Piston engines and propellers

The Tern's propellers are turned by electric motors, but most propellers in the world are still turned by piston engines, and the basic ideas of propeller performance were worked out on them. Prairiewind Aerospace keeps a small fleet of Plover two-seat trainers at Wichita's airport for its pilots' currency flying, each with a 134 kW (180 hp) four-cylinder engine and a two-blade propeller. When the flying club asked whether a turbocharged engine would let the Plover keep its climb rate on summer days at altitude, Dr. Nakamura gave the question to Arjun as a warm-up. It covers both halves of this lesson: how a piston engine's power changes with altitude, and how a propeller turns shaft power into thrust.

The four-stroke engine in brief

An aircraft piston engine is a spark-ignition engine running on the Otto cycle: intake, compression, power and exhaust strokes, with one power stroke for every two crankshaft revolutions in each cylinder. Most light aircraft engines are horizontally opposed, air cooled, with four or six cylinders, two spark plugs per cylinder from two independent magnetos, and a fixed or controllable-pitch propeller bolted directly to the crankshaft. Their power is roughly proportional to the mass of air they can draw into the cylinders each minute, because the fuel burned is limited by the oxygen available. That one fact explains how they behave with altitude.

Brake power, the power at the crankshaft, can be written as

where \(V_d\) is the displacement, \(N\) the revolutions per second, \(\eta_v\) the volumetric efficiency, \(\rho\) the density of the intake air and \(A/F\) the air-fuel ratio. Everything on the right except the density is set by the engine and how it is operated. As the aircraft climbs, density falls and so does the power.

Power lapse with altitude

A normally aspirated engine loses power faster than density alone suggests, because friction and accessory losses stay roughly constant while the indicated power falls. A widely used empirical form is the Gagg and Ferrar relation,

where \(\sigma = \rho/\rho_0\) is the density ratio. At 3000 m (about 10,000 ft), where \(\sigma\) is 0.74, the engine gives about 71 percent of its sea-level power. Pilots meet the same effect on hot days at high airports: density altitude, not field elevation, decides the power available.

A turbocharger fights the lapse. Exhaust gas drives a small turbine on a shaft with a centrifugal compressor that pressurizes the intake air. A wastegate bypasses exhaust around the turbine at low altitude, so the manifold pressure is held at its sea-level value; as the aircraft climbs, the wastegate closes to send more exhaust through the turbine. At the critical altitude the wastegate is fully closed and above it the power falls again. The cost is weight, heat, complexity and careful engine management: the pilot must watch manifold pressure and turbine inlet temperature, and cool the engine down gently after flight.

Propellers: momentum theory

A propeller is a rotating wing. Its blades produce lift whose forward component is thrust, and drag whose torque the engine must overcome. The simplest model ignores the blades altogether and treats the propeller as an actuator disc that adds a pressure jump to the air passing through it. Applying mass, momentum and energy conservation to the stream tube through the disc gives three results. The velocity at the disc is the average of the far upstream and far downstream speeds, so half the final velocity increase happens before the disc. The induced velocity at the disc, \(v_i\), satisfies

and the ideal efficiency is the Froude efficiency of Lesson 1 written for the disc, \(\eta_i = V_0/(V_0 + v_i)\). For a given thrust, a bigger disc means a smaller induced velocity and a higher ideal efficiency. That is the propeller's version of the large, slow stream. It also shows why efficiency is poor at low speed and high thrust, such as the take-off roll: there the induced velocity is large compared with the flight speed.

Propeller coefficients and blade angle

Real propellers are described by non-dimensional coefficients. With rotational speed \(n\) in revolutions per second and diameter \(D\):

The advance ratio \(J\) is the distance the aircraft moves in one revolution, divided by the diameter. It sets the angle at which the air meets each blade section. For a fixed blade angle there is one advance ratio at which efficiency peaks; at lower \(J\) the blades are at a high angle of attack, at higher \(J\) the angle of attack falls toward zero and thrust vanishes. A fixed-pitch propeller is a compromise between climb and cruise. A constant-speed propeller changes blade angle with a governor to hold the engine speed the pilot selects, so it rides the top of the family of efficiency curves.

Blade element theory fills in what momentum theory leaves out. Each blade is divided into strips; at each strip the relative velocity is the vector sum of the flight speed plus induced velocity and the rotational speed \(2\pi n r\). Airfoil lift and drag data give the force on the strip, and integrating along the blade gives thrust and torque. Profile drag, swirl in the slipstream, tip losses and compressibility at the tips all reduce efficiency below the momentum value. Good propellers reach about 85 to 88 percent at cruise.

Compressibility sets a hard limit. The blade tips move along a helix whose speed combines rotation and flight speed; when the helical tip Mach number exceeds about 0.85 to 0.9, shock waves form, efficiency drops and noise rises sharply. Large slow-turning propellers, reduction gearboxes on turboprops and swept, thin blade tips are all ways of keeping the tips subsonic.

