The Diesel cycle is the thermodynamic cycle which approximates the pressure and volume of the combustion chamber in a Diesel engine. In the Diesel cycle, the working medium in the combustion chamber is

  1. compressed adiabatically from V1, p1 to V2 < V1, p2 > p1,
  2. expanded isobarically at p2 from V2 to V3 > V2,
  3. expanded adiabatically from V3, p2 to V1 and p4 > p1,
  4. cooled isochorically at V1 from p4 to the initial pressure p1.

The heat absorbed by the medium is given by Q=23CpdT. Assuming the working medium is an ideal monatomic gas, show that the efficiency of the Diesel cycle can be written in the form

η=1γ1R31γR21γR31R21

P-V Diagram for idealized Diesel cycle.

P-V Diagram for idealized Diesel cycle.

Efficiency of the Diesel Cycle

Because the process 12 and 34 are adiabatic, heat is not lost during those processes, so the efficiency can therefore be written in terms of the heat gained and lost, QH and QL, as indicated on the diagram.

From the equation given above we can say that the heat gained QH is given in terms of specific heat and temperature by:

(1)QH=Cp(T3T2)

We can likewise deduce the heat lost as Q=41CVdT such that

(2)QL=CV(T1T4)

The thermal efficiency is defined as1 η=QH+QLQH, which we can express using (1) and (2) in the form of specific heats and temperatures (3)η=1+CV(T1T4)Cp(T3T2)

Using the ideal gas law PV=nRT where the ns and Rs cancel, and the adiabatic index γ=CpCV we can further write

η=1+γ1P1V1P4V4P3V3P2V2

From the diagram it can be seen that the volume V is the same in state 4 as in 1, i.e. V4=V1. Likewise for pressure, P3=P2. Therefore we can reduce our number of variables so that

η=1+γ1V1(P1P4)P3(V3V2)

Dividing the fraction through by P3V1:

η=1+γ1P1P3P4P3R31R21

where R31:=V3V1 and R21:=V2V1.

For an adiabatic expansion, PVγ is constant. Exploiting this fact, we can see from the diagram that the following relations hold:

P1P3=(V2V1)γ=R21γP4P3=(V3V1)γ=R31γ

Finally, putting these back in and playing with minus signs:

η=1γ1R31γR21γR31R21;


  1. See MIT’s brief page about the Diesel cycle. It’s also whence I nicked my diagram. ↩︎