Problem — Ranking three Carnot engines

Problem

Three Carnot engines operate between:

• A: $T_c = 400$ K, $T_f = 300$ K;
• B: $T_c = 600$ K, $T_f = 300$ K;
• C: $T_c = 600$ K, $T_f = 200$ K.

Rank the efficiencies from smallest to largest.

Solution

Formula. For a Carnot engine the efficiency depends only on the two temperatures:

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

Computation for each.

$\eta_A = 1 - \frac{300}{400} = 0{,}25$

$\eta_B = 1 - \frac{300}{600} = 0{,}50$

$\eta_C = 1 - \frac{200}{600} = \frac{2}{3} \approx 0{,}667$

Ranking. Comparing the three values $0{,}25 < 0{,}50 < 0{,}667$:

$\ev{\eta_A < \eta_B < \eta_C}$

Physical note: going from A to B raises $T_c$ (efficiency up); going from B to C lowers $T_f$ (efficiency up again). Both moves widen the ratio $T_c/T_f$.

Links

Topics: Macchine termiche Concepts: Rendimento · Ciclo di Carnot Skills: Ragionamento di ranking Objects: Macchina termica