Audit a heat engine’s efficiency

Start with the incoming energy

At the default 300 kPa, 30 L setting, the gas receives 3 kJ on A → B and 15 kJ on B → C. Total heat input is therefore 18 kJ. It rejects 9 kJ on C → D and 5 kJ on D → A, a total rejected magnitude of 14 kJ. The remaining 4 kJ leaves as net work.

Keep two notations separate. Each leg’s Q is signed, positive into the gas. Qin is the sum of all positive heat entries. Qout is the positive magnitude of all rejected heat. Thus net heat = Qin − Qout, and the cyclic balance is Wnet = Qin − Qout. All three quantities here are energies per cycle.

Define thermal efficiency

Thermal efficiency eta = Wnet/Qin asks what fraction of incoming heat becomes net work. Eta is dimensionless; multiply by 100 for percent. For the default cycle, eta = 4/18 ≈ 0.22222, or 22.222%. The rejected fraction is 14/18 ≈ 77.778%. These fractions add to one because the cycle restores internal energy and the model includes only heat and work.

The OpenStax heat-engine reference describes the distinction between engine work, heat input and rejected heat. The rejected energy has not disappeared. Thermal efficiency counts only net work as the useful output; a different service definition, such as useful heat delivery, needs its own clearly defined accounting.

Three tempting shortcuts fail. Dividing 4 kJ by net heat 4 kJ reports 100%, hiding the 14 kJ rejection. Dividing expansion work 6 kJ by 18 kJ ignores 2 kJ of work needed for return compression. Dividing by only the 15 kJ expansion-leg heat leaves out the 3 kJ of heating at fixed volume. Use the complete ledger.

Compare designs with more than one metric

Capture the default cycle, then set upper pressure 200 kPa and larger volume 40 L. Its work is (200−100)(40−10) = 3000 J. Heat input is 1500 J during fixed-volume heating plus 15000 J during expansion: 16500 J. Efficiency is 3000/16500 ≈ 18.182%. It has a larger volume swing but lower work and lower efficiency than the default. Volume swing alone does not rank engines.

Now compare two cycles with equal work. At 200 kPa/30 L, Wnet = 2000 J and Qin = 11500 J, giving 17.391%. At 300 kPa/20 L, Wnet is also 2000 J, but Qin = 10500 J, giving 19.048%. Equal enclosed areas can therefore hide different heat requirements. Record the values and justify the comparison from individual legs.

Do not confuse work with power

The diagram has no cycle duration. If a separately specified machine delivered 4 kJ net work per cycle and ran at two cycles per second, its average net power would be 8 kJ/s = 8 kW. The number is conditional on that operating rate; the quasistatic model does not predict or validate it. An ideal reversible limit does not establish finite power, and auxiliary devices or friction would require additional accounting.

The rectangle is not a Carnot cycle. Gas temperature varies during all nonzero legs, and thermal contacts must be controlled along those changes. It is not a specification of just one hot and one cold reservoir. Do not label its efficiency as 1 minus the ratio of minimum to maximum gas temperature. Temperature-based limits require a separate second-law argument and explicitly defined thermal contacts.

Handle zero-output limits honestly

At upper pressure 100 kPa and larger volume 30 L, the path retraces a line. Work is zero but heat input is 5000 J, so thermal efficiency is 0%. At 100 kPa/10 L all transfers vanish: the expression is 0/0, which is undefined, not 0% or 100%. The lab displays this difference.

Audit a short performance claim

An engine receives 24 kJ per cycle and delivers 6 kJ net work. A report says it rejects 12 kJ and has efficiency 50%. Correct the report, then calculate average power if an independently specified rate is three cycles per second.

Check the efficiency transfer

Rejected heat must be 24−6 = 18 kJ per cycle to close the cyclic energy account. Efficiency is 6/24 = 25%. At the stated rate, average power is 6 × 3 = 18 kW. The energy ledger alone supplies no rate, and the efficiency claim does not prove the machine is feasible under unspecified reservoir conditions.