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Technical Documents - Documentos Técnicos:Heat engines, Entropy and the second law of Thermodynamics

The first law of thermodynamics, is a statement of conservation of energy, generalized to include internal energy. This law states that a change in internal energy in a system can occur as a result of energy transfer by heat or by work, or by both. The law makes no distinction between the results of heat and the results of work—either heat or work can cause a change in internal energy. However, an important distinction between the two is not evident from the first law. One manifestation of this distinction is that it is impossible to convert internal energy completely to mechanical energy by taking a substance through a thermodynamic cycle such as in a heat engine, a device we study in this chapter.

Although the first law of thermodynamics is very important, it makes no distinction between processes that occur spontaneously and those that do not. However, we find that only certain types of energy-conversion and energy-transfer processes actually take place. The second law of thermodynamics, which we study in this page, establishes which processes do and which do not occur in nature. The following are examples of processes that proceed in only one direction, governed by the second law:

• When two objects at different temperatures are placed in thermal contact with each other, energy always flows by heat from the warmer to the cooler, never from the cooler to the warmer.

• A rubber ball dropped to the ground bounces several times and eventually comes to rest, but a ball lying on the ground never begins bouncing on its own.

• An oscillating pendulum eventually comes to rest because of collisions with air molecules and friction at the point of suspension. The mechanical energy of the system is converted to internal energy in the air, the pendulum, and the suspension; the reverse conversion of energy never occurs.

All these processes are irreversible—that is, they are processes that occur naturally in one direction only. No irreversible process has ever been observed to run backward—if it were to do so, it would violate the second law of thermodynamics.( Although we have never observed a process occurring in the time-reversed sense, it is possible for it to occur. As we shall see later in the page, however, such a process is highly improbable.From this viewpoint, we say that processes occur with a vastly greater probability in one direction than in the opposite direction )

From an engineering standpoint, perhaps the most important implication of the second law is the limited efficiency of heat engines. The second law states that a machine capable of continuously converting internal energy completely to other forms of energy in a cyclic process cannot be constructed.

HEAT ENGINES AND THE SECOND LAW OF THERMODYNAMICS

A heat engine is a device that converts internal energy to mechanical energy. For instance, in a typical process by which a power plant produces electricity, coal or some other fuel is burned, and the high-temperature gases produced are used to convert liquid water to steam. This steam is directed at the blades of a turbine, setting it into rotation. The mechanical energy associated with this rotation is used to drive an electric generator. Another heat engine—the internal combustion engine in an automobile—uses energy from a burning fuel to perform work that results in the motion of the automobile.

A heat engine carries some working substance through a cyclic process during which

(1) the working substance absorbs energy from a high-temperature energy reservoir,

(2) work is done by the engine, and

(3) energy is expelled by the engine to a lower-temperature reservoir. As an example, consider the operation of a steam engine ( e.g. a steam-driven locomotive), in which the working substance is water. The water in a boiler absorbs energy from burning fuel and evaporates to steam, which then does work by expanding against a piston. After the steam cools and condenses, the liquid water produced returns to the boiler and the cycle repeats.

It is useful to represent a heat engine schematically as in the following figure :

Fig. 1 - Schematic representation of a heat engine. The engine absorbs energy Qh from the hot reservoir, expels energy Qc to the cold reservoir, and does work W.

The engine absorbs a quantity of energy Qh from the hot reservoir, does work W, and then gives up a quantity of energy Qc to the cold reservoir. Because the working substance goes through a cycle, its initial and final internal energies are equal, and so ΔEint=0 . Hence, from the first law of thermodynamics  ΔEint= Q - W, and with no change in internal energy, the net work W done by a heat engine is equal to the net energy Qnet flowing through it. As we can see from Fig. 1 Qnet = Qh - Qc , therefore,

W = Qh - Qc     (1)

In this expression and in many others throughout this pages, to be consistent with traditional treatments of heat engines, we take both Qh and Qc to be positive quantities, even though Qc represents energy leaving the engine. In discussions of heat engines, we shall describe energy leaving a system with an explicit minus sign, as in equation (1). Also note that we model the energy input and output for the heat engine as heat, as it often is; however, the energy transfer could occur by another mechanism.

