Aircraft performance basics

An airline sales director visits Prairiewind Aerospace and asks three questions about the Tern. How far can it fly with nine passengers? How fast does it climb on a hot day with one motor out? How long a runway does it need? These are performance questions, and they decide whether the aircraft can do the job customers want. Élise gives the questions to Arjun as a learning exercise, using the drag polar from Lesson 3 and the powertrain data from Lesson 5. This lesson covers the core ideas behind his answers: power required and available, climb, range and endurance, and take-off and landing.

Thrust required and power required

In steady level flight, thrust equals drag. Combining the lift equation with the drag polar gives the thrust required at any speed. At low speed, the lift coefficient must be high to support the weight, so induced drag dominates; at high speed, parasite drag dominates. The total drag curve therefore has a minimum, at the speed for maximum L/D found in Lesson 3. For propeller aircraft, power is more natural than thrust, because engines and motors produce shaft power. The power required is the drag times the speed, and it also has a minimum, at a lower speed than the minimum-drag speed.

The minimum-power speed occurs where induced drag is three times parasite drag, at \(C_L = \sqrt{3C_{D0}/k}\). At that speed the aircraft uses the least energy per unit time, so it gives the longest endurance for a propeller aircraft or an electric one. The minimum-drag speed, where \(C_L = \sqrt{C_{D0}/k}\), gives the most distance per unit of energy for a propeller aircraft, because energy used per kilometer is proportional to drag. Neither is where airlines normally fly: they cruise faster than the best-range speed, trading a little efficiency for shorter trip times and therefore lower crew and ownership costs per trip.

# Power required, stall and best speeds, and Breguet range for the Tern
import math
rho, S, m, CD0, k = 0.9093, 28.0, 6500.0, 0.028, 0.0386
W = m * 9.81
def power_required(V):
    q = 0.5 * rho * V**2
    CL = W / (q * S)
    return q * S * (CD0 + k * CL**2) * V
print(" V (km/h)  P_req (kW)")
for kmh in (180, 220, 260, 300, 340, 380, 420):
    print(f"{kmh:8d} {power_required(kmh/3.6)/1000:10.0f}")
V_mp = (2 * W / (rho * S) * math.sqrt(k / (3 * CD0))) ** 0.5
V_md = (2 * W / (rho * S) * math.sqrt(k / CD0)) ** 0.5
print(f"Minimum power speed {V_mp*3.6:.0f} km/h (best endurance), minimum drag speed {V_md*3.6:.0f} km/h (best range, propeller)")

# Breguet range for a propeller aircraft: R = (eta/c) * (L/D) * ln(W0/W1)
eta, c_kg_per_J = 0.82, 0.30 / 3.6e6       # propeller efficiency; specific fuel consumption 0.30 kg/kWh
LD = 0.5 / math.sqrt(k * CD0)               # maximum L/D
W0, fuel = m, 900.0
R = eta / (c_kg_per_J * 9.81) * LD * math.log(W0 / (W0 - fuel))
print(f"Max L/D {LD:.1f}; Breguet range with {fuel:.0f} kg fuel: {R/1000:.0f} km")
# Electric aircraft: range does not depend on a weight change
e, eta_t, m_b = 250 * 3600, 0.80, 1120.0
R_e = e * m_b * eta_t * LD / (m * 9.81)
print(f"Battery-electric range with {m_b:.0f} kg of cells: {R_e/1000:.0f} km")

 V (km/h)  P_req (kW)
     180        291
     220        283
     260        305
     300        354
     340        431
     380        536
     420        672
Minimum power speed 210 km/h (best endurance), minimum drag speed 276 km/h (best range, propeller)
Max L/D 15.2; Breguet range with 900 kg fuel: 2274 km
Battery-electric range with 1120 kg of cells: 192 km

Climb performance

Excess power, the difference between power available and power required, is what lets an aircraft climb. Climbing at a rate \(RC\) raises the aircraft's potential energy at the rate \(W \times RC\), which must come from excess power. The best rate of climb is obtained near the speed where excess power is greatest, roughly the minimum-power speed for a propeller aircraft. The best angle of climb, important for clearing obstacles after take-off, occurs at a lower speed where excess thrust, not excess power, is greatest.

