Otto Cycle

Why Otto and not something simpler

Before Nikolaus Otto, internal-combustion engines were curiosities. Étienne Lenoir's 1860 engine drew in a fuel-air mixture, ignited it at atmospheric pressure, and let it expand — no compression at all. Its thermal efficiency was around 3 percent. Otto's 1876 patent on the four-stroke cycle introduced the single change that put internal combustion on the road: compress the charge before you light it. Compression raises the temperature ratio between heat addition and rejection, exactly the lever the Carnot theorem identifies as the source of all engine efficiency. The four-stroke arrangement — intake, compression, power, exhaust — exploits that lever using a single piston by spreading one thermodynamic loop across two crankshaft revolutions. Within a decade, Otto's machines were running European factories. Within thirty years, Daimler and Benz had shrunk the cycle into something that fit between a horse and a cart.

The cycle's longevity has very little to do with elegance and almost everything to do with the kinematic accident that a slider-crank produces a near-sinusoidal piston motion ideal for both the rapid compression and rapid expansion the cycle needs. The same mechanism can intake, compress, fire, expand, and exhaust at thousands of cycles per minute with only one moving part exposed to the hot gas. No turbine, no rotary, no free-piston experimental design has yet beaten the four-stroke slider-crank on the joint axes of efficiency, cost, manufacturability, and noise — a remarkable defensive record for a patent that expired in 1890.

The four idealised processes

The closed thermodynamic loop is conventionally numbered 1 → 2 → 3 → 4 → 1, with state points at the dead centres of the piston travel:

Process	Path	What happens	Heat	Work
1 → 2	Isentropic compression	Piston rises BDC → TDC; PV^γ = const	Q = 0	W < 0 (work in)
2 → 3	Constant-volume heat addition	Spark fires; combustion releases Q_in at fixed V_TDC	Q_in > 0	W = 0
3 → 4	Isentropic expansion	Piston falls TDC → BDC; the power stroke	Q = 0	W > 0 (work out)
4 → 1	Constant-volume heat rejection	Exhaust valve opens; pressure drops at fixed V_BDC	Q_out < 0	W = 0

Two of the four processes (1 → 2 and 3 → 4) are work exchanges with the piston; the other two (2 → 3 and 4 → 1) are heat exchanges that happen so fast the piston is treated as motionless. The mechanical four-stroke arrangement (intake → compression → power → exhaust) needs two extra strokes — intake and exhaust — to refresh the working gas, but in the idealised closed cycle those strokes lie on top of one another at ambient pressure and cancel as a pair of overlapping line segments. That's why the textbook Otto diagram shows only four corners despite the engine making four strokes per cycle.

Where the efficiency formula comes from

Treat the working gas as a calorically perfect ideal gas with constant c_v and γ. The first law applied around the closed loop gives

W_net = Q_in − |Q_out|
η     = W_net / Q_in = 1 − |Q_out|/Q_in

The two heat-transfer legs both occur at constant volume, so Q_in = m c_v (T₃ − T₂) and |Q_out| = m c_v (T₄ − T₁). Substituting,

η = 1 − (T₄ − T₁) / (T₃ − T₂)

The isentropic legs T V^(γ−1) = constant give T₂/T₁ = (V₁/V₂)^(γ−1) = r^(γ−1) and the same ratio T₃/T₄ = r^(γ−1). After factoring,

η_Otto = 1 − 1 / r^(γ−1)

The result is striking: only the compression ratio and γ appear. The temperatures, pressures, and absolute amount of heat released drop out entirely. A 0.5-litre lawn-mower and a 6-litre V8 with the same r have the same air-standard efficiency.

