Substituting in Equation 13.25, the cruise range, Rcruise, can be written as:

As mentioned previously, over the cruise range, changes in the sfc and L/D typically are minor. If the midcruise values are taken as an average, then they may be treated as constant and are taken outside the integral sign. Then, Equation

This equation is known as the Breguet range formula, originally derived for propeller-driven aircraft that had embedded propeller parameters (jet propulsion was not yet invented). √

The LRC is carried out at the best sfc and at the maximum value of CL/CD (i.e., L/D) to maximize range. Typically, the best L/D occurs at the midcruise condition. For a very high LRC (i.e., 2,500 nm or more), the aircraft weight difference from initial to final cruise is significant. It is beneficial if cruise is carried out at a higher altitude when the aircraft becomes lighter, which can be done either in a stepped altitude or by making a gradual, shallow climb that matches the gradual lightening of the aircraft. Sometimes a mission may demand HSC to save time, in which case Equation 13.29 is still valid but not operating for the best range.

13.5 Aircraft Performance Substantiation: Worked-Out

Examples (Bizjet)

This section computes aircraft performance to substantiate capabilities as required by the FAR and the operators. Table 13.4 gives the speed schedules appropriate to an aircraft takeoff; aircraft drag polar is given in Figure 9.1. The wing area, SW = 30 m2 (323 ft2) and the MTOM = 9,400 kg (20,723 lb). Assuming that 20 kg of fuel is consumed during the taxi, the MTOM for the takeoff estimation = 9,380 kg (20,680 lb), and wing-loading, W/SW = 64 lb/ft2. The known stalling speed is computed by using the following:

Table 13.8 provides the Bizjet aircraft data generated thus far. To make the best use of the available data, all computations are in the FPS system. The results subsequently can be converted to the SI system.

13.5.1 Takeoff Field Length (Bizjet)

Three decision speeds are worked out to establish the V1 for the BFL computation. Equation 13.2 gives average acceleration as:

a¯ = g[(T/ W − µ) − (CLSq/ W)(CD/CL − µ)]

Table 13.8. Bizjet performance parameters (takeoff and landing)

Takeoff at 8- and 20-deg flaps. Landing at 35to 40-deg flap, engines at idle, and Vstall at aircraft landing weight of 15,800 lb.

The average acceleration a¯ is at 0.7V of the segment of operation. For each segment of the BFL, a¯ is computed.

The lift coefficient during the ground run is changing with speed gain and is not easy to determine. During the ground run, the angle of attack is low, even when the aircraft reaches the stall speed, Vstall. Up to the decision speed V1, only a fraction of the aircraft weight is taken up by the wing as a result of lift generation. Liftoff is not achieved until a pilot rotates the aircraft just above the Vstall. There is a rapid gain in lift generation because the angle of attack increases rapidly with rotation. Table 13.9 lists typical CL and CD/CL values.

Segment A: All Engines Operating up to the Decision Speed V1

Using Equation 13.2 and data from Table 13.6, the average acceleration becomes:

a¯ = 32.2 × [(0.34 − 0.03) − (CLq/64)(0.1 − 0.03)] = 32.2 × (0.31 − CLq/914.3)

At a representative speed of 0.7V1, the average q = 0.5 × 0.002378 × 0.49V12 = 0.0006 × V12. At this segment, the average CL = 0.5 (yet to reach the full value). Then:

a¯ = 32.2 × (0.31 − CLq/914.3)

= 32.2 × (0.31 − 0.0003 × V12/914.3) ft/s2

Table 13.9. Bizjet takeoff aerodynamic coefficients (from experiments and statistics)

One engine inoperative

Table 13.10. Segment A: Bizjet all-engine 0 to V1 (8-deg flap)

Equation 13.3 gives the ground distance covered as:

SG = Vave × (Vf − Vi )/a¯

Table 13.10 computes the ground distance covered for all engines operating up to V1.

Segment B: One-Engine Inoperative Acceleration from V1 to Liftoff Speed, VLO

Because one engine is inoperative, there is a loss of power by half (T/W = 0.17) plus an asymmetric drag rise (CD/CL = 0.102). As the speed increases, the average CL increases to 0.8, making the weight on the wheels lighter; therefore, the ground friction, µ, is reduced to 0.025. The acceleration Equation 13.2 is rewritten as follows:

a¯ = 32.2 × [(0.17 − 0.025) − (CLq/64)(0.102 − 0.025)] = 32.2

× (0.145 − 0.8 × q/831.2)

The velocity that would give the average acceleration is:

V0.7 = 0.7 × (VLo − V1) + V1

a¯ = 32.2 × (0.145 − 0.000951 × V0.72/831.2 = 32.2 × (0.145 − 0.00000114 × V02.7)

Equation 13.3 gives the ground distance covered as:

SG = Vave × (Vf − Vi )/a¯

Table 13.11 computes the ground distance covered from V1 to VLO for the two flap settings.

Table 13.11. Segment B: Bizjet one-engine ground distance V1 to VLO (8-deg flap)

Segment C: Flaring Distance with One Engine Inoperative from VLO to V2

The flaring distance reaches V2 from VLO; from statistics, the time to flaring is 3 s. Table 13.12 computes the ground distance covered from VLO to V2 with one engine inoperative for the two flap settings. In this segment, an aircraft is airborne; hence, there is no ground friction. Taking the average velocity between V2 and VLO gives the distance covered during flare.

The next step is to compute the stopping distance with the maximum application of brakes.

Segment D: Distance Covered in 1 s as Pilot-Recognition

Time and 2 s for Brakes to Act from V1 to VB (Flap Settings

Are of Minor Consequence)

Table 13.13 computes the ground distance covered from V1 to VB.

Segment E: Braking Distance from VB to Zero Velocity (Flap Settings Are of Minor Consequence)

The reaction time to apply the brakes, after the decision speed, V1, is 3 s. The aircraft continues to accelerate during the 3 s.

For an aircraft in full braking with µB = 0.4, all engines shut down, and the average CL = 0.5, Equation 13.2 for average acceleration, based on 0.7VB (≈ 0.7V1), reduces to:

a¯ = 32.2 × [(−0.4) − (CLq/64)/(0.1 − 04)] = 32.2 × [−0.4 + (0.15q/64)]

= 32.2 × [−0.4 + q/426.7)] = 0.075q − 12.88

Table 13.14 computes the ground distance covered from VB to stopping.

The TOFL (see Table 13.12, Segments A + B + C) and the stopping distance (see Table 13.14, Segments A + D + E) are plotted in Figure 13.11 to obtain the BFL for a flap setting of 8 deg and summarized in Table 13.16. It satisfies the specified TOFL requirement of 4,400 ft.

Table 13.13. Segment D: Bizjet failure-recognition distance