Prev
Unit 1 · Lesson 1-3
Fuel Consumption and Range
2h 30m1300 XP
Next

Fuel Consumption and Range

Thrust specific fuel consumption, specific thrust, overall efficiency and the Breguet range equation

New York to Sydney, 18–20 October 2019
On the evening of 18 October 2019 a brand-new Boeing 787-9, registered VH-ZNI, takes off from New York's JFK as Qantas flight QF7879. On board are about fifty people, most of them crew and researchers. The aircraft has full tanks, no cargo and almost no luggage. 19 hours and 16 minutes later it lands in Sydney, the first non-stop flight between the two cities. It was a research flight: Qantas wanted to know whether people, and aircraft, could manage regular flights of that length (Airline Geeks at the gate). Why did it have to fly almost empty? Because how far an aircraft can go is a product of three numbers — one for the engines, one for the wing and one for the structure — and on that route even the best 787 ran out of all three.
Engine makers and airlines measure fuel economy with one figure: the thrust specific fuel consumption (TSFC, often just SFC) — the fuel mass flow needed per unit of thrust. It is a rate: kilograms of fuel per second per newton, or more conveniently grams per kilonewton-second. American data sheets still quote pounds of fuel per hour per pound of thrust: 1 lb/(lbf·h) = 28.33 g/(kN·s).
Lower is better. A modern high-bypass turbofan in cruise uses about 0.5–0.55 lb/(lbf·h), roughly 15 g/(kN·s); a turbojet of the 1950s used twice as much.
TSFC=m˙fF,η0=V0TSFC  hPR\mathrm{TSFC} = \frac{\dot m_f}{F}, \qquad \eta_0 = \frac{V_0}{\mathrm{TSFC}\; h_{PR}}

Thrust specific fuel consumption, and the overall efficiency it implies at flight speed V0. At a given speed, TSFC and overall efficiency carry the same information.

Derivation
TSFC and overall efficiency are the same figure
Start from the overall efficiency of Lesson 1-2, thrust power over fuel power:
η0=F V0m˙f hPR.\eta_0 = \frac{F\,V_0}{\dot m_f\, h_{PR}} .
Divide top and bottom by FF and recognise m˙f/F\dot m_f/F as the TSFC:
η0=V0(m˙f/F) hPR=V0TSFC  hPR.\eta_0 = \frac{V_0}{(\dot m_f/F)\, h_{PR}} = \frac{V_0}{\mathrm{TSFC}\; h_{PR}} .
So at a fixed flight speed a lower TSFC is a higher overall efficiency. Comparing TSFCs at different speeds is misleading: an engine's TSFC naturally rises with flight speed (ram drag grows) even when its overall efficiency improves. That is why take-off and cruise TSFCs in a data sheet differ by a factor of two.
📐From a data sheet to an efficiency
A turbofan has a cruise TSFC of 0.55 lb/(lbf·h) at V0=250V_0 = 250 m/s (about Mach 0.85 at 11 km).
  • In SI: 0.55×28.33=15.580.55 \times 28.33 = 15.58 g/(kN·s) =15.58×10−6= 15.58 \times 10^{-6} kg/(N·s)
  • Overall efficiency: η0=250/(15.58×10−6×43×106)=0.373\eta_0 = 250 / (15.58 \times 10^{-6} \times 43 \times 10^6) = 0.373 — 37 %
  • At 60 kN of cruise thrust it burns 15.58×60×3600/1000=336515.58 \times 60 \times 3600 / 1000 = 3365 kg of fuel an hour
Lab 1 does these conversions with pint, so that a pound never meets a newton unannounced.
From fuel to distance. In steady cruise the engines' thrust equals the drag, F=DF = D, and the wing's lift equals the weight, L=WL = W. So the thrust needed is F=W/(L/D)F = W/(L/D): a better lift-to-drag ratio means less thrust for the same weight. The fuel flow is TSFC×F\mathrm{TSFC} \times F, and the aircraft gets lighter as it burns it. Following that weight down over the whole cruise gives the range.
R=V0g  TSFC  LD  ln⁡WiWf=η0 hPRg  LD  ln⁡WiWfR = \frac{V_0}{g\;\mathrm{TSFC}}\;\frac{L}{D}\;\ln\frac{W_i}{W_f} = \frac{\eta_0\, h_{PR}}{g}\;\frac{L}{D}\;\ln\frac{W_i}{W_f}

The Breguet range equation, for cruise at constant speed, TSFC and L/D. Wi and Wf are the weights at the start and end of cruise.

