1. Best-range C_L = √(C_D0/(3K)) = √(0.018/0.135) = 0.3651; C_D = (4/3) × 0.018 = 0.0240.
2. C_L^(1/2)/C_D = 0.60428/0.0240 = 25.18.
3. √(2/(ρS)) = √(2/(0.4127 × 150)) = 0.17974; √W₀ − √W₁ = 774.60 − 670.82 = 103.78 N^½.
4. R = (2/c_t)·√(2/(ρS))·(C_L^(1/2)/C_D)·(√W₀ − √W₁) = 12 000 × 0.17974 × 25.18 × 103.78 = 5.636 × 10⁶ m = 5636 km.
5. Check the speeds: V = √(2W/(ρSC_L)) falls from 230.4 m/s (M 0.77) to 199.5 m/s (M 0.67), acceptably subsonic.
6. Endurance at (L/D)_max = 1/(2√(0.045 × 0.018)) = 17.57: E = (1/c_t)·(L/D)_max·ln(W₀/W₁) = 6000 × 17.57 × 0.28768 = 30 320 s = 8.42 h.

Common mistakes

• Using TSFC in h⁻¹ with V in m/s. Convert c_t to s⁻¹ first.
• Applying the propeller range result (best at (L/D)_max) to a jet; a jet's best range is at the Carson speed, C_L^(1/2)/C_D maximum.
• Using the cruise-climb formula for a constant-altitude flight with large weight change; the two programmes give different answers.
• Assuming wind changes endurance.
• Forgetting the Mach limit: the 1/√ρ gain in range stops near the drag-divergence Mach number.

For GATE AE

Expect Breguet range and endurance numericals for jets with TSFC conversion, questions on the optimum C_L (C_L^(1/2)/C_D for range, C_L/C_D for endurance) and the corresponding speed ratios, the effect of altitude on jet range versus propeller range, and comparisons of flight programmes. Practise the constant-altitude formula with √W terms and the derivation from dW/dt = −c_t·D.

Quick check

1. At what aerodynamic condition is a jet's endurance a maximum?
2. For a jet at constant altitude, what is C_D at the best-range condition in terms of C_D0?
3. Convert c_t = 0.72 h⁻¹ to s⁻¹.
4. Does the jet's best range at constant C_L increase or decrease with altitude?

Answers: 1. (L/D)_max. 2. (4/3)·C_D0. 3. 2.0 × 10⁻⁴ s⁻¹. 4. It increases (∝ 1/√ρ), until Mach effects intervene.