13.1 Rankine and Brayton cycles

Rankine Cycle

Components of the Rankine Cycle

The Rankine cycle is the foundation of steam power plants, including coal-fired and nuclear facilities. It uses water as its working fluid, taking advantage of the large energy transfers that happen during phase changes (liquid to vapor and back).

Four components make up the cycle:

• Boiler: Heats the working fluid (water) to produce high-pressure steam. This is where thermal energy enters the cycle.
• Turbine: Expands the high-pressure steam, converting thermal energy into mechanical work that drives a generator for electricity production.
• Condenser: Rejects heat to the environment (often a cooling tower or body of water), condensing the low-pressure exhaust steam back into liquid.
• Pump: Pressurizes the liquid water and sends it back to the boiler, completing the loop.

Processes in the Rankine Cycle

Each component corresponds to one of four thermodynamic processes. On a T-s (temperature-entropy) diagram, these trace out a closed loop:

1. Isentropic compression (Pump): The pump raises the pressure of the liquid water with minimal entropy change. Because liquids are nearly incompressible, the pump work is small compared to the turbine output.
2. Isobaric heat addition (Boiler): At constant pressure, the boiler heats the water until it becomes superheated steam. This step accounts for the largest energy input in the cycle.
3. Isentropic expansion (Turbine): The high-pressure, high-temperature steam expands through the turbine, producing work. Entropy stays roughly constant in the ideal case.
4. Isobaric heat rejection (Condenser): The low-pressure steam releases heat at constant pressure, condensing back into a liquid so the pump can handle it again.

Steam power plants (coal, nuclear, biomass, and concentrated solar) all rely on this cycle as their core thermodynamic framework.

Brayton Cycle

Components of the Brayton Cycle

The Brayton cycle is the basis for gas turbines, including jet engines and natural-gas power plants. Unlike the Rankine cycle, the working fluid (air) stays in the gas phase throughout.

• Compressor: Draws in ambient air and compresses it to high pressure. This step requires significant work input.
• Combustion chamber: Burns fuel (natural gas, kerosene, etc.) to heat the compressed air at roughly constant pressure.
• Turbine: Expands the hot, high-pressure gases, extracting mechanical work. Part of this work drives the compressor; the rest is useful output.
• Heat exchanger / exhaust (optional): In a simple open cycle, exhaust gases are released to the atmosphere. In a closed cycle or one with a regenerator, a heat exchanger preheats the compressed air using exhaust heat, improving efficiency.

Processes in the Brayton Cycle

1. Isentropic compression (Compressor): Air is compressed to high pressure with ideally no change in entropy. Unlike the Rankine pump, the compressor consumes a large fraction of the turbine's output because compressing a gas takes much more work than pressurizing a liquid.
2. Isobaric heat addition (Combustion chamber): Fuel is burned at constant pressure, raising the gas temperature significantly.
3. Isentropic expansion (Turbine): The hot gases expand through the turbine, producing work.
4. Isobaric heat rejection: Heat is rejected at constant pressure, either to the atmosphere (open cycle) or through a heat exchanger (closed cycle).

Applications include jet engines for aircraft propulsion and gas turbine power plants for electricity generation. Gas turbines are also commonly paired with steam turbines in combined-cycle plants to capture waste heat.

Rankine vs. Brayton Cycle Efficiency

Both cycles share the same four process types: isentropic compression, isobaric heat addition, isentropic expansion, and isobaric heat rejection. The key performance difference comes down to heat rejection temperature. The Rankine cycle rejects heat in a condenser that operates near ambient temperature, while the Brayton cycle exhausts gases that are still quite hot. Since thermal efficiency improves when you minimize the temperature at which heat is rejected, Rankine cycles generally achieve higher thermal efficiencies than simple Brayton cycles operating between similar temperature limits.

Rankine efficiency depends primarily on the maximum and minimum temperatures of the working fluid. Brayton efficiency is strongly governed by the pressure ratio across the compressor, along with the peak and minimum cycle temperatures.

Calculations for Thermodynamic Cycles

Rankine Cycle Efficiency

Net work is the difference between what the turbine produces and what the pump consumes:

$W_{net} = W_t - W_p$

The heat input $Q_b$ is the energy added in the boiler. Thermal efficiency is then:

$\eta_{th} = \frac{W_{net}}{Q_b} = \frac{W_t - W_p}{Q_b}$

Because pump work is small relative to turbine work, Rankine efficiency is often close to $W_t / Q_b$ as a rough estimate. In practice, modern coal plants achieve around 33–45% thermal efficiency, and nuclear plants around 30–35%.

Brayton Cycle Efficiency

The same structure applies, but the compressor replaces the pump:

$W_{net} = W_t - W_c$

where $W_c$ is the compressor work, which is much larger relative to $W_t$ than pump work is in the Rankine cycle. Thermal efficiency is:

$\eta_{th} = \frac{W_{net}}{Q_{in}} = \frac{W_t - W_c}{Q_{in}}$

For an ideal Brayton cycle with constant specific heats, efficiency depends only on the pressure ratio $r_p$ (the ratio of compressor outlet pressure to inlet pressure):

$\eta_{th} = 1 - \frac{1}{r_p^{(\gamma - 1)/\gamma}}$

Here $\gamma$ is the specific heat ratio ($c_p / c_v$), which equals about 1.4 for air. This equation tells you that higher pressure ratios yield higher efficiency. For example, a pressure ratio of 10 with $\gamma = 1.4$ gives an ideal efficiency of about 48%. Modern simple-cycle gas turbines typically achieve 30–40% in practice, while combined-cycle plants can exceed 60%.