Sizing for Takeoff Field Length
TOFL is the field length (i. e., runway plus clearway; Figure 11.1) required to clear a 35-ft (10-m) obstacle in the clearway while maintaining a specified minimum climb
gradient, y, with one engine inoperative and flaps and undercarriage extended. The FAR requirement for a two-engine aircraft minimum climb gradient is 1.2 (see Table 13.3 for aircraft with more than two engines).
For sizing, field-length calculations are at the sea-level standard day (no wind) and at a zero airfield gradient of paved runway. For further simplification, drag changes are ignored during the transition phase of liftoff to clear the obstacle (flaring after liftoff takes less than 3 s); in other words, the equations applied to Vuft-off are extended to V2 = 1.2Vstan. This gives V22 = [2 x 1.44 x (W/S)]/(pCLstaii). An aircraft stalls at CLmax. Chapter 13 addresses takeoff performance in detail.
A simplified expression for all engines is:
where dV/dt = a and V and a are instantaneous velocity and acceleration of the aircraft on the ground encountering friction (coefficient p = 0.025 for a paved, metaled runway). Average acceleration, a, is taken at 0.7V2. By replacing V2 in terms of CLstall, Equation 11.1 reduces to:
fV2 „ 1 44W/S
TOFL = (1 /a) VdV = (V22/2a) = ——(11.2)
In terms of wing-loading, Equation 11.2 can be written as:
(W/S) = (TOFL x p x a x CLstaii)/1.44 (11.3)
where average acceleration, a = F/m and applied force F = (T – D) – p(W-L). Until liftoff is achieved, W > L and F is the average value at 0.7 V2. Therefore:
a = [(T – D) – p(W – L)]g/ W = g(T/ W)[1 – D/ T – pW/ T + pL/ T] (11.4)
Substituting Equation 11.3, it becomes:
(W/S) = (TOFL x p x [g(T/ W)(1 – D/ T – pW/T + pL/ T)] x CLstaii)/1.44
In the FPS system, it can be written as p = 0.00238 slugs and g = 32.2 ft/s2. Therefore:
(W/S) = (TOFL x (T/ W)(1 – D/ T – pW/ T + pL/ T)] x CLai)/18.85 (11.6a)
In the SI system, it becomes p = 1.225 kg/m3 and g = 9.81 m/s2. Therefore,
(W/S) = 8.345 x (TOFL x (T/ W)(1 – D/ T – pW/ T + pL/ T)] x CLstaii)
where W/S is in Newton/m2 to remain in alignment with the units of thrust in Newtons.
Checking of the second-segment climb gradient occurs after aircraft drag estimation, which is explained in Sections 13.5.1 and 13.5.3. If it falls short, then the TSLS must be increased. In general, TOFL requirements are not generous; therefore, satisfying the TOFL is also likely to satisfy the second-segment climb gradient.