Entropy and Efficiency Ceilings

The best coal plant ever built converts about 45% of its fuel energy to electricity. The best combined-cycle gas plant reaches about 63%. Neither is close to 100%, and neither ever will be. The ceiling is not engineering; it is entropy.

The Carnot efficiency (η = 1 - T_cold/T_hot, temperatures in kelvin) is the maximum work any heat engine can extract from a temperature difference. It is a direct consequence of entropy: converting all heat to work would decrease total entropy, which the second law forbids. The Carnot limit depends only on the hot source and cold sink temperatures, not on fuel type, materials, or engineering skill.

Coal's 600°C steam caps it near 65% and gas turbines' 1,400°C combustion caps them near 82%, the same Carnot ceilings established earlier. Real plants fall 15-25 percentage points below Carnot due to friction, heat leakage, and turbine imperfections. The gap between real and Carnot is where engineering improvement happens. The gap between Carnot and 100% is permanent.

Worked Example

A geothermal plant uses 150°C water (423 K) as its heat source with ambient cooling at 25°C (298 K).

• Calculate Carnot limit. η = 1 - 298/423 = 29.6%.
• The actual plant achieves 12% efficiency.

Why is geothermal efficiency so low despite "free" fuel?

The source temperature is low (150°C vs. 1,400°C for gas). The Carnot ceiling is only 29.6%, and real-world losses pull the actual efficiency well below that. Geothermal can still be economical because the fuel cost is zero, but the physics limits how much electricity each unit of heat produces.

The Carnot limit is why higher combustion temperatures are the universal goal in thermal power engineering, and why solar PV and wind (which are not heat engines) face fundamentally different efficiency constraints.

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