6.4 The Carnot cycle and Carnot principles

The Carnot cycle is a theoretical model that defines the maximum efficiency any heat engine can achieve when operating between two temperature limits. It's the gold standard for the Second Law of Thermodynamics because it tells you exactly how much useful work you can ever extract from a given temperature difference.

Carnot Cycle Components

Reversible Processes

The entire Carnot cycle is built on reversible processes, which are idealized processes where the system stays in equilibrium with its surroundings at every instant. That means all changes in pressure, volume, and temperature happen infinitesimally slowly, so the process could be reversed at any point with no net change to the surroundings.

The Carnot cycle consists of four reversible processes, executed in order:

1. Isothermal expansion (constant $T_H$)
2. Adiabatic expansion (no heat transfer, temperature drops)
3. Isothermal compression (constant $T_C$)
4. Adiabatic compression (no heat transfer, temperature rises back to $T_H$)

No real process is truly reversible. Reversibility is an idealization, which is exactly why the Carnot cycle sets a theoretical ceiling rather than a practical target.

Thermal Reservoirs

The cycle operates between two thermal reservoirs:

• A high-temperature reservoir (heat source) at temperature $T_H$, which supplies heat during isothermal expansion
• A low-temperature reservoir (heat sink) at temperature $T_C$, which absorbs heat during isothermal compression

Both reservoirs are assumed to have infinite heat capacity, so their temperatures remain constant no matter how much heat flows in or out. This is another idealization that simplifies the analysis.

Isothermal Processes

During isothermal expansion (Process 1), the working fluid absorbs heat $Q_H$ from the hot reservoir while its temperature stays constant at $T_H$. The fluid expands and does work on the surroundings. Because internal energy doesn't change for an ideal gas at constant temperature, all the absorbed heat converts directly into work output.

During isothermal compression (Process 3), the opposite happens. The working fluid rejects heat $Q_C$ to the cold reservoir at constant temperature $T_C$. Work is done on the fluid to compress it, and that energy leaves as heat to the cold reservoir.

Adiabatic Processes

During adiabatic expansion (Process 2), the fluid continues to expand but with no heat exchange ($Q = 0$). The fluid does work at the expense of its own internal energy, so its temperature drops from $T_H$ to $T_C$.

During adiabatic compression (Process 4), the fluid is compressed with no heat exchange. Work is done on the fluid, raising its temperature from $T_C$ back to $T_H$, which returns the system to its initial state and completes the cycle.

Carnot Cycle Efficiency

Theoretical Maximum Efficiency

The Carnot cycle produces the highest possible efficiency for any heat engine operating between two given temperatures. Two features make this result powerful:

• The efficiency depends only on $T_H$ and $T_C$. It doesn't matter what working fluid you use or how the engine is designed.
• No real or hypothetical engine operating between those same reservoirs can exceed the Carnot efficiency. This is a direct consequence of the Second Law.

Limitations of Real Heat Engines

Real engines always fall short of Carnot efficiency because they contain irreversibilities: friction between moving parts, finite-rate heat transfer across temperature differences, mixing, unrestrained expansion, and other losses. These effects generate entropy and destroy the potential for useful work.

The Carnot cycle serves as a benchmark. If a real engine operating between 300 K and 600 K achieves 35% efficiency, you know the Carnot limit is 50%, so there's room for improvement. If it's already at 48%, you're close to the theoretical wall and further gains will be extremely difficult.

Applying Carnot Principles

Heat Engine Efficiency

The Carnot principles (sometimes called Carnot's theorem) make two key claims:

1. No heat engine operating between two reservoirs can be more efficient than a reversible engine operating between the same reservoirs.
2. All reversible engines operating between the same two reservoirs have the same efficiency.

The Carnot efficiency for a heat engine is:

$\eta_C = 1 - \frac{T_C}{T_H}$

where $T_C$ and $T_H$ are the absolute temperatures (in Kelvin or Rankine) of the cold and hot reservoirs.

Notice what this formula tells you: to increase efficiency, you either raise $T_H$ or lower $T_C$. This is why power plants use superheated steam (high $T_H$) and why condensers cool the exhaust steam as much as possible (low $T_C$).

Refrigerators and Heat Pumps

The Carnot principles apply to reversed cycles too. For these devices, performance is measured by the coefficient of performance (COP) rather than thermal efficiency.

• Carnot COP for a refrigerator: $COP_R = \frac{T_C}{T_H - T_C}$
• Carnot COP for a heat pump: $COP_{HP} = \frac{T_H}{T_H - T_C}$

A useful relationship to remember: $COP_{HP} = COP_R + 1$. This holds for any Carnot (reversible) cycle. Also notice that as $T_H$ and $T_C$ get closer together, both COPs increase. That's why heat pumps work most efficiently when the indoor-outdoor temperature difference is small.

Carnot Engine Efficiency Calculation

Efficiency Formula

The efficiency of a Carnot engine can be defined two equivalent ways:

• From an energy balance: $\eta_C = \frac{W_{net}}{Q_H}$, where $W_{net}$ is the net work output and $Q_H$ is the heat absorbed from the hot reservoir.
• From reservoir temperatures: $\eta_C = 1 - \frac{T_C}{T_H}$

Since $W_{net} = Q_H - Q_C$ for a cycle (First Law), you can also write $\eta_C = 1 - \frac{Q_C}{Q_H}$. For a Carnot cycle specifically, $\frac{Q_C}{Q_H} = \frac{T_C}{T_H}$, which is what connects the two expressions.

Temperature Conversion

You must use absolute temperatures in these formulas. Using Celsius or Fahrenheit will give you wrong answers.

• Kelvin: $T(K) = T(°C) + 273.15$
• Rankine: $T(°R) = T(°F) + 459.67$

Example Calculation

Suppose a Carnot engine operates between a hot reservoir at 600 K and a cold reservoir at 300 K.

1. Identify the temperatures: $T_H = 600 \text{ K}$, $T_C = 300 \text{ K}$

2. Apply the formula: $\eta_C = 1 - \frac{T_C}{T_H} = 1 - \frac{300}{600} = 0.50$

3. Convert to percentage: $\eta_C = 50\%$

This means at most 50% of the heat absorbed from the hot reservoir can be converted to work. The other 50% must be rejected to the cold reservoir. No clever engineering can change that for these temperature limits.

Practical Considerations

Real engines operating between these same reservoirs might achieve 30-40% efficiency. The gap between actual and Carnot efficiency tells you how much room exists for design improvements like reducing friction, improving insulation, or minimizing pressure drops.

However, there are diminishing returns. As you eliminate the easy sources of irreversibility, each additional percentage point of efficiency becomes harder and more expensive to achieve. The Carnot efficiency is an asymptotic limit that real engines can approach but never reach.