Figure 11.1. Sizing for takeoff

These three parameters are primarily dependent on control-surface sizing (as well as engine sizing, to an extent), which is not addressed in this book. It is assumed that engine size for fast initial climb rates are sufficient and that enough control surface is available to perform the g requirements. A lower aspect ratio for the wing is considered for higher roll rates to reduce the wing-root bending moments.

As mentioned previously, an aircraft must simultaneously satisfy the takeoff field length, initial climb rate, initial maximum cruise speed–altitude capabilities, and LFL. Low wing-loading (i.e., a larger wing area) is required to sustain low speed at liftoff and touchdown (for a pilot’s ease), whereas high wing-loading (i.e., a low wing area) is suitable at cruise because high speeds generate the required lift on a smaller wing area. The large wing area for takeoff and landing results in excess wing for high-speed cruise. To obtain the minimum wing area and satisfy all requirements, a compromise for sizing of the wing area must be found; this may require suitable high-lift devices to keep the wing area smaller. The wing area is sized in conjunction with a matched engine for takeoff, climb, cruise, and landing; landing is performed at the idle-engine rating.

In general, W/S varies with time as fuel is consumed and T/W is throttledependent. Therefore, a reference design condition of the MTOW and TSLS at ISA + SL are used for sizing considerations. This means that the MTOW, TSLS, and SW are the only parameters considered for aircraft sizing and engine matching. In general, wing-size variations are associated with changes in all other affecting parameters (e.g., AR, λ, and wing sweep). However, at this stage, they are kept invariant – that is, the variation in wing size only scales the wing span and chord, leaving the general planform unaffected (like zoom in/zoom out).

At this point in the discussion, readers require knowledge of aircraft perfor- mance, and the important derivations of the equations used are provided in Chapter 13. References [2] through [6] are textbook sources for the detailed derivation of the performance equations. Other proven semi-empirical relations are in [4]. Although the methodology described herein is the same, the industry practice is more detailed and involved in order to maintain a high degree of accuracy.

Worked-out examples continue with the Learjet 45 Bizjet class for civil aircraft and the BAe Hawk class for military aircraft. Throughout this chapter, wing-loading (W/S) in the SI system is in N/m2 to align with the thrust (in Newtons) in thrust loading (TSLS/W) as a nondimensional parameter.

11.3.1 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, γ , 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 Vlift-off are extended to V2 = 1.2Vstall. This gives V22 = [2 × 1.44 × (W/S)]/(ρCLstall). An aircraft stalls at CLmax. Chapter 13 addresses takeoff performance in detail.

A simplified expression for all engines is:

0

where dV/dt = a and V and a are instantaneous velocity and acceleration of the aircraft on the ground encountering friction (coefficient µ = 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:

where average acceleration, a¯ = F/m and applied force F = (T − D) − µ(W–L). Until liftoff is achieved, W > L and F is the average value at 0.7V2. Therefore:

a¯ = [(T − D) − µ(W − L)]g/ W = g(T/ W)[1 − D/ T − µW/ T + µL/ T] (11.4) Substituting Equation 11.3, it becomes:

(W/S) = (TOFL × ρ × [g(T/ W)(1 − D/ T − µW/ T + µL/ T)] × CLstall)/1.44 (11.5)

In the FPS system, it can be written as ρ = 0.00238 slugs and g = 32.2 ft/s2. Therefore:

(W/S) = (TOFL × (T/ W)(1 − D/ T − µW/ T + µL/ T)] × CLstall)/18.85 (11.6a) In the SI system, it becomes ρ = 1.225 kg/m3 and g = 9.81 m/s2. Therefore,

(W/S) = 8.345 × (TOFL × (T/ W)(1 − D/ T − µW/ T + µL/ T)] × CLstall) (11.6b)

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.

tion 11.4 is minimal and can be omitted at this stage for the sizing calculation. In addition, for the one-engine inoperative condition after the decision speed (V1; see Section 13.4), the acceleration slows down, making the TOFL longer than the all- engines-operative case. Therefore, in the sizing computations to produce the specified TOFL, further simplification is possible by applying a semi-empirical correction factor primarily to compensate for loss of an engine. The correction factors are as follows (see [4]); all sizing calculations are performed at the MTOW and with

TSLS:

For two engines, use a factor of 0.5 (loss of thrust by a half). Then, Equation 11.6

For three engines, use a factor of 0.66 (loss of thrust by a third). Then, Equation 11.6 in the FPS system reduces to:

For four engines, use a factor of 0.75 (loss of thrust by a fourth). Then, Equation 11.6 in the FPS system reduces to:

Military Aircraft Design: Takeoff

Because military aircraft mostly have a single engine, there is no requirement for one engine being inoperative; ejection is the best solution if the aircraft cannot be landed safely. Therefore, Equation 11.6 can be directly applied (for a multiengine design, the one-engine inoperative case generally uses measures similar to the civilaircraft case).

In the FPS system, this can be written as:

Military aircraft have a thrust, TSLS/W, that is substantially higher than civil aircraft, which makes (D/T – µW/T + µL/T) even smaller. Therefore, for a single-engine