The expansion ratio R is the ratio of

Where v₁ = volume at the beginning of expansion and v₂ = volume at the end of expansion.

Concept:

For the Diesel cycle:

Processes involved in compression engine (diesel cycle) are:

3-4: Reversible adiabatic compression

4-1: Constant pressure heat addition

1-2: Reversible adiabatic expansion

2-3: Constant volume of heat rejection

Now 

The compression ratio is given by

\(r = \frac{{{V_3}}}{{{V_4}}}=\frac{V_2}{V_4}\).....(V2 = V3)

The cut-off ratio is given by

\(r_c = \frac{{{V_1}}}{{{V_4}}}\)

The expansion ratio is given by 

\(r_e = \frac{{{V_2}}}{{{V_1}}}\)

\(r_c × r_e=\frac{V_1}{V_4} × \frac{V_2}{V_1}=\frac{V_2}{V_4}\)

but we know that, 

\(r = \frac{{{V_1}}}{{{V_2}}}=\frac{V_4}{V_2}\)

∴ rc × re = r