Gas turbines: the Brayton cycle with regeneration, intercooling and reheat

Kofi Mensah's second option for the top of Northgate's plant is a single gas turbine of about 6 MW. A gas turbine compresses air, burns fuel in it at constant pressure and expands the hot gas through a turbine that drives both the compressor and a generator. It is compact, runs for long periods between overhauls, and rejects nearly all its waste heat in one hot exhaust stream, which is ideal for raising steam. Its weaknesses are a lower efficiency than the best engines at this size and a strong sensitivity to the temperature of the air it breathes. The Brayton cycle, the air-standard model of the gas turbine, explains both.

The ideal Brayton cycle

In the open-cycle gas turbine, fresh air enters the compressor and exhaust leaves the turbine. The air-standard model closes the loop with a heat exchanger that rejects heat at constant pressure. The ideal Brayton cycle then has four internally reversible processes: (1-2) isentropic compression, (2-3) heat addition at constant pressure, (3-4) isentropic expansion, and (4-1) heat rejection at constant pressure. Each component is a steady-flow device, so with constant specific heats

With the pressure ratio \(r_p = P_2/P_1\), the isentropic relations give \(T_2/T_1 = T_3/T_4 = r_p^{(k-1)/k}\), and the thermal efficiency becomes

As in the Otto cycle, efficiency depends only on a ratio and on \(k\). Raising the pressure ratio raises the average temperature at which heat is added and lowers the average temperature at which it is rejected. But the turbine inlet temperature \(T_3\) is fixed by what the turbine's first-stage blades can survive, and that sets a limit on the net work. For fixed \(T_1\) and \(T_3\), the ideal net work per kilogram of air is greatest when

At very high pressure ratios the compressor work approaches the turbine work and the net work shrinks toward zero, even though the efficiency keeps rising. A gas turbine designer chooses the pressure ratio between the value for maximum work, which gives the smallest machine for a given power, and the higher value for maximum efficiency.

The back work ratio and real components

The feature that sets gas turbines apart from steam plants is the back work ratio, the fraction of the turbine's work used to drive the compressor. Compressing a gas takes far more work than pumping a liquid, and in a gas turbine the back work ratio is typically 0.4 to 0.6. Every percent of compressor or turbine inefficiency therefore costs several percent of net output. With isentropic efficiencies \(\eta_c\) and \(\eta_t\),

Early gas turbines of the 1930s and 1940s, with component efficiencies near 70 percent, barely produced net work. Modern machines reach polytropic efficiencies of about 90 percent in both compressor and turbine and turbine inlet temperatures far above the melting point of the blade alloys, made possible by internal air cooling, thermal-barrier coatings and single-crystal superalloys. The cooling air is bled from the compressor and bypasses part of the expansion, which is one reason real machines fall short of the simple cycle prediction.

Northgate's candidate turbine

The candidate machine has a pressure ratio of 16, a turbine inlet temperature of 1400 K, a compressor efficiency of 0.86 and a turbine efficiency of 0.89. The program models it with cold-air-standard properties at three inlet temperatures: the ISO rating condition of 15 °C, a Minneapolis winter day at −30 °C and a summer day at 30 °C. It then sweeps the pressure ratio and tests regeneration.

# Brayton cycle: Northgate's candidate gas turbine, pressure ratio, ambient temperature and regeneration
cp, k = 1.005, 1.4
e = (k - 1) / k

def brayton(rp, T1, T3, eta_c=1.0, eta_t=1.0, regen=0.0):
    T2 = T1 * (1 + (rp ** e - 1) / eta_c)
    T4 = T3 * (1 - eta_t * (1 - rp ** -e))
    wc, wt = cp * (T2 - T1), cp * (T3 - T4)
    T5 = T2 + regen * (T4 - T2)          # air leaving the regenerator (regen = effectiveness)
    qin = cp * (T3 - T5)
    return dict(T2=T2, T4=T4, wc=wc, wt=wt, wnet=wt - wc, bwr=wc / wt, eta=(wt - wc) / qin, T5=T5)

rp, T3 = 16, 1400.0
print("pressure ratio 16, turbine inlet 1400 K, compressor 0.86, turbine 0.89")
for T1, label in ((288.15, "ISO day 15 C"), (243.15, "winter -30 C"), (303.15, "summer 30 C")):
    c = brayton(rp, T1, T3, 0.86, 0.89)
    m = 6000 / c['wnet']
    print(f"  {label:13s} T2 = {c['T2']:5.0f} K  T4 = {c['T4']:5.0f} K  wnet = {c['wnet']:5.1f} kJ/kg"
          f"  bwr = {c['bwr']:.2f}  eta = {c['eta']:.3f}  air for 6 MW = {m:4.1f} kg/s")

