Thermodynamic Cycles — Carnot, Otto, Diesel & Rankine

Thermodynamic cycles describe machines that repeat the same thermodynamic loop. After one loop, the working fluid returns to its starting state, so the fluid itself is reset even though the cycle has produced a net effect such as work output or heat transfer.

For fast understanding, keep this picture in mind: heat goes in, part of it becomes work, and the rest is rejected so the cycle can close. Carnot is the ideal upper limit. Otto and Diesel are idealized engine cycles. Rankine is the standard steam power-plant cycle.

What Makes A Thermodynamic Process A Cycle

A cycle ends where it started in state space. The pressure, volume, temperature, and other state variables of the working fluid come back to their initial values after one loop.

In the usual closed-system treatment with negligible kinetic and potential energy changes, that means the net change in internal energy over a full cycle is zero:

$$\Delta U_{cycle} = 0.$$

Under that condition, the first law reduces to a simple bookkeeping result for one full cycle:

$$W_{net} = Q_{in} - Q_{out}.$$

That equation is useful because it tells you what the cycle is really doing. A cycle is not valuable because the fluid ends up hotter or larger. It is valuable because the loop produces a net work output, or in reverse-cycle cases, uses work to move heat.

Carnot, Otto, Diesel, And Rankine At A Glance

Carnot Cycle

The Carnot cycle is the ideal reversible heat-engine cycle between a hot reservoir at temperature $T_h$ and a cold reservoir at temperature $T_c$. Its main purpose is not to describe a practical engine but to define the maximum possible thermal efficiency for that temperature pair.

If the engine is reversible and both temperatures are absolute temperatures in Kelvin, then

$$\eta_{Carnot} = 1 - \frac{T_c}{T_h}.$$

No real heat engine operating between the same two reservoir temperatures can exceed that efficiency.

Otto Cycle

The Otto cycle is the standard ideal model for spark-ignition engines such as gasoline engines. In the air-standard version, it has two isentropic processes and heat addition at constant volume.

A common ideal result is

$$\eta_{Otto} = 1 - \frac{1}{r^{\gamma - 1}},$$

where $r$ is the compression ratio and $\gamma$ is the heat-capacity ratio. That formula is not a universal engine law. It comes from the ideal-air model with simplifying assumptions.

Diesel Cycle

The Diesel cycle is the standard ideal model for compression-ignition engines. It is similar in spirit to the Otto cycle, but its idealized heat-addition step occurs at constant pressure instead of constant volume.

That difference matters when you compare ideal efficiencies. Under the usual air-standard comparison with the same compression ratio, the ideal Otto cycle is more efficient than the ideal Diesel cycle. Real diesel engines often run at higher compression ratios, so you should not carry that ideal result over to real engines without stating the conditions.

Rankine Cycle

The Rankine cycle is the basic ideal model for steam power plants. Instead of compressing a gas through a full piston-engine style loop, it pumps liquid water, adds heat in a boiler, expands steam through a turbine, and then condenses it back to liquid.

That is why Rankine appears in thermal power stations rather than Otto or Diesel. It is built for phase change and turbine-based power production.

The Main Distinction: Limit Vs Engine Model

Students often group these four cycles together as if they were direct competitors. They are related, but they answer different questions.

Carnot is a benchmark. It gives the ceiling set by the second law for a reversible engine between two temperatures.

Otto and Diesel are idealized internal-combustion engine cycles. They help explain how reciprocating engines convert fuel energy into shaft work.

Rankine is the standard steam-cycle model for large-scale power generation.

If you keep that one distinction clear, most confusion disappears.

Worked Example: Carnot Cycle Efficiency

Suppose an ideal reversible heat engine operates between a hot reservoir at $T_h = 600\ \mathrm{K}$ and a cold reservoir at $T_c = 300\ \mathrm{K}$.

Because this is a Carnot engine and the temperatures are given in Kelvin, you can use

$$\eta_{Carnot} = 1 - \frac{T_c}{T_h}.$$

Substitute the values:

$$\eta_{Carnot} = 1 - \frac{300}{600} = 1 - 0.5 = 0.5.$$

So the maximum possible efficiency under those conditions is $50\%$.

That number does not mean every engine between those temperatures will reach $50\%$. It means no engine can do better if it operates only between those two thermal reservoirs. Real engines fall below this value because real heat transfer is not perfectly reversible and real machines have friction, pressure losses, finite temperature differences, and other irreversibilities.

Common Mistakes With Thermodynamic Cycles

Treating the Carnot cycle as a real engine design

Carnot is primarily a theoretical limit. It is useful because it tells you what cannot be exceeded, not because engineers literally build standard Carnot engines.

Comparing ideal cycle efficiencies under different conditions

An Otto-cycle formula and a Diesel-cycle formula are based on specific ideal assumptions. If the compression ratio, heat-addition model, or working-fluid model changes, the comparison changes too.

Using Celsius in a Carnot formula

The Carnot efficiency formula requires absolute temperature. You should use Kelvin, not Celsius.

Forgetting what returns to its initial state

The working fluid returns to its initial state after one loop. That does not mean heat transfer and work transfer are zero. It means the state is restored while the net effect over the loop remains.

Where Each Thermodynamic Cycle Is Used

Carnot appears in theory, especially when you study the second law and efficiency limits.

Otto appears in introductory discussions of gasoline engines. Diesel appears in compression-ignition engine analysis. Rankine appears in steam turbines, fossil-fuel plants, geothermal systems, and many thermal power-generation settings.

Even if you never design an engine, these cycles matter because they teach three durable ideas: energy balance, efficiency limits, and the importance of assumptions.

Try A Similar Cycle Question

Try your own version of the Carnot example by changing $T_h$ or $T_c$ and checking how the efficiency limit moves. Then compare that benchmark with the ideal Otto or Diesel idea that efficiency also depends on cycle assumptions, not just on one temperature pair.

If you want to go one step further, solve a similar cycle problem with explicit assumptions and sign conventions. A step-by-step tool such as GPAI Solver can help you check the setup, but the core skill is the same: state the conditions before you trust the formula.