Carnot Efficiency

In 1824 Sadi Carnot, a young French engineer, conceived of an idealized heat engine. This ideal engine had an efficiency given by

e = 1 - T_c/T_h = (T_h - T_c)/T_h

where

e = the so-called Carnot efficiency (no units)

T_c = the temperature of the waste heat reservoir (in degrees Kelvin, °K[4])

T_h = the temperature of the heat source (in °K)

Unhappily, T_c cannot be made zero (and e therefore made equal to 1, which is 100% efficiency). Physicists have shown absolute zero to be unattainable, although they have approached to within a hundredth of a degree in the laboratory.

Waste heat, since it must be rejected to the surrounding atmosphere, outer space, or water (rivers, the ocean, etc.), must be rejected at T_c greater than 300°K. The reason for this is that these physical reservoirs have average temperatures around 300°K (about 80°F) themselves. The fact that T_c must be 300°K or more is a basic limitation on the Carnot efficiency. The loss in efficiency with increased T_c explains why a jet plane has a harder job taking off on a hot day.

One way to improve the Carnot efficiency, which is the maximum efficiency for any heat engine, is to raise T_h as high as possible without melting the engine. For a coal-fired electrical power plant, T_h = 600°K and T_c = 300°K, so that

e = 1 - 300/600 = 0.5 = 50%

The actual efficiency is somewhat less than this ideal value because some power is diverted to pumps and other equipment and to unavoidable heat losses. Later on, we shall see that magnetohydrodynamic (MHD) generators hold prospects for increasing T_h by hundreds of degrees.

Everything that has been said about the Second Law of Thermodynamics (You can’t even break even) applies to heat engines, where we begin with thermal energy. Suppose instead that we start with kinetic or chemical energy and convert it into electricity without turning it into heat first. We can then escape the Carnot efficiency strait jacket. Chemical batteries perform this trick. So do fuel cells, solar cells, and many other direct conversion devices we shall discuss. Thus, we circumvent the Carnot efficiency limitation by using processes to which it does not apply.

Problem 3

Some space power plants contemplate using the space cabin heat (T_h = 300°K) to drive a heat engine which rejects its waste heat to the liquid-hydrogen rocket fuel stored at T_c = 20°K. What would be the Carnot efficiency of this engine?