Lesson 5 · 40 min

Mission Profiles

A range requirement says how far; a mission profile says how. Climb to what altitude, cruise at what speed, loiter for how long, fight where, with what held in reserve. Sizing starts from the mission profile, segment by segment, so writing it down correctly is the first real calculation of a new design.

Learning objectives

Describe a mission as numbered segments and use the weight notation \(W_0, W_1, \dots, W_x\).
Sketch the typical profiles of transport, combat, strike, patrol and ferry missions, and distinguish range from radius.
Compute the time and distance of climb and descent, the cruise distance that remains, and the block time of a flight.
Explain how a profile becomes a fuel fraction, using historical segment fractions and Breguet.

A mission as a list of segments

A mission profile divides the flight into segments, each simple enough to analyze on its own: warmup and takeoff, climb, cruise, loiter, combat, descent, landing. The weights are numbered at the segment boundaries. The aircraft starts at the takeoff weight \(W_0\), weighs \(W_1\) at the end of segment 1, \(W_2\) at the end of segment 2, and \(W_x\) at the end of the last segment \(x\). Each segment is then described by its weight fraction \(W_i/W_{i-1}\), which is less than 1 because fuel is burned (or weapons dropped).

Mission weight fraction

\[ \frac{W_x}{W_0} = \frac{W_1}{W_0}\cdot\frac{W_2}{W_1}\cdots\frac{W_x}{W_{x-1}} = \prod_{i=1}^{x}\frac{W_i}{W_{i-1}} \]

The fraction of the takeoff weight burned on the whole mission is \(1 - W_x/W_0\) (when nothing is dropped).

A profile is drawn as altitude against distance (or time), not to scale. Its job is to list every segment, with the numbers each one needs: distance or time, altitude, speed, and what is carried or dropped.

Figure 5.1 Typical mission profiles, after Raymer's examples (schematic, not to scale). Choose a mission, then a segment to read what happens in it. The numbers at the segment ends are the weight indices: the aircraft weighs \(W_i\) at the end of segment \(i\).

Civil and military missions

Transport

Takeoff, climb, cruise the design range, descend, land, with reserves for a diversion and holding. The airline version adds an explicit diversion to an alternate airport (Lesson 6).

Combat radius

Out to a combat area, minutes of combat at full thrust (perhaps dropping weapons), and back. The requirement is a radius, the one-way distance, so the aircraft flies at least twice it.

Strike and patrol

A strike profile flies high where cruise is efficient and low where it must hide (hi-lo-lo-hi). A patrol profile is sized by the endurance on station, not distance.

The distinction between range and radius trips up students every year. An aircraft with a 1000 km combat radius flies 2000 km of cruise plus its combat and reserves, and it flies the way home lighter (less fuel, and maybe no weapons) than the way out. A radius requirement is therefore much more demanding than the same number as a range.

Time and distance: building the timeline

Before any fuel is estimated, the profile must be consistent: the segments must add up to the required distance, and their times give the block time that the airline schedules and the crew is paid for. The geometry is simple:

Climb, descent and cruise

\[ t_{\text{climb}} = \frac{h}{\overline{RC}}, \quad d_{\text{climb}} = \overline{V}_{\text{climb}}\,t_{\text{climb}} \] \[ d_{\text{descent}} = \frac{h}{\tan\gamma_d} \] \[ d_{\text{cruise}} = R - d_{\text{climb}} - d_{\text{descent}} \]

\(h\) is the cruise altitude, \(\overline{RC}\) the average rate of climb, \(\overline{V}_{\text{climb}}\) the average horizontal speed in the climb, \(\gamma_d\) the descent angle (about \(3^\circ\) for airliners). Block time adds taxi out and taxi in to the time in the air.

Example 5.1 — The timeline of a 2000 km flight

A jet transport flies a \(2000\ \text{km}\) trip. It taxis out for \(12\ \text{min}\), climbs to \(11\,000\ \text{m}\) at an average of \(9\ \text{m/s}\) and \(600\ \text{km/h}\), cruises at \(850\ \text{km/h}\), descends at \(3^\circ\) at an average of \(600\ \text{km/h}\), and taxis in for \(6\ \text{min}\). Find the cruise distance and the block time.

