The Carnot efficiency increases with the temperature difference between the heat source and sink. This relationship highlights the importance of maximizing the source temperature (Ti) while minimizing the sink temperature (To) for better performance.

The efficiency formula μC = (Ti - To) / Ti shows that even a small increase in source temperature can lead to a significant efficiency gain, especially when the source temperature is relatively low.

Practical Temperature Limits

In real-world applications, both the upper and lower temperatures are constrained:

Lower Temperature (Sink) Limits:

Typically limited by available environmental heat sinks (atmosphere, ocean, rivers)
Usually ranges between 10°C (283 K) and 20°C (293 K) for ambient conditions
Cooling water from rivers or lakes might be slightly cooler

Upper Temperature (Source) Limits:

Constrained by material properties and metallurgical strength
Higher temperatures require more expensive, specialized materials
Engineering compromises between efficiency and material costs

Real-World Efficiency Considerations

While the Carnot cycle represents the theoretical maximum efficiency for heat engines operating between two temperature limits, actual engines achieve lower efficiencies due to:

Irreversibilities (friction, unrestrained expansion)
Heat losses to surroundings
Incomplete heat transfer
Non-ideal gas behavior

Understanding the Carnot efficiency provides a crucial benchmark for evaluating and improving real heat engine designs.

Theoretical Basis: Why Carnot Efficiency is the Maximum

The Carnot cycle represents a theoretical ideal because it is internally reversible. All processes are carried out infinitely slowly, allowing the system to remain in thermodynamic equilibrium. This eliminates all sources of irreversibility like friction, turbulence, or finite temperature differences during heat transfer.

This reversibility is directly tied to the Second Law of Thermodynamics. The Second Law states that no real heat engine can be more efficient than a reversible one operating between the same two temperature reservoirs. Therefore, the Carnot efficiency, μC, sets the absolute upper limit for the thermal efficiency of any heat engine operating between a high-temperature source at Ti and a low-temperature sink at To.

Any real engine, due to unavoidable irreversibilities, will have an efficiency less than μC. The difference between the Carnot efficiency and a real engine's efficiency is a measure of how far the engine deviates from the ideal reversible process.

Example Calculation

To demonstrate the Carnot efficiency formula in practice, consider a geothermal power plant where the hot reservoir temperature ( $T_i$ ) is 150°C (423.15 K) and the cold reservoir (sink) temperature ( $T_o$ ) is the ambient ground temperature of 15°C (288.15 K).

The theoretical maximum (Carnot) efficiency for this system would be calculated as: $\mu_C = \frac{T_i - T_o}{T_i} = \frac{423.15\,K - 288.15\,K}{423.15\,K} \approx 0.319 \text{ or } 31.9\%$

This result underscores a key principle: even with a substantial temperature difference of 135°C, the fundamental limit imposed by the Second Law means over two-thirds of the input heat energy must be rejected to the sink.

Practical Sink Temperature Context

The theoretical efficiency is highly sensitive to the absolute temperature of the heat sink. The extracted text specifies that if the sink is a natural body like an ocean or river, the lowest available temperature is typically in the 10–20°C range (283–293 K). This practical constraint is critical for real-world power cycle design and explains why plants in colder climates or using deep ocean water can achieve marginally higher theoretical efficiencies than those in warmer environments, all else being equal.

Carnot Cycle Description

The Carnot cycle, proposed by Sadi Carnot, is an ideal reversible thermodynamic cycle that achieves the maximum possible efficiency for a heat engine operating between two temperature reservoirs.

The cycle consists of four distinct processes:

Isothermal Expansion: The engine absorbs heat $Q_H$ from a hot reservoir at temperature $T_i$ (in Kelvin) while expanding at constant temperature.
Adiabatic Expansion: The engine continues to expand without heat exchange, causing the temperature to drop to $T_o$ .
Isothermal Compression: The engine rejects heat $Q_C$ to a cold reservoir at temperature $T_o$ while being compressed at constant temperature.
Adiabatic Compression: The engine is compressed without heat exchange, raising the temperature back to $T_i$ to complete the cycle.

The efficiency of this cycle, known as Carnot efficiency, is given by:

$\mu_C = \frac{T_i - T_o}{T_i}$

where $\mu_C$ is the efficiency, $T_i$ is the absolute temperature of the hot reservoir, and $T_o$ is the absolute temperature of the cold reservoir. This efficiency represents the upper limit for any heat engine operating between $T_i$ and $T_o$ , as dictated by the second law of thermodynamics.

References

Original Source

Source: engineeringtoolbox.com