Brayton cycle analysis

Every gas turbine, from the Tern's 665 kW range extender to the largest airliner engine, runs on the Brayton cycle: compress air, add heat at nearly constant pressure, expand the hot gas through a turbine, and reject heat to the atmosphere through the exhaust. Thermodynamics II already derived the ideal cycle and looked at it as a power plant. This lesson takes the aircraft engine designer's view. Dr. Nakamura asks Arjun to explain to the Swift 900 program managers why the two engine proposals they received differ so much: one has a core pressure ratio of 16 and a turbine inlet temperature of 1450 K, the other 26 and 1650 K. To do it he needs the real cycle, with component losses, and its two key outputs: how much work each kilogram of air delivers and how efficiently the fuel is used.

The ideal cycle in brief

In the ideal Brayton cycle the compression and expansion are isentropic and the gas is perfect with constant specific heats. States 1 to 4 are compressor inlet, compressor exit, turbine inlet and turbine exit. With the pressure ratio \(\pi = p_2/p_1\) and \(\gamma\) the ratio of specific heats, the thermal efficiency is

It depends only on the pressure ratio. The net specific work, the shaft work per kilogram of air, also depends on the turbine inlet temperature \(T_3\):

For a fixed \(T_3\) the work rises with pressure ratio, reaches a maximum, and falls back to zero when the compressor exit temperature reaches \(T_3\) and there is no room left to add heat. The maximum occurs where \(T_2 = T_4\), at a pressure ratio of \((T_3/T_1)^{\gamma/(2(\gamma-1))}\).

Real components

Real compressors and turbines generate entropy. Their performance is described by isentropic efficiencies, which compare the actual temperature change with the isentropic one between the same pressures:

The compressor needs more work than the ideal one, and the turbine gives less. Add a few percent of total pressure loss through the combustor and ducts, and use different gas properties for the hot side (about 1150 J/(kg K) and \(\gamma = 1.33\) for combustion products, against 1005 and 1.4 for air), and the cycle changes character. Because the turbine work and compressor work are two large numbers whose difference is the net output, small losses in each take a large bite out of the net work. A compressor efficiency of 0.86 and a turbine efficiency of 0.90 typically cut the net work by a quarter or more.

What the real cycle shows

The real cycle brings in a second important parameter: the cycle temperature ratio \(T_3/T_1\). In the ideal cycle, efficiency depends on pressure ratio alone. In the real cycle, a higher turbine inlet temperature makes the losses a smaller fraction of the work, so efficiency rises with temperature too. And the best pressure ratio for efficiency is much higher than the best pressure ratio for specific work. That gives designers a real choice.

# Brayton cycle with component efficiencies: thermal efficiency and specific work
# against pressure ratio, ideal and real, for a turbine inlet temperature of 1500 K.
cp, g = 1005.0, 1.4; cpg, gg = 1150.0, 1.33
T1 = 288.15
def cycle(pr, T3, eta_c=0.86, eta_t=0.90, loss=0.04, ideal=False):
    if ideal:
        T2 = T1 * pr ** ((g - 1) / g); T4 = T3 / pr ** ((g - 1) / g)
        w = cp * (T3 - T4) - cp * (T2 - T1); q = cp * (T3 - T2)
        return w / 1000, w / q
    T2 = T1 * (1 + (pr ** ((g - 1) / g) - 1) / eta_c)
    pr_t = pr * (1 - loss)                         # combustor and duct pressure losses
    T4 = T3 * (1 - eta_t * (1 - pr_t ** (-(gg - 1) / gg)))
    w = cpg * (T3 - T4) - cp * (T2 - T1); q = cpg * (T3 - T2)
    return w / 1000, w / q
print(" pr   ideal w kJ/kg  ideal eta   real w kJ/kg  real eta")
best = (0, 0)
for pr in (4, 8, 12, 16, 20, 25, 30, 40):
    wi, ei = cycle(pr, 1500, ideal=True); wr, er = cycle(pr, 1500)
    best = max(best, (wr, pr))
    print(f"{pr:3d}     {wi:6.1f}       {ei:5.3f}       {wr:6.1f}      {er:5.3f}")
print(f"\nreal-cycle specific work peaks near pressure ratio {best[1]} ({best[0]:.0f} kJ/kg)")
for T3 in (1300, 1500, 1700):
    print(f"TIT {T3} K, pr 20: real w {cycle(20, T3)[0]:5.1f} kJ/kg, eta {cycle(20, T3)[1]:.3f}")