Compression ratio r	η (γ = 1.4)	Typical use
6	51 %	Pre-1950 cars, two-stroke string trimmers
8	56 %	1960s muscle cars on leaded fuel
9.5	59 %	Naturally aspirated regular-grade gasoline (RON 91)
10	60 %	Premium-grade port-injection (RON 95-98)
12	63 %	Modern direct-injection turbo (Skyactiv-G, Mazda)
14	65 %	Hybrid Atkinson (Toyota A25A, Honda LFA1)
20	70 %	Diesel territory — knock not relevant, fuel sprayed in

Why a 60-percent engine returns 30-percent fuel economy

The air-standard η of 60 percent at r = 10 is a ceiling no engine has ever approached. The gap between ideal Otto and brake thermal efficiency comes from a stack of physical effects, each shaving a few percentage points:

• Finite-time combustion. The flame propagates at 10–40 m/s, taking 1–3 ms to cross the bore. During that time the piston has already moved several millimetres past TDC, so heat is partly released into an expanding cylinder rather than at constant V. This rounds off the 2–3 corner on the P-V diagram, costing 3–6 points.
• Heat transfer to coolant. Peak gas temperatures hit 2500 K against cylinder walls held at 400 K by the coolant. Roughly 20–30 percent of fuel energy is conducted away through the cylinder walls, head, and piston. This is irrecoverable in a conventional design — though regenerators and adiabatic ceramic engines have tried.
• Friction. The piston-ring assembly slides on a thin oil film at speeds up to 25 m/s. Bearing friction, valve-train friction, and oil-pump parasitic load together absorb 7–12 percent of indicated power, dropping further with low-viscosity oils and DLC coatings.
• Pumping work. Naturally aspirated spark-ignition engines throttle the intake to control load — and the work to suck air past a closed butterfly is negative. At part load this costs 5–10 percent. Diesel and direct-injection lean-burn engines avoid most of it.
• Variable γ. γ = c_p/c_v drops from 1.40 at intake to 1.25–1.30 at peak combustion (more atoms with vibrational modes). The cycle's analytic 60 percent assumes a constant 1.4, so the real polytropic spread is narrower.
• Incomplete combustion. A few percent of fuel exits unburned as CO, HC, or soot — chemical energy that never becomes heat. Catalytic converters clean it up but cannot recover its work.

Adding it all up, naturally aspirated port-injection gasoline engines convert about 25–30 percent of fuel chemical energy to brake horsepower. The best production engines today — Toyota A25A in the Camry hybrid, Honda's LFA1 in the Accord hybrid, Hyundai's Smartstream G2.5 — hit 40–41 percent brake thermal efficiency using direct injection, Atkinson timing, high compression (r ≈ 13), and cooled EGR.

Knock — the wall at r = 12-14

The blunt thermodynamic answer to "make engines more efficient" is "raise r". The blunt practical answer is "knock won't let you." Knock is autoignition of the end-gas: the unburned charge furthest from the spark plug gets compressed and heated by the advancing flame front, and if its temperature crosses the autoignition threshold before the flame arrives, it self-ignites. Multiple flame fronts then race toward each other, producing pressure oscillations at 5–7 kHz that ring the cylinder like a bell — audible as a metallic ping from the engine bay. The pressure spikes are violent enough to erode piston crowns, melt ring lands, and blow head gaskets within a few hours of sustained operation.

Knock resistance is quantified by the research and motor octane numbers (RON/MON) for gasoline. The "anti-knock index" posted at U.S. pumps is (RON + MON)/2. Regular U.S. unleaded is AKI 87 (≈ RON 91); premium is AKI 93 (≈ RON 98); European premium 95 RON. Higher numbers tolerate higher r before knocking. The empirical ceiling for naturally aspirated port-injection on premium pump gasoline sits around r = 11. To push beyond, engine designers exploit three tricks:

• Direct injection. Spraying fuel directly into the cylinder during the compression stroke evaporates it inside the combustion chamber, cooling the charge by 20–30 K and buying ~1 point of compression-ratio headroom. The cooling effect — "charge cooling" — is the single biggest reason modern engines run higher r than 1990s engines on the same fuel.
• Knock-sensor feedback. A piezoelectric accelerometer on the block listens for the 5–7 kHz signature. When detected, the ECU pulls ignition timing 1–2 crank degrees retarded, which lowers peak cylinder temperature. Continuous closed-loop control lets the engine operate within 1° of the knock boundary at all times.
• Cooled EGR. Recirculating cooled exhaust gas displaces fresh charge and dilutes the mixture without throttling, lowering combustion temperatures and suppressing knock while improving part-load efficiency.
• Atkinson / Miller valve timing. Delayed intake-valve closing reduces the effective compression ratio while leaving the geometric expansion ratio high — knock margin without sacrificing expansion work. The cornerstone of hybrid engines.

