Brayton Rankine

Brayton and Rankine power cycles: efficiency, net work and processes. Theory, formulas, worked examples and interactive graphs.

Conceptual context

The Brayton and Rankine cycles are the two major power cycles underpinning modern electricity generation. The Brayton cycle is the thermodynamic foundation of gas turbines, jet engines and combined-cycle power plants. The Rankine cycle is the steam-water cycle that drives turbines in coal, nuclear and concentrated solar power plants. Understanding both cycles means understanding how most of the energy consumed by humanity is converted into electricity.

Both cycles answer the same fundamental question: given a temperature difference between a hot source and a cold sink, how much useful work can be extracted per joule of heat supplied? The answer is limited by the second law of thermodynamics, and both cycles represent engineering solutions optimized for different temperature ranges and different working fluids.

The comparison between Brayton and Rankine is not merely academic: every year, investment decisions worth billions of euros are made based precisely on which cycle offers higher efficiency for a given application.

🟢 Essential level

Imagine a pressure cooker connected to a turbine. Heat causes the fluid to expand forcefully, pushing the turbine blades and producing mechanical work. At the end, the fluid must be returned to the start of the cycle at the lowest possible cost. That is the central idea of any power cycle.

The Brayton cycle uses gas (typically air) as the working fluid. Gas enters the compressor, is compressed to high pressure, absorbs heat in a combustion chamber and expands through the turbine producing work. Hot exhaust gases are released to the atmosphere. What makes the Brayton special is that both the compressor and turbine always work with gas.

The Rankine cycle uses steam. Liquid water is pumped to high pressure, heated until it becomes steam in the boiler, the steam expands through the turbine producing work, and is finally condensed back to liquid in the condenser. The key advantage of Rankine is that pumping liquid requires far less work than compressing gas: the density of liquid is much higher.

$\eta_{th}$ measures in both cycles how much of the energy supplied is converted into useful work. That fraction is always below the theoretical limit of the Carnot cycle operating between the same temperatures.

🔵 Formal level

Basic formulas

In the ideal Brayton cycle with constant gas properties, efficiency depends only on the pressure ratio $r_p$ and the adiabatic index $\gamma$ of the gas:

Thermal efficiency of the Brayton cycle

This expression shows that $\eta_{th}$ increases with $r_p$ in a sub-linear fashion. For air with $\gamma$ approximately equal to 1.40, a pressure ratio of ten yields a theoretical efficiency of around 48 %, while $r_p$ of twenty raises it to 57 %. However, increasing $r_p$ indefinitely is not the optimal strategy: although $\eta_{th}$ grows, $W_{net}$ per unit mass may decrease if the compressor outlet temperature approaches the maximum cycle temperature.

The energy balance of the cycle is the direct expression of the first law:

Energy balance of the cycle

All absorbed heat $Q_H$ is distributed between net work $W_{net}$ and rejected heat $Q_C$. The second law guarantees that $Q_C$ can never be zero.

Real student questions

Why does the Rankine cycle use a condenser if that means rejecting heat? Because condensing steam to liquid requires far less volume than keeping it as gas. The pump that returns the liquid to its initial state consumes a tiny fraction of the turbine work. Without a condenser, a closed cycle would not be energetically viable.

Why is the Carnot cycle not used in practice? The Carnot cycle requires isothermal heat absorption and rejection processes, which in practice demand infinite heat exchanger areas or process rates approaching zero. It is not technically or economically feasible.

What is the difference between the ideal and real $\eta_{th}$? Irreversibilities in the compressor, turbine and heat exchangers reduce the real $\eta_{th}$ by 20 % to 40 % compared to the ideal. The key data are the isentropic efficiencies of each component.

How does $\gamma$ affect Brayton cycle efficiency? Higher $\gamma$ means temperature rises more in adiabatic compression. This increases compression work but also the expansion potential in the turbine. For a fixed $r_p$, higher $\gamma$ produces higher $\eta_{th}$.

Final synthesis

The Brayton and Rankine cycles transform high-temperature heat into mechanical work through optimized sequences of thermodynamic processes. $\eta_{th}$ of Brayton depends on $r_p$ and $\gamma$; that of Rankine depends on $T_H$, $T_C$ and improvements such as reheating and regeneration. In both cases, $W_{net}$ is the difference between turbine work and compressor or pump work, and $Q_C$ is the heat inevitably rejected to the environment by mandate of the second law.