print("\neffect of pressure ratio at 15 C (ideal and actual)")
for r in (4, 8, 12, 16, 20, 24, 30):
    i, a = brayton(r, 288.15, T3), brayton(r, 288.15, T3, 0.86, 0.89)
    print(f"  rp = {r:2d}: ideal eta = {i['eta']:.3f} wnet = {i['wnet']:5.1f}   actual eta = {a['eta']:.3f} wnet = {a['wnet']:5.1f}")
print(f"  pressure ratio for maximum ideal net work = {(T3 / 288.15) ** (k / (2 * (k - 1))):.1f}")

print("\nregeneration (effectiveness 0.80) at 15 C")
for r in (4, 6, 8, 16):
    plain, reg = brayton(r, 288.15, T3, 0.86, 0.89), brayton(r, 288.15, T3, 0.86, 0.89, 0.80)
    gain = "helps" if reg['eta'] > plain['eta'] else "no gain: T4 < T2"
    print(f"  rp = {r:2d}: T2 = {plain['T2']:4.0f} K, T4 = {plain['T4']:4.0f} K, eta {plain['eta']:.3f} -> {reg['eta']:.3f} ({gain})")

# Ideal two-stage intercooling and reheat with perfect regeneration, overall pressure ratio 16
ri = 16 ** 0.5
wc = 2 * cp * 288.15 * (ri ** e - 1); wt = 2 * cp * T3 * (1 - ri ** -e)
qin_noregen = cp * (T3 - 288.15 * ri ** e) + cp * (T3 - T3 * ri ** -e)
print(f"\nideal intercool + reheat, rp = 16: wnet = {wt - wc:.1f} kJ/kg, eta without regenerator = {(wt - wc) / qin_noregen:.3f},"
      f" with ideal regenerator = {1 - 288.15 / T3 * ri ** e:.3f}")

pressure ratio 16, turbine inlet 1400 K, compressor 0.86, turbine 0.89
  ISO day 15 C  T2 =   693 K  T4 =   718 K  wnet = 278.3 kJ/kg  bwr = 0.59  eta = 0.392  air for 6 MW = 21.6 kg/s
  winter -30 C  T2 =   585 K  T4 =   718 K  wnet = 341.8 kJ/kg  bwr = 0.50  eta = 0.417  air for 6 MW = 17.6 kg/s
  summer 30 C   T2 =   729 K  T4 =   718 K  wnet = 257.1 kJ/kg  bwr = 0.62  eta = 0.381  air for 6 MW = 23.3 kg/s

effect of pressure ratio at 15 C (ideal and actual)
  rp =  4: ideal eta = 0.327 wnet = 319.4   actual eta = 0.258 wnet = 245.9
  rp =  8: ideal eta = 0.448 wnet = 395.3   actual eta = 0.341 wnet = 287.7
  rp = 12: ideal eta = 0.508 wnet = 415.8   actual eta = 0.375 wnet = 288.4
  rp = 16: ideal eta = 0.547 wnet = 419.9   actual eta = 0.392 wnet = 278.3
  rp = 20: ideal eta = 0.575 wnet = 417.2   actual eta = 0.400 wnet = 264.4
  rp = 24: ideal eta = 0.597 wnet = 411.1   actual eta = 0.402 wnet = 249.0
  rp = 30: ideal eta = 0.622 wnet = 398.9   actual eta = 0.399 wnet = 225.2
  pressure ratio for maximum ideal net work = 15.9

regeneration (effectiveness 0.80) at 15 C
  rp =  4: T2 =  451 K, T4 =  992 K, eta 0.258 -> 0.474 (helps)
  rp =  6: T2 =  512 K, T4 =  901 K, eta 0.310 -> 0.477 (helps)
  rp =  8: T2 =  560 K, T4 =  842 K, eta 0.341 -> 0.466 (helps)
  rp = 16: T2 =  693 K, T4 =  718 K, eta 0.392 -> 0.403 (helps)

ideal intercool + reheat, rp = 16: wnet = 638.8 kJ/kg, eta without regenerator = 0.445, with ideal regenerator = 0.694

The effect of air temperature

The program's first block shows why gas turbines love cold weather. At −30 °C the compressor needs much less work, because work per kilogram is proportional to the inlet temperature, and the denser air means more mass flows through the same machine. The net work per kilogram rises from 278 to 342 kJ/kg and the efficiency from 0.392 to 0.417. At 30 °C both fall. Because a real turbine's compressor swallows a nearly fixed volume of air per second, the mass flow also changes with density, and the power output swings even more than the work per kilogram. Manufacturers publish curves showing output falling by roughly 0.5 to 0.8 percent for every degree Celsius above the rating temperature.