The combined effect: Mazda's Skyactiv-G runs r = 13 on premium 91 RON; Toyota's Dynamic Force engines run r = 13 (Atkinson) on regular; the experimental Mazda Skyactiv-X reaches r = 16 using compression-ignited stratified charge (HCCI hybrid). Above r ≈ 16 the autoignition delay becomes too short for any spark-controlled flame front to win, and the engine becomes a Diesel.

Variants — Diesel, Atkinson, Miller

Three cycle variants address Otto's two-decade-old design constraints in different ways.

Cycle	Heat addition	Compression r	Spark?	Used in
Otto	Constant V	8 – 14	Yes	Gasoline cars, motorcycles, small aviation
Diesel	Constant P	15 – 22	No (compression ignition)	Trucks, ships, locomotives, generators
Atkinson	Constant V	geo 12-14, eff. ~8-10	Yes	Hybrids (Prius, Camry hybrid, Accord hybrid)
Miller	Constant V	geo 11-13, eff. ~9	Yes	Supercharged premium gas (BMW B58, Mazda Skyactiv-G)
Dual	Const V then Const P	16 – 20	No	High-speed Diesels (better real-cycle model)

Diesel. Rudolf Diesel's 1893 cycle replaces the spark with compression ignition. Air alone is compressed to r = 15–22, reaching 700–900 K — well above the autoignition temperature of diesel fuel. Fuel is sprayed in at TDC and burns as fast as it is sprayed, approximating constant-pressure heat addition. The ideal Diesel efficiency is actually lower than Otto at the same r (the cutoff-ratio term r_c penalises it), but Diesel's much higher allowable r turns the comparison around in practice: modern truck Diesels hit 45 percent brake thermal efficiency, beating any gasoline engine.

Atkinson. James Atkinson's 1882 patent used a complex linkage to give the piston a longer power stroke than compression stroke — netting more expansion work per cycle. The mechanical linkage was unreliable and the cycle was forgotten for a century. Modern "Atkinson-cycle" engines (every Toyota hybrid since the 1997 Prius) achieve the same effect with delayed intake-valve closing: the intake valve stays open ~60 crank degrees past BDC, pushing some of the charge back into the manifold, so the effective compression starts late while the expansion remains full. The net is an asymmetric loop with expansion work exceeding compression work by 5–10 percent. The trade-off is reduced peak power per displacement — fine in a hybrid, where the electric motor supplies the missing torque.

Miller. Ralph Miller's 1947 patent is operationally identical to Atkinson but historically associated with supercharged engines: the boost compensates for the volumetric inefficiency of the late-intake trick. Mazda's Skyactiv-G, BMW's B58, Audi's 2.0 TFSI, and most modern downsized turbo engines run Miller cycles at part load and switch to full Otto under boost. The valve-event difference between Atkinson and Miller is mostly marketing — both delay intake-valve closing relative to BDC.

Worked example: a 2.0-litre direct-injection engine at full load

Consider a four-cylinder 2.0-litre engine with compression ratio r = 12, running at 6000 rpm on stoichiometric gasoline (lower heating value LHV = 43 MJ/kg, stoichiometric air-fuel ratio AFR = 14.7). Compute the rough brake power and efficiency.

Air drawn per cycle (one cylinder) at intake conditions (T₁ = 320 K, P₁ = 100 kPa accounting for volumetric efficiency 0.92 and slight intake heating):

V_swept_per_cyl = 0.0005 m³ (0.5 L)
m_air per cycle = P₁ V_swept / (R T₁) = 100000 × 0.0005 × 0.92 / (287 × 320)
                ≈ 5.0 × 10⁻⁴ kg
m_fuel per cycle = m_air / AFR = 5.0e-4 / 14.7 ≈ 3.4 × 10⁻⁵ kg
Q_in per cycle  = m_fuel × LHV = 3.4e-5 × 43e6 ≈ 1463 J

Per cycle work output assuming real-cycle indicated efficiency of 0.42 (close to the best DI turbo engines):

