5.1 Heat engines and the Carnot cycle

Heat Engines and the Carnot Cycle

Heat engines convert thermal energy into mechanical work by exploiting temperature differences between a hot source and a cold sink. They're central to the Second Law of Thermodynamics because they reveal a fundamental truth: you can never convert all the heat you put in into useful work. The Carnot cycle formalizes this limit, giving you the absolute maximum efficiency any heat engine can achieve between two given temperatures.

Components of Heat Engines

A heat engine works by moving energy from a high-temperature reservoir to a low-temperature reservoir, extracting useful work along the way. Not all the heat gets converted to work; some must always be rejected to the cold reservoir.

Every heat engine has four basic components:

• Working substance: The material (usually a gas or steam) that undergoes a cyclic process of expansion and compression
• High-temperature reservoir (heat source): Supplies thermal energy to the working substance (e.g., a furnace or combustion chamber)
• Low-temperature reservoir (heat sink): Absorbs the rejected heat (e.g., the atmosphere or cooling water)
• Mechanical mechanism: Converts the expansion and compression of the working substance into useful work (e.g., a piston-cylinder assembly or turbine)

The energy balance for one complete cycle is:

$Q_H = W_{net} + Q_C$

where $Q_H$ is the heat absorbed from the hot reservoir, $W_{net}$ is the net work output, and $Q_C$ is the heat rejected to the cold reservoir.

The Carnot Cycle

The Carnot cycle is a theoretical, fully reversible cycle that represents the best possible heat engine operating between two temperatures. No real engine can match it, but it sets the benchmark.

It consists of four processes, carried out on an ideal working substance:

1. Isothermal expansion at $T_H$: The working substance absorbs heat $Q_H$ from the hot reservoir while expanding slowly at constant temperature $T_H$. Because the temperature stays constant, all the absorbed heat goes into doing work on the surroundings.

2. Adiabatic expansion: The working substance is insulated (no heat exchange) and continues to expand. With no heat input, the expansion cools the substance down from $T_H$ to $T_C$.

3. Isothermal compression at $T_C$: The working substance is compressed at constant temperature $T_C$ while in contact with the cold reservoir. Heat $Q_C$ is rejected to the cold reservoir during this step.

4. Adiabatic compression: The substance is insulated again and compressed further. This raises its temperature back from $T_C$ to $T_H$, returning it to its initial state and completing the cycle.

The Carnot cycle is significant beyond just engine design. It helped establish the concept of entropy and gave a precise statement of the Second Law: there are fundamental, unavoidable limits on how much heat can be converted to work.

Efficiency of a Carnot Engine

The thermal efficiency of any heat engine is the fraction of input heat that becomes useful work:

$\eta = \frac{W_{net}}{Q_H} = 1 - \frac{Q_C}{Q_H}$

For a Carnot engine specifically, the ratio of heats equals the ratio of absolute temperatures. This gives the remarkably simple result:

$\eta_{Carnot} = 1 - \frac{T_C}{T_H}$

• $T_H$ = absolute temperature of the hot reservoir (in Kelvin)
• $T_C$ = absolute temperature of the cold reservoir (in Kelvin)

You must use Kelvin for this formula. Using Celsius or Fahrenheit will give a wrong answer.

Example: A steam power plant operates between a boiler at 500°C (773 K) and a condenser at 30°C (303 K). The maximum possible efficiency is:

$\eta_{Carnot} = 1 - \frac{303}{773} = 1 - 0.392 = 0.608$

That's about 60.8%. No real engine operating between these temperatures can exceed this value.

To maximize Carnot efficiency, you need to increase the gap between $T_H$ and $T_C$:

• Raise $T_H$: Use higher-temperature heat sources or better insulation to maintain high temperatures
• Lower $T_C$: Use colder cooling fluids or more effective heat exchangers

Notice that $\eta_{Carnot} = 1$ (100% efficiency) only if $T_C = 0 \text{ K}$, which is physically unattainable. This reinforces that perfect conversion of heat to work is impossible.

Why Real Engines Fall Short

The Carnot cycle assumes a perfectly reversible process with zero irreversibilities. Real engines always have them, which is why actual efficiencies are significantly lower than the Carnot limit.

Common sources of irreversibility include:

• Friction in moving parts (pistons, bearings, turbine blades), which converts useful work into waste heat
• Heat losses through engine walls, exhaust gases, and imperfect insulation
• Incomplete combustion of fuel, leaving chemical energy unconverted
• Non-ideal working fluids that deviate from ideal gas behavior, especially at high pressures or low temperatures
• Finite-rate processes: Real engines operate quickly, so heat transfer and expansion/compression never happen as slowly and reversibly as the Carnot cycle requires

The Second Law guarantees that no heat engine, no matter how cleverly designed, can exceed the Carnot efficiency for the same $T_H$ and $T_C$. This isn't an engineering limitation you can overcome with better technology; it's a law of nature.