Checking an Engine Claim Against the Carnot Limit
A workshop technician tests a small heat engine that is claimed to run between a hot reservoir at 500 K and a cold reservoir at 300 K. In a 10-minute trial the engine takes in 8.0×10^4 J from the hot reservoir, rejects 4.2×10^4 J to the cold reservoir, and delivers 3.8×10^4 J of shaft work. The technician also measures a cycle time of 0.50 s and claims the engine is 'more efficient than a Carnot engine because the pistons are polished'. A student is asked to check the first-law energy balance, compute the actual efficiency, compare it with the Carnot limit, and decide whether the claim can be valid. The reservoirs are large and remain at fixed temperatures; the working substance returns to its initial state each cycle. The workshop wants to know whether to invest in the design or report a measurement error.
1. Check the first-law energy balance and calculate the thermal efficiency of the engine from the trial data.
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First law for one cycle: Q_h = W + Q_c, assuming no other losses. Measured: Q_h = 8.0×10^4 J, Q_c = 4.2×10^4 J, W = 3.8×10^4 J. Then W + Q_c = 3.8×10^4 + 4.2×10^4 = 8.0×10^4 J, so the trial data balance internally.
The thermal efficiency is η = W/Q_h = 3.8×10^4/8.0×10^4 = 0.475 = 47.5%.
The cycle time of 0.50 s gives an average power of 3.8×10^4/0.50 = 7.6×10^4 W = 76 kW, but cycle time does not affect the efficiency comparison.
2. Calculate the Carnot efficiency for the two reservoirs and decide whether the claimed performance is physically possible. Identify the most likely measurement problem if the engine is not fraudulent.
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Carnot efficiency between T_h = 500 K and T_c = 300 K: η_C = 1 − T_c/T_h = 1 − 300/500 = 0.40 = 40%.
The measured 47.5% exceeds the Carnot limit for the stated reservoirs. That is not a small experimental error; it would require the engine to convert heat to work with an entropy decrease, which the second law forbids for a cyclic device. Polished pistons reduce friction but cannot raise the efficiency above the Carnot bound.
The most likely explanations are a measurement error in W (too high) or in Q_c (too low), or that the hot reservoir is actually hotter than 500 K. The design should not be funded on these data.
3. Keeping Q_h = 8.0×10^4 J, what is the maximum work the engine could deliver reversibly, and what rejected heat would accompany it? Use this to judge whether the 3.8×10^4 J work figure or the 4.2×10^4 J heat figure is more suspect.
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Keeping Q_h = 8.0×10^4 J, the reversible engine would reject Q_c,rev = Q_h(T_c/T_h) = 8.0×10^4 × 0.60 = 4.8×10^4 J and deliver W_rev = Q_h − Q_c,rev = 3.2×10^4 J.
The measured work exceeds this by 0.6×10^4 J, while the measured rejected heat is below the reversible value by the same 0.6×10^4 J. The discrepancy is therefore 0.6×10^4 J, not a rounding effect. If the work measurement is correct, the engine is impossible; if the heat rejection is correct, the work must be about 3.2×10^4 J for a reversible engine, and less for a real one. The data must be re-measured before any efficiency claim is accepted.