Problem — Ranking four refrigerators

Problem

Four ideal refrigerators: (A) $T_f = 270$ K, $T_c = 300$ K; (B) $T_f = 250$ K, $T_c = 310$ K; (C) $T_f = 200$ K, $T_c = 300$ K; (D) $T_f = 270$ K, $T_c = 320$ K. Order the COPs from minimum to maximum.

Solution

Formula. For a Carnot refrigerator the coefficient of performance is:

$\beta = \frac{T_f}{T_c - T_f}$

Calculation for each.

$\beta_A = \frac{270}{300 - 270} = \frac{270}{30} = 9{,}0$

$\beta_B = \frac{250}{310 - 250} = \frac{250}{60} \approx 4{,}17$

$\beta_C = \frac{200}{300 - 200} = \frac{200}{100} = 2{,}0$

$\beta_D = \frac{270}{320 - 270} = \frac{270}{50} = 5{,}4$

Ordering. Since $2{,}0 < 4{,}17 < 5{,}4 < 9{,}0$:

$\ev{\beta_C < \beta_B < \beta_D < \beta_A}$

The COP grows when the thermal gap $T_c - T_f$ is small and $T_f$ is high: pumping heat across a narrow thermal difference costs little work (case A). By contrast, C must extract heat from $200$ K and dump it at $300$ K, a large gap, hence a low COP.

Collegamenti

Argomenti: Heat engines Concetti: Refrigerator and heat pump · Carnot cycle Competenze: Ranking reasoning Oggetti: Refrigerator