Heat Engines and Efficiency
Simple Explanation
A heat engine takes in heat from a hot reservoir, converts some of it into useful work, and expels the rest as waste heat to a cold reservoir. Its efficiency β the fraction of heat energy actually converted into work β is Ξ·=W/Qβ=1βQc/Qβ.
Why Do We Need It?
No real heat engine can be 100% efficient β some waste heat is always expelled β and understanding why, and how to calculate exactly how efficient an engine is, is central to designing every engine, power plant, and refrigerator.
See It
A box labelled Engine, with an arrow flowing in from above labelled Qh (heat from the hot reservoir), an arrow flowing out below labelled Qc (waste heat to the cold reservoir), and an arrow flowing out to the side labelled W (useful work output)
Formula
Heat Engine Efficiency
Ξ· = W/Qβ = 1 β Qc/Qβ
A heat engine's efficiency is the fraction of the heat energy it takes in from a hot reservoir that it actually converts into useful work, rather than expelling as waste heat to a cold reservoir.
- Ξ·
- β efficiency (a fraction from 0 to 1, or a percentage)
- W
- β useful work output, in joules (J)
- Qβ
- β heat absorbed from the hot reservoir, in joules (J)
- Qc
- β heat expelled to the cold reservoir, in joules (J)
When to use it: Whenever the efficiency of a heat engine needs to be found from the heat it takes in and the heat it expels (or the work it produces).
Worked Example
Find a heat engine's efficiency
A heat engine absorbs 5000 J from its hot reservoir and expels 3000 J to its cold reservoir. Find its efficiency.
Why Does This Work?
By energy conservation, all the heat taken in (Qβ) must either become useful work (W) or be expelled as waste heat (Qc) β so W=QββQc always holds, and dividing by Qβ gives the fraction of the input heat that actually became useful work, which is exactly the definition of efficiency.
Real-Life Example
A car engine's efficiency
A typical gasoline car engine converts only about 25-30% of the fuel's chemical energy into useful motion, expelling the rest as waste heat through the exhaust and radiator.
This is exactly the kind of efficiency this formula measures β the fraction of energy input that becomes useful work, with the rest lost as waste heat, exactly as no real heat engine avoids.
Practice
A heat engine absorbs 8000 J and expels 5000 J to its cold reservoir. Find its efficiency as a percentage.
HardCommon mistake
Assuming efficiency can reach 100% with good enough engineering β the second law of thermodynamics guarantees some waste heat (Qc) must always be expelled, so Ξ· is always strictly less than 1 for any real heat engine.
Quick Review
- Ξ· = W/Qβ = 1βQc/Qβ.
- W = QββQc, by energy conservation.
- No real heat engine reaches 100% efficiency.