Application Note: Engines, Refrigerators, and Real Systems

Heat Engine

A heat engine converts heat to mechanical work in a cyclic process.

Working:

Absorbs heat $Q_{1}$ from hot reservoir at $T_{1}$ (K)
Converts part of it to useful work W
Rejects remaining heat $Q_{2}$ to cold reservoir at $T_{2}$ (K)

$\eta = \frac{W}{Q_1} = \frac{Q_1 - Q_2}{Q_1} = 1 - \frac{T_2}{T_1}$

Examples: Steam turbine, internal combustion engine, jet engine.

Practical efficiencies: Car engine ≈ 25%, steam turbine ≈ 35–40%, gas turbine ≈ 45%.

Carnot Engine

The ideal (theoretical) engine operating between $T_{1}$ and $T_{2}$ . No real engine can exceed its efficiency.

Carnot cycle: two isothermal + two adiabatic processes
All processes are reversible
η_Carnot = 1 − $T_{2}$ / $T_{1}$ is the maximum attainable efficiency

Refrigerator / Heat Pump

Operates in reverse — work is supplied to transfer heat from cold to hot.

$\text{COP}_{refrigerator} = \frac{Q_2}{W} = \frac{T_2}{T_1 - T_2}$

$\text{COP}_{heat pump} = \frac{Q_1}{W} = \frac{T_1}{T_1 - T_2} = \text{COP}_{ref} + 1$

Real-world example: Air conditioner with $T_{1}$ = 310 K (outside), $T_{2}$ = 295 K (room): $\text{COP} = \frac{295}{310 - 295} = \frac{295}{15} \approx 19.7$

For every 1 J of electrical work, 19.7 J of heat is removed from the room — which is why air conditioners are more efficient than electric heaters.

Comparison Table

Parameter	Heat Engine	Refrigerator
Energy input	$Q_{1}$ from hot reservoir	Work W
Useful output	Work W	$Q_{2}$ from cold reservoir
Figure of merit	η = W/ $Q_{1}$ ≤ 1	COP = $Q_{2}$ /W > 1 possible
Ideal limit	η → 1 only if $T_{2}$ → 0 K	COP → ∞ if $T_{1}$ = $T_{2}$
Second Law constraint	η < 1 (cannot extract all $Q_{1}$ )	W > 0 required (heat can't flow cold → hot for free)