The Otto Cycle Explained: Thermodynamics of the Gasoline Engine
The Otto cycle is the thermodynamic blueprint every gasoline engine follows: four strokes, two isentropic processes, two constant-volume heat exchanges. How efficiency, compression ratio, and knock connect.
Every gasoline engine in a car, motorcycle, or generator follows the same thermodynamic script. Nikolaus Otto codified it in 1876. The script has four acts: two do work and two set up the next cycle. Once you know it, you know why compression ratio matters, why high-octane fuel exists, where engine efficiency comes from, and where it stops.
What Is the Otto Cycle?
The Otto cycle is an idealized thermodynamic cycle describing how a spark-ignition piston engine converts chemical energy into mechanical work. "Idealized" means it strips away friction, heat loss through cylinder walls, and the messy gas dynamics of real valves, leaving only the core thermodynamic logic.
It consists of four processes operating on a fixed mass of gas trapped in a cylinder:
| Process | Strokes | What happens |
|---|---|---|
| 1→2 Isentropic compression | Compression stroke | Piston rises, gas compressed adiabatically: no heat exchange, entropy constant |
| 2→3 Isochoric heat addition | Power stroke (ignition) | Spark fires, fuel burns, pressure spikes at constant volume |
| 3→4 Isentropic expansion | Power stroke (expansion) | Hot gas pushes piston down, again adiabatic |
| 4→1 Isochoric heat rejection | Exhaust/intake | Burned gas exhausted, fresh charge taken in, modeled as a constant-volume pressure drop |
"Isentropic" means adiabatic and reversible: no friction, no heat leak. "Isochoric" means constant volume: the piston is not moving during heat exchange.
Real engines violate all of these idealizations. The Otto cycle earns its place because it predicts the direction of each efficiency trade-off and gives a clean analytical formula for why compression ratio is the dominant lever.
The Efficiency Formula
Thermal efficiency of the ideal Otto cycle depends on one variable: compression ratio.
η = 1 − 1 / r^(γ − 1)
Where:
- r = compression ratio (swept volume + clearance volume) / clearance volume
- γ = ratio of specific heats (Cp/Cv) ≈ 1.35 for a gasoline-air mixture
At r = 10: η = 1 − 1/10^0.35 ≈ 59%. At r = 13: η ≈ 65%. Real engines capture about half of that ideal, 30–40% brake thermal efficiency, because of the losses the ideal cycle ignores. The shape of the curve holds: compression ratio pays diminishing returns past about 12–13, and the gains steepen below 8.
Try It: Live Otto Cycle
The chart below is a real simulation with the engine running. Drag the compression ratio slider and watch the P-V loop stretch upward. A taller loop encloses more area, which is more work extracted per cycle. The ideal efficiency number updates in real time.
CYLINDER 1 THERMO
in-cylinder pressure, live
Ideal Otto efficiency η = 1 − 1/rγ−1 (γ = 1.35). Drag the CR slider to see how compression ratio drives efficiency — and where knock limits the gain.
Above CR 13 the loop shape becomes harder to sustain without knock. In a real engine the end gas would autoignite before the flame front arrives, collapsing the controlled burn into a destructive pressure spike. The ideal cycle has no knock model; real engines do, and that is the ceiling.
The Four Processes in Detail
1→2: Isentropic Compression
The piston rises from bottom dead center (BDC) to top dead center (TDC) with both valves closed. No heat enters or leaves, so the work of compression goes into raising the gas temperature and pressure.
The temperature ratio across compression is:
T₂ / T₁ = r^(γ − 1)
At r = 10 and an intake temperature of 300 K: T₂ ≈ 300 × 10^0.35 ≈ 672 K. Gas that entered the cylinder at 27°C leaves the compression stroke at nearly 400°C, before combustion begins. Diesel engines ignite fuel without a spark on this principle: push compression to 16:1 and T₂ exceeds 800 K, above any fuel's autoignition temperature.
2→3: Isochoric Heat Addition
The spark fires a few degrees before TDC. In the ideal cycle, the fuel releases all its chemical energy at once at constant volume, with the piston stationary during combustion. In a real engine the flame front takes a few milliseconds to sweep the chamber, so engineers calibrate ignition timing to center the pressure rise near TDC rather than at it.
The pressure ratio across combustion:
p₃ / p₂ = T₃ / T₂
Peak cycle temperature T₃ runs 2,000–2,800 K in a real engine. This is the thermodynamic constraint on exhaust emissions: NOₓ formation accelerates above roughly 1,800 K, so EGR, lean burn, and cooled charge strategies all aim at limiting T₃.
3→4: Isentropic Expansion
The high-pressure, high-temperature gas drives the piston back to BDC. This is the only stroke that delivers work to the crankshaft. The expansion is adiabatic in the ideal cycle: all the enthalpy of the gas converts to mechanical work.
The expansion ratio equals the compression ratio in the symmetric Otto cycle. This symmetry is the source of a real inefficiency: the exhaust valve opens before BDC, releasing blowdown energy that has not yet been converted to work. Atkinson-cycle and Miller-cycle engines break the symmetry with a longer expansion stroke than compression stroke, recovering some of that blowdown.