Climb performance falls with altitude because the air is thinner: power required at a given equivalent airspeed rises with true airspeed, and air-breathing engines lose power. The altitude where the best rate of climb falls to 0.5 m/s (100 ft/min) is the service ceiling. For multi-engine aircraft, the climb with one engine inoperative is often the critical case. Certification rules require a positive climb gradient with one engine out after take-off, and the Tern's batteries are partly sized for this: if one motor fails, the other must deliver enough power for the aircraft to climb away safely on a hot day.

Range and the Breguet equation

How far an aircraft can fly depends on how efficiently it turns energy into distance. For a fuel-burning propeller aircraft, a classic result derived by Louis Breguet in the early twentieth century connects range to propeller efficiency, specific fuel consumption, lift-to-drag ratio and the fraction of the aircraft's weight that is fuel. The derivation assumes the aircraft flies at constant L/D and efficiency while it gets lighter as fuel burns.

Here \(c\) is the specific fuel consumption in kilograms of fuel per joule of shaft energy, \(W_0\) the initial weight and \(W_1\) the final weight. The equation shows the three ways to increase range: a more efficient propulsor (higher \(\eta/c\)), a cleaner aircraft (higher L/D) and a larger fuel fraction. A jet version replaces \(\eta/c\) with \(V/c_T\), where \(c_T\) is the thrust-specific fuel consumption; jets therefore achieve their best range at the speed that maximizes \(V \times L/D\), higher than the propeller aircraft's best speed.

Battery-electric aircraft do not get lighter as they fly, so the logarithm disappears. Range is proportional to battery specific energy, the battery's share of total mass, the efficiency of the whole drivetrain and L/D. The program compares the two for the Tern: 900 kg of fuel takes it over 2000 km at best L/D in the idealized calculation, while 1120 kg of batteries takes it under 200 km. That gap is the whole reason for the hybrid design.

Endurance

Endurance is how long an aircraft can stay airborne. It matters for patrol, surveillance, training and holding while waiting to land. For propeller and electric aircraft, the best endurance comes from flying at the minimum-power speed, because energy is used at the slowest rate there. For jets, which burn fuel in proportion to thrust, the best endurance is at the minimum-drag speed. Drones designed for surveillance often have long, slender wings to push up L/D and fly slowly for many hours.

Take-off and landing

The take-off is divided into a ground roll, during which the aircraft accelerates to its rotation speed and lifts off, and an airborne segment to a screen height, usually 35 ft (10.7 m) for transport-category aircraft and 50 ft (15 m) for smaller ones. The ground roll depends on weight, wing area, maximum lift coefficient, thrust and air density. A simple estimate assumes the aircraft accelerates at a roughly constant average rate \(\bar{a}\) to a lift-off speed of about 1.2 times the stall speed, giving a ground roll of \(s = V_{LOF}^2/(2\bar{a})\).

Because lift-off speed squared is proportional to weight divided by density, the ground roll grows roughly with the square of weight and inversely with density, which is why hot, high and heavy conditions are so demanding. Runways that are wet, contaminated with snow or slush, uphill or gravel-covered lengthen the run further. For multi-engine aircraft, certification takes the engine failure case into account: the take-off distance must allow the crew either to continue after an engine failure at the decision speed or to stop on the remaining runway.

Landing is similar in reverse. The aircraft approaches at a reference speed of about 1.23 to 1.3 times its stall speed in the landing configuration, crosses the screen height, flares and touches down, then brakes to a stop using wheel brakes, propeller reverse or ground spoilers. Landing distance grows with approach speed squared, which is why high-lift flaps are so valuable. For a commuter aircraft aiming to use short northern or rural strips, the Tern's landing distance and its performance on gravel are among the most important numbers in its specification.