Heat Engines: Kelvin and Clausius Statements

"Computer Science is no more about computers than astronomy is about telescopes."

E. W. Dijkstra

As I have indicated, much of the development of thermodynamics took place during the nineteenth century as it became important to develop more and more efficient "heat engines" (initially steam engines, later internal combustion and diesel engines). Where we use the term heat engine to describe any device which, working in cycles between hot and cold reservoirs (whose temperatures remain constant), converts heat into work. A schematic representation of a heat engine is shown at right. Note that in each cycle the net result of the engine is to convert an amount of heat Q_H - Q_C to an equal amount of work, W.

A refrigerator can be thought of as a heat engine operating in reverse (reverse all the arrows in the diagram). By using an amount of work W, heat Q_C is extracted from the cold reservoir.

KELVIN STATEMENT:

It is impossible to extract an amount of heat Q_H from a hot reservoir and use it all to do work W . Some amount of heat Q_C must be exhausted to a cold reservoir. This precludes a perfect heat engine.

CLAUSIUS STATEMENT:

It is not possible for heat to flow from a colder body to a warmer body without any work having been done to accomplish this flow. Energy will not flow spontaneously from a low temperature object to a higher temperature object. This precludes a perfect refrigerator.

By combining two heat engines (refrigerators) it can be shown that these two, apparently different, statements are equivalent.

The efficiency of a heat engine is defined as follows,

Since Q_H > Q_C and Q_C is not zero, e < 1.

For refrigerators we define the coefficient of performance

Note that the EER (energy efficiency rating) on air conditioners and refrigerators is the c.o.p in mixed units - (Btu/hr)/Watts. Since 1 W = 3.413 Btu/hr, then EER = 3.413 x c.o.p.

Schematically a heat pump is identical to a refrigerator, but in this case maximizing Q_H is paramount. For a heat pump the c.o.p is defined as Q_H/W.

Early morning Physics class filled with slightly dazed freshmen. Eager beaver postdoc teaching the class asks "The wavelength of the Sodium yellow line. What is it? You there!" Fortunately, he has his eagle eye on the guy next to me, who mutters and replies "A hundred and one?"
"Hah!" says the postdoc "A hundred and one what?"
"Um, a hundred and one, point two?"

Dr. C. L. Davis
Physics Department
University of Louisville
email: c.l.davis@louisville.edu