Derivation
The Breguet range equation
The weight falls at the rate the fuel is burned:
dWdt=−g m˙f=−g  TSFC  F.\frac{dW}{dt} = -g\,\dot m_f = -g\;\mathrm{TSFC}\;F .
In cruise F=DF = D and L=WL = W, so F=W/(L/D)F = W/(L/D):
dWdt=−g  TSFCL/D W.\frac{dW}{dt} = -\frac{g\;\mathrm{TSFC}}{L/D}\,W .
The distance flown in dtdt is dR=V0 dtdR = V_0\,dt. Eliminate dtdt:
dR=−V0 (L/D)g  TSFC  dWW.dR = -\frac{V_0\,(L/D)}{g\;\mathrm{TSFC}}\;\frac{dW}{W} .
With V0V_0, TSFC and L/DL/D constant, integrate from the initial weight WiW_i to the final weight WfW_f:
R=V0g  TSFC  LD  ln⁡WiWf.R = \frac{V_0}{g\;\mathrm{TSFC}}\;\frac{L}{D}\;\ln\frac{W_i}{W_f} .
Using η0=V0/(TSFC hPR)\eta_0 = V_0/(\mathrm{TSFC}\,h_{PR}) gives the second form. The equation carries the name of the French aviation pioneer Louis Breguet; its logarithm is the same one as in the rocket equation, and for the same reason: the fuel must carry the fuel.
📐A long-haul twin
Take V0=250V_0 = 250 m/s, TSFC = 15.6 g/(kN·s), L/D=20L/D = 20, and a cruise that burns 37.5 % of the starting weight (Wi/Wf=1/0.625=1.6W_i/W_f = 1/0.625 = 1.6): R=2509.807×15.6×10−6×20×ln⁡1.6=15 361 km.R = \frac{250}{9.807 \times 15.6\times10^{-6}} \times 20 \times \ln 1.6 = 15\,361 \text{ km}. That is about the distance from New York to Sydney — with nothing left for climb, reserves or headwinds, which is exactly why QF7879 flew almost empty. Lab 2 shows the sensitivities: 1 % better TSFC or L/D buys 1 % more range; and a 10 % better TSFC buys 11 %, because the range goes as 1/TSFC1/\mathrm{TSFC}.
Now, aviation brings a new concept of time and distance to the affairs of men.
– Charles Lindbergh, Reader's Digest, November 1939 (Wikiquote)
Specific thrust and engine size. TSFC is not the only figure on the sheet. Specific thrust, F/m˙0F/\dot m_0, sets how big the engine must be for its thrust: low specific thrust means a slow jet and high propulsive efficiency, but a large fan, a heavy engine and a big nacelle with its own drag and weight — which the Breguet equation charges for through L/DL/D and Wi/WfW_i/W_f. Lab 3 shows the real trend in public engine data: as bypass ratios climbed, the thrust delivered per square metre of fan face fell. Engine design is a negotiation between TSFC and size.
💡Where you will use this
TSFC and specific thrust are the two outputs of every cycle analysis in Units 3 and 4, and the axes of the turbofan carpet plot in Lesson 3-2. Lesson 8-2 shows how both change with altitude, speed and air temperature. In aircraft design courses the Breguet equation sizes the fuel tanks.
📚Further reading
  • Daniel P. Raymer, Aircraft Design: A Conceptual Approach (6th ed., AIAA, 2018), chapter 17, on range and the Breguet equation.
  • Jack D. Mattingly and Keith M. Boyer, Elements of Propulsion: Gas Turbines and Rockets (2nd ed., AIAA, 2016), chapter 1, on TSFC and the aircraft–engine link.
  • Breguet range equation, Wikipedia — the derivation and its jet and propeller forms.
↓But there is a problem
Thrust, efficiency, fuel, range: every result in this unit assumed that we knew the jet velocity V9V_9 and the exit pressure p9p_9. We don't. Inside the engine, air is squeezed, heated and expanded at speeds where it stops behaving like the gentle fluid of first-year mechanics. On 14 October 1947, over the Mojave, a rocket plane flew into that region of speed on purpose — and the pilots of the 1940s had good reason to call it a barrier.