ReferenceThe First Estimate of Takeoff Weight
Formula Sheet
Every key result from the module in one place. Statistics are from Raymer, Chapter 3; check the values against your edition.
Fits on one page of Letter or A4. For a digital copy, choose “Save as PDF” as the printer.
The sizing equation
Weight build-up and sizing
\[ \Wo = W_{\text{crew}} + W_{\text{payload}} + \Wf + \We \] \[ \Wo = \frac{W_{\text{crew}} + W_{\text{payload}}}{1 - \Wf/\Wo - \We/\Wo} \]- Solve by iteration: guess \(\Wo\), evaluate \(\We/\Wo\), compute a new \(\Wo\), repeat. Graphically, where the trend meets the affordable curve \(1 - \Wf/\Wo - (W_{\text{crew}} + W_{\text{payload}})/\Wo\).
- Crew and payload from the requirements (this module: 90 kg per crew member, 100 kg per passenger with baggage).
More in Lesson 1
Empty-weight fraction
Statistical empty-weight trend
\[ \frac{\We}{\Wo} = A\,\Wo^{\,C}K_{vs}K_{\text{tech}}, \qquad A_{\text{kg}} = A_{\text{lb}}(0.4536)^{-C} \]- Jet transport \(1.02,\ -0.06\); twin turboprop \(0.96,\ -0.05\); GA single \(2.36,\ -0.18\); GA twin \(1.51,\ -0.10\); jet trainer \(1.59,\ -0.10\); jet fighter \(2.34,\ -0.13\); military cargo or bomber \(0.93,\ -0.07\) (\(A\) for lb).
- \(K_{vs} = 1.04\) for variable sweep; \(K_{\text{tech}}\) stated (about 0.9 for extensive composites). Real aircraft scatter about ±15%.
More in Lesson 2
\(L/D\) from the sketch
Wetted area, \(\CDz\) and \(\LDmax\)
\[ \Swet \approx S_{\text{exp}}\left(1.977 + 0.52\,\tfrac{t}{c}\right) \] \[ \Swet \approx 3.4\,\frac{A_{\text{top}} + A_{\text{side}}}{2} \] \[ \CDz = C_{fe}\frac{\Swet}{\Sref}, \qquad A_{\text{wet}} = \frac{b^2}{\Swet} \] \[ \LDmax = \frac12\sqrt{\frac{\pi e A_{\text{wet}}}{C_{fe}}} \]- \(C_{fe}\): transport 0.0030, military cargo and Air Force fighter 0.0035, Navy fighter 0.0040, light single 0.0055, light twin 0.0045.
- Jets: cruise \(0.866\,\LDmax\), loiter \(\LDmax\). Propellers: cruise \(\LDmax\), loiter \(0.866\,\LDmax\).
More in Lesson 3
Fuel consumption
Jets and propellers
\[ C\ (1/\text{h}) = \frac{c_T\ [\text{mg/(N s)}]}{28.33} \] \[ C = \frac{c_P\,g\,V}{1000\,\eta_p}, \qquad \frac{V}{C} = \frac{3600\,\eta_p}{c_P\,g}\ \text{km} \]- Jet \(C\) cruise / loiter: turbojet 0.9 / 0.8, low-bypass 0.8 / 0.7, high-bypass 0.5 / 0.4 (1/h).
- Propeller \(c_P\): piston 0.4 / 0.5, turboprop 0.5 / 0.6 lb/(hp h); \(\eta_p\) about 0.8. \(1\ \text{lb/(hp h)} = 0.6083\ \text{kg/(kW h)}\); \(V\) in m/s.
More in Lesson 4
Mission fuel fraction
Segment fractions and the product
\[ \text{cruise } e^{-RC/(V\,L/D)}, \qquad \text{loiter } e^{-EC/(L/D)} \] \[ \text{combat } 1 - C\,\frac{T}{W}\,t \] \[ \frac{W_x}{W_0} = \prod\frac{W_i}{W_{i-1}}, \qquad \frac{\Wf}{\Wo} = 1.06\left(1 - \frac{W_x}{W_0}\right) \]- Historical: takeoff 0.970, climb 0.985, landing 0.995. \(R\) in km, \(V\) in km/h, \(E\) and \(t\) in hours.
- Payload drop: follow \(W_i\) through the mission from a guess, subtract the drop, \(\Wf = 1.06(\Wo - W_{\text{drop}} - W_x)\), new \(\Wo = (W_{\text{crew}} + W_{\text{payload}} + \Wf)/(1 - \We/\Wo)\); iterate.
Sensitivities
- Find a sensitivity by changing one input and resizing. Growth factor \(= \Delta\Wo/\Delta W\); with fixed fractions \(1/(1 - \Wf/\Wo - \We/\Wo)\), a little more than the resized value.
- The business jet: +1% \(L/D\) or −1% \(C\) gives about −1% \(\Wo\); the empty-weight trend ±10% gives −25% / +45%.
More in Lesson 7
Common mistakes
- Pound \(A\) with kilograms. Convert \(\Wo\) or \(A\) first.
- Wrong \(L/D\). Jet cruise is \(0.866\,\LDmax\); propeller cruise is \(\LDmax\).
- Minutes with \(C\) per hour. 45 min is 0.75 h.
- km/h in the propeller \(C\). \(V\) in m/s there.
- Adding fractions. Multiply the segment fractions.
- Stopping at one iteration. Iterate until \(\Wo\) stops changing.
- Crew as payload, or payload left out. Both go in the numerator, crew counted once.