Which is the more effective way to increase the thermal efficiency of a Carnot engine: to increase \(T_{H}\), keeping \(T_{L}\) constant; or to decrease \(T_{L}\), keeping \(T_{H}\) constant?

The Carnot cycle is a theoretical thermodynamic cycle that defines the maximum possible efficiency any heat engine can achieve. It consists of four reversible processes: two isothermal (constant temperature) processes and two adiabatic (no heat transfer) processes. The Carnot engine, which operates on this cycle, has the highest efficiency possible under the given temperature limits of its heat reservoirs.
In the Carnot cycle:

• Heat is absorbed from a high-temperature reservoir during the isothermal expansion process.
• Then, the engine undergoes an adiabatic expansion where the gas does work.
• Heat is released to a low-temperature reservoir during the isothermal compression process.
• Finally, the gas undergoes an adiabatic compression back to its original state.

The efficiency of a Carnot engine is determined solely by the temperatures of the hot and cold reservoirs, making it an ideal benchmark for real engines.

Thermal efficiency is a measure of how well a heat engine converts the heat energy it receives into work. The efficiency of a Carnot engine is given by the formula:
\[ \text{η} = 1 - \frac{T_{L}}{T_{H}} \]
where \(T_{H}\) is the temperature of the hot reservoir and \(T_{L}\) is the temperature of the cold reservoir.

This formula shows that thermal efficiency depends on the ratio of the temperatures of the heat reservoirs.
Key facts:

• To increase the thermal efficiency, \(T_{H}\) should be increased or \(T_{L}\) should be decreased.
• Decrease in \(T_{L}\) makes more significant improvements to efficiency than the same increase in \(T_{H}\) because \(T_{L}\) is in the numerator of the fraction.

Overall, achieving higher thermal efficiency involves managing these temperature levels effectively.

Thermodynamic temperature is the absolute measure of temperature in thermodynamics, typically measured in Kelvin (K). It's important because the efficiency calculations for a Carnot engine use absolute temperature values.
In terms of the Carnot engine efficiency formula \( \text{η} = 1 - \frac{T_{L}}{T_{H}} \):

• \(T_{H}\) is the absolute temperature of the hot reservoir, and


• \(T_{L}\) is the absolute temperature of the cold reservoir.

These temperatures directly impact the overall performance of the engine. Absolute temperatures ensure consistency in thermodynamic calculations and are essential for correctly determining the thermal efficiency of heat engines. Make sure that when calculating the values of efficiency or other heat engine parameters, you always convert temperatures to Kelvin to avoid any inaccuracies.