Question

Is it possible to develop \((a)\) an actual and \((b)\) a reversible heat-engine cycle that is more efficient than a Carnot cycle operating between the same temperature limits? Explain.

Short answer

Answer: (a) No, an actual heat-engine cycle will always be less efficient than a Carnot cycle due to losses such as heat losses to the environment, friction, and imperfect components. (b) No, it is possible to develop a reversible heat-engine cycle that has the same efficiency as a Carnot cycle, but not one that is more efficient.

Step-by-step solution

Understanding the Carnot Cycle

The Carnot cycle is an idealized cycle of a heat engine operating between two thermal reservoirs at constant temperatures, T_H (hot temperature) and T_C (cold temperature). The Carnot cycle consists of a series of four reversible processes: two isothermal processes and two adiabatic processes. It is considered the most efficient cycle because it assumes no dissipative processes (such as friction) occur during the cycle. The efficiency of a Carnot cycle is given by the formula: Efficiency = 1 - (T_C / T_H)

(a) Actual Heat-Engine Cycle

An actual heat-engine cycle is one that experiences losses from various sources, such as heat losses to the environment, friction, and imperfect components. Due to these losses, an actual heat-engine cycle's efficiency will always be less than the efficiency of a Carnot cycle operating between the same temperature limits. Consequently, it is not possible to develop an actual heat-engine cycle that is more efficient than a Carnot cycle.

(b) Reversible Heat-Engine Cycle

A reversible heat-engine cycle is an idealized cycle that consists of only reversible processes. Since the Carnot cycle is composed of reversible processes, it is considered a reversible heat-engine cycle. One important result from Thermodynamics is that the efficiency of any reversible heat engine operating between the same temperature limits T_H and T_C will be the same as the efficiency of the Carnot engine operating between those same temperature limits. Thus, it is possible to develop a reversible heat-engine cycle that has the same efficiency as a Carnot cycle, but it is not possible to develop one that is more efficient than a Carnot cycle operating between the same temperature limits.

Key concepts

Heat-Engine Cycle

A heat-engine cycle represents a sequence of processes that a working fluid undergoes to convert heat energy from a high-temperature source into mechanical work, and then discharge the residual heat to a low-temperature sink. These cycles are the fundamental principles that govern the operation of engines and refrigerators. The performance of such a cycle is characterized by its efficiency, which is the ratio of work output to the heat input from the high-temperature source.

When we discuss the efficiency of heat-engine cycles, we look at how well they convert heat into work. No actual heat-engine cycle can surpass the Carnot cycle's efficiency when operating between the same temperature limits. This is because in the real world, imperfections such as friction and heat losses are unavoidable. Despite the idealized models offering insights, engineers strive to design cycles that approach, but can never exceed, the Carnot efficiency due to these practical limitations.

Reversible Processes

Reversible processes are hypothetical transformations where the system always remains infinitesimally close to an equilibrium state, and thus the process can proceed in either direction. In other words, these processes are idealizations in which no energy is dissipated as waste heat or friction. Reversible processes play a central role in determining the maximum efficiency of heat-engine cycles, such as the Carnot cycle. Because these processes do not exist in the real world—they are too idealized—the efficiencies of actual cycles will always be less than that of a cycle which consists solely of reversible processes.

It's important to note that we use reversible processes as a benchmark in thermodynamics. Although it's impossible to achieve these perfect conditions in practice, understanding reversible processes helps us identify where irreversibilities (such as friction and turbulence) in real systems contribute to losses and how we might reduce them.

Thermal Reservoirs

Thermal reservoirs are hypothetically infinite energy sources or sinks which their temperature remains constant during heat transfer processes. A high-temperature reservoir provides the heat input, while a low-temperature reservoir absorbs the waste heat. In the Carnot cycle, these thermal reservoirs are essential because they allow the isothermal expansion and compression parts of the cycle to occur.

In a real-world scenario, no reservoir is truly infinite; however, large bodies of water or the atmosphere can often be approximated as reservoirs due to their large thermal capacity. Understanding the concept of thermal reservoirs is crucial because it sets a theoretical limit on the efficiency of heat engines—the efficiency depends on the temperature difference between the high-temperature and low-temperature reservoirs.

Adiabatic Processes

Adiabatic processes are those where no heat is transferred to or from the working fluid during the process. In essence, all of the energy transfer is in the form of work done on or by the system. For an adiabatic process to occur, a system must be insulated so that no heat can enter or leave. In the Carnot cycle, adiabatic processes are responsible for the working fluid's temperature changes without heat transfer, linking the hot and cold isothermal processes.

In practice, achieving perfectly adiabatic conditions is challenging due to thermal losses. However, fast processes, where there is insufficient time for heat to transfer, or those involving very good insulators, can be approximated as adiabatic. When studying the Carnot cycle, adiabatic processes are critical because they allow us to connect the thermal reservoirs without altering their temperatures, which is essential for maintaining the cycle's theoretical efficiency.

Isothermal Processes

Isothermal processes are characterized by the constant temperature of the working fluid during the process, despite the heat transfer that occurs. In a Carnot cycle, the isothermal expansion occurs when the system is in thermal contact with a hot reservoir, absorbing heat while doing work. Conversely, during isothermal compression, the system is in contact with a cold reservoir, releasing heat while work is done on it.

Implementing a true isothermal process requires the heat transfer to occur sufficiently slowly so that the system's temperature remains constant. While challenging to achieve in practice because of external and internal temperature gradients, isothermal processes are essential in thermodynamic analysis because they serve as ideal models by which we can measure the efficiency of real processes. By maintaining a constant temperature throughout the process, isothermal steps in the Carnot cycle help achieve reversible conditions and set the standard for maximum theoretical efficiency.