Problem — Carnot with three reservoirs

Problem

What changes if a Carnot engine can exchange heat with three reservoirs at temperatures $T_1 > T_2 > T_3$ instead of two? Can it have an efficiency greater than the two-reservoir Carnot efficiency between $T_1$ and $T_3$?

Solution

What sets the limit. The maximum efficiency of a heat engine depends only on the highest temperature at which it absorbs heat and the lowest at which it rejects it:

$\eta_\text{max} = 1 - \frac{T_\text{cold}}{T_\text{hot}}$

The intermediate reservoir doesn’t help. Adding a reservoir at intermediate $T_2$ does not extend this range: the hottest source remains $T_1$ and the coldest remains $T_3$. Any exchange with $T_2$ either has no effect on the limit, or (if the heat passes through $T_2$ in two stages) is equivalent to putting two Carnot engines in series. Two Carnot engines in series between $T_1$-$T_2$ and $T_2$-$T_3$ give exactly the same overall limit:

$\eta_\text{serie} = 1 - \frac{T_2}{T_1}\cdot\frac{T_3}{T_2} \cdot(\dots) \;\Rightarrow\; 1 - \frac{T_3}{T_1}$

that is, the heat rejected at the bottom is always at $T_3$ and the heat absorbed at the top is always at $T_1$.

Conclusion.

$\ev{\eta_\text{max} = 1 - \frac{T_3}{T_1}\ \text{unchanged}}$

The intermediate reservoir does not raise the Carnot limit. Only widening the range — raising $T_1$ or lowering $T_3$ — can beat it. Any redistribution of the reservoirs within $[T_3, T_1]$ leaves the theoretical ceiling where it was.

Collegamenti

Argomenti: Macchine termiche Concetti: Ciclo di Carnot · Rendimento · Secondo principio della termodinamica Competenze: Analisi di casi limite e fantafisica