When to use ideal gas law in fluid mechanics?

The ideal gas law (aka the equation of state) is given by

$$ p/\rho_N = k_BT, $$ where $\rho_N$ is number density. When am I allowed to use this to describe a fluid?

Answer

As it turns out, you can use it for the vast majority of the time. The ideal gas law is considered "ideal" because it assumes interactions between the gas molecules only occur due to collisions and no long