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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. |
