Problem — Ranking four Carnot engines (2)

Problem

Four ideal engines operate between: (A) $T_c = 400$ K, $T_f = 300$ K; (B) $T_c = 600$ K, $T_f = 300$ K; (C) $T_c = 1000$ K, $T_f = 400$ K; (D) $T_c = 2000$ K, $T_f = 300$ K. Order the efficiencies from smallest to largest.

Solution

Formula. For a Carnot engine:

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

Calculation for each.

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

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

$\eta_C = 1 - \frac{400}{1000} = 0{,}60$

$\eta_D = 1 - \frac{300}{2000} = 0{,}85$

Ordering. Since $0{,}25 < 0{,}50 < 0{,}60 < 0{,}85$:

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

What matters is the ratio $T_f/T_c$: the smaller it is, the higher the efficiency. Engine D, with the hot reservoir at $2000$ K, exploits the widest thermal gap.

Collegamenti

Argomenti: Heat engines Concetti: Efficiency · Carnot cycle Competenze: Ranking reasoning Oggetti: Heat engine