06 / CHAPTER 08
Heat engines and refrigerators
Compare cycles by coefficient of performance and efficiency.
ANCHOR EQUATION
Carnot efficiency
In symbols: η = 1 − Tc/Th
No heat engine operating between two temperatures can exceed the Carnot limit.
A heat engine converts a temperature gap imperfectly
An engine absorbs heat from a hot reservoir, rejects some to a cold reservoir, and delivers the difference as work. Efficiency is limited because entropy must not decrease overall.
Use both first and second laws
For one cycle, ΔU = 0, so W = Q_h − Q_c. Reversible entropy balance gives Q_h/T_h = Q_c/T_c, so Q_c = Q_h T_c/T_h. Therefore η = W/Q_h = 1 − Q_c/Q_h = 1 − T_c/T_h. This is the Carnot upper bound for any engine between those temperatures.
For T_h = 600 K and T_c = 300 K, η_max = 1 − 0.5 = 0.50. An engine absorbing 2 kJ can deliver at most 1 kJ in the reversible limit.
Real losses lower efficiency
Friction, finite temperature differences, fluid resistance, and incomplete combustion produce entropy, so real efficiency is below Carnot. A higher hot temperature or lower cold temperature increases the limit but may create material constraints.
Work it: A refrigerator moves 800 J from a 270 K interior while consuming 250 J of work. Find heat rejected and coefficient of performance for cooling, Q_c/W.