Thermal Efficiency of a Heat Engine

First-law efficiency is the definition η = W_net/Q_in, sometimes called thermal efficiency or energy efficiency. It is purely an energy accounting ratio: how much of the heat you put in came out as work? Second-law efficiency — sometimes called exergy efficiency or effectiveness — compares the actual efficiency to the Carnot limit: η_II = η / η_Carnot. First-law efficiency tells you how well the engine converts fuel to work. Second-law efficiency tells you how close the engine is to the thermodynamic ideal for its operating temperatures, and therefore how much room for improvement remains. A 40% efficient engine at 800 K / 300 K has a second-law efficiency of 64%, meaning there is 36% of the ideal still theoretically available.

Three main reasons. First, diesels run at higher compression ratios — typically 14–23:1 versus 8–12:1 for gasoline — which raises the effective expansion ratio and extracts more work from the hot gases. Second, diesels are unthrottled: gasoline engines throttle the intake air to control power, which costs pumping work, while diesels control power by varying the fuel injected into a full air charge. Third, diesels run lean overall, which reduces the specific heat ratio penalty. The result is a typical diesel efficiency of 35–42% versus 25–30% for a comparable gasoline engine. The trade-offs are higher NOx and particulate emissions, higher weight, and higher initial cost.

Heat rate is the reciprocal of efficiency, expressed as fuel energy per unit of useful output: HR = 3600/η in kJ/kWh, or HR = 3412/η in BTU/kWh. It is the quantity that matters for fuel cost, because it directly tells you how much fuel you burn per kilowatt-hour generated. A plant with η = 40% has HR = 9000 kJ/kWh, while a plant with η = 45% has HR = 8000 kJ/kWh — an 11% reduction in fuel consumption for the same output. Utilities track heat rate obsessively because fuel is by far the largest operating cost of a thermal power plant, typically 60–80% of the total. A one-point improvement in efficiency on a 500 MW plant saves millions of dollars per year.

By using two engines in series. A gas turbine burns fuel at very high temperature (T_H around 1600 K) and produces work, but its exhaust is still hot — around 850 K. Instead of rejecting that heat to the environment, the plant routes the exhaust through a heat-recovery steam generator, which raises steam that drives a steam turbine. The steam turbine then converts additional work from heat that would otherwise have been wasted. The combined efficiency is roughly the sum of the two cycles minus overlap: 40% from the gas turbine plus 20% from the steam bottoming cycle gives about 58–62% overall. This is why combined-cycle plants dominate new natural gas power generation — they are the most efficient heat engines ever built at scale.

For a given engine cycle and given reservoir temperatures, the theoretical efficiency does not depend on the fuel — it depends only on T_H and T_C. In practice, however, different fuels have different combustion temperatures, flame speeds, and energy densities, which affect the achievable T_H and the losses. Natural gas burns hotter and cleaner than coal, allowing higher turbine inlet temperatures in a gas turbine. Hydrogen burns hotter still, with no carbon emissions, but its storage and handling are more difficult. The fuel's energy content (higher heating value versus lower heating value) also affects the quoted efficiency, since HHV efficiency is always lower than LHV efficiency for the same engine — a fact that matters when comparing engines from different manufacturers who may use different conventions.

No, for a heat engine operating in a cycle between two thermal reservoirs, efficiency cannot exceed the Carnot limit. If you calculate an efficiency higher than Carnot, either the reservoir temperatures you entered are wrong, or the device is not a heat engine — for example, a fuel cell converts chemical energy directly to electricity without going through a heat cycle, so it is not bound by the Carnot limit. Similarly, a heat pump or refrigerator has a coefficient of performance that can be greater than 1, because it uses work to move heat rather than converting heat into work. The Carnot limit applies specifically to devices that take heat in and produce work out.

Any consistent energy or power units work, because η is a ratio. If you are analysing a single cycle, use kJ for both W_net and Q_in — that is the convention this calculator uses. If you are analysing a continuous engine running at steady state, use kW for both the net power output and the heat input rate. Other valid pairs include BTU and BTU/h, kcal and kcal/h, or MJ and MJ/s (which is the same as kW). The only rule is that both quantities must be in the same unit. Mixing units — for example kJ for work and kW for heat — is the most common source of confusion in hand calculations.