4→1: Isochoric Heat Rejection
In the ideal model, the piston reaches BDC and pressure drops back to atmospheric at constant volume as heat is "rejected" to the environment. In the real engine this is the exhaust and intake strokes, the two gas-exchange strokes that the ideal cycle collapses into a single step.
The rejected heat Q_out leaves through the exhaust pipe as heat, and the fraction that does not leave becomes useful work. Efficiency is:
η = 1 − Q_out / Q_in = 1 − T₁/T₂ = 1 − 1/r^(γ−1)
The two expressions are identical. The compression ratio formula is the temperature ratio in disguise.
Why Compression Ratio Has a Ceiling
The ideal cycle says higher compression ratio always improves efficiency. Real gasoline engines cap out at 12–13:1 because of knock, the spontaneous autoignition of the end gas ahead of the flame front.
End-gas temperature at the point of ignition is approximately:
T_endgas ≈ T_intake × (p_cylinder / p_intake)^((γ−1)/γ)
Gasoline's autoignition threshold is roughly 700–750 K (depending on octane rating). At CR 10.5, a typical engine clears that threshold with a margin of around 20 K on a cool day. At CR 14, that margin is gone. The end gas detonates, pressure oscillates at 5–9 kHz, and those oscillations strip the protective boundary layer from the piston crown and cylinder walls.
Octane rating measures how high that threshold sits. Each octane point above or below 95 RON moves the threshold by roughly 3.5 K. Compression ratio, intake temperature, boost pressure, and ignition advance all push T_endgas toward the threshold from the other direction. The ECU's knock feedback loop lives in that margin, advancing timing for efficiency and retreating when the knock sensors detect the characteristic ring.
For a closer look at the knock mechanism, see What Is Engine Knock? Causes, Sounds, and How to Stop It.
Real Otto vs. Ideal Otto
The gap between ideal (59% at CR 10) and real (30–35% brake) efficiency comes from losses the cycle ignores:
| Loss | Typical magnitude |
|---|---|
| Heat transfer through cylinder walls | 6–8 percentage points |
| Combustion phasing (finite burn duration, non-constant volume) | 4–6 pp |
| Gas exchange pumping work | 2–4 pp |
| Friction (rings, bearings, accessories) | 4–6 pp |
| Incomplete combustion / crevice volumes | 1–2 pp |
| Blowdown before EVO | 1–2 pp |
The most recoverable losses in modern engine development are combustion phasing (direct injection, cooled EGR, variable compression) and pumping work (cylinder deactivation, Atkinson cycle, variable valve timing). Heat transfer losses are mostly the cost of having a piston, which is why ceramic thermal barrier coatings show up in racing and heavy-diesel applications.
From Otto Cycle to Real Dyno Curve
The ideal cycle predicts efficiency but not power. Power output depends on how much air the engine can move per cycle, its volumetric efficiency, which peaks at a specific RPM set by intake runner resonance. Below that RPM, the runners do not charge effectively. Above it, gas velocity and valve timing limit filling.
Torque peaks where volumetric efficiency peaks (3,000–5,000 RPM for most naturally-aspirated engines), and power continues rising past that because more cycles per second outweighs declining torque, until volumetric efficiency collapses and both fall together. You can watch this in the engine simulator: run a dyno sweep on the I4 and the inflection point in the torque curve tracks the runner tuning.
Otto Cycle FAQs
Why doesn't the ideal cycle have an intake or exhaust stroke?
The ideal cycle models only the thermodynamic conversion of heat to work, not the gas-exchange machinery. It treats intake and exhaust as a single isochoric heat rejection (4→1). The math works out the same as if the burned gas teleported out and fresh charge teleported in at BDC.
What's the difference between the Otto cycle and the Diesel cycle?
The Diesel cycle uses isobaric (constant-pressure) heat addition instead of isochoric (constant-volume). Because injection and combustion happen over a finite crank travel rather than at TDC, the pressure stays roughly constant while the piston moves. At the same compression ratio, the Diesel cycle is a little less efficient than Otto. Diesel engines use much higher CR (16–23:1) than gasoline engines can, which is why diesel thermal efficiency (40–45%) beats gasoline (30–35%) in practice.
What is the Atkinson cycle and how does it differ?
The Atkinson cycle uses a longer expansion stroke than compression stroke, breaking the symmetry of the Otto cycle. More of the gas enthalpy converts to work before the exhaust valve opens, improving efficiency at the cost of lower peak power density. Modern hybrids (the Toyota Prius, for instance) run a modified Atkinson cycle using late intake valve closing to reduce effective compression while keeping a long expansion stroke.
What does γ represent?
γ = Cp/Cv is the ratio of specific heats at constant pressure and constant volume. For a diatomic gas (air, roughly) γ ≈ 1.4. A fuel-air mixture with higher molecular complexity runs a little lower (≈ 1.3–1.35) because combustion products (CO₂, H₂O) have more degrees of freedom. The value matters for efficiency: higher γ means a steeper compression temperature rise and more work extracted per unit compression.
Why does ignition timing affect efficiency?
The ideal cycle assumes instantaneous combustion at TDC. Real flame propagation takes time, so if you fire the spark at TDC the pressure peak arrives well after, when the piston has already descended and the crank has lost its most efficient leverage angle. Advancing timing centers the pressure rise at TDC and extracts more work. Advance too far and you are compressing an already-burning charge, raising T_endgas and causing knock.