Problem — An engine that is too efficient

Problem

An engine claims to have an efficiency of $0{,}60$ while operating between $T_c = 400$ K and $T_f = 300$ K. Is it credible? Why?

Solution

Carnot’s limit. No heat engine, however perfect, can exceed the efficiency of the Carnot cycle working between the same two temperatures. This is a direct consequence of the second law of thermodynamics:

$\eta \le \eta_{\text{C}} = 1 - \frac{T_f}{T_c}$

Computing the ceiling. With $T_f = 300$ K and $T_c = 400$ K:

$\eta_{\text{C}} = 1 - \frac{300}{400} = 1 - 0{,}75 = 0{,}25$

Comparison. The claimed efficiency is $0{,}60$, i.e. more than double the maximum allowed $0{,}25$. Since $0{,}60 > 0{,}25 = \eta_{\text{C}}$, the engine would violate the second law.

$\ev{\text{Not credible: } \eta = 0{,}60 > \eta_{\text{C}} = 0{,}25}$

No technological trick can get around this wall: raising the efficiency beyond $0{,}25$ would require lowering $T_f$ or raising $T_c$, not improving the engine.

Links

Topics: Macchine termiche Concepts: Rendimento · Ciclo di Carnot · Secondo principio della termodinamica Skills: Analisi di casi limite e fantafisica Objects: Macchina termica