Lift the bonnet of any petrol car and you are looking at a machine whose performance was decided long before the first bolt was tightened. The combustion happening thousands of times a minute inside each cylinder follows a thermodynamic recipe, and the single most important ingredient in that recipe is how tightly the engine packs the air-fuel mixture before it ignites. Two engines of the same displacement can differ in fuel economy by a wide margin, and the compression ratio is often the reason.
This article explains the air-standard Otto cycle, the idealised model behind every spark-ignition engine. We will work through where its efficiency formula comes from, run a concrete example, and look at why engineers cannot simply crank the compression ratio up without limit.
Why this calculation matters
The Otto cycle is the reference model for petrol engines in cars, motorcycles, generators, lawn equipment, and small aircraft. Even though a real engine has friction, heat loss, incomplete combustion, and finite valve timing, the air-standard cycle gives a clean upper bound on what the geometry alone allows. If the ideal cycle says 60 percent and your engine delivers 35, you know the gap is losses you might attack. If the ideal cycle itself is low, no amount of tuning will rescue it.
That makes the efficiency formula a design tool, not just a textbook exercise. It tells an engine designer how much is to be gained by raising the compression ratio, and it frames the central trade-off of spark-ignition design: efficiency wants a high compression ratio, but the fuel and the combustion chamber will only tolerate so much. Getting that number early shapes the bore, the stroke, the chamber shape, and the octane rating the engine will demand at the pump.
The core formula
The air-standard Otto cycle treats the working fluid as an ideal gas with constant specific heats and idealises the engine into four reversible processes:
1 -> 2 : adiabatic (isentropic) compression
2 -> 3 : constant-volume heat addition (combustion)
3 -> 4 : adiabatic (isentropic) expansion (power stroke)
4 -> 1 : constant-volume heat rejection (exhaust)
Heat goes in at constant volume during the spark-driven burn, and heat leaves at constant volume as the exhaust blows down. Work comes out of the expansion stroke and goes into the compression stroke; the difference is the net work.
When you push the constant-volume heat terms through the first law and use the isentropic relations between temperature and volume, almost everything cancels. What survives is strikingly simple:
eta = 1 - 1 / r^(gamma - 1)
Here eta is the thermal efficiency, r is the compression ratio (the cylinder volume at bottom dead centre divided by the volume at top dead centre), and gamma is the ratio of specific heats of the working gas. For air at moderate temperatures, gamma is about 1.4.
Two things are worth noticing. First, the efficiency depends only on r and gamma. It does not depend on how much heat you add, on the engine size, or on the peak temperature. Add more fuel and you get more work, but the fraction of heat converted to work stays the same. Second, because gamma is greater than one, raising r always raises efficiency. The catch is that the curve flattens: the first few units of compression ratio buy a lot, and each additional unit buys less.
A worked example
Take a petrol engine with a compression ratio of r = 10, a typical figure for a modern naturally aspirated unit, and treat the working fluid as air with gamma = 1.4.
Step 1 — find the exponent.
gamma - 1 = 1.4 - 1 = 0.4
Step 2 — raise the compression ratio to that power.
r^(gamma - 1) = 10^0.4 = 2.512
Step 3 — apply the efficiency formula.
eta = 1 - 1 / 2.512
eta = 1 - 0.398
eta = 0.602
So the ideal Otto cycle for this engine has a thermal efficiency of about 60 percent. That is the ceiling the geometry sets. A real engine of this compression ratio will land considerably lower, often in the low-to-mid 30s as a brake thermal efficiency, because friction, heat transfer to the cylinder walls, pumping losses, and non-instantaneous combustion all erode the ideal figure.
Notice what the formula tells you about design direction. Drop the compression ratio to 8 and the efficiency falls to about 56 percent; push it to 12 and it climbs to roughly 63 percent. The compression ratio is a genuine lever on fuel economy — which is exactly why it is one of the most carefully chosen numbers in an engine. The reason it is not pushed higher is engine knock: too much compression heats the end-gas enough to auto-ignite before the spark, and that uncontrolled detonation can wreck pistons.
Common mistakes
Confusing compression ratio with pressure ratio. The Otto cycle is built on the volume ratio between bottom and top dead centre. The Brayton cycle uses a pressure ratio, and its efficiency formula has a different exponent. Mixing the two gives a wrong answer and the wrong intuition.
Expecting efficiency to depend on the fuel or the heat added. It does not, in the air-standard model. Burning a richer mixture raises peak temperature and net work, but the efficiency stays pinned by r and gamma. More fuel is more output, not better conversion.
Forgetting that gamma drifts. Real combustion gases are hot and contain carbon dioxide and water vapour, so their gamma is below the 1.4 of cool air — often nearer 1.3. Using 1.4 throughout flatters the prediction. The air-standard number is an idealisation, useful as a bound rather than a forecast.
Treating the ideal efficiency as achievable. The 60 percent figure ignores every real loss. Quoting it as the engine's efficiency is a classic error. It is the thermodynamic potential of the cycle, not the performance of the hardware.
Assuming higher compression is always free. On paper the curve only rises. In practice knock, mechanical stress, and emissions set a hard ceiling, which is why production petrol engines cluster in a fairly narrow compression-ratio band.
Try the interactive NovaSolver calculator
Working the exponent by hand is fine once, but seeing how the cycle responds as you move the inputs builds far better intuition. The Otto Cycle Simulator — Thermal Efficiency of a Spark-Ignition Engine on NovaSolver lets you set the compression ratio, the specific heat ratio, the intake temperature, and the heat input, then returns the thermal efficiency, the end-of-compression temperature, the after-combustion temperature, and the net work — so you can watch efficiency climb then flatten as you raise the compression ratio.
Related calculators
- Carnot Cycle Simulator — the absolute efficiency ceiling set by the hot and cold reservoir temperatures, against which any real cycle is judged.
- Brayton Cycle Simulator — Thermal Efficiency of a Gas Turbine — the gas-turbine counterpart, where a pressure ratio rather than a volume ratio governs efficiency.
- Rankine & Refrigeration Cycle Simulator — the steam power cycle behind most thermal power stations, for comparing combustion engines with vapour-based plants.
You can browse the full set in the thermal engineering tools hub.
Closing note
The Otto cycle distils a complex machine into one clean idea: pack the charge tighter and you convert a larger share of the fuel's energy into work. The efficiency formula, eta = 1 - 1/r^(gamma-1), depends only on the compression ratio and the gas, which is why the compression ratio is treated as a prized engine parameter. The reason engines do not simply chase ever-higher numbers is knock, the practical ceiling that classical thermodynamics cannot see. Run the formula for your own compression ratio, treat the result as a bound rather than a promise, and you will understand why two similar engines can return very different fuel economy.
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