Show solution

\[ t_{\text{climb}} = \frac{11\,000}{9} = 1222\ \text{s} = 20.4\ \text{min}, \qquad d_{\text{climb}} = 600\,\frac{1222}{3600} = 203.7\ \text{km} \] \[ d_{\text{descent}} = \frac{11.0\ \text{km}}{\tan 3^\circ} = 209.9\ \text{km}, \qquad t_{\text{descent}} = \frac{209.9}{600} = 0.350\ \text{h} = 21.0\ \text{min} \] \[ d_{\text{cruise}} = 2000 - 203.7 - 209.9 = 1586.4\ \text{km} \] \[ t_{\text{cruise}} = \frac{1586.4}{850} = 1.866\ \text{h} = 112.0\ \text{min} \] \[ t_{\text{block}} = 12 + 20.4 + 112.0 + 21.0 + 6 = 171.4\ \text{min} \approx 2\ \text{h}\ 51\ \text{min} \]

Timeline of Example 5.1
Segment	Distance (km)	Time (min)
Taxi out and takeoff	0	12.0
Climb	203.7	20.4
Cruise	1586.4	112.0
Descent	209.9	21.0
Taxi in	0	6.0
Total	2000.0	171.4

A fifth of the trip is climb and descent. On short flights that share is larger still, which is why short-haul aircraft are sized differently from long-haul ones.

Figure 5.2 A transport mission with true distances and altitudes, both in kilometres. The vertical scale is stretched more than fifty times: drawn with equal scales, the profile would be almost flat. Change the trip and the climb and descent to see how the cruise distance and the block time change. Set the trip short enough and there is no cruise left at all.

From profile to fuel: a first look

Module 3 turns the profile into a takeoff weight. The idea is already within reach. For segments with little fuel and a lot of variety (warmup and takeoff, climb, landing), sizing uses historical weight fractions; for cruise and loiter it uses Breguet:

Historical mission-segment weight fractions (source: Raymer, Table 3.2)
Segment	\(W_i/W_{i-1}\)
Warmup and takeoff	0.970
Climb	0.985
Landing	0.995
Cruise	\(\exp\!\left[-\dfrac{R\,C}{V\,(L/D)}\right]\) (Breguet)
Loiter	\(\exp\!\left[-\dfrac{E\,C}{L/D}\right]\) (endurance, Lesson 6)

Sizing then allows 6% extra for reserve and trapped fuel: \(\Wf/\Wo = 1.06\,(1 - W_x/W_0)\).

Example 5.2 — The fuel fraction of Example 5.1

For the mission of Example 5.1, take \(L/D = 16\) and \(C = 0.55/\text{h}\) in cruise. Estimate \(W_x/W_0\) and \(\Wf/\Wo\), counting descent with the landing.

Show solution

\[ \frac{W_3}{W_2} = \exp\left[-\frac{1586.4(0.55)}{850(16)}\right] = e^{-0.06416} = 0.9379 \] \[ \frac{W_x}{W_0} = 0.970(0.985)(0.9379)(0.995) = 0.8916, \qquad \frac{\Wf}{\Wo} = 1.06(1 - 0.8916) = 0.115 \]

About 11.5% of the takeoff weight is fuel. Notice that only the cruise fraction depends on the trip distance; the climb credited 204 km of the range, which is why the cruise is 1586 km, not 2000.

Check your understanding

Key takeaways

A mission profile is a numbered list of segments; the aircraft weighs \(W_i\) at the end of segment \(i\), and \(W_x/W_0\) is the product of the segment fractions.
Transport missions are sized by range, combat missions by radius (out and back), patrol missions by endurance.
Climb: \(t = h/\overline{RC}\), \(d = \overline{V}t\); descent: \(d = h/\tan\gamma_d\); cruise is what remains of the range.
Historical fractions: takeoff 0.970, climb 0.985, landing 0.995; cruise and loiter from Breguet; \(\Wf/\Wo = 1.06(1 - W_x/W_0)\).
Next, Lesson 6 fills in the reserves, the winds and what "design range" really means.