W_ind per cycle per cyl = 0.42 × 1463 ≈ 614 J
Cycles per second per cyl (4-stroke) = 6000 / 60 / 2 = 50 Hz
P_ind per cyl = 614 × 50 ≈ 30700 W = 30.7 kW
P_ind total (4 cyl) ≈ 123 kW = 165 hp
P_brake after 10% friction/pumping loss ≈ 110 kW = 148 hp

This is within a few percent of the actual rated output of the Volkswagen EA888 evo4 (2.0 TSI) at 140-150 kW. The mismatch between ideal Otto (η ≈ 63 percent at r = 12) and observed 38–40 percent BTE is exactly the stack of losses catalogued in the previous section.

A short history of the cycle

• 1862. Alphonse Beau de Rochas patents the conceptual four-stroke principle (compression before ignition) but never builds an engine. His patent is in the public domain by 1876.
• 1876. Nikolaus Otto and his partner Eugen Langen at the Deutsche Gasmotorenfabrik in Cologne build and patent the first practical four-stroke engine, running on city gas. It dominates stationary power for a decade.
• 1885–1886. Otto's employees Gottfried Daimler and Wilhelm Maybach miniaturise the engine to run on gasoline. Karl Benz independently builds the first automobile.
• 1886. Otto's patent is voided by the German courts on the basis of Beau de Rochas's prior art. Competitors flood the market.
• 1923. Tetraethyl lead introduced as an anti-knock additive, enabling r > 6.
• 1957. First production direct-injection gasoline engine (Mercedes 300 SL) — a technology that would only become mainstream 50 years later.
• 1975. Catalytic converters require unleaded fuel; compression ratios drop temporarily as anti-knock chemistry is rebuilt around aromatics and ethanol.
• 1997. Toyota Prius reintroduces the Atkinson cycle for series-hybrid duty, eventually pushing the technology into every modern hybrid.
• 2011. Mazda Skyactiv-G launches at r = 14 on regular gasoline using cooled EGR and combustion-chamber geometry optimisation — the modern high-water mark for naturally aspirated spark ignition.
• 2018. Mazda Skyactiv-X ships with HCCI-augmented combustion at r = 16. Still less than 1 percent of global engine production but a glimpse of where gasoline efficiency may go.

Common pitfalls

• Confusing geometric and effective compression ratios. Geometric r is V_BDC/V_TDC, set by piston travel and combustion chamber volume. Effective r is what the gas actually experiences — which on Atkinson/Miller engines is much smaller because the intake valve closes late. Quoting one for the other gives wildly wrong knock and efficiency estimates.
• Assuming Otto efficiency scales with mass or displacement. The η formula has no dependence on either. A 0.05-L drone engine at r = 9 has the same air-standard efficiency as a 6-L V8 at r = 9. Differences come from friction scaling, surface-to-volume ratios for heat loss, and pumping losses — not from the cycle itself.
• Treating γ as exactly 1.4. The textbook 60 percent at r = 10 uses γ = 1.4 (cold air). Real combustion gases sit at γ ≈ 1.27–1.30 at peak temperatures, dropping the analytic efficiency to ~52 percent before any losses. Many introductory texts skip this correction.
• Confusing the Otto cycle (thermodynamic) with the four-stroke arrangement (mechanical). Two-stroke engines also approximate the Otto cycle thermodynamically; they just compress and exhaust differently. Rotary (Wankel) engines run the Otto cycle in a non-reciprocating geometry. The cycle and the mechanism are separable.
• Ignoring the pumping loop. The textbook closed cycle has only four corners. Real spark-ignition engines have a small "pumping loop" of negative work near atmospheric pressure caused by the intake throttle. At idle this loop is the dominant loss; at wide-open throttle it nearly vanishes. Diesel and lean-burn DI gasoline engines unthrottle their intakes and eliminate it.
• Reading the P-V diagram as time-resolved. A Cartesian P-V plot collapses the time dimension. The 2 → 3 vertical line at constant V is in reality a ~2 ms event; the 1 → 2 line is ~10 ms. Plotting P vs crank angle (indicator diagram) instead shows the real-time evolution.