Lesson 6 · 40 min

The Modern Transport and the Breguet Range Equation

A 707 of 1959 and a twin-engine airliner of today look alike: a swept wing, engines in pods, a tube fuselage. The shape barely changed; the numbers inside it did. One equation, Breguet's, shows which numbers matter and why every major advance since 1960 has attacked one of its three factors.

Learning objectives

Write the Breguet range equation in Raymer's form and identify its aerodynamic, propulsive and structural factors.
Use it to estimate range or the fuel fraction needed for a cruise, and explain why jets cruise near \(0.866\,\LDmax\).
Explain how the high-bypass turbofan, supercritical wing, winglets, fly-by-wire and composite structures each improve one of the factors.
Read the growth in size of transport aircraft from 1928 to 1982 in Loftin's data.

The Breguet range equation

In steady cruise, lift equals weight and thrust equals drag. The fuel flow, by weight, is the thrust times the thrust-specific fuel consumption \(C\) (weight of fuel per unit thrust per unit time), so the weight falls as \(\dd W/\dd t = -CT = -CW/(L/D)\). Dividing by \(V = \dd x/\dd t\) and integrating from the start weight \(W_{i-1}\) to the end weight \(W_i\) at constant \(V\), \(C\) and \(L/D\) gives the range attributed to Louis Breguet:

Breguet range (jet; Raymer's form)

\[ R = \colV{\frac{V}{C}}\,\colL{\frac{L}{D}}\,\ln\frac{W_{i-1}}{W_i} \]

\(C\) in 1/h (lb of fuel per lb of thrust per hour, or N/(N·h)) with \(V\) in km/h gives \(R\) in km. For a propeller aircraft, \(V/C\) is replaced by \(\eta_p/C_{\text{bhp}}\) in consistent units: the propeller efficiency over the power-specific fuel consumption.

Aerodynamics: \(L/D\)

Lower \(\CDz\), higher span and aspect ratio. A jet gets its best range at \(L/D = 0.866\,\LDmax\) (Raymer), a little faster than the speed for \(\LDmax\).

Propulsion: \(V/C\)

Fly fast, burn little. Typical cruise values of \(C\) used in initial sizing are about 0.9/h for a turbojet, 0.8/h for a low-bypass turbofan and 0.5/h for a modern high-bypass turbofan.

Structure: \(\ln(W_{i-1}/W_i)\)

The more of the takeoff weight is fuel, the farther it goes, but only logarithmically. A lighter empty weight leaves more room for fuel and payload.

Example 6.1 — How much fuel did the 707 need?

Loftin gives the 707-320B a range of \(6240\ \text{mi}\) (\(10\,040\ \text{km}\)) with maximum payload and no reserves, cruise at \(886\ \text{km/h}\), and \(\LDmax\) of 19 to 19.5. Take \(\LDmax = 19.25\), cruise at \(0.866\,\LDmax\), and \(C = 0.8/\text{h}\) for its low-bypass JT3D engines. What fraction of the starting weight is burned in cruise? Compare with the published weights: \(\Wo = 336\,000\ \text{lb}\), \(\We = 147\,000\ \text{lb}\), payload \(53\,900\ \text{lb}\).

Show solution

\[ \frac{L}{D} = 0.866(19.25) = 16.67, \qquad \ln\frac{W_0}{W_1} = \frac{RC}{V(L/D)} = \frac{10\,040(0.8)}{886(16.67)} = 0.544 \] \[ \frac{W_1}{W_0} = e^{-0.544} = 0.580, \qquad \text{fuel burned} = 1 - 0.580 = 0.42\ \text{of}\ W_0 \]

The weights leave room for at most \(1 - 147\,000/336\,000 - 53\,900/336\,000 = 1 - 0.438 - 0.160 = 0.40\) of \(\Wo\) as fuel (less still after crew, takeoff, climb and landing). The simple estimate is within a few percent: good enough to size an aircraft, and a reminder of how sensitive range is to every factor.

