SME
Aircraft Design · Problem type
Mission fuel fraction (Breguet cruise)
Use this for “A jet cruises 2000 km at 245 m/s with cₜ = 0.7 per hour and L/D = 15. Its other segments’ ratio is 0.95. Find the fuel fraction.”
Chart
Wc(2,000) ≈ 0.8996
This is an example. Type your own values to start a new problem.
Wc
=
exp
−
R (km)
×
cₜ (h⁻¹)
V (m/s)
×
Wf/W₀
=
1.06
×
(1
−
Wₒ
×
Wc
)
Wₓ/W₀
Mission weight ratio
calculated
Formulas
Wc
=
e
−
R
× cₜ ÷ (
V
× (L/D))
0.8996
=
e
−2,000,000 × 0.0001944 ÷ (245 × 15)
Wₓ/W₀ = Wₒ × Wc
0.8546 = 0.95 × 0.8996
Wf/W₀ = 1.06 × (1 − (Wₓ/W₀))
0.1541 = 1.06 × (1 − 0.8546)
Formulas are worked in m, s⁻¹. The step-by-step shows the conversions.
Assumptions
•
The Breguet range equation, solved for the cruise weight ratio Wc = Wend ÷ Wₛₜₐᵣₜ.
•
Wₒ, the takeoff, climb, descent and landing ratios multiplied, comes from the course table.
•
The fuel fraction adds 6% to the fuel burned, for reserve and trapped fuel.
Step-by-step
WE KNOW
R = 2,000 km, V = 245 m/s, cₜ = 0.7 h⁻¹, L/D = 15, Wₒ = 0.95
FIND
cruise weight ratio (Wc), mission weight ratio (Wₓ/W₀), fuel fraction (Wf/W₀)
Step 1 · Convert to the formula’s units
The formulas don’t work directly in the units you chose, so convert first (to m, s⁻¹).
R = 2,000 km = 2,000,000 m (1 km = 1000 m)
cₜ = 0.7 h⁻¹ = 0.0001944 s⁻¹ (1 s⁻¹ = 3600 h⁻¹)
Step 2 · Find cruise weight ratio (Wc)
Use Wc = e^(−R × cₜ ÷ (V × (L/D)))
Breguet’s
R
=
(
V
cₜ
)(L/D)
ln(
Wₛₜₐᵣₜ
Wend
),
solved
for
Wend
Wₛₜₐᵣₜ
.
Wc
=
e
−2,000,000 × 0.0001944 ÷ (245 × 15)
Wc
=
e
−2,000,000 × 0.0001944 ÷ 3,675
Wc
=
e
−388.8 ÷ 3,675
Wc
=
e
−0.1058
Wc = 0.8996
Step 3 · Find mission weight ratio (Wₓ/W₀)
Use Wₓ/W₀ = Wₒ × Wc
Each segment’s ratio is its end weight over its start weight, so the ratios multiply.
Wₓ/W₀ = 0.95 × 0.8996
Wₓ/W₀ = 0.8546
Step 4 · Find fuel fraction (Wf/W₀)
Use Wf/W₀ = 1.06 × (1 − (Wₓ/W₀))
The weight burned is the share 1 − Wₓ/W₀ of W₀; 6% more covers reserve and trapped fuel.
Wf/W₀ = 1.06 × (1 − 0.8546)
Wf/W₀ = 1.06 × 0.1454
Wf/W₀ = 0.1541
Check
Put the numbers back in. Both sides should match.
0.8996
=
e
−2,000,000 × 0.0001944 ÷ (245 × 15)
✓
0.8546 = 0.95 × 0.8996 ✓
0.1541 = 1.06 × (1 − 0.8546) ✓
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Takeoff weight with an empty-weight fit
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