Dec 072012

In my previous articles, I gave an introduction to Otto cycle and derived the air-standard efficiency of Otto cycle. In this article, I will derive the mean effective pressure (m.e.p) of Otto cycle.

What is Mean Effective Pressure?

Mean effective pressure is the ratio of work done (W) during the working stroke(s) of a cycle to the stroke volume or swept volume (Vs) of the cylinder. It is denoted by ‘pm‘ and its unit is N/m2.


In order to derive an expression for mean effective pressure of Otto cycle, we have to find out an expression for work done and stroke volume of Otto cycle.

From my previous article, p-V and T-s diagrams of Otto cycle are as follows:

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

From my previous article,

Compression ratio,



Pressure ratio,


In process 1-2 (isentropic compression),

{p_2/p_1}~=~(V_1/V_2)^ gamma~=~r^ gamma

In process 3-4 (isentropic expansion),

{p_3/p_4}~=~(V_4/V_3)^ gamma ~=~r^ gamma

Also from the p-V diagram above,

V1 = V4 and V2 = V3

Let Vc = V2 = V3 = 1

Work done during Otto cycle,

W = Work done during isentropic expansion (process 3-4) – Work done during isentropic compression (process 1-2)

W~=~{{p_3V_3~-~p_4V_4}/{ gamma ~-~1}}~-~{{p_2V_2~-~p_1V_1}/{ gamma ~-~1 }}

W~=~1/{ gamma ~-~1 }[(p_3V_3~-~p_4V_4)~-~(p_2V_2~-~p_1V_1)]

W~=~1/{ gamma ~-~1 }[V_3 lbrace p_3~-~p_4(V_4/V_3) rbrace ~-~V_2 lbrace p_2~-~p_1(V_1/V_2) rbrace ]

W~=~1/{ gamma ~-~1 }[V_3(p_3~-~p_4 r)~-~V_2(p_2~-~p_1 r)]~[Since,~{V_4/V_3}={V_1/V_2}=r]

W~=~1/{ gamma ~-~1 }[(p_3~-~p_4 r)~-~(p_2~-~p_1 r)]~~~[Since,~V_3~=~V_2~=~1]

W~=~1/{ gamma ~-~1 }[p_4 r({p_3/p_4 r}~-~1)~-~p_1 r({p_2/p_1 r}~-~1)]

W~=~1/{ gamma ~-~1 }[p_4 r({r^ gamma /r}~-~1)~-~p_1 r({r^ gamma /r}~-~1)]~[Since,~{p_3/p_4}={p_2/p_1}=r^ gamma ]

W~=~r/{ gamma ~-~1 }[p_4(r^{ gamma - 1}~-~1)~-~p_1(r^{ gamma - 1}~-~1)]

W~=~{p_1 r/}{ gamma ~-~1 }[(p_4/p_1)(r^{ gamma - 1}~-~1)~-~(r^{ gamma - 1}~-~1)]

W~=~{p_1 r}/{ gamma ~-~1 }[k(r^{ gamma - 1}~-~1)~-~(r^{ gamma - 1}~-~1)]~~~[Since,~{p_4/p_1}={p_3/p_2}=k]

W~=~{p_1 r}/{ gamma ~-~1 }(r^{ gamma - 1}~-~1)(k~-~1)

This is the expression for work done (W) in terms of r, k, γ and p1.

Now, let us derive an expression for stroke volume Vs in terms of r.

We know that in Otto cycle,

Vs = V1 – V2       (See p-V diagram above)



Now, Mean effective pressure,


p_m~=~{p_1 r}/{ gamma ~-~1 }(r^{ gamma - 1}~-~1){(k~-~1)/(r~-~1)}

The above expression can be written as

p_m~=~{p_1 r}{(r^{ gamma - 1}~-~1)/(r~-~1)}{(k~-~1)/( gamma ~-~1 )}

which is the required expression for mean effective pressure of otto cycle :)

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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

  • meegoda

    very clear thank you :)

  • Abhishek Deshpande

    one doubt:
    how come V3=V2=1?

    • Surjeet S

      In order to simplify the derivation, we assume the clearance volume (Vc) as 1.
      This is the reason why Vc=V3 =V2=1

  • ashok burnwal

    how work done -ve and +ve taken

    • Surjeet S

      In general, work done on a thermodynamic system is taken as negative work while work done by the system is taken as positive work. Here, in Otto Cycle, work is done on the system in process 1-2 and work is done by the system in process 3-4.

  • Denver Cheddie

    Thanks for this wonderful presentation. Question: is p3/p2 = p4/p1?
    2-3 is the combustion stage, so I would expect that the pressure increases because combustion increases the moles of gas species in the system. Process 4-1 is just an exhaust back to atmospheric conditions.

  • Revathi Bachinappa

    what does gross m.e.p and pumping m.e.p mean?

  • reshmi

    p3/p2 = p4/p1?