Thermodynamic Cycles — Carnot, Otto, Diesel & Rankine

Carnot, Otto, Diesel, and Rankine are all thermodynamic cycles, but they answer different questions: Carnot is a theoretical efficiency ceiling, Otto and Diesel are idealized internal-combustion engine models, and Rankine is the standard steam power-plant cycle. Sorting them by that one distinction clears up most of the confusion.

A thermodynamic cycle is a machine that repeats 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. Heat goes in, part of it becomes work, and the rest is rejected so the cycle can close.

The Four Cycles Side By Side

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

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

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. 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 Energy Balance Every Cycle Shares

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, the net change in internal energy over a full cycle is zero:

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

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.

When To Reach For Each Cycle

• Use Carnot when the question is about the ceiling set by the second law for a reversible engine between two temperatures.
• Use Otto or Diesel when modeling how reciprocating internal-combustion engines convert fuel energy into shaft work.
• Use Rankine when the system is a large-scale steam power plant with a phase-changing working fluid.

Applying It: 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

Substitute the values:

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.

Points That Trip Students Up

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. 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.

Even if you never design an engine, these cycles teach three durable ideas: energy balance, efficiency limits, and the importance of stating assumptions before you trust a formula.