Ideal turbojet

The turbojet is the simplest jet engine and the one in which the cycle and the thrust are most directly linked: compressor, combustor, a turbine that drives the compressor and nothing else, and a nozzle that turns all the remaining energy into jet velocity. Pure turbojets have nearly vanished from civil aviation because, as Lesson 1 showed, their fast jets waste energy at subsonic speeds. They remain the best teaching model, because every later engine is a turbojet core with something added. Dr. Nakamura asks Arjun to build an ideal turbojet model at the Swift 900 cruise condition, Mach 0.80 at 12 500 m, as the base for the turbofan study in Lesson 5.

Stations and total quantities

Engine analysis uses standard station numbers: 0 for the free stream, 2 for the compressor face, 3 for the compressor exit, 4 for the turbine inlet, 5 for the turbine exit and 9 for the nozzle exit. It also uses total (stagnation) temperature and pressure, the values the flow would reach if brought to rest isentropically. Flight speed raises them before the air even reaches the engine:

At Mach 0.80 the ram temperature ratio is 1.128 and the ram pressure ratio 1.52. Ram compression is free work done by the aircraft's motion, and at supersonic speeds it becomes large enough that a ramjet needs no compressor at all.

The ideal turbojet equations

In the ideal engine every component is isentropic except the combustor, which adds heat at constant total pressure; the nozzle expands to ambient pressure; and the fuel mass is neglected in the momentum balance. Define the compressor temperature ratio \(\tau_c = \pi_c^{(\gamma-1)/\gamma}\) and the burner parameter \(\tau_\lambda = T_{t4}/T_0\). The turbine must supply exactly the compressor work, which fixes its temperature ratio:

The nozzle then turns what is left into velocity. In terms of the ambient speed of sound \(a_0\), the specific thrust and fuel-air ratio are

and the fuel consumption follows as \(S = f/(F/\dot m_0)\). The thermal efficiency of the ideal turbojet is that of the Brayton cycle with the total pressure ratio \(\pi_r\pi_c\):

These few lines are the core of parametric cycle analysis, the method engine companies use at the start of a design to map out which pressure ratios and temperatures are worth pursuing, before any component exists.

# Ideal turbojet: specific thrust and fuel consumption against compressor pressure ratio,
# at the Swift 900 cruise point (M 0.80, 12 500 m) and at sea-level static conditions.
import math
g, R, cp, hPR = 1.4, 287.05, 1005.0, 43.2e6
def turbojet(M0, T0, pi_c, Tt4):
    a0 = math.sqrt(g * R * T0); V0 = M0 * a0
    tau_r = 1 + (g - 1) / 2 * M0**2               # ram temperature ratio
    tau_l = Tt4 / T0                              # burner exit / ambient total-enthalpy ratio
    tau_c = pi_c ** ((g - 1) / g)
    tau_t = 1 - tau_r / tau_l * (tau_c - 1)       # turbine drives the compressor
    if tau_t <= 0: return None
    Ve_a0 = math.sqrt(2 / (g - 1) * tau_l / (tau_r * tau_c) * (tau_r * tau_c * tau_t - 1))
    Fs = a0 * (Ve_a0 - M0)                        # specific thrust, N per kg/s of air
    f = cp * T0 / hPR * (tau_l - tau_r * tau_c)   # fuel-air ratio
    S = f / Fs                                    # kg fuel per N per s
    eta_th = 1 - 1 / (tau_r * tau_c)
    eta_p = 2 * M0 / (Ve_a0 + M0) if M0 > 0 else 0
    return Fs, S * 1e6, f, eta_th, eta_p
T_cr = 216.65                                     # K, ISA above 11 km
print("cruise M 0.80 at 12 500 m, Tt4 = 1500 K")
print(" pi_c   F/mdot N s/kg   TSFC mg/(N s)    f      eta_th  eta_p")
for pc in (2, 5, 10, 15, 20, 30, 40):
    r = turbojet(0.80, T_cr, pc, 1500)
    if r: print(f"{pc:4d}      {r[0]:7.1f}        {r[1]:6.2f}      {r[2]:.4f}   {r[3]:.3f}   {r[4]:.3f}")
tau_c_opt = math.sqrt(1500 / T_cr) / (1 + 0.2 * 0.80**2)
print(f"pressure ratio for maximum specific thrust: {tau_c_opt ** 3.5:.1f}")
print("\nsea-level static, Tt4 = 1500 K")
for pc in (5, 10, 20):
    r = turbojet(0.0, 288.15, pc, 1500)
    print(f"pi_c {pc:3d}: F/mdot {r[0]:6.1f} N s/kg, TSFC {r[1]:5.2f} mg/(N s)")