Figure 6.1 The Breguet range equation as three factors. The bars show the range factor \((V/C)(L/D)\) and the range for the fuel burned in cruise. The 707 preset uses Loftin's data and \(C = 0.8\)/h; the other presets are illustrative values, not data for a particular aircraft. Try improving one factor at a time by 20%: each gives the same 20% more range.

Propulsion: the high-bypass turbofan

A turbojet makes thrust by accelerating a small mass of air to a very high speed, which wastes kinetic energy in the exhaust. A turbofan passes most of its air around the core through a large fan (the bypass ratio is the ratio of bypass to core airflow), accelerating more air less: a higher propulsive efficiency and a lower \(C\), and much less noise. The first high-bypass engines flew on the Lockheed C-5A (1968, GE TF39) and the Boeing 747 (1969, P&W JT9D). Bypass ratios have since grown from about 5 to more than 10.

Bigger, more efficient engines also changed the number of engines. From the mid-1980s, regulators allowed twin-engine airliners to fly long routes far from diversion airports (extended-range twin operations), and long-haul flying moved from four engines (707, 747) and three (DC-10, L-1011) to two.

Size: wide bodies and economies of scale

Larger aircraft carry more payload for each unit of drag and of crew cost. The 747-200B's gross weight, 379 t, is 62 times the Ford Trimotor's. Figure 6.2 shows the growth.

Figure 6.2 Gross weight of transport aircraft in Loftin's tables, by year (logarithmic scale). Each step up in size came with a new technology: metal monoplanes (1930s), four-engine pressurized airliners (1940s–50s), jets (1950s), wide bodies with high-bypass turbofans (1969–70).

Figure 6.3 Jet transports, to scale (grid squares are 10 m), with the DC-3 for comparison. Engine positions changed as designers traded wing bending relief and ground clearance (pods under the wing) against a clean wing and quiet cabin (engines at the tail).

Aerodynamics, control and structures since 1970

Loftin's tables stop in 1982. The improvements since then are less visible than the jump from props to jets, but each acts on a Breguet factor:

Supercritical airfoils (Richard Whitcomb, NASA, late 1960s; flight-tested on an F-8 in 1971). A flatter upper surface keeps the supersonic region over the wing weak, so the shock is weaker and the drag rise comes later. Designers spent the gain on thicker wings (lighter, more fuel volume), less sweep or a higher cruise Mach number.
Winglets (also Whitcomb, 1970s). Small near-vertical surfaces at the tips recover some of the energy in the tip vortex, reducing induced drag without the full weight of a longer span.
Fly-by-wire and relaxed stability. Electrical signals and computers between the pilot and the control surfaces: the F-16 (1974) was designed with relaxed static stability, and the Airbus A320 (1987) brought digital fly-by-wire to airliners. A smaller tail and less trim drag improve \(L/D\); computers also keep the aircraft inside its flight envelope.
Composite structures. Carbon-fiber composites moved from control surfaces in the 1970s to the primary structure: about half the structural weight of the Boeing 787 (2009) and Airbus A350 (2013). Lighter structure and higher allowable cabin pressure and humidity; the weight saving improves the structural factor.
Digital design. The Boeing 777 (1994) was designed entirely as 3D computer models. Computational fluid dynamics, finite-element analysis and multidisciplinary optimization now let a design be analyzed in depth before metal is cut.

Check your understanding

Key takeaways

Breguet: \(R = (V/C)(L/D)\ln(W_{i-1}/W_i)\). Range is proportional to the product of a propulsive, an aerodynamic and a (logarithmic) structural factor.
Jets get their best range near \(L/D = 0.866\,\LDmax\).
High-bypass turbofans cut \(C\); supercritical wings and winglets raise \(L/D\) or allow higher speed; fly-by-wire trims drag and tail size; composites cut structural weight.
Transport gross weight grew about sixty-fold between 1928 and 1970, with each step tied to a new technology.
Next, Lesson 7 steps back from the aircraft to the process that designs them, and shows how historical data become the first estimate of a new design.