Gas Turbine and Combined-Cycle Plants

The gas turbine runs on the Brayton cycle, and it powers both jet engines and modern combined-cycle power stations. It is the third cycle family on the Industrial and Power Plant Engineering paper, and it rewards a clean grasp of one efficiency formula, one temperature relation, and the idea of back-work. Work in SI: temperatures in kelvin, work in kJ/kg, using air with k = 1.4 and cp = 1.005 kJ/kg K.

The Brayton cycle

The ideal Brayton cycle has four processes, all in steady flow:

1. Compressor (1 to 2): air is compressed isentropically, raising its pressure and temperature.
2. Combustor (2 to 3): fuel burns, adding heat at constant pressure and raising the temperature to the turbine-inlet temperature.
3. Turbine (3 to 4): the hot gas expands isentropically, producing work.
4. Exhaust (4 to 1): heat is rejected at constant pressure (in an open cycle, the gas simply leaves and fresh air enters).

The key parameter is the pressure ratio, rp = P2/P1. The ideal thermal efficiency depends only on rp and k:

eta_Brayton = 1 minus 1/rp^((k minus 1)/k)

Worked example: with rp = 8 and k = 1.4, the exponent (k minus 1)/k = 0.2857. Then 8^0.2857 = 1.811, so eta = 1 minus 1/1.811 = 1 minus 0.552 = 0.448, or 44.8%. Raising rp to 10 gives 48.2%; efficiency climbs with pressure ratio.

Temperatures around the cycle

The isentropic relation ties temperatures to the pressure ratio:

T2/T1 = rp^((k minus 1)/k)      and      T3/T4 = rp^((k minus 1)/k)

Worked example: with T1 = 300 K and rp = 8, the compressor exit is T2 = 300 x 1.811 = 543 K. With a turbine inlet of T3 = 1,200 K, the turbine exit is T4 = 1,200/1.811 = 662 K. The turbine-inlet temperature T3 is the metallurgical limit that caps the whole cycle: hotter gas means more work, so blade cooling and materials set the ceiling.

The back-work ratio

Unlike a steam plant, a gas turbine spends a large share of its turbine output just driving its own compressor. That share is the back-work ratio:

BWR = compressor work / turbine work = (T2 minus T1) / (T3 minus T4)

Worked example: from the temperatures above, compressor work is proportional to 543 minus 300 = 243 K and turbine work to 1,200 minus 662 = 538 K, so BWR = 243/538 = 0.45. Nearly half the turbine's output goes back to the compressor. This is why gas turbines need high turbine-inlet temperatures: a cool turbine would leave almost nothing as net work.

The net work per kg is the difference:

wnet = cp x [(T3 minus T4) minus (T2 minus T1)] = 1.005 x (538 minus 243) = about 295 kJ/kg

Regeneration, intercooling, and reheat

Three modifications sharpen the Brayton cycle:

• Regeneration: a heat exchanger uses the hot turbine exhaust to preheat the compressed air before the combustor. Less fuel is then needed, raising efficiency. It helps only at low pressure ratios, where the exhaust is still hotter than the compressed air.
• Intercooling: cooling the air between compressor stages lowers the total compressor work, cutting the back-work ratio and raising net work.
• Reheat: reheating the gas between turbine stages raises the total turbine work, again increasing net work.

Intercooling and reheat both raise net work but, on their own, slightly lower thermal efficiency because they add heat at a lower average temperature. Combined with regeneration, all three together give the classic high-output, high-efficiency gas turbine.

Combined-cycle plants

A combined-cycle plant is the most efficient way to burn fuel for power. A gas turbine (Brayton, topping cycle) exhausts gas at 500 to 600 degrees C; that hot exhaust passes through a heat recovery steam generator (HRSG) to raise steam for a steam turbine (Rankine, bottoming cycle). One fuel input drives two cycles.

The overall efficiency combines the two so that the waste heat of the first becomes the input of the second:

eta_combined = 1 minus (1 minus eta_gas)(1 minus eta_steam)

Worked example: a gas turbine of 40% efficiency topping a steam cycle of 30% efficiency gives eta_combined = 1 minus (1 minus 0.40)(1 minus 0.30) = 1 minus (0.60)(0.70) = 1 minus 0.42 = 0.58, or 58%. Neither cycle alone reaches that; the gain comes from reusing the gas turbine's rejected heat instead of dumping it.

Exam-day strategy

• Brayton efficiency needs only rp and k: eta = 1 minus 1/rp^((k minus 1)/k). For air the exponent is 0.2857.
• Temperatures scale as T2/T1 = rp^0.2857; the same factor sets T3/T4.
• Back-work ratio (T2 minus T1)/(T3 minus T4) is large for gas turbines; a high turbine-inlet temperature is what keeps net work positive and useful.
• Regeneration raises efficiency at low pressure ratios; intercooling and reheat raise net work but not, by themselves, efficiency.
• Combined-cycle efficiency is 1 minus (1 minus eta_gas)(1 minus eta_steam): the topping cycle's waste heat feeds the bottoming cycle.