In a Diesel engine, atmospheric air is quickly compressed to about 1/20 of its original volume. Estimate the temperature of the air after compression, and explain why a Diesel engine does not require spark plugs.

Atmospheric air is quickly compressed to about 1/20 of its original volume.

The expression below explains the relationship between V and T as

$V{T}^{fR2}=Cons\tan t..............................\left(1\right)$

Where V = volume and T = Temp

Consider the air molecules are diatomic, so its degree of freedom is

f=5( 3 translation + 2 rotational).

Write the equation for final and initial conditions and equate them to find final temp as below

$$\begin{gathered} V_i{T}_i^{\left(\frac{f}{2}\right)}=V_f{T}_f^{\left(\frac{f}{2}\right)} \\ {T}_f^{\left(\frac{f}{2}\right)}=\frac{V_i}{V_f}{T}_i^{\left(\frac{f}{2}\right)}...................................\left(2\right) \\ {T}_f={\left(\frac{V_i}{V_f}\right)}^{\left(\frac{2}{f}\right)}{T}_i.....................................\left(3\right) \end{gathered}$$

Substitute values and calculate to get final temp

$$\begin{gathered} T_f={\left(\frac{V_o}{\left(\frac{V_o}{20}\right)}\right)}^{\left(\frac{2}{5}\right)}293K \\ {T}_f=971.14\mathrm{K} \end{gathered}$$

We know that the auto ignition temperature for diesel is 210^oC

So the fuel will automatically ignite when the air is compressed as temp is much above the auto ignition temp.

For this reason diesel engines don't need any spark plugs.