b/A heat engine. Hot air
from the candles rises through
the fan blades and makes the
angels spin.c/Sadi Carnot (1796-1832)5.3.2 Heat engines
Heat may be more useful in some forms than in others, i.e., there
are different grades of heat energy. In figure a/1, the difference in
temperature can be used to extract mechanical work with a fan
blade. This principle is used in power plants, where steam is heated
by burning oil or by nuclear reactions, and then allowed to expand
through a turbine which has cooler steam on the other side. On a
smaller scale, there is a Christmas toy, b, that consists of a small
propeller spun by the hot air rising from a set of candles, very much
like the setup shown in figure a.
In figure a/2, however, no mechanical work can be extracted
because there is no difference in temperature. Although the air in
a/2 has the same total amount of energy as the air in a/1, the heat
in a/2 is a lower grade of energy, since none of it is accessible for
doing mechanical work.
In general, we define a heat engine as any device that takes heat
from a reservoir of hot matter, extracts some of the heat energy to do
mechanical work, and expels a lesser amount of heat into a reservoir
of cold matter. The efficiency of a heat engine equals the amount of
useful work extracted,W, divided by the amount of energy we had
to pay for in order to heat the hot reservoir. This latter amount of
heat is the same as the amount of heat the engine extracts from the
high-temperature reservoir,QH. By conservation of energy, we have
QH=W+QL, whereQLis the amount of heat expelled into the
low-temperature reservoir, so the efficiency of a heat engine,W/QH,
can be rewritten as
efficiency = 1−QL
QH
. [efficiency of any heat engine]
(As described on p. 315, we takeQL,QH, andWall to be positive.)
It turns out that there is a particular type of heat engine, the
Carnot engine, which, although not 100% efficient, is more efficient
than any other. The grade of heat energy in a system can thus be
unambiguously defined in terms of the amount of heat energy in it
that cannot be extracted even by a Carnot engine.
How can we build the most efficient possible engine? Let’s start
with an unnecessarily inefficient engine like a car engine and see
how it could be improved. The radiator and exhaust expel hot
gases, which is a waste of heat energy. These gases are cooler than
the exploded air-gas mixture inside the cylinder, but hotter than
the air that surrounds the car. We could thus improve the engine’s
efficiency by adding an auxiliary heat engine to it, which would
operate with the first engine’s exhaust as its hot reservoir and the
air as its cold reservoir. In general, any heat engine that expels
heat at an intermediate temperature can be made more efficient by
changing it so that it expels heat only at the temperature of the
cold reservoir.
Section 5.3 Entropy as a macroscopic quantity 321