15.9

Carnot’s Principle and the Carnot Engine

What is it that allows a heat engine to operate with maximum efficiency? The French engineer Sadi Carnot (1796–1832) proposed that a heat engine has maximum efficiency when the processes within the engine are reversible. A reversible process is one in which both the system and its environment can be returned to exactly the states they were in before the process occurred.
In a reversible process, both the system and its environment can be returned to their initial states. Therefore, a process that involves an energy-dissipating mechanism, such as friction, cannot be reversible because the energy wasted due to friction would alter the system or the environment or both. There are also reasons other than friction why a process may not be reversible. For instance, the spontaneous flow of heat from a hot substance to a cold substance is irreversible, even though friction is not present. For heat to flow in the reverse direction, work must be done, as we will see in Section 15.10. The agent doing such work must be located in the environment of the hot and cold substances, and, therefore, the environment must change while the heat is moved back from cold to hot. Since the system and the environment cannot both be returned to their initial states, the process of spontaneous heat flow is irreversible. In fact, all spontaneous processes are irreversible, such as the explosion of an unstable chemical or the bursting of a bubble. When the word “reversible” is used in connection with engines, it does not just mean a gear that allows the engine to operate a device in reverse. All cars have a reverse gear, for instance, but no automobile engine is thermodynamically reversible, since friction exists no matter which way the car moves.
Today, the idea that the efficiency of a heat engine is a maximum when the engine operates reversibly is referred to as Carnot’s principle.

| Carnot’s Principle: An Alternative Statement of the Second Law of Thermodynamics
| No irreversible engine operating between two reservoirs at constant temperatures can have a greater efficiency than a reversible engine operating between the same temperatures. Furthermore, all reversible engines operating between the same temperatures have the same efficiency.

Carnot’s principle is quite remarkable, for no mention is made of the working substance of the engine. It does not matter whether the working substance is a gas, a liquid, or a solid. As long as the process is reversible, the efficiency of the engine is a maximum. Furthermore, Carnot’s principle does not state, or even imply, that a reversible engine has an efficiency of 100%.
It can be shown that if Carnot’s principle were not valid, it would be possible for heat to flow spontaneously from a cold substance to a hot substance, in violation of the second law of thermodynamics. In effect, then, Carnot’s principle is another way of expressing the second law.
No real engine operates reversibly. Nonetheless, the idea of a reversible engine provides a useful standard for judging the performance of real engines. Figure 15-13 shows a reversible engine, called a Carnot engine, that is particularly useful as an idealized model. An important feature of a Carnot engine is that all input heat () originates from a hot reservoir at a single temperature TH and all rejected heat () goes into a cold reservoir at a single temperature TC. This important feature is emphasized in Problem 56, which focuses on a pressure-versus-volume plot for a Carnot engine that utilizes an ideal gas as its working substance.

Figure 15-13 A Carnot engine is a reversible engine in which all input heat |QH| originates from a hot reservoir at a single temperature TH, and all rejected heat |QC| goes into a cold reservoir at a single temperature TC. The work done by the engine is |W|.

Carnot’s principle implies that the efficiency of a reversible engine is independent of the working substance of the engine, and therefore can depend only on the temperatures of the hot and cold reservoirs. Since efficiency is according to Equation 15.13, the ratio |QC|/|QH| can depend only on the reservoir temperatures. This observation led Lord Kelvin to propose a thermodynamic temperature scale. He proposed that the thermodynamic temperatures of the cold and hot reservoirs be defined such that their ratio is equal to |QC|/|QH|. Thus, the thermodynamic temperature scale is related to the heats absorbed and rejected by a Carnot engine, and is independent of the working substance. If a reference temperature is properly chosen, it can be shown that the thermodynamic temperature scale is identical to the Kelvin scale introduced in Section 12.2 and used in the ideal gas law. As a result, the ratio of the magnitude of the rejected heat |QC| to the magnitude of the input heat |QH| is

| (15.14)

where the temperatures TC and TH must be expressed in kelvins.
The efficiency eCarnot of a Carnot engine can be written in a particularly useful way by substituting Equation 15.14 into Equation 15.13 for the efficiency, :

| (15.15)

This relation gives the maximum possible efficiency for a heat engine operating between two Kelvin temperatures TC and TH, and the next example illustrates its application.

The physics of extracting work from a warm ocean.

Problem solving insight When determining the efficiency of a Carnot engine, be sure the temperatures TC and TH of the cold and hot reservoirs are expressed in kelvins; degrees Celsius or degrees Fahrenheit will not do.

| Example 7 | | | A Tropical Ocean as a Heat Engine

Water near the surface of a tropical ocean has a temperature of 298.2 K (25.0 °C), whereas water 700 m beneath the surface has a temperature of 280.2 K (7.0 °C). It has been proposed that the warm water be used as the hot reservoir and the cool water as the cold reservoir of a heat engine. Find the maximum possible efficiency for such an engine.
Reasoning The maximum possible efficiency is the efficiency that a Carnot engine would have (Equation 15.15) operating between temperatures of TH = 298.2 K and TC = 280.2 K.
Solution Using TH = 298.2 K and TC = 280.2 K in Equation 15.15, we find that

In Example 7 the maximum possible efficiency is only 6.0%. The small efficiency arises because the Kelvin temperatures of the hot and cold reservoirs are so close. A greater efficiency is possible only when there is a greater difference between the reservoir temperatures. However, there are limits on how large the efficiency of a heat engine can be, as Conceptual Example 8 discusses.

| Conceptual Example 8 | | | Natural Limits on the Efficiency of a Heat Engine

Consider a hypothetical engine that receives 1000 J of heat as input from a hot reservoir and delivers 1000 J of work, rejecting no heat to a cold reservoir whose temperature is above 0 K. Decide whether this engine violates the first or the second law of thermodynamics, or both.
Reasoning and Solution The first law of thermodynamics is an expression of energy conservation. From the point of view of energy conservation, nothing is wrong with an engine that converts 1000 J of heat into 1000 J of work. Energy has been neither created nor destroyed; it has only been transformed from one form (heat) to another (work). This engine does, however, violate the second law of thermodynamics. Since all of the input heat is converted into work, the efficiency of the engine is 1, or 100%. But Equation 15.15, which is based on the second law, indicates that the maximum possible efficiency is 1 − TC/TH, where TC and TH are the temperatures of the cold and hot reservoirs, respectively. Since we know that TC is above 0 K, it is clear that the ratio TC/TH is greater than zero, so the maximum possible efficiency is less than 1, or less than 100%. It is important to understand that the first and second laws of thermodynamics address different aspects of nature. It is the second law, not the first law, that limits the efficiencies of heat engines to values less than 100%.

Example 8 has emphasized that even a perfect heat engine has an efficiency that is less than 1.0 or 100%. In this regard, we note that the maximum possible efficiency, as given by Equation 15.15, approaches 1.0 when TC approaches absolute zero (0 K). However, experiments have shown that it is not possible to cool a substance to absolute zero (see Section 15.12), so nature does not allow a 100%-efficient heat engine to exist. As a result, there will always be heat rejected to a cold reservoir whenever a heat engine is used to do work, even if friction and other irreversible processes are eliminated completely. This rejected heat is a form of thermal pollution. The physics of thermal pollution. The second law of thermodynamics requires that at least some thermal pollution be generated whenever heat engines are used to perform work. This kind of thermal pollution can be reduced only if society reduces its dependence on heat engines to do work.

Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.