Oct 162012

In my previous article, I gave a short introduction to Otto cycle and discussed the processes in it. In this article, I will derive the air-standard efficiency of Otto cycle.

p-V and T-s Diagrams of Otto Cycle:

p-V Diagram T-s Diagram
Otto Cycle p-V Diagram Otto Cycle T-s Diagram

Basic terms used in derivation of air-standard efficiency of Otto cycle:

Total Cylinder Volume:

It is the total volume (maximum volume) of the cylinder in which Otto cycle takes place. In Otto cycle,

Total cylinder volume = V1 = V4 = Vc + Vs     (Refer p-V diagram above)


Vc → Clearance Volume

Vs → Stroke Volume

Clearance Volume (Vc):

At the end of the compression stroke, the piston approaches the Top Dead Center (TDC) position. The minimum volume of the space inside the cylinder, at the end of the compression stroke, is called clearance volume (Vc). In Otto cycle,

Clearance Volume, Vc = V2 (See p-V diagram above)

Stroke Volume (Vs):

In Otto cycle, stroke volume is the difference between total cylinder volume and clearance volume.

Stroke Volume, Vs = Total Cylinder Volume – Clearance Volume = V1 – V2 = V4 – V3

Compression Ratio:

Compression ratio (r) is the ratio of total cylinder volume to the clearance volume.



Now that we know the basic terms, let us derive expressions for T2 and T3. These expressions will be useful for us to derive the expression for air-standard efficiency of otto cycle. For finding T2, we take process 1-2 and for finding T3, we take process 3-4.

Process 1-2:

This process is an isentropic (reversible adiabatic) process. For this process, the relation between T and V is as follows:

{T_2/T_1}~=~{(V_1/V_2)^{ gamma ~-~1}}=~r^{ gamma ~-1}~~~~~(Since,~{V_1/V_2}~=~r)

T_2~=~T_1~*~r^{ gamma ~-~1}~. . . . .~(i)

Process 3-4:

This is also an isentropic process. The relation between T and V in this process is similar to the relation between T and V in process 1-2:


{T_3/T_4}~=~{(V_4/V_3)^{ gamma ~-~1}}=~r^{ gamma ~-1}~~~~~(Since,~{V_4/V_3}~=~r)

T_3~=~T_4~*~r^{ gamma ~-~1}~. . . . .~(ii)

Air-standard efficiency of Otto cycle:

It is defined as the ratio between work done during Otto cycle to the heat supplied during Otto cycle.

Air-Standard Efficiency (thermal efficiency) of Otto cycle,

{eta _th}~=~{Work~Done}/{Heat~Supplied}

From my previous article,


eta_th~=~1~-~{(T_4~-~T_1)/(T_4(r)^{ gamma ~-~1}~-~T_1(r)^{ gamma ~-~1})} ~~[From ~ (i) ~ and ~ (ii)]

eta_th~=~1~-~(T_4~-~T_1)/{(T_4~-~T_1)(r)^{ gamma ~-~1}}

eta_th~=~1~-~{1/{r^( gamma ~-~1)}}

If you have any ideas or suggestions about air-standard efficiency of Otto cycle, you can comment on this article.

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I am a mechanical engineer with a passion for technical stuff. I am the founder and former editor-in-chief of Mechteacher.com.

  • http://ibnumalik.my ibnumalik

    thanks for posting this article, it really help me to understand air standard efficiency clearly :)

    • Surjeet S

      You are welcome :) Stay tuned with mechteacher.com for more useful articles…

  • Zaid

    Thanks for posting this derivation
    And you have made this derivation is very easy

  • sara

    could you help me please ? Are You an Engineer

  • aamina

    thanks for the derivation.. and i need help regarding thermal engineering

  • Hoax

    It should be 1 – r ^ (1 – gamma), not 1 – 1 / r ^ (gamma – 1); the work done should equal T1 – T4 and not the other way around

    • Hoax

      I’m sorry… the work done should equal |T1 – T4| (absolute value of T1 – T4) and since T4 > T1, |T1 – T4| = T4 -T1; forgot that the input energy is always positive

      It can still be rewritten to 1 – r ^ (1 -gamma), though