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.

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