The math checks out: a bit of theory in practice

When we had an HVAC service person out a few months ago, he pointed out that whoever installed our system did something right: on a long run of duct with vents along the way, it should get narrower as it goes on in order to keep the pressure up.

This is fairly intuitive, but the math also checks out. I was thinking about this the other day and remembered the ideal gas law from high school chemistry.

$$pV = nRT$$

$p$

Pressure

$V$

Volume of gas

$n$

Amount of gas

$R$

Ideal gas constant

$T$

Temperature

$R$ is a constant, and $T$ is essentially constant, too, so we can ignore them and state that

$$pV \propto n$$

In my HVAC scenario, we have a long duct with vents along the way. Air escapes through those vents, meaning $n$ is decreasing. Pressure is what we’re looking to maintain, so looking at it in terms of pressure,

$$p \propto \frac{n}{V}$$

So in order to hold pressure constant as $n$ is decreasing, volume has to come down, too. Good thing they made the tube narrower!