Otto Cycle

The four strokes

Step	Process	Volume	Pressure	Temperature	Heat / Work
1 → 2	Adiabatic compression	V₁ → V₂ (= V₁/r)	P₁ → P₂ = P₁ r^γ	T₁ → T₂ = T₁ r^(γ−1)	W on gas, Q = 0
2 → 3	Isochoric heat addition (combustion)	V₂ const	P₂ → P₃	T₂ → T₃	Q_in = c_v(T₃ − T₂)
3 → 4	Adiabatic expansion (power stroke)	V₂ → V₁	P₃ → P₄ = P₃ / r^γ	T₃ → T₄ = T₃ / r^(γ−1)	W by gas, Q = 0
4 → 1	Isochoric heat rejection (exhaust)	V₁ const	P₄ → P₁	T₄ → T₁	Q_out = c_v(T₄ − T₁)

In a real four-stroke engine there are additionally two atmospheric strokes — intake (drawing fresh mixture in at roughly constant pressure) and exhaust (pushing burnt gas out). The thermodynamic Otto cycle proper compresses these into the single isochoric heat-rejection step 4 → 1, treating the cylinder gas as a fixed mass of air refreshed instantaneously.

Efficiency formula

Net work = Q_in − Q_out. Air-standard analysis with constant c_v gives:

η_Otto = 1 − (T₄ − T₁) / (T₃ − T₂)
       = 1 − T₁/T₂   (because both adiabats share the same temperature ratio)
       = 1 − 1/r^(γ−1)
       = 1 − r^(1−γ)

Efficiency depends only on r and γ — not on heat input, displacement, or any other parameter. That is the central insight of the air-standard cycle.

Compression ratio r	η for γ = 1.4	η for γ = 1.33 (rich mixture)	Engine type
4	42.6%	34.9%	Very old engines (1920s)
6	51.2%	43.0%	Pre-octane gasoline (1950s)
8	56.5%	48.4%	Standard 1970s gasoline
10	60.2%	52.4%	Modern naturally aspirated
12	63.0%	55.5%	High-performance, direct injection
14	65.2%	58.0%	Mazda Skyactiv-X, premium-grade fuel
16	67.0%	60.1%	Diesel-only territory

Worked example — a 2-litre, r = 10 engine

Air-standard Otto cycle with V₁ = 0.5 L (cylinder size in a 4-cylinder, 2 L engine at BDC), r = 10, γ = 1.4, intake at 300 K and 100 kPa, peak temperature 2400 K.

State	V (cm³)	P (MPa)	T (K)
1 — BDC, cool mixture	500	0.1	300
2 — TDC, after compression	50	2.51	753
3 — TDC, after spark	50	8.00	2400
4 — BDC, after power stroke	500	0.319	957

Net work per cycle: η × Q_in. Q_in = c_v · m · (T₃ − T₂) ≈ 717 J/(kg·K) × 5.8 × 10⁻⁴ kg × 1647 K ≈ 686 J per cylinder per cycle. η = 60.2 % gives 413 J of useful work per cylinder per cycle. At 3000 rpm and 4 cylinders, that is 4 × 413 J × 25 cycles/s = 41 kW (55 hp), in the right range for a 2 L midsize engine at cruise.

Variants

• Diesel cycle. Same adiabats and exhaust isochor, but heat addition is isobaric (constant pressure) instead of isochoric. Efficiency η = 1 − (1/r^(γ−1)) · (r_c^γ − 1)/(γ(r_c − 1)), where r_c is the cutoff ratio. Lower than Otto at the same r, but Diesel can use r = 14–22 because no fuel is present during compression.
• Atkinson and Miller cycles. Late intake valve closing or early intake valve closing reduces effective compression ratio while keeping the expansion ratio high — useful for hybrid powertrains because the engine trades peak power for better thermal efficiency.
• HCCI (homogeneous-charge compression ignition). Compresses a lean fuel-air mixture until it autoignites without a spark — Otto-like P-V trace, but the heat-addition stroke is a controlled volumetric autoignition. Reaches near-Diesel efficiency with near-Otto emissions.
• Two-stroke Otto. The same four thermodynamic processes happen, but they overlap — compression and exhaust share a stroke. Higher power-to-weight, worse emissions, used in chainsaws and small motorcycles.
• Wankel rotary. Three combustion chambers around a triangular rotor execute Otto-like cycles in sequence; per revolution, output per displacement is much higher than a piston engine, but sealing losses and surface-to-volume ratio reduce efficiency.
• Stirling engine. Two isothermals and two isochors instead of two adiabats and two isochors. Theoretically Carnot-efficient, mechanically much harder; used in submarine air-independent propulsion, cryocoolers, and concentrating-solar prototypes.

