Tire Pressure Math

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Introduction

I woke up this morning to a rather brisk temperature of -23 °F (-31 °C). This is cold. At work, we had a short hallway discussion on the impact of temperature on tire pressure. I mentioned the following "rule of thumb" for tire pressure versus temperature (one of many sources):

Tire pressure decreases by 1 psi for each 10 °F decrease temperature.

This rule of thumb is easily derived and I thought I would put out a quick post on the subject. According to this rule, my tire pressure has dropped 6 psi since I filled my tires last week. The rule as stated assumes a nominal tire pressure of ~40 psi. You will see the rule changed to 2 psi for 10 °F of temperature change for a commercial truck tire, which are often inflated to 80 psi or more. As you will see below, the nominal tire pressure affects the amount of pressure variation with temperature.

Background

My derivation will be based on the ideal gas law. I will also make the following assumptions:

• The ideal gas law is valid for air under pressure.

  The ideal gas law works well for low-pressure gases and gas mixtures, but it becomes less accurate as pressures increase. It turns out that at typical tire pressures, the error is less than 1% (see Appendix A).

• The volume of the tire changes insignificantly with pressure.

  Tires are pretty stiff (e.g. they may have Kevlar or steel cords embedded in them). I know that severe under-inflation is obvious because the tire material is sagging. However, tires near full inflation vary little in volume as pressure changes.

Analysis

General Result

Given the assumptions above, we can derive an expression that states that the percentage change of air pressure in a tire is inversely proportional to the tire air temperature (Equation 1).

Eq. 1

where

• T is the absolute temperature of the tire air.
• p(T) is the tire air pressure as a function of temperature.
• is the percentage tire pressure change with temperature.

We can derive Equation 1 as shown in Figure 1.

Figure 1: Derivation of Equation 1.

Specific Example

Figure 2 shows how we can apply Equation 1 to a common tire pressure scenario to give us the rule of thumb. The tire pressure scenario is:

• Tire air temperature of 20 °C (68 °F).
• Sea level air pressure of 14. 7 psi.
• Nominal tire pressure of 40 psi (gauge pressure)

I should note that Mathcad automatically converts the temperatures to an absolute scale (e.g. Kelvin). So when you see an expression like "(-20)°F", understand that this temperature is being converted into Kelvin.

Figure 2: Derivation of the Rule of Thumb for a Tire Pressure of 40 psi.

We see in the example that the change in tire pressure with temperature is about 1 psi for every 10 °F change in temperature for a 40 psi tire pressure at 68 °F. In Figure 2, I look at two different air temperatures. You will notice that for a given tire pressure, the pressure change with temperature increases with lower reference temperature.

Figure 3 shows a graph of how the tire pressure (40 psi @ 68 °C) varies with temperature.

Figure 3: Tire Pressure Graph.

Conclusion

Straightforward application of the ideal gas law. Pretty simple post.

Appendix A: Compressibility Factor for Air

The compressibility factor for air is shown in Table 1 (Source). The compressibility factor is the ratio of the molar volume of a gas to the molar volume of an ideal gas at the same temperature and pressure. The compressibility factor can be used as a correction factor for the ideal gas law when applied to real gases.

As you can see from the table, the compressibility factor deviates from 1 by a fraction of a one percent even at 60 psi (~5 bar absolute). This means that using the ideal gas law will result in an error of less than 1%.