 pr   ideal w kJ/kg  ideal eta   real w kJ/kg  real eta
  4      352.3       0.327        277.0      0.230
  8      440.3       0.448        343.1      0.317
 12      466.9       0.508        357.8      0.359
 16      474.9       0.547        357.4      0.385
 20      475.0       0.575        350.9      0.402
 25      469.7       0.601        338.9      0.417
 30      461.4       0.622        325.0      0.427
 40      440.8       0.651        295.2      0.438

real-cycle specific work peaks near pressure ratio 12 (358 kJ/kg)
TIT 1300 K, pr 20: real w 243.4 kJ/kg, eta 0.379
TIT 1500 K, pr 20: real w 350.9 kJ/kg, eta 0.402
TIT 1700 K, pr 20: real w 458.5 kJ/kg, eta 0.416

The program shows the trade clearly. In the ideal cycle at 1500 K, specific work peaks near a pressure ratio of 16 to 20 at 475 kJ/kg and efficiency keeps rising. With real components, the specific work peaks earlier, near a pressure ratio of 12, at only 358 kJ/kg, while efficiency still climbs, from 0.36 at 12 to 0.44 at 40. Raising the turbine inlet temperature from 1300 to 1700 K at a pressure ratio of 20 nearly doubles the specific work, from 243 to 459 kJ/kg, and lifts the efficiency from 0.38 to 0.42.

Arjun's explanation to the program managers follows directly. The higher-pressure, hotter engine will have better cycle efficiency, which means lower fuel burn, and more work per kilogram of air, which means a smaller, lighter core for the same thrust. The price is turbine materials and cooling for 1650 K (Lesson 10), more compressor stages or higher blade speeds for a pressure ratio of 26 (Lesson 8), and typically higher maintenance cost. Whether the fuel saving pays for those depends on how the Swift 900's customers fly. Business jets fly fewer hours per year than airliners, which makes the fuel saving worth less and the maintenance cost relatively more important.

From the shaft cycle to aircraft engines

A shaft engine, like the Tern's turboshaft, takes all its useful output as turbine work and leaves the exhaust slow. A jet engine uses the turbine only to drive the compressor and lets the remaining energy accelerate the exhaust through a nozzle. The thermodynamic cycle is the same; what changes is where the useful energy is taken out. Lessons 4 to 6 treat the turbojet, turbofan and turboprop as different ways of dividing the cycle's energy between shaft and jet. In each, the designer's first decision is the core pressure ratio and turbine inlet temperature, which set the cycle, and the second is how to turn the cycle's energy into thrust most efficiently, which sets the propulsive efficiency.

Two practical additions modify the simple cycle. A recuperator, a heat exchanger that heats compressor exit air with turbine exhaust, raises efficiency at low pressure ratios and is used in some ground turbines and small experimental aircraft engines, but it is too heavy and bulky for most aircraft. Intercooling between compressor stages saves compressor work, but again the heat exchanger weight usually rules it out. For aircraft, pressure ratio and temperature are the main levers.

Off-design operation

An engine spends little of its life at its design point. At part power the turbine inlet temperature falls, the compressor runs at lower speed and pressure ratio, and the cycle efficiency drops. A gas turbine at 30 percent power may use 40 to 60 percent more fuel per unit of output than at full power. That is one reason aircraft cruise engines are sized so that cruise falls near their best efficiency, and why the Tern's turboshaft was sized for cruise rather than take-off: the battery handles the peaks, and the turbine runs close to its design point for most of the flight.

Ambient conditions matter too. On a hot day the compressor inlet temperature rises, which lowers the cycle temperature ratio \(T_3/T_1\) and with it both specific work and efficiency at a given turbine inlet temperature. To protect the turbine, engines are flat rated: they deliver full take-off thrust up to a certain ambient temperature, often around ISA plus 15 to 30 K, by running cooler than their limit on cold days, and lose thrust above it. That is why take-off performance charts show a corner temperature, and why hot, high airports limit payload.

Engine makers compare cycles on a carpet plot, a chart of thermal efficiency against specific work with lines of constant pressure ratio and constant turbine inlet temperature. The best engines sit toward the upper right, and the plot shows at a glance what a given increase in temperature or pressure ratio buys. Arjun's slides to the Swift 900 program managers end with one: the two proposals sit on different corners of it, and the choice between them is a business decision as much as a technical one.