Propeller and jet aircraft characteristics

Lesson 3 described what the Tern needs. This lesson describes what its propulsion system can supply, and how that differs between propeller aircraft and jets. The airline's evaluation team is comparing the Tern with a used 76-seat regional jet for some of its longer routes, so Arjun prepares a side-by-side explanation. The difference in how the two produce thrust explains almost every difference in how they should be flown: their best speeds, their best altitudes, and how their performance changes with height.

Power available and thrust available

Performance comes from comparing what is required with what is available. For a propeller aircraft, the engine or motor delivers shaft power \(P_{shaft}\) to the propeller, and the propeller converts it into thrust power with an efficiency \(\eta_p\). The thrust power available is \(\eta_p P_{shaft}\), and the thrust available is that power divided by the true airspeed. For a jet, the engine produces thrust directly, and its thrust power is thrust times speed.

The two idealizations, constant power for propeller aircraft and constant thrust for jets, are only approximations, but they capture the essential difference. A propeller aircraft's thrust falls with speed, because the same power spread over a higher speed gives less force. A jet's thrust changes much more slowly with speed. On a graph of thrust against speed, the propeller aircraft's available thrust looks like a falling hyperbola and the jet's looks like a nearly flat line; on a graph of power against speed, the propeller aircraft's line is nearly flat and the jet's rises steadily.

How propellers work and why their efficiency changes

A propeller is a rotating wing. Each blade section meets the air at a combination of the forward speed and its own rotational speed, and produces lift that has a forward component, the thrust. Propeller efficiency, the ratio of thrust power to shaft power, depends mainly on the advance ratio \(J = V/(nD)\), where \(n\) is the rotational speed in revolutions per second and \(D\) the diameter. At zero forward speed the efficiency is zero by definition, because no thrust power is produced, though the propeller still makes a large static thrust. As speed rises, efficiency climbs to a peak of about 0.80 to 0.88 for a well-designed variable-pitch propeller, then falls as the blades reach their pitch limits and their tips approach the speed of sound.

Variable-pitch, constant-speed propellers keep the efficiency near its peak over a wide speed range by changing blade angle as speed changes. The Tern's propellers are of this type, and they can also feather and reverse. Momentum theory gives a useful upper bound: an ideal propeller accelerating a stream of air has a propulsive efficiency of \(2/(1 + V_j/V)\), where \(V_j\) is the speed of the air behind it. Large propellers, which accelerate a lot of air by a small amount, are therefore efficient. That same principle explains why high-bypass turbofans are more efficient than early turbojets, as a later section shows.

Engines, motors and altitude

Different power sources behave differently with altitude. A normally aspirated piston engine loses power roughly in proportion to air density, because each intake stroke takes in less air. A turbocharged piston engine holds its sea-level power up to a critical altitude. A turboprop engine, a gas turbine driving a propeller through a gearbox, loses power with altitude too, though less sharply on cold days; its power is often flat-rated, held to a fixed value up to a certain temperature or height by design. Electric motors are different: their power does not depend on air density at all, only on the electrical power fed to them and on cooling, which does get harder in thin air.

The Tern's hybrid layout puts these effects together. Its two 450 kW electric motors are rated for continuous power regardless of height, with a five-minute 600 kW contingency rating each. The power they can draw comes from two sources: the turbine generator, which produces about 800 kW at sea level falling to about 700 kW at 3000 m, and the 600 kg battery pack, which can supply up to 600 kW for a few minutes. After motor and inverter losses, the generator alone delivers about 665 kW of shaft power at 3000 m. That figure, not the motor rating, sets the Tern's maximum continuous cruise speed.

# Maximum level speed: the Tern (power-producing) and a regional jet (thrust-producing)
import math
g = 9.81
def isa_rho(h):
    T = 288.15 - 0.0065 * min(h, 11000)
    p = 101325 * (T / 288.15) ** 5.2559
    if h > 11000:
        p *= math.exp(-g * (h - 11000) / (287.05 * 216.65))
    return p / (287.05 * T), math.sqrt(1.4 * 287.05 * T)
def bisect(f, a, b):
    for _ in range(60):
        c = (a + b) / 2
        if f(a) * f(c) <= 0: b = c
        else: a = c
    return (a + b) / 2
# Tern: shaft power from the motors, propeller efficiency rises with speed
S, CD0, k, m = 28.0, 0.028, 0.039, 6500.0
def eta(V): return 0.84 * (1 - math.exp(-V / 30))          # simple fit to propeller data
def P_req(V, rho):
    q = 0.5 * rho * V**2; CL = m * g / (q * S)
    return q * S * (CD0 + k * CL**2) * V
rho, a = isa_rho(3000)
for label, Pshaft in (("generator only, 665 kW", 665e3), ("generator + battery, 900 kW", 900e3)):
    Vmax = bisect(lambda V: eta(V) * Pshaft - P_req(V, rho), 60, 200)
    print(f"Tern at 3000 m, {label}: V_max = {Vmax*3.6:.0f} km/h ({Vmax/0.5144:.0f} kt), eta = {eta(Vmax):.2f}")
# Regional jet: thrust lapses with density and Mach; wave drag above M_crit (Lock's rule)
mj, Sj, CD0j, kj, T_sl, Mcr, Mmo = 38000.0, 72.0, 0.020, 0.042, 2 * 60e3, 0.72, 0.82
def jet_drag(M, rho, a):
    V = M * a; q = 0.5 * rho * V**2; CL = mj * g / (q * Sj)
    cdw = 20 * (M - Mcr)**4 if M > Mcr else 0.0
    return q * Sj * (CD0j + kj * CL**2 + cdw)
def jet_thrust(M, rho):
    return 0.80 * T_sl * (rho / 1.225) ** 0.75 * (1 - 0.35 * M)   # maximum cruise rating
for h in (6000, 9000, 11000):
    rho, a = isa_rho(h)
    Mmax = bisect(lambda M: jet_thrust(M, rho) - jet_drag(M, rho, a), 0.5, 1.0)
    D78 = jet_drag(0.78, rho, a)
    print(f"Jet at {h} m: thrust-limited M_max = {Mmax:.3f} (limited in service to M_MO {Mmo}); drag at M0.78 = {D78/1000:.1f} kN")

