Pressure and temperature influence on the performance of a naturally aspirated engine.

The ideal gas law

The ideal gas law is the equation of state of a hypothetical ideal gas. It is a good approximation to the behavior of many gases under many conditions, including air. This law is expressed as:

$PV = nRT$

Where $n$ is the amount of substance of gas, $R$ is the ideal gas constant, $P$ is the absolute pressure, $V$ is the volume and T is the temperature. In SI units, $n$ is measured in moles. $R$ has the value 8.314 J·K−1·mol−1. The temperature used in the equation of state is an absolute temperature: in the SI system of units kelvins (K).
As the amount of substance could be given in mass instead of moles, an alternative form of the ideal gas law is useful. The number of moles ($n$) is equal to the mass ($m$) divided by the molar mass ($M$):

$n = {m \over M}$

By replacing $n$, and with density $ρ = m/V$, we get:

$PV = {m \over M}RT$

$P = ρ{R \over M}T$

So the density of air is proportional to P/T.

Pressure influence

With the assumption of a constant filling index the mass of air entering the cylinder during the intake stroke is proportional to the density of air, since the volume remains the same. The density is at its turn proportional to the absolute pressure which leads to the following conclusion at a constant ambient temperature:

Power at $P_1 = {P_1 \over P_0}$ * Power at $P_0$

Temperature influence

With the assumption of a constant filling index the mass of air entering the cylinder during the intake stroke is proportional to the density of air, since the volume remains the same. The density is at its turn proportional to the absolute temperature (K) which leads to the following conclusion at a constant pressure:

Power at $P_1 = {T_0 \over T_1 }$ * Power at $P_0$

However, this equation should be altered for the effects of heating of the air. As the air flows in the cylinder its temperature will increase because of heat transfer from the cylinder walls to the air. As the temperature of the air increases, this heat transfer will decrease. This means that the density of the air in the cylinder will not decrease proportional to T but will decrease less than that. A commonly used formula is as follows:

Power at $P_1 = \sqrt{T_0 \over T_1 }$ * Power at $P_0$

Experimental Torque/Power curves at different atmospheric pressures at 35°C atmospheric temperature: