# Adiabatic compression in a diesel engine

Reference: Daniel V. Schroeder, An Introduction to Thermal Physics, (Addison-Wesley, 2000) – Problem 1.37.

As an example of adiabatic compression of an ideal gas, consider the compression of air in a diesel engine. Atmospheric air (at a temperature of, say, ${10^{\circ}\mbox{C}=283\mbox{ K}}$) is quickly compressed to ${\frac{1}{20}}$ of its original volume. From the relation

$\displaystyle VT^{f/2}=\mbox{constant} \ \ \ \ \ (1)$

where ${f}$ is the number of degrees of freedom of a gas molecule, we can estimate the temperature of the air after compression. As most air molecules are diatomic, we can take ${f=5}$ (3 translational + 2 rotational degrees of freedom; this assumes that vibrational modes are frozen out, although I’m not sure that’s true for higher temperatures), so the temperature ${T_{f}}$ after compression is

 $\displaystyle T_{f}$ $\displaystyle =$ $\displaystyle \left(\frac{V_{i}}{V_{f}}\right)^{2/f}T_{i}\ \ \ \ \ (2)$ $\displaystyle$ $\displaystyle =$ $\displaystyle 20^{2/5}\times283\ \ \ \ \ (3)$ $\displaystyle$ $\displaystyle =$ $\displaystyle 938\mbox{ K}\ \ \ \ \ (4)$ $\displaystyle$ $\displaystyle =$ $\displaystyle 665^{\circ}\mbox{ C} \ \ \ \ \ (5)$

The autoignition temperature for diesel is ${256^{\circ}\mbox{ C}}$ so the fuel will automatically ignite when the air is compressed, which is why diesel engines don’t need spark plugs.