For Northgate this is good news: the district heating load peaks in a Minneapolis winter, exactly when the turbine is most capable. In summer, when output sags, the plant's heat load is small anyway. Plants in hot climates often cool the inlet air with evaporative coolers or chillers to recover summer output; Lesson 11 analyzes evaporative cooling.

Pressure ratio

The second block of output sweeps the pressure ratio. For the ideal cycle, efficiency rises steadily and net work peaks near \(r_p = 16\), matching the formula \((1400/288.15)^{1.75} = 15.9\). For the actual cycle, both curves are flatter and peak earlier: the net work is greatest near a pressure ratio of 10 to 12, and the efficiency peaks near 24 and then falls, because component losses grow with the work they handle. The candidate machine's ratio of 16 sits between the two peaks.

Regeneration

In a simple-cycle gas turbine with a low pressure ratio, the exhaust leaves hotter than the compressed air. A regenerator, or recuperator, is a heat exchanger that uses the exhaust to preheat the compressed air before the combustor, so less fuel is needed for the same turbine inlet temperature. Its effectiveness compares the actual temperature rise of the air with the largest possible rise:

Regeneration helps only when \(T_4 > T_2\). The program shows the effect for an effectiveness of 0.80: at a pressure ratio of 4 the efficiency jumps from 0.258 to 0.474, at 6 it rises from 0.310 to 0.477, and at 16 it barely moves, because the exhaust (718 K) is only slightly hotter than the compressed air (693 K). This is why recuperated gas turbines, such as microturbines of 30 to 300 kW, use low pressure ratios of about 3 to 5. A recuperator adds cost, weight and pressure loss, and it lowers the exhaust temperature, which is a disadvantage when the exhaust heat is wanted for cogeneration. For Northgate, Elena's team concludes that a recuperator makes no sense: the exhaust heat is the point.

Intercooling and reheat

The work to compress a gas between two pressures falls if the gas is cooled during compression, and the work from a turbine rises if heat is added during expansion. Practical machines approximate these ideas by compressing in stages with intercoolers between them and expanding in stages with reheat combustors between them. For two stages with perfect intercooling back to \(T_1\) and reheating back to \(T_3\), the net work is greatest when each stage has the same pressure ratio, \(\sqrt{r_p}\).

Intercooling and reheat greatly increase the net work, by about half again in the program's example, but on their own they lower the efficiency, because the extra heat is added at a lower average temperature and the exhaust leaves hotter. Combined with regeneration, the picture changes: the hotter exhaust preheats the air, and as the number of stages grows the cycle approaches the Ericsson cycle, with isothermal compression and expansion and an efficiency approaching Carnot. The program finds 0.445 without a regenerator and 0.694 with an ideal one for two stages. Aeroderivative turbines with intercooling exist, and some large heavy-duty turbines use sequential (reheat) combustion, but most machines in Northgate's size range are simple cycle.

Deviations from the ideal cycle

Beyond component inefficiencies, real gas turbines differ from the air-standard model in several ways. The working fluid after the combustor is a mixture of nitrogen, carbon dioxide, water vapor and excess oxygen with a higher specific heat than air; its \(c_p\) also rises with temperature. The fuel adds about 2 percent to the mass flow through the turbine. Pressure drops in the inlet filter, the combustor and the exhaust system reduce the expansion ratio available to the turbine; an exhaust back-pressure of a few kilopascals, typical of a heat-recovery steam generator, costs about a percent of output. And cooling air bled from the compressor bypasses part of the turbine. Lesson 12's combustion analysis will let us replace the air-standard heat addition with an actual fuel flow.

The air-standard model still earns its place. It shows which variables matter, gives correct trends with ambient temperature and pressure ratio, and lets the team check a manufacturer's performance figures for reasonableness. Tomás builds the candidate machine's predicted output against outdoor temperature from the program and finds that it agrees with the vendor's published curve within a few percent after he applies the vendor's stated efficiency at the rating point.

In US practice, gas turbine performance is quoted at ISO conditions (59 °F, 14.696 psia and 60 percent relative humidity), and heat rate is quoted in Btu/kWh, usually on the higher heating value of the fuel for utilities and on the lower heating value for turbine ratings. A 34 percent efficient machine (on LHV) has a heat rate of 3412/0.34 = 10,040 Btu/kWh. The cycle calculations work the same way in US customary units with \(c_p = 0.240\) Btu/lbm·R and temperatures in rankine: the rating inlet temperature of 59 °F is 518.7 R.