Otto Cycle Efficiency

Equivalent forms

\eta = 1 - \frac{T_1}{T_2} = 1 - r^{1-\gamma}

Two adiabats and two isochores collapse into a one-parameter efficiency law — the whole gasoline engine in r^(γ−1).

Symbol	Name	SI unit	Dimension	Range
$η$	Thermal efficiencyoutput Fraction of heat converted to work	dimensionless	1	0 – 1
$r$	Compression ratio Ratio of largest to smallest cylinder volume	dimensionless	1	2 – 14
$γ$	Heat capacity ratio Cp/Cv of the working gas (≈1.4 for air)	dimensionless	1	1.2 – 1.67

Unit systems

SI:: all dimensionless
natural:: all dimensionless
CGS:: all dimensionless

Where it holds

Air-standard idealization (ideal gas, reversible adiabats, instantaneous combustion); real engines reach

\sim 25

–35% due to friction, heat loss, and finite burn time.

Discovery

Nicolaus Otto / analysis by Beau de Rochas · 1876

Nicolaus Otto built the first practical four-stroke engine in 1876; Alphonse Beau de Rochas had patented the idealized cycle in 1862, and its air-standard efficiency formula bears Otto's name.

Try this

Why can't your car engine just be made arbitrarily efficient?

The efficiency of a gasoline engine depends on one number — the compression ratio. Crank it up and efficiency rises… until the fuel ignites early and the engine knocks itself apart.

Research status: stable

Common misconceptions

Otto efficiency depends only on compression ratio, NOT on how much heat you add — more fuel gives more power but the same thermal efficiency. It is also always below the Carnot limit for the same temperature extremes.

Derivation

The cycle is: adiabatic compression

1\to 2

, constant-volume heat addition

2\to 3

, adiabatic expansion

3\to 4

, constant-volume heat rejection

4\to 1

.

Heat in

Q_in = C_v(T_3-T_2)

, heat out

Q_out = C_v(T_4-T_1)

.

The adiabats give

T_2/T_1 = T_3/T_4 = r^{\gamma -1}

,

so \eta = 1 - Q

_out/

Q_in = 1 - T_1/T_2 = 1 - r^{1-\gamma }

.

Limiting cases

r = 1

⟶

\eta = 0

: no compression, no work extracted

r \to \infty

⟶

\eta \to 1

, but knock and material limits cap real engines near

r \approx 10

–12

\gamma \to 1

⟶

\eta \to 0

: a gas with no temperature rise on compression does no useful cycle

Adiabatic Process→Carnot Efficiency→First Law of Thermodynamics→Heat Capacity at Constant Volume→