The net work done in a cyclic process is the area enclosed by the curve representing the process on a PV diagram. This is shown for an arbitrary cyclic process in Figure 2

Fig. 2 - PV diagram for an arbitrary cyclic process. The value of the net work done equals the area enclosed by the curve.

The thermal efficiency e of a heat engine is defined as the ratio of the net work done by the engine during one cycle to the energy absorbed at the higher temperature during the cycle:

(2)

We can think of the efficiency as the ratio of what you get (mechanical work) to what you give (energy transfer at the higher temperature). In practice, we find that all heat engines expel only a fraction of the absorbed energy as mechanical work and that consequently the efficiency is less than 100%. For example, a good automobile engine has an efficiency of about 20%, and diesel engines have efficiencies ranging from 35% to 40%.

Equation (2) shows that a heat engine has 100% efficiency (e = 1) only if Qc = 0  - that is, if no energy is expelled to the cold reservoir. In other words, a heat engine with perfect efficiency would have to expel all of the absorbed energy as mechanical work. On the basis of the fact that efficiencies of real engines are well below 100%, the Kelvin–Planck form of the second law of thermodynamics states the following:

 It is impossible to construct a heat engine that, operating in a cycle, produces no effect other than the absorption of energy from a reservoir and the performance of an equal amount of work.

This statement of the second law means that, during the operation of a heat engine, W can never be equal to Qh , or, alternatively, that some energy Qc must be rejected to the environment. Figure 3 is a schematic diagram of the impossible “perfect” heat engine.

Fig. 3 - Schematic diagram of a heat engine that absorbs energy Qh from a hot reservoir and does an equivalent amount of work. It is impossible to construct such a perfect engine.

The first and second laws of thermodynamics can be summarized as follows:

 The first law specifies that we cannot get more energy out of a cyclic process by work than the amount of energy we put in, and the second law states that we cannot break even because we must put more energy in, at the higher temperature, than the net amount of energy we get out by work

Refrigerators and Heat Pumps

Refrigerators and heat pumps are heat engines running in reverse. Here, we introduce them briefly for the purposes of developing an alternate statement of the second law.

Fig.4 - Schematic diagram of a refrigerator, which absorbs energy Qc from a cold reservoir and expels energy Qh to a hot reservoir. Work W is done on the refrigerator. A heat pump, which can be used to heat or cool a building, works the same way.

Fig.5 - Schematic diagram of an impossible refrigerator or heat pump  - that is, one that absorbs energy Qc from a cold reservoir and expels an equivalent amount of energy to a hot reservoir with W = 0.

In a refrigerator or heat pump, the engine absorbs energy Qc from a cold reservoir and expels energy Qh to a hot reservoir (Fig. 4 ). This can be accomplished only if work is done on the engine. From the first law, we know that the energy given up to the hot reservoir must equal the sum of the work done and the energy absorbed from the cold reservoir. Therefore, the refrigerator or heat pump transfers energy from a colder body (for example, the contents of a kitchen refrigerator or the winter air outside a building) to a hotter body (the air in the kitchen or a room in the building). In practice, it is desirable to carry out this process with a minimum of work. If it could be accomplished without doing any work, then the refrigerator or heat pump would be “perfect” (Fig. 5). Again, the existence of such a device would be in violation of the second law of thermodynamics, which in the form of the Clausius statement ( First expressed by Rudolf Clausius (1822–1888))states:

It is impossible to construct a cyclical machine whose sole effect is the continuous transfer of energy from one object to another object at a higher temperature without the input of energy by work.

In simpler terms, energy does not flow spontaneously from a cold object to a hot object. For example, we cool homes in summer using heat pumps called air conditioners. The air conditioner pumps energy from the cool room in the home to the warm air outside. This direction of energy transfer requires an input of energy to the air conditioner, which is supplied by the electric power company.

