Question

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.

Short answer

The temperature of air compression is 971.14 K.

Compressed temp is much above the auto ignition temp of diesel so diesel engines don't need any spark plugs.

Step-by-step solution

Given information

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

Explanation

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.