Ideal Otto cycle diagrams and efficiency

The ideal Otto cycle is the air-standard cycle of the spark ignition engine.

Diagrams

 P-v diagram                      T-s diagram
 P                                T
 |   3                            |             3
 |   |\                           |            /|
 |   | \                          |           / |
 |   |  '.                        |          /  |
 |   |    '.                      |        .'   |
 |   2      '-.                   |    2 .'     |
 |    \        '-.                |    |        4
 |     '.         '-.             |    |       /
 |       '-.         '-. 4        |    |     .'
 |          '-._         |        |    |   .'
 |              '-.._    |        |    1.-'
 |                   '-- 1        |
 +-----------------------> v      +--------------------> s
    v2                   v1
 1-2 isentropic compression       2-3 constant-v heat addition
 3-4 isentropic expansion         4-1 constant-v heat rejection

Derivation

Let r = v₁/v₂ be the compression ratio. Treat air as an ideal gas with constant cv and γ = cp/cv.

Heat added at constant volume (2-3): q_in = cv (T₃ − T₂) Heat rejected at constant volume (4-1): q_out = cv (T₄ − T₁)

η = 1 − q_out/q_in = 1 − (T₄ − T₁)/(T₃ − T₂)

Isentropic compression 1-2 and expansion 3-4, with v₂ = v₃ and v₁ = v₄:

T₂/T₁ = (v₁/v₂)^(γ−1) = r^(γ−1) T₃/T₄ = (v₄/v₃)^(γ−1) = r^(γ−1)

Hence T₂/T₁ = T₃/T₄, which gives T₄/T₁ = T₃/T₂, so

(T₄ − T₁)/(T₃ − T₂) = T₁/T₂ = 1/r^(γ−1)

η_Otto = 1 − 1/r^(γ−1)

The efficiency depends only on the compression ratio r and on γ; it rises as r increases. Example: r = 8, γ = 1.4: η = 1 − 1/8^0.4 = 56.5 %. In practice r is limited to about 10 by knocking.