The Clausius and Kelvin–Planck statements of the second law of thermodynamics appear, at first sight, to be unrelated, but in fact they are equivalent in all respects. Although we do not prove so here, if either statement is false, then so is the other.

REVERSIBLE AND IRREVERSIBLE PROCESSES

In the next section we discuss a theoretical heat engine that is the most efficient possible. To understand its nature, we must first examine the meaning of reversible and irreversible processes. In a reversible process, the system undergoing the process can be returned to its initial conditions along the same path shown on a PV diagram, and every point along this path is an equilibrium state. A process that does not satisfy these requirements is irreversible.

All natural processes are known to be irreversible. From the endless number of examples that could be selected, let us examine the adiabatic free expansion of a gas and show that it cannot be reversible.

The system that we consider is a gas in a thermally insulated container, as shown in Figure 6.

Fig. 6 - Adiabatic free expansion of a gas.

A membrane separates the gas from a vacuum. When the membrane is punctured, the gas expands freely into the vacuum. As a result of the puncture, the system has changed because it occupies a greater volume after the expansion. Because the gas does not exert a force through a distance on the surroundings, it does no work on the surroundings as it expands. In addition, no energy is transferred to or from the gas by heat because the container is insulated from its surroundings. Thus, in this adiabatic process, the system has changed but the surroundings have not.

For this process to be reversible, we need to be able to return the gas to its original volume and temperature without changing the surroundings. Imagine that we try to reverse the process by compressing the gas to its original volume. To do so, we fit the container with a piston and use an engine to force the piston inward.

During this process, the surroundings change because work is being done by an outside agent on the system. In addition, the system changes because the compression increases the temperature of the gas. We can lower the temperature of the gas by allowing it to come into contact with an external energy reservoir. Although this step returns the gas to its original conditions, the surroundings are again affected because energy is being added to the surroundings from the gas. If this energy could somehow be used to drive the engine that we have used to compress the gas, then the net energy transfer to the surroundings would be zero. In this way, the system and its surroundings could be returned to their initial conditions, and we could identify the process as reversible. However, the Kelvin–Planck statement of the second law specifies that the energy removed from the gas to return the temperature to its original value cannot be completely converted to mechanical energy in the form of the work done by the engine in compressing the gas. Thus, we must conclude that the process is irreversible.

We could also argue that the adiabatic free expansion is irreversible by relying on the portion of the definition of a reversible process that refers to equilibrium states. For example, during the expansion, significant variations in pressure occur throughout the gas. Thus, there is no well-defined value of the pressure for the entire system at any time between the initial and final states. In fact, the process cannot even be represented as a path on a PV diagram. The PV diagram for an adiabatic free expansion would show the initial and final conditions as points, but these points would not be connected by a path. Thus, because the intermediate conditions between the initial and final states are not equilibrium states, the process is irreversible.

Although all real processes are always irreversible, some are almost reversible. If a real process occurs very slowly such that the system is always very nearly in an equilibrium state, then the process can be approximated as reversible. For example, let us imagine that we compress a gas very slowly by dropping some grains of sand onto a frictionless piston, as shown in Figure 7.

Fig. 7 - A gas in thermal contact with an energy reservoir is compressed slowly as individual grains of sand drop onto the piston. The compression is isothermal and reversible.

We make the process isothermal by placing the gas in thermal contact with an energy reservoir, and we transfer just enough energy from the gas to the reservoir during the process to keep the temperature constant. The pressure, volume, and temperature of the gas are all well defined during the isothermal compression, so each state during the process is an equilibrium state. Each time we add a grain of sand to the piston, the volume of the gas decreases slightly while the pressure increases slightly. Each grain we add represents a change to a new equilibrium state. We can reverse the process by slowly removing grains from the piston.

A general characteristic of a reversible process is that no dissipative effects (such as turbulence or friction) that convert mechanical energy to internal energy can be present. Such effects can be impossible to eliminate completely. Hence, it is not surprising that real processes in nature are irreversible.

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