Problem — Carnot compared with other cycles

Problem

Why is the Carnot efficiency higher than that of any other reversible cycle (for example Otto or Diesel) operating between the same extreme temperatures?

Solution

The special feature of the Carnot cycle. The Carnot cycle exchanges heat with the outside only at two temperatures: it absorbs all of $Q_c$ at the maximum temperature $T_c$ (hot isotherm) and releases all of $Q_f$ at the minimum temperature $T_f$ (cold isotherm). The other two transformations are adiabatic, with no heat exchange. Its efficiency is therefore the absolute maximum:

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

The other cycles. In an Otto or Diesel cycle, heat is absorbed along transformations (isochoric, isobaric) in which the temperature varies continuously. Part of $Q_c$ is thus absorbed at intermediate temperatures, lower than $T_c$, and part of $Q_f$ is released at temperatures higher than $T_f$.

Why this lowers efficiency. Absorbing heat at an average temperature $\bar{T}_c < T_c$ and releasing it at an average $\bar{T}_f > T_f$ is equivalent to working across a narrower effective thermal gap. Since efficiency grows with the separation of the exchange temperatures, every exchange at an intermediate temperature is less efficient. It follows that:

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

Only by exchanging heat at the two extreme temperatures — as Carnot does — is the maximum possible work extracted. Any other cycle “muddies” the exchanges with intermediate temperatures and pays for it in efficiency.

Collegamenti

Argomenti: Heat engines Concetti: Carnot cycle · Efficiency · Second law of thermodynamics Competenze: Symbolic setup Oggetti: Heat engine