Lesson 5 · 40 min
Mission Segment Weight Fractions
Module 2 wrote the mission down as numbered segments. Now each segment gets its weight fraction: historical values where little fuel is burned and the details are messy, the Breguet equations where most of the fuel goes. Multiplied together, they give the fuel fraction that the sizing equation needs.
Learning objectives
- Assign a weight fraction to every segment of a mission: historical, cruise, loiter or combat.
- Compute cruise and loiter fractions with the right \(L/D\) and \(C\) for the engine type.
- Multiply the segment fractions into \(W_x/W_0\), and apply the allowance for reserves and trapped fuel.
A fraction for every segment
The aircraft weighs \(W_0\) at takeoff and \(W_i\) at the end of segment \(i\). Each segment is described by \(W_i/W_{i-1}\):
Segment weight fractions
\[ \text{Warmup and takeoff: } 0.970, \qquad \text{climb: } 0.985, \qquad \text{landing: } 0.995 \] \[ \text{Cruise: } \frac{W_i}{W_{i-1}} = \exp\left[-\frac{R\,C}{V\,(L/D)}\right], \qquad \text{loiter: } \frac{W_i}{W_{i-1}} = \exp\left[-\frac{E\,C}{L/D}\right] \] \[ \text{Combat at full thrust for time } t:\ \ \frac{W_i}{W_{i-1}} = 1 - C\,\frac{T}{W}\,t \]\(R\) in km, \(V\) in km/h, \(E\) and \(t\) in hours, \(C\) in 1/h. \(L/D\) by engine type (Lesson 3): jets cruise at \(0.866\,\LDmax\) and loiter at \(\LDmax\); propeller aircraft the other way round.
The historical fractions (source: Raymer, Table 3.2) cover segments whose fuel depends on details not yet known (engine start, taxi, the climb schedule), and that burn a few percent at most. The combat fraction is direct: at full thrust the engines burn \(CT\) of fuel weight per hour, so a time \(t\) burns \(CTt\), a fraction \(C(T/W)t\) of the weight.
Mission fraction and fuel fraction
\[ \frac{W_x}{W_0} = \prod_{i=1}^{x}\frac{W_i}{W_{i-1}}, \qquad \frac{\Wf}{\Wo} = 1.06\left(1 - \frac{W_x}{W_0}\right) \]The factor 1.06 allows 6% for reserve and trapped (unusable) fuel. If the mission already includes an explicit reserve, such as the business jet's loiter, the allowance still covers the trapped fuel and a margin.
The business jet's mission
Example 5.1 — From the profile to \(\Wf/\Wo\)
The business jet flies: warmup and takeoff; climb; cruise of \(4600\ \text{km}\) at \(850\ \text{km/h}\) with \(C = 0.65/\text{h}\); a 45-minute loiter with \(C = 0.55/\text{h}\); landing. Its \(\LDmax = 16.9\) (Lesson 3). Find each fraction, \(W_x/W_0\) and \(\Wf/\Wo\).
Show solution
Cruise at \(0.866(16.9) = 14.64\); loiter at \(16.9\).
\[ \frac{W_3}{W_2} = \exp\left[-\frac{4600(0.65)}{850(14.64)}\right] = e^{-0.2404} = 0.7864 \] \[ \frac{W_4}{W_3} = \exp\left[-\frac{0.75(0.55)}{16.9}\right] = e^{-0.02441} = 0.9759 \]| \(i\) | Segment | \(W_i/W_{i-1}\) | \(W_i/W_0\) |
|---|---|---|---|
| 1 | Warmup and takeoff | 0.9700 | 0.9700 |
| 2 | Climb | 0.9850 | 0.9555 |
| 3 | Cruise | 0.7864 | 0.7514 |
| 4 | Loiter (reserve) | 0.9759 | 0.7332 |
| 5 | Landing | 0.9950 | 0.7295 |
The cruise burns about three quarters of the fuel (0.204 of the 0.271 of \(W_0\) burned); every other segment together, the rest. That is why \(L/D\) and \(C\), which enter only the cruise and loiter, dominate the fuel fraction.
Combat and other segments
Example 5.2 — Five minutes of combat
A fighter fights for 5 minutes at full afterburner, with \(\TW = 0.9\) at that weight and an afterburning \(C = 1.8/\text{h}\) (given for this example). Find the combat fraction.
Show solution
Five minutes of combat burn 13.5% of the weight, more than an hour of cruise. Combat time is one of the most expensive numbers in a fighter's requirements.
Other segments are built the same way. A low-level dash is a cruise at the dash speed with the low-altitude \(L/D\) and \(C\). A descent is usually counted in the landing fraction. When something is dropped (weapons, cargo, sonobuoys), the weight falls without fuel being burned, and the fractions after the drop depend on the takeoff weight: Lesson 6 shows how to handle that.
Check your understanding
Key takeaways
- Historical fractions: takeoff 0.970, climb 0.985, landing 0.995.
- Cruise \(\exp[-RC/(V\,L/D)]\), loiter \(\exp[-EC/(L/D)]\), combat \(1 - C(\TW)t\).
- \(W_x/W_0\) is the product; \(\Wf/\Wo = 1.06(1 - W_x/W_0)\).
- The business jet: \(\Wf/\Wo = 0.2867\), three quarters of it burned in cruise.
- Next, Lesson 6 puts the fractions together and closes the sizing.