Otto

Otto cycle: theoretical efficiency of spark-ignition engines as a function of compression ratio

Conceptual context

The Otto cycle is the idealized thermodynamic model describing the operation of spark-ignition engines, commonly known as gasoline engines. The central problem it solves is determining what fraction of the thermal energy released during combustion can be converted into useful mechanical work, and how that fraction depends exclusively on cylinder geometry (compression ratio) and gas properties (adiabatic index). The Otto cycle is the simplest of all power cycles because its efficiency does not depend on the amount of heat supplied, which fundamentally differentiates it from Diesel.

🟢 Essential level

The Otto cycle consists of four processes: adiabatic compression of the gas, heat addition at constant volume (simulating instantaneous combustion of the air-fuel mixture), adiabatic expansion producing work, and heat rejection at constant volume (simulating exhaust).

The thermal efficiency $\eta_O$ depends exclusively on two parameters: the compression ratio $r$ and the adiabatic index $\gamma$. Unlike the Diesel cycle, Otto efficiency does not depend on the amount of heat supplied $Q_H$. Higher $r$ means higher efficiency, but with diminishing marginal returns. Higher $\gamma$ means higher efficiency because the adiabats are steeper and the cycle approaches Carnot more closely.

This is why $r$ alone, without knowing $\gamma$, is not enough to predict efficiency precisely: the same compression ratio yields a noticeably different result for cold air than for hot combustion gases.

As $r$ increases, the fraction of $Q_H$ converted into $W_{neto}$ grows and the fraction rejected as $Q_C$ decreases. The practical limit of $r$ in gasoline engines (8 to 12) is imposed by premature fuel detonation, not by thermodynamics. Net work $W_{neto}$ is always the difference between $Q_H$ and $Q_C$, and efficiency is always the ratio of $W_{neto}$ to $Q_H$.

🔵 Formal level

Basic formulas

The thermal efficiency of the Otto cycle is defined as the fraction of absorbed heat $Q_H$ converted into net work $W_{neto}$:

Efficiency as energy ratio

The explicit efficiency formula is:

Otto cycle thermal efficiency

The energy balance of the cycle is:

Otto cycle energy balance

All three expressions describe the same cycle from complementary angles: definition, explicit geometry-based form, and energy closure.

Real student questions

Question one: why does Otto efficiency not depend on $Q_H$? Because both heat exchange processes occur at constant volume. Temperature rises proportionally to $Q_H$ both during combustion and exhaust, and the fraction converted to work depends only on the prior compression.

Question two: why do gasoline engines not use higher compression ratios? Because at compression ratios above 12, the air-gasoline mixture self-ignites by compression before the spark fires (detonation or knocking), causing mechanical damage and efficiency losses.

Question three: can an Otto engine exceed Carnot efficiency? Never. Carnot between the extreme cycle temperatures will always be higher. The difference measures the irreversibilities of the isochoric combustion mode.

Final synthesis

The Otto cycle establishes the maximum theoretical efficiency of spark-ignition engines as a function of two parameters: the compression ratio and the adiabatic index. Its formula is the simplest among all power cycles because efficiency does not depend on the amount of heat supplied. The compression ratio is the only design parameter controlling efficiency, and its practical limit is imposed by fuel detonation, not by thermodynamics.