JavaScript — Otto cycle thermodynamics

function ottoEfficiency(r, gamma = 1.4) {
  return 1 - Math.pow(r, 1 - gamma);
}

console.log(`r=8:  η = ${(ottoEfficiency(8) * 100).toFixed(1)}%`);   // 56.5
console.log(`r=10: η = ${(ottoEfficiency(10) * 100).toFixed(1)}%`);  // 60.2
console.log(`r=12: η = ${(ottoEfficiency(12) * 100).toFixed(1)}%`);  // 63.0

// Full state trace, air-standard
function ottoStates({ V1, P1, T1, r, T3, gamma = 1.4 }) {
  const V2 = V1 / r;
  const P2 = P1 * Math.pow(r, gamma);
  const T2 = T1 * Math.pow(r, gamma - 1);
  const P3 = P2 * (T3 / T2);
  const T4 = T3 / Math.pow(r, gamma - 1);
  const P4 = P3 * (T4 / T3);  // isochoric: P4/P3 = T4/T3
  return {
    1: { V: V1, P: P1, T: T1 },
    2: { V: V2, P: P2, T: T2 },
    3: { V: V2, P: P3, T: T3 },
    4: { V: V1, P: P4, T: T4 },
  };
}

const trace = ottoStates({ V1: 5e-4, P1: 1e5, T1: 300, r: 10, T3: 2400 });
console.log(trace);
console.log(`η = ${(ottoEfficiency(10) * 100).toFixed(1)}%`);

// Per-cycle work for a single cylinder
function ottoWorkPerCycle({ m_air = 5.8e-4, cv = 717, T2, T3, T4, T1 }) {
  const Q_in  = cv * m_air * (T3 - T2);
  const Q_out = cv * m_air * (T4 - T1);
  return { Q_in, Q_out, W: Q_in - Q_out };
}

const { T1, T2, T3, T4 } = {
  T1: trace[1].T, T2: trace[2].T, T3: trace[3].T, T4: trace[4].T,
};
console.log(ottoWorkPerCycle({ T1, T2, T3, T4 }));

Where the Otto cycle matters

• Cars. 99% of internal combustion engines in passenger vehicles run the Otto cycle (or an Atkinson/Miller variant) on gasoline.
• Small power equipment. Lawn mowers, leaf blowers, snowblowers, portable generators — overwhelmingly four-stroke Otto.
• Motorcycles and outboards. Mid-displacement bikes use four-stroke Otto with r ≈ 10–13; outboards have moved from two-stroke to four-stroke Otto for emissions reasons.
• Light aircraft. Lycoming and Continental flat-six engines on small aircraft use Otto with r ≈ 8.5 to fit avgas's anti-knock rating.
• Hybrid cars. The internal-combustion side of a Toyota Prius is Atkinson-cycle Otto with effective r ≈ 13, optimized for cruising efficiency, with the electric motor covering peak power.
• Education and design. The Otto cycle is the canonical undergraduate engineering thermodynamics problem; every internal-combustion-engine textbook anchors on it before introducing Diesel, Brayton, and combined cycles.
• Engine simulation. Modern 1-D engine simulators (GT-Power, AVL Boost, Ricardo Wave) start from the air-standard Otto trace and add correction layers: variable specific heats, finite combustion duration, heat transfer, gas exchange.

Common mistakes

• Confusing compression ratio with pressure ratio. r = V_max / V_min, the volume ratio. Pressure ratio = r^γ — for r = 10, pressure rises ~25-fold during compression.
• Using η = 1 − T_c/T_h. That is Carnot. Otto efficiency is set by compression ratio, not by source and sink temperatures alone.
• Forgetting γ depends on the mixture. Lean mixtures have γ closer to 1.4 (air-like); rich mixtures and hot combustion products are γ ≈ 1.30. Variable-γ engine simulators correct for this.
• Ignoring volumetric efficiency. The air-standard Otto cycle assumes fresh charge fills the cylinder completely. Real engines reach 70 to 95% volumetric efficiency depending on intake design and RPM.
• Mixing up indicated, brake, and thermal efficiency. Indicated efficiency (η_i) is from P-V trace inside the cylinder, brake efficiency (η_b) subtracts friction losses, thermal efficiency includes fuel energy not converted to crankshaft work. They differ by 10–20% in real engines.
• Treating high r as a free lunch. Beyond a point, knock, thermal stress, friction, and emissions all rise. Modern engines use direct injection, variable valve timing, and 91+ octane fuel to safely raise r — but there is no realistic path to r > 16 in spark-ignition.

Performance — efficiency vs reality

Brake thermal efficiency in production cars hovers around 35–40% at the engine's best-BSFC operating point and as low as 15% at idle or part-throttle. The gap to ideal Otto is dominated by: heat transfer through cylinder walls (15–25%), pumping losses to draw in air and push out exhaust (5–10%), friction (5–8%), and finite-rate combustion (3–5%). A modern Atkinson hybrid like the Prius's 1.8L reaches 41% peak brake thermal efficiency; Mazda Skyactiv-X claims 47% under best-spot conditions; Volvo–Geely's 3-cyl prototype hit 48% in 2025. The thermodynamic ceiling (Otto at r = 14) is 65%, so ~70% of that is now actually delivered at the crankshaft — a remarkable achievement compared to the ~25% of a 1970s engine.