# Piston engine power lapse with altitude, and propeller performance by momentum theory
# and the non-dimensional coefficients, for the Plover trainer and the Tern.
import math

def isa(h):                                     # troposphere, h in m
    T = 288.15 - 0.0065 * h
    p = 101325 * (T / 288.15) ** 5.2559
    return T, p, p / (287.05 * T)

P0 = 134.0                                      # kW, Plover's normally aspirated engine at sea level
print("altitude m   sigma   Gagg-Ferrar P kW   turbocharged P kW (flat to 4500 m)")
for h in (0, 1500, 3000, 4500, 6000):
    T, p, rho = isa(h); sigma = rho / 1.225
    P_na = P0 * (1.132 * sigma - 0.132)
    P_tc = P0 if h <= 4500 else P0 * (1.132 * (rho / isa(4500)[2]) - 0.132)
    print(f"{h:8d}    {sigma:5.3f}      {P_na:7.1f}            {P_tc:7.1f}")

def momentum(T_N, V, D, rho):
    A = math.pi * D**2 / 4
    vi = -V / 2 + math.sqrt((V / 2)**2 + T_N / (2 * rho * A))   # induced velocity at the disc
    eta_i = V / (V + vi)                                           # ideal (Froude) efficiency
    return vi, eta_i, T_N * (V + vi) / 1000                       # ideal power, kW

print("\nTern propeller, D = 2.6 m, 1700 rpm, cruise at 3000 m")
T, p, rho = isa(3000)
for V, F in ((45.0, 6000.0), (70.0, 4200.0), (105.0, 2310.0)):
    vi, eta_i, Pi = momentum(F, V, 2.6, rho)
    n = 1700 / 60; J = V / (n * 2.6)
    CT = F / (rho * n**2 * 2.6**4)
    eta_real = 0.85 * eta_i if V > 60 else 0.80 * eta_i          # profile and swirl losses, illustrative
    P_shaft = F * V / eta_real / 1000
    CP = P_shaft * 1000 / (rho * n**3 * 2.6**5)
    print(f"V {V:5.1f} m/s  J {J:4.2f}  CT {CT:5.3f}  CP {CP:5.3f}  ideal eta {eta_i:4.2f}  "
          f"realistic eta {eta_real:4.2f}  shaft power {P_shaft:5.0f} kW")
tip = math.hypot(math.pi * 2.6 * 1700 / 60, 105.0) / math.sqrt(1.4 * 287.05 * T)
print(f"\nhelical tip Mach number at cruise {tip:.2f}")

altitude m   sigma   Gagg-Ferrar P kW   turbocharged P kW (flat to 4500 m)
       0    1.000        134.0              134.0
    1500    0.864        113.3              134.0
    3000    0.742         94.9              134.0
    4500    0.634         78.5              134.0
    6000    0.539         64.0              111.1

Tern propeller, D = 2.6 m, 1700 rpm, cruise at 3000 m
V  45.0 m/s  J 0.61  CT 0.180  CP 0.171  ideal eta 0.80  realistic eta 0.64  shaft power   421 kW
V  70.0 m/s  J 0.95  CT 0.126  CP 0.152  ideal eta 0.92  realistic eta 0.79  shaft power   374 kW
V 105.0 m/s  J 1.43  CT 0.069  CP 0.119  ideal eta 0.98  realistic eta 0.83  shaft power   291 kW

helical tip Mach number at cruise 0.77

The program answers the flying club's question first. The Plover's normally aspirated engine gives 95 kW at 3000 m and 79 kW at 4500 m; a turbocharger flat rated to 4500 m would keep the full 134 kW up there, a large gain for climb on hot days and for flying over the Rockies west of Colorado. Dr. Nakamura's comment was that the flying club also needs to price the overhaul cost and the training for turbocharger management before deciding.

For the Tern propeller, the momentum results show the classic pattern. At 45 m/s in the initial climb, the ideal efficiency is 0.80 and the realistic value, with profile and swirl losses, about 0.64, so 6.0 kN of thrust needs 421 kW of shaft power, close to the motor's 450 kW rating. At 105 m/s cruise the ideal efficiency is 0.98 and the realistic value 0.83, and 2.31 kN needs 291 kW. The helical tip Mach number at cruise is 0.77, safely below the range where compressibility losses begin. The realistic efficiencies use fixed loss factors and are illustrative; the company's propeller supplier provides full performance maps.

Engine operation and fuel

Piston engines in light aircraft are leaned in cruise: the pilot reduces the fuel flow with a mixture control as altitude increases, because the thinner air needs less fuel for the best mixture. Most still burn leaded aviation gasoline, and the transition to unleaded fuels approved for existing engines is a major issue for the general aviation industry in both Canada and the United States. Diesel and jet-fuel piston engines, and electric trainers, are alternatives now in service in small numbers.