Tern at 3000 m, generator only, 665 kW: V_max = 382 km/h (206 kt), eta = 0.82
Tern at 3000 m, generator + battery, 900 kW: V_max = 438 km/h (236 kt), eta = 0.83
Jet at 6000 m: thrust-limited M_max = 0.836 (limited in service to M_MO 0.82); drag at M0.78 = 33.3 kN
Jet at 9000 m: thrust-limited M_max = 0.845 (limited in service to M_MO 0.82); drag at M0.78 = 25.3 kN
Jet at 11000 m: thrust-limited M_max = 0.844 (limited in service to M_MO 0.82); drag at M0.78 = 22.5 kN

Maximum speed

Maximum level speed is where the power available equals the power required, at the high-speed crossing of the two curves. The program finds it for the Tern at 3000 m: 382 km/h(206 kt) on generator power alone, almost exactly the airline's chosen cruise speed, and 438 km/h with the battery boosting the motors to their full continuous rating. The battery cannot sustain that for long, so the useful maximum cruise speed is the generator-limited one. A lower-speed crossing also exists on the back side of the power curve, but for most aircraft it lies below the stall and has no practical meaning.

For the jet, maximum speed is where thrust available equals drag. The program's simple model gives a thrust-limited maximum near Mach 0.84 at 9000 to 11,000 m, but in service the jet may not exceed its maximum operating Mach number \(M_{MO}\) of 0.82, a structural and handling limit set at certification. Jets therefore normally cruise below their thrust-limited speed, and their maximum speed in practice is a limit, not a performance capability.

Jets: thrust, Mach number and fuel

A turbofan engine produces thrust by accelerating two streams of air: a core stream that passes through the compressor, combustor and turbine, and a bypass stream moved by the large fan. The bypass ratio, the mass flow in the bypass stream divided by that in the core, ranges from about 5 for regional jets to above 10 for the newest airliner engines. Higher bypass ratios accelerate more air by less, which raises propulsive efficiency, exactly as for a large propeller. Their thrust falls faster with forward speed than a pure turbojet's, but they use much less fuel per unit of thrust.

Jet fuel use is described by the thrust-specific fuel consumption \(c_T\), the mass of fuel burned per unit of thrust per unit of time, typically quoted in kilograms per newton per hour or pounds per pound-force per hour. A modern regional jet engine at cruise has a TSFC of about 0.06 kg/(N·h)(about 0.6 lb/(lbf·h)). Thrust falls with altitude roughly in proportion to \(\sigma^{0.75}\), but drag at the minimum-drag point does not change with altitude, so jets climb high to fly faster through thinner air for the same thrust and fuel flow.

Above about Mach 0.7, compressibility becomes important. As the aircraft approaches its critical Mach number, local airflow over the wing reaches the speed of sound, and shock waves form. Above the drag-divergence Mach number, wave drag rises steeply. A simple empirical rule, often credited to the British aerodynamicist C. N. H. Lock, models the wave drag as \(20(M - M_{cr})^4\) above the critical Mach number. Swept, thin, supercritical wings push the critical Mach number up, which is why jet airliners cruise near Mach 0.78 to 0.85 and why their wings look so different from the Tern's straight, high-aspect-ratio wing.

Comparing the two aircraft

Arjun's side-by-side comparison for the airline shows why the two aircraft occupy different niches. The regional jet flies about twice as fast and three times as high, carries eight times as many passengers and burns much more fuel per trip. Per seat-kilometer it is competitive on long, busy routes, but on short, thin routes to smaller communities it burns fuel climbing to an altitude where it spends little time, and its runway needs exclude many airports. The Tern is slower but uses far less energy per trip on those routes, and it can use short and gravel runways.