21.7 - Maximum engine efficiency and reservoir temperatures
Heat engines are everywhere in the modern world. They convert heat into useful work.
These engines always consist of a “hot” source reservoir from which thermal energy is
taken í for example, the coal burning firebox in an old-fashioned train engine í and a
“cold” exhaust reservoir where thermal energy is expelled. In the case of a train’s steam
engine, that cold reservoir is the atmosphere.
Higher efficiency í turning more of the heat into work í is what heat engine designers
strive for. This results in less fuel consumption, which saves money, extends natural
resources and reduces pollution.
We can state an inequality, shown on the right, expressing the maximum efficiency of
any engine as determined by the temperatures of its hot and cold reservoirs. In this
section, we derive this inequality. To do so, we consider the engine, including its
reservoirs, as an isolated system, and assume the reservoirs are large enough that the
heat flow does not appreciably change their temperatures.
During an engine cycle, the entropy of the hot reservoir decreases as heat flows out of it
and the entropy of the cold reservoir increases as heat flows into it. The entropy of the
gas does not change after the completion of a cycle because the engine returns to its
initial state.
The second law of thermodynamics dictates that the entropy of an isolated system must
increase or stay the same during any process, including a complete engine cycle. This
means the decrease in entropy of the hot reservoir must be matched or exceeded by
the increase in entropy of the cold reservoir. The net entropy increases.
Variables
What is the strategy?
- Calculate the change in entropy for the hot and cold reservoirs. The net change in entropy, which cannot be negative, is the sum of
these two quantities. Write this as an inequality, and then rearrange the inequality so the ratio of reservoir temperatures is on one side. - Apply the definition of engine efficiency to the inequality.
Physics principles and equations
The definition of entropy
The entropy statement of the second law
ǻS 0
An equation for engine efficiency
Maximum efficiency of heat
engine
e = efficiency
Tc = temperature of cold reservoir (K)
Th = temperature of hot reservoir (K)
engine efficiency e
net change in entropy ǻS
hot reservoir cold reservoir
temperature Th Tcheat transferred Qh Qc
change in entropy ǻSh ǻSc