Stirling Engine Efficiency, Entropy, and an Over-Unity Proposal
Students are characterizing a small Stirling engine in an energy-conversion lab. The hot cylinder sits in a beaker of boiling water held at 373 K; the cold cylinder sits in an insulated ice-water bath at 273 K. An electric heater in the hot bath is measured to deliver 300 J of energy per engine cycle (12.0 V at 2.50 A for 10.0 s). A prony brake on the flywheel measures a torque of 0.72 N·m through 10.0 revolutions during the same cycle, which the students convert to about 45 J of mechanical work. The engine runs smoothly and repeats cycle after cycle. A teammate proposes a redesigned engine that, between the same two reservoirs, would absorb the same 300 J per cycle and deliver 90 J of work; she argues that this is only a modest improvement and should be achievable with better seals and a lighter flywheel. A third student asks what will happen if the ice bath warms to 313 K during a long run.
1. Calculate the engine's efficiency, compare it with the Carnot limit, and use an entropy calculation to decide whether the device is behaving lawfully.
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Thermal efficiency is η = W/Q_h = 45/300 = 0.150, or 15.0%. The Carnot limit between 373 K and 273 K is 1 − T_c/T_h = 1 − 273/373 = 0.268, so at most about 80 J of work could be extracted from 300 J. Since 15.0% < 26.8%, no law is violated; the engine is simply irreversible.
The heat rejected is Q_c = Q_h − W = 255 J. Check the second law with entropy: the hot reservoir loses 300/373 = 0.804 J/K, and the cold reservoir gains 255/273 = 0.933 J/K, so the universe gains 0.129 J/K per cycle. A positive total is exactly what a real engine must produce, because friction, finite-rate heat transfer and gas viscosity all generate entropy. Note also why the ice bath is a good reservoir: while ice remains, melting pins the temperature at 273 K regardless of how much heat arrives, so T_c is fixed and the Carnot limit is well defined.
2. Evaluate the teammate's proposed 90 J-per-cycle design. What measurements would you check before declaring it impossible?
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As stated, 90/300 = 30.0%, which exceeds the 26.8% Carnot limit for 373 K and 273 K, so it is impossible: better seals and a lighter flywheel reduce irreversibility but cannot exceed the reversible limit. Entropy confirms it. Delivering 90 J leaves Q_c = 210 J, so ΔS_universe = −300/373 + 210/273 = −0.804 + 0.769 = −0.035 J/K, a decrease. That cannot happen.
Before rejecting the claim, however, check the two inputs on which the verdict rests. If the cold side is actually a salt-ice mixture near 253 K, Carnot becomes 1 − 253/373 = 32.2%, and 30% (93% of Carnot) is allowed, though very hard to achieve. Also, 300 J is the heater's electrical input, not necessarily the heat the gas absorbs: some warms the beaker, insulation and seals, so the true Q_h could be lower and the apparent efficiency higher. Room heat leaking into the cold side would reduce the measured Q_c. Thermometer and independent calorimetry come before any second-law accusation.
3. Predict the effect of the ice bath warming to 313 K, with reasoning, and state what the students should record to document it.
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At T_c = 313 K the Carnot limit falls to 1 − 313/373 = 16.1%. At 273 K the engine achieved only 56% of its Carnot limit, so expect its work to fall roughly in proportion: 45 J × (16.1/26.8) ≈ 27 J per cycle, possibly less if irreversibility grows.
The physical reason is entropy accounting. Each joule rejected to a warmer cold reservoir removes less entropy, so the entropy the engine generates internally becomes a larger fraction of what the cold side can absorb, and less of the absorbed heat can be converted to work. Equivalently, the temperature difference driving the cycle has shrunk from 100 K to 60 K, so the pressure swings that turn the flywheel are smaller. Practically, the engine will slow and the brake torque will drop. The students should log T_c continuously with a stirred bath, plot work per cycle against T_c, and flag any point at 313 K where efficiency exceeds 16.1%: that would indicate a measurement error, not a better engine.