Heat Engine Efficiency

A heat engine is any device that takes heat from a hot source, converts some of it into useful work, and dumps the rest into a cold sink. Every car engine, every power plant, every steam turbine, every gas turbine — they're all heat engines. They just differ in what the hot source is (burning gasoline, nuclear fission, solar concentrator, geothermal heat) and what the cold sink is (the atmosphere, a cooling tower, a river).

Every heat engine is constrained by the same fundamental law: it cannot convert all of its input heat into work. Some fraction must always be wasted to the cold reservoir. This isn't a technological limitation — it's the second law of thermodynamics. You can't cheat it no matter how clever your engine design is.

Defining efficiency

The efficiency of a heat engine is simply:

$\eta = \dfrac{\text{useful output}}{\text{energy input}} = \dfrac{W_{\text{net}}}{Q_{h}}$

where $W_{\text{net}}$ is the net work output and $Q_{h}$ is the heat absorbed from the hot reservoir.

By conservation of energy, whatever heat goes in must either come out as work or as heat to the cold reservoir:

$Q_{h} = W_{\text{net}} + Q_{c}$

So the efficiency can be rewritten:

$\eta = \dfrac{Q_{h} - Q_{c}}{Q_{h}} = 1 - \dfrac{Q_{c}}{Q_{h}}$

This form is useful: it says efficiency is "1 minus the fraction of heat you dump as waste."

The Carnot upper bound

There's a theoretical maximum for efficiency set by the second law of thermodynamics. It was discovered by Sadi Carnot in 1824 and depends only on the two reservoir temperatures (in kelvin):

$\eta_{\text{Carnot}} = 1 - \dfrac{T_{c}}{T_{h}}$

No real engine can exceed this. Any engine claiming to is either broken or fraudulent. It's called a "first-law" vs "second-law" limitation: the first law (energy conservation) would allow 100% efficiency in principle, but the second law says that's only possible if $T_{c} = 0$ (absolute zero), which is unreachable.

Worked example: the default problem

$Q_{h} = 1000\;\text{J}$, $T_{h} = 600\;\text{K}$, $T_{c} = 300\;\text{K}$.

Step 1: Compute the Carnot maximum efficiency.

$\eta_{\text{max}} = 1 - \dfrac{T_{c}}{T_{h}} = 1 - \dfrac{300}{600} = 0.5 = 50\%$

Step 2: Compute the maximum work output.

$W_{\text{max}} = \eta_{\text{max}} \cdot Q_{h} = 0.5 \cdot 1000 = 500\;\text{J}$

Step 3: Compute the minimum waste heat.

$Q_{c,\text{min}} = Q_{h} - W_{\text{max}} = 1000 - 500 = 500\;\text{J}$

So the best this engine can do is output 500 J of useful work while dumping 500 J to the cold reservoir. Notice that half the input energy has to become waste heat — there's no way to recover it without violating the second law.

Comparing real engines to the Carnot ceiling

Real engines achieve some fraction of the Carnot efficiency. Let's call the ratio second-law efficiency:

$\eta_{\text{2nd}} = \dfrac{\eta_{\text{actual}}}{\eta_{\text{Carnot}}}$

A well-designed engine might have $\eta_{\text{2nd}} = 0.6$, meaning it's capturing 60% of the theoretically available work.

For our problem, if the real engine achieves only 30% efficiency (instead of the 50% Carnot limit), then:

$\eta_{\text{2nd}} = \dfrac{0.30}{0.50} = 0.60$

$W_{\text{real}} = 0.30 \cdot 1000 = 300\;\text{J}$

$Q_{c,\text{real}} = 1000 - 300 = 700\;\text{J}$

The engine is dumping 700 J to the cold reservoir instead of the minimum 500 J. Those extra 200 J are lost opportunity — they could have been work but instead became waste heat due to irreversibilities (friction, fast heat transfer, incomplete combustion, etc.).

Why real engines fall short of Carnot

The Carnot efficiency assumes the engine runs a reversible cycle — infinitely slowly, with no friction, with heat transfers happening at infinitesimal temperature differences. Real engines are nothing like this:

• Fast combustion. Real gasoline combustion happens in milliseconds, not quasi-statically. The combustion gas temperature is much hotter than the cylinder walls for most of the stroke, so heat transfer is very irreversible.
• Friction. Every moving part has friction. Pistons sliding in cylinders, bearings turning, gears meshing — every one of these bleeds energy into waste heat.
• Imperfect insulation. Heat leaks out of the engine block while it's supposed to be running the working fluid, reducing the effective temperature difference.
• Exhaust losses. Real engines don't run the working fluid all the way down to ambient temperature. The exhaust gas leaves hot, carrying away wasted heat.
• Finite-time operation. An engine has to run fast enough to be useful. A Carnot cycle is reversible only in the limit of infinite cycle time, which produces zero power output. Real engines trade efficiency for power.

A real gasoline engine achieves about 25-35% thermal efficiency. A modern combined-cycle gas turbine (gas turbine + steam turbine in tandem) approaches 60%. A nuclear power plant's steam cycle runs around 33-37%. All of these are well below the Carnot ceiling set by their operating temperatures.