cruise M 0.80 at 12 500 m, Tt4 = 1500 K
 pi_c   F/mdot N s/kg   TSFC mg/(N s)    f      eta_th  eta_p
   2        609.2         45.90      0.0280   0.273   0.437
   5        783.9         33.03      0.0259   0.440   0.376
  10        847.0         28.24      0.0239   0.541   0.358
  15        863.0         26.15      0.0226   0.591   0.354
  20        865.8         24.85      0.0215   0.623   0.353
  30        857.7         23.17      0.0199   0.665   0.355
  40        843.3         22.04      0.0186   0.691   0.359
pressure ratio for maximum specific thrust: 19.4

sea-level static, Tt4 = 1500 K
pi_c   5: F/mdot  879.2 N s/kg, TSFC 27.61 mg/(N s)
pi_c  10: F/mdot  956.1 N s/kg, TSFC 22.96 mg/(N s)
pi_c  20: F/mdot  974.5 N s/kg, TSFC 19.62 mg/(N s)

The program reproduces the Brayton trade from Lesson 3, now in thrust terms. Specific thrust peaks at 866 N s/kg near a compressor pressure ratio of 19, the value given by the closed-form optimum \(\tau_c = \sqrt{\tau_\lambda}/\tau_r\). TSFC keeps falling, from 46 mg/(N s) at a pressure ratio of 2 to 22 at 40, as the thermal efficiency rises from 0.27 to 0.69. The propulsive efficiency hardly changes, staying near 0.35, because the jet is always far faster than the aircraft: about 2.5 times the flight speed even at the best point. That flat, low propulsive efficiency is the turbojet's basic problem at subsonic speed.

At sea level and zero speed the specific thrust is higher, about 975 N s/kg at a pressure ratio of 20, because there is no ram drag and the ambient air is denser and warmer relative to the turbine temperature. Thrust lapse with speed and altitude is one of the main things an engine model must capture, because the aircraft's take-off, climb and cruise each need different thrusts and the engine must be sized for whichever is hardest.

Why specific thrust matters

Specific thrust sets engine size. The air mass flow needed for a thrust \(F\) is \(F\) divided by the specific thrust, and the inlet and fan diameters scale with the square root of the mass flow. A turbojet with 866 N s/kg needs only 4.2 kg/s of air for the Swift 900's 3.6 kN cruise thrust, a very small engine. But the same engine burns about 25 mg/(N s) at cruise, far more than the 16 to 17 of a modern turbofan. The turbofan of Lesson 5 trades away specific thrust, accepting a larger engine, to buy propulsive efficiency.

At supersonic speeds the trade reverses. The flight speed is high enough that a fast jet no longer wastes so much energy, and a small, high-specific-thrust engine has less drag. That is why supersonic aircraft use turbojets or low-bypass turbofans, often with afterburners that add a second combustor behind the turbine to raise jet temperature and thrust for short periods at a large cost in fuel.

From ideal to real

The ideal model overstates performance. Real inlets lose a few percent of total pressure; compressors and turbines have efficiencies of 0.85 to 0.92; the combustor loses 3 to 5 percent of total pressure and does not burn every molecule of fuel; the turbine must also drive accessories and supply bleed air to the cabin and anti-ice systems; and the nozzle does not always expand to ambient. Non-ideal cycle analysis carries each of these losses through the same station-by-station calculation, with different gas properties on the hot side. The result is typically 20 to 30 percent less specific thrust and 10 to 20 percent more fuel consumption than the ideal model. The ideal model's value is that it shows the trends correctly and quickly.

Thrust lapse and engine sizing

Specific thrust depends strongly on flight condition. As altitude rises, the air gets thinner, so the mass flow through a given engine falls roughly with the ambient pressure and the thrust falls with it, although the colder air partly compensates by raising \(\tau_\lambda\). As flight speed rises at fixed altitude, ram drag grows and specific thrust falls, but ram compression increases the mass flow per unit inlet area, so the installed thrust falls more slowly. Engine makers supply this information in performance decks, computer programs that return thrust and fuel flow for any altitude, Mach number, temperature and power setting.

An aircraft designer sizes the engine for its hardest requirement. For an airliner this is usually the top of climb, where the engine must still give enough thrust to climb at a minimum rate at cruise altitude; for a business jet it may be the take-off field length at a hot, high airport, or the one-engine-inoperative climb gradient. The Swift 900's sizing case, Arjun finds from the performance deck, is the top of climb at 12 500 m, where each engine must give 4.1 kN at Mach 0.78 to hold a residual climb rate of 1.5 m/s , 14 percent more than the cruise thrust.

The ideal turbojet also shows why turbine inlet temperature is so valuable. At a fixed pressure ratio of 20, raising the turbine inlet temperature from 1500 to 1700 K raises the ideal specific thrust by more than 10 percent, so the same thrust comes from a smaller engine. The cost, again, is turbine cooling and life, the subject of Lesson 10.