Question

To increase the thermal efficiency of a reversible power cycle operating between reservoirs at \(T_{\mathrm{H}}\) and \(T_{C}\), would you increase \(T_{\mathrm{H}}\) while keeping \(T_{\mathrm{C}}\) constant, or decrease \(T_{\mathrm{C}}\) while keeping \(T_{\mathrm{H}}\) constant? Are there any natural limits on the increase in thermal efficiency that might be achieved by such means?

Short answer

Increase \(T_H\) or decrease \(T_C\) to improve thermal efficiency, but natural material limits and absolute zero are constraints.

Step-by-step solution

Understanding Thermal Efficiency

The thermal efficiency (\texteta) of a reversible power cycle can be expressed using the Carnot efficiency equation: \[ \texteta = 1 - \frac{T_{C}}{T_{H}} \]Where \(T_H\) is the temperature of the hot reservoir, and \(T_C\) is the temperature of the cold reservoir.

Effect of Increasing \(T_H\)

Let's analyze the effect of increasing \(T_H\) while keeping \(T_C\) constant. From the equation \[ \texteta = 1 - \frac{T_{C}}{T_{H}} \], one can see that as \(T_H\) increases, the fraction \( \frac{T_C}{T_H} \) becomes smaller, thus making \(1 - \frac{T_C}{T_H} \) larger. This means that the thermal efficiency increases as \(T_H\) increases.

Effect of Decreasing \(T_C\)

Now, let's see the effect of decreasing \(T_C\) while keeping \(T_H\) constant. If \(T_C\) decreases, the fraction \( \frac{T_C}{T_H} \) also becomes smaller. Consequently, \(1 - \frac{T_C}{T_H} \) increases, meaning that the thermal efficiency improves as \(T_C\) decreases.

Determining Natural Limits

In practical terms, both increasing \(T_H\) and decreasing \(T_C\) to improve thermal efficiency have natural limits. For \(T_H\), materials and components can withstand only so much heat before they degrade. For \(T_C\), absolute zero (0 Kelvin) is a theoretical limit where \(T_C\) cannot be lower. Thus, while efficiency can be improved by adjusting these temperatures, physical and practical limitations will always exist.

Conclusion

To maximize thermal efficiency, it is effective to either increase \(T_H\) while keeping \(T_C\) constant or decrease \(T_C\) while keeping \(T_H\) constant. However, practical limitations related to material and environmental constraints must be taken into account.

Key concepts

Carnot Cycle

The Carnot cycle is a theoretical model that defines the maximum possible efficiency a heat engine can achieve. Named after French physicist Sadi Carnot, this model is essential for understanding the principles behind thermodynamics. It operates between two temperature reservoirs: a hot one (with temperature \(T_H\)) and a cold one (with temperature \(T_C\)). The cycle consists of four stages: two isothermal (constant temperature) processes and two adiabatic (no heat exchange) processes.
During the isothermal expansion, the system absorbs heat from the hot reservoir, while in isothermal compression, it releases heat to the cold reservoir. The adiabatic processes ensure there is no heat exchange while the temperature changes, moving the system between the two isothermal processes.
The Carnot cycle is highly significant because it establishes an upper limit to the efficiency of all real heat engines. This idealized efficiency is given by the equation: \[ \texteta = 1 - \frac{T_C}{T_H} \] which shows that the efficiency depends solely on the temperatures of the hot and cold reservoirs.

Reversible Power Cycle

A reversible power cycle is one where all processes can be reversed without any net change to the system or surroundings. In other words, reversing a reversible cycle returns the system to its initial state along the same path. This concept eliminates energy loss due to irreversibilities like friction, unrestrained expansion, and heat transfer through a finite temperature difference.
Because reversible power cycles are idealized, they serve as a benchmark for real-world engines. The Carnot cycle, for example, is a type of reversible cycle, and its efficiency formula provides a theoretical maximum limit for engine efficiency. Practically, no real engine can achieve this ideal state, but engineers strive to minimize losses to approach this ideal as closely as possible.
Reversible power cycles are crucial because they provide engineers with insights into how to optimize real-life systems. By identifying where and why energy losses occur, improvements can be made to make practical cycles as efficient as possible.

Temperature Reservoirs

Temperature reservoirs are bodies with a relatively large thermal capacity that can absorb or provide finite amounts of heat without undergoing a significant temperature change. In the context of a heat engine, there are typically two kinds: a hot reservoir and a cold reservoir. Understanding these reservoirs is essential for analyzing and designing heat engines and other thermodynamic cycles.
The hot reservoir is at a higher temperature \(T_H\), and it supplies heat to the engine. This energy input drives the cycle's work-producing steps. The cold reservoir, at a lower temperature \(T_C\), absorbs the waste heat discharged from the engine. This necessary step ensures the cycle can continue.
The efficiency of heat engines is closely tied to the temperature difference between these reservoirs. A larger temperature differential between \(T_H\) and \(T_C\) generally leads to higher efficiency. However, there are natural limits to this improvement.
Increasing \(T_H\) is limited by the materials used in engine construction, as excessive temperatures can cause breakdowns. Conversely, reducing \(T_C\) is constrained by the lower limit of absolute zero (0 Kelvin), which is theoretically unattainable in practice.