Why you can't just crank up $T_{h}$

If Carnot efficiency grows with $T_{h}/T_{c}$, why not just build engines that run at 5000 K to get 94% efficiency?

• Materials. At 5000 K, steel melts, aluminum boils away, and even tungsten softens. You'd need exotic ceramics or cooled blade designs, which are expensive and fragile.
• Cooling. The hotter the source, the more heat you need to dump to the cold sink, and cooling capacity is limited. Power plants are often sited next to rivers or oceans for cooling water; desert plants struggle.
• Lifecycle. Higher temperatures accelerate every failure mode — corrosion, creep, oxidation, thermal cycling fatigue. Engines running at the edge of their material limits don't last long.
• Emissions. Higher combustion temperatures in internal combustion engines produce more nitrogen oxides (NOx), which are regulated air pollutants. Aggressive combustion for efficiency trades off against emissions.

There's always an engineering sweet spot between efficiency and practicality.

What about the cold side?

Similarly, you can't just drop $T_{c}$ to zero. The cold reservoir is usually the atmosphere, a body of water, or a cooling tower. Its temperature is fixed by environmental conditions — typically ~300 K for a cooling tower, ~280 K for a river, ~250 K for a sub-zero winter night.

Some specialized engines use colder sinks:

• Cryogenic engines for rocket fuels use liquid oxygen or nitrogen as the cold side (~90 K or ~77 K), giving them extraordinary Carnot ceilings. The "cold" side is used as the oxidizer.
• Space-based systems radiate heat to the cosmic microwave background (~3 K!), enabling extremely high Carnot efficiencies in principle. The catch: radiative cooling is slow, so the engine must be small or run slowly.

Heat pumps and refrigerators — the reverse engine

A heat pump or refrigerator runs a heat engine "in reverse": it takes work as input and pumps heat from cold to hot. Instead of efficiency, we measure coefficient of performance (COP):

$\text{COP}_{\text{ref}} = \dfrac{Q_{c}}{W}$

For a Carnot refrigerator, the maximum COP is:

$\text{COP}_{\text{ref, Carnot}} = \dfrac{T_{c}}{T_{h} - T_{c}}$

For a fridge cooling food to 275 K in a 300 K kitchen: $\text{COP} = 275 / 25 = 11$. That's spectacular — every joule of electrical input moves 11 joules of heat. Real fridges achieve about 2-4.

A heat pump for heating has $\text{COP} = Q_{h}/W = T_{h}/(T_{h} - T_{c})$. In a mild climate you can get COP = 3-5, meaning your electric bill runs at 1/3 to 1/5 the cost of resistance heating.

Real-world engine efficiencies (approximate)

A rough comparison (these numbers depend heavily on load, operating point, and measurement method):

• Incandescent lightbulb (as a heater, not a light source) — 100% efficient at converting electricity to heat, but only 2-5% at producing visible light.
• Gasoline car engine — 25-35%. Carnot limit at ~2500 K combustion, 800 K exhaust: ~68%.
• Diesel engine — 35-45%. Higher compression ratio → higher peak temperature → higher Carnot ceiling.
• Combined-cycle gas turbine — 55-62%. Uses the gas turbine's hot exhaust to run a steam turbine downstream, capturing heat that would otherwise be wasted.
• Coal-fired steam plant — 35-45%. Limited by the maximum steam temperature and pressure materials can withstand.
• Nuclear steam plant — 33-37%. Lower steam temperatures than coal (for safety reasons) → lower Carnot ceiling.
• Solar thermal — 15-25%. Concentrated solar collectors heat fluid; fluid drives a turbine.
• Photovoltaic solar — not technically a heat engine. 15-22% for commercial panels.
• Fuel cell — 40-60%. Also not a heat engine; converts chemical energy to electricity directly, so not subject to the Carnot limit.
• Human muscle — ~25% (converts food chemical energy to mechanical work). Your body isn't a heat engine either — it extracts chemical energy directly via metabolism.

Try it

• Drag $T_{h}$ higher and see the Carnot maximum rise toward 100%. Notice the absolute limit is at $T_{h} = \infty$, which is obviously unachievable.
• Drag $T_{c}$ toward $T_{h}$ and see the efficiency drop to zero. When the reservoirs are at the same temperature, there's no temperature difference to exploit.
• Try $T_{c} = 0$: Carnot says 100% efficient, but the third law of thermodynamics says you can't actually reach absolute zero (it would take infinite work and infinite time).
• Use the "actual efficiency" slider to model a real engine that's less efficient than Carnot. Watch the "wasted work" indicator — that's the gap between what the engine outputs and what it could output at the Carnot ceiling.
• Try a modest $\Delta T$ (like 100 K difference). The efficiency collapses quickly — that's why low-grade heat (waste heat, solar thermal, geothermal) is hard to convert to work.
• Compare ideal vs real: even at the theoretical Carnot maximum, half the energy is "wasted" as heat to the cold reservoir. Real engines do worse. The fundamental lesson is that some waste heat is unavoidable.