Question

A hot thermal reservoir is separated from a cold thermal reservoir by a cylindrical rod insulated on its lateral surface. Energy transfer by conduction between the two reservoirs takes place through the rod, which remains at steady state. Using the Kelvin-Planck statement of the second law, demonstrate that such a process is irreversible.

Short answer

The process is irreversible because reversing it would violate the Kelvin-Planck statement, requiring forbidden work input.

Step-by-step solution

- Understand the System

The system consists of a hot thermal reservoir and a cold thermal reservoir connected by a cylindrical rod. The rod is insulated on its lateral surface, ensuring that energy transfer occurs only longitudinally through the rod.

- Steady-State Energy Transfer

At steady state, the energy transfer by conduction through the rod remains constant over time. This implies that the energy entering the rod from the hot reservoir is equal to the energy leaving the rod to the cold reservoir.

- Kelvin-Planck Statement

The Kelvin-Planck statement of the second law of thermodynamics states that it is impossible for any device operating in a cycle to extract heat from a single heat reservoir and produce an equivalent amount of work without any other effects. This essentially means that heat cannot flow spontaneously from a cold body to a hot body.

- Identify the Reversibility

If the process were reversible, it would imply that the system could return to its initial state without any net changes occurring in the surroundings. However, in this scenario, the energy transfer is unidirectional—from the hot reservoir to the cold reservoir.

- Implication of Irreversibility

Due to the unidirectional energy transfer, reversing the process would require an external work input to transfer energy from the cold reservoir back to the hot reservoir, which is forbidden by the Kelvin-Planck statement. This demonstrates that the process cannot be reversed without violating the second law of thermodynamics.

Key concepts

Kelvin-Planck Statement

The Kelvin-Planck statement is a fundamental principle of the Second Law of Thermodynamics. It declares that it is impossible to construct an engine that operates in a cycle and produces no effect other than the extraction of heat from a single reservoir and the performance of an equivalent amount of work. This indicates that thermal energy naturally flows from a hot body to a cold body and not the other way around.

This principle is crucial in understanding why certain processes are irreversible. In the context of the exercise, it implies that energy transfer in a thermodynamic cycle cannot be completely converted into work without some waste heat being transferred to a cooler reservoir. The key takeaway is that heat engines cannot achieve 100% efficiency due to this inherent limitation.

Remember, the Kelvin-Planck statement highlights why processes naturally move towards equilibrium and cannot revert without external work. This concept is essential for understanding the irreversibility in the given scenario where heat flows from the hot to the cold reservoir through a rod.

Second Law of Thermodynamics

The Second Law of Thermodynamics asserts that entropy, a measure of disorder, always increases in an isolated system. This law underlies why some processes are naturally irreversible. It dictates that systems tend to progress towards a state of maximum entropy or disorder.

In practical terms, this means that energy transfer processes, like the one described in the exercise, cannot be reversed without doing work. The heat from the hot reservoir transfers to the cold reservoir, increasing the overall entropy of the system. Trying to revert this energy transfer without external energy input would violate the Second Law of Thermodynamics.

In summary, the Second Law provides a direction for thermodynamic processes and explains why certain energy transfers, like conduction through a rod, are naturally irreversible due to the increase in entropy.

Energy Transfer by Conduction

Conduction is one of the three modes of heat transfer. It occurs when heat moves through a material without the material itself moving. In solids, conduction happens through the vibration and collision of particles.

In the given problem, a cylindrical rod conducts heat from the hot thermal reservoir to the cold thermal reservoir. The rod is insulated on its sides to ensure that heat transfer occurs only longitudinally. The steady-state condition implies that the amount of heat entering one end of the rod is equal to the amount exiting the other end. This results in a constant rate of energy transfer over time.

Understanding conduction is essential for analyzing the rate and direction of heat flow. Factors influencing this can include material properties like thermal conductivity, cross-sectional area, temperature gradient, and the rod’s length.

Steady-State Process

A steady-state process is one where the properties within the system do not change over time. In the context of energy transfer by conduction, this means that the amount of heat entering and leaving the system remains constant.

For the cylindrical rod in the exercise, achieving a steady state implies a constant temperature gradient along the length of the rod. This ensures that the energy flow from the hot reservoir to the cold reservoir is unchanging over time.

At steady state, the system's equilibrium allows for predictable analysis and design. Engineers and scientists utilize these conditions to calculate energy transfer rates and optimize thermal management systems. Understanding steady-state processes is crucial for accurately predicting long-term behavior in thermodynamic systems.