⚙️ Carnot Heat Engine — Efficiency & PV Diagram

The temperature-entropy (T-S) diagram displays the Carnot cycle as a perfect rectangle: heat absorbed Q_H equals T_H × ΔS (the top horizontal line), heat rejected Q_C equals T_C × ΔS (the bottom line), and the enclosed area equals net work W_net = Q_H − Q_C. This makes the efficiency immediately visible as the ratio of rectangle height difference to total height, which is not as intuitive from the curved PV loop.

Real engines suffer from friction, finite-rate heat transfer (which requires a temperature gradient, introducing irreversibility), turbulence, heat leakage through walls, and imperfect combustion. The Carnot cycle assumes every process is perfectly quasi-static and reversible — infinitely slow — which is impossible in practice. The friction loss slider in this simulator lets you see how even small losses dramatically cut actual efficiency below the Carnot limit.

A higher compression ratio increases the volume swept per cycle, so both Q_H and W_net scale roughly with ln(compression ratio). However, it also raises the maximum pressure, demanding stronger (heavier, more expensive) cylinder walls. Petrol engines typically use compression ratios of 9–12:1, while diesel engines run 14–23:1 to compensate for the lower reactivity of diesel fuel.

If T_C could equal 0 K (absolute zero), the Carnot efficiency would reach 100%. However, the third law of thermodynamics states that absolute zero is unattainable in a finite number of steps, so perfect efficiency is also unattainable. This is not merely a technological limitation — it is a fundamental constraint of nature encoded in the second law of thermodynamics.

The number of moles n scales the work done in each isothermal stroke: W = nRT·ln(V₂/V₁). So doubling n doubles both Q_H and W_net while leaving the efficiency unchanged, because efficiency depends only on the temperature ratio. In engineering terms, adding more working fluid increases the power output of an engine without altering its thermal efficiency.

Yes. The Carnot theorem is universal — it applies to any reversible heat engine, whether the working fluid is an ideal gas, steam, a magnetocaloric material, or even photons. Real steam-turbine cycles (Rankine cycles) and gas-turbine cycles (Brayton cycles) are approximations to the Carnot ideal. The simulator models an ideal gas, but the efficiency formula η = 1 − T_C/T_H holds for any reversible cycle between the same two temperatures.

Entropy S is a state function measuring the dispersal of energy among microstates. During isothermal expansion the gas absorbs Q_H at T_H, gaining entropy ΔS = Q_H/T_H. During isothermal compression it rejects Q_C at T_C, losing ΔS = Q_C/T_C. Because the cycle is reversible, these entropy changes are equal, so ΔS_net = 0. The T-S diagram rectangle makes this balance visually explicit: the area added on the top equals the area removed on the bottom.

Every thermal power station — coal, gas, nuclear — is limited by the Carnot bound. Increasing the steam temperature T_H from 500 K to 700 K raises the maximum efficiency from about 40% to 57%, which is why modern ultra-supercritical coal plants operate at very high steam pressures and temperatures. Ocean thermal energy conversion (OTEC) uses the small temperature difference between warm surface water (~300 K) and cold deep water (~275 K), giving a Carnot efficiency of only about 8%, illustrating how critical a large ΔT is.