Lesson 6 · 35 min
Fuel Reserves, Winds and Design Range
Every airliner lands with fuel it hoped never to need. The rules decide how much, and on a long flight it is several tonnes: weight that must be lifted and carried all the way. Add the winds, which make the same route longer one way than the other, and the "design range" in a requirement turns out to be quite a bit more than the distance on the map.
Learning objectives
- Explain what reserve fuel is for, and summarize the main fuel rules for general aviation and airlines.
- Compute holding fuel with the Breguet endurance equation, and total reserves with a fuel rule.
- Convert a ground distance flown against a wind into an equivalent still-air distance.
- Build a design range from a route: great circle, routing allowance, winds, and reserves carried as fuel.
What reserves are for
The flight plan assumes forecast winds, the planned route and altitude, and a landing on arrival. Reality differs. Reserve fuel covers:
- Contingency: stronger headwinds, a lower or longer route given by air traffic control, a heavier aircraft than planned.
- Diversion: weather or a closed runway at the destination: a missed approach, then a climb and cruise to an alternate airport.
- Holding: circling in a holding pattern while waiting for a landing slot.
In sizing, reserves are carried on every flight but (normally) never burned, so they behave like payload that cannot be sold. They are accounted for either as explicit mission segments (a diversion, a loiter) or with a simple allowance, the factor 1.06 in \(\Wf/\Wo = 1.06(1 - W_x/W_0)\) of Lesson 5, which covers reserves and trapped fuel together.
The fuel rules
| Operation | Fuel required beyond the trip to the destination |
|---|---|
| Visual flight rules (aeroplanes) | 30 min at normal cruising speed by day, 45 min at night |
| Instrument flight rules | Fuel to the alternate airport (when one is required), then 45 min at normal cruising speed |
| US airlines, domestic | Fuel to the most distant alternate, then 45 min at normal cruising fuel consumption |
| US airlines, international (turbine aircraft other than turboprops) | 10% of the total flight time at normal cruising fuel consumption, then fuel to the most distant alternate, then 30 min of holding at 1500 ft above the alternate |
| ICAO and European rules (turbine aeroplanes) | Contingency fuel (typically 5% of the trip fuel), alternate fuel, and a final reserve of 30 min of holding at 1500 ft (450 m); plus taxi fuel and any extra the captain decides |
Canada's Aviation Regulations have equivalent rules. Real operations add many details (when an alternate is required, reduced contingency with better flight planning, fuel for extended operations over water), but for conceptual design these rules set the size of the reserve, and a customer's requirement usually names one: "5000 km with international reserves".
Holding fuel and the endurance equation
Holding is flown for time, not distance. In Breguet's derivation, keep the time instead of dividing by the speed, and the result is the endurance equation. A jet holds at the speed for \(\LDmax\), where the drag, and so the fuel flow, is least:
Breguet endurance (jet), and holding fuel
\[ E = \frac{1}{C}\,\colL{\frac{L}{D}}\,\ln\frac{W_{i-1}}{W_i} \qquad\Longleftrightarrow\qquad \frac{W_i}{W_{i-1}} = \exp\left[-\frac{E\,C}{L/D}\right] \]\(E\) in hours with \(C\) in 1/h. For loiter, a high-bypass turbofan has \(C\) of about 0.4/h (0.5/h in cruise). The fuel burned is \(W_{i-1}\left(1 - W_i/W_{i-1}\right)\).
Example 6.1 — Thirty minutes in the hold
A narrow-body starts a 30-minute hold at \(62\,000\ \text{kg}\), with \(\LDmax = 18\) and \(C = 0.4/\text{h}\). How much fuel does it burn?
Show solution
Over a short time the exponential is almost linear: \(W\,EC/(L/D) = 62\,000(0.5)(0.4)/18 = 689\ \text{kg}\). That is simply drag times \(C\) times time: a fuel flow of about \(1380\ \text{kg/h}\). Real holding fuel flows are often higher, since aircraft hold at low altitude, where engines are less efficient, and not always at the best speed: treat this as a first estimate.
Example 6.2 — An international reserve
A wide-body plans a 7.0-hour flight burning an average of \(5600\ \text{kg/h}\) in cruise. Its alternate is \(400\ \text{km}\) from the destination and needs \(2900\ \text{kg}\) of fuel; holding burns \(4400\ \text{kg/h}\). Find the reserve under the US international rule, and compare it with the ICAO rule with 5% contingency (take the trip fuel as \(7.0 \times 5600\ \text{kg}\)).
Show solution
Either way, the reserve is about a fifth of the trip fuel of \(39\,200\ \text{kg}\), and it must be carried, and paid for, on every flight.
Winds: the equivalent still-air distance
Breguet gives range through the air. Against a headwind \(w\), the ground speed is \(V - w\), so the flight takes longer, and during that time the aircraft flies a longer distance through the air. The equivalent still-air distance (ESAD) is the air distance that uses the same fuel:
Equivalent still-air distance
\[ t = \frac{d_{\text{ground}}}{V - w}, \qquad d_{\text{air}} = V\,t = d_{\text{ground}}\,\frac{V}{V - w} \]\(V\) is the true airspeed; \(w\) is the average headwind component (negative for a tailwind).
Example 6.3 — Westbound across the Atlantic
A \(5600\ \text{km}\) route is flown at a true airspeed of \(850\ \text{km/h}\). Westbound, the average headwind component is \(90\ \text{km/h}\) (about 50 kt). Eastbound, the same wind is a tailwind. Find the still-air distance each way.
Show solution
The westbound flight needs 12% more range than the map distance, and 24% more than the eastbound flight. Average headwinds are not symmetric either: the jet streams blow from the west at cruise altitudes, strongest in winter.
Route studies do not use the average day. They use a wind that is not exceeded on most days of the year (85% of days is a common choice), so that the aircraft can fly the route nonstop with a full load almost every day.
From a route to a design range
Putting Lessons 2, 5 and 6 together, the range requirement for a route is built in steps:
- Great-circle distance between the airports (Lesson 2).
- Routing allowance: airways, departure and arrival procedures and air traffic control make the flown distance a few percent longer.
- Winds: convert to the still-air distance for the design wind, in the more demanding direction.
- Reserves by the operating rule: carried as fuel, on top of the trip fuel, not added to the range.
Example 6.4 — London to Ottawa, westbound
The great-circle distance is \(5347\ \text{km}\) (Example 2.2). Allow 4% for routing and an average headwind of \(60\ \text{km/h}\) at \(850\ \text{km/h}\). What still-air range must the aircraft have, with reserves on top?
Show solution
About \(6000\ \text{km}\) of still-air range, 12% more than the map distance, with the reserve fuel of the chosen rule carried in addition. A manufacturer that wanted this city pair (and the many like it) would set the design range to at least this, at the design payload.
Check your understanding
Key takeaways
- Reserves cover contingency, diversion to an alternate and holding; they are carried, not planned to be burned.
- Rules: IFR 45 min; US international airlines 10% of flight time + alternate + 30 min hold; ICAO 5% contingency + alternate + 30 min final reserve.
- Endurance: \(E = (1/C)(L/D)\ln(W_{i-1}/W_i)\); jets hold at \(\LDmax\).
- Still-air distance \(= d_{\text{ground}}\,V/(V - w)\): headwinds cost more than tailwinds save.
- Design range = great circle + routing + winds, at the design payload, with reserves carried as fuel.
- Next, Lesson 7 asks what all these requirements cost, and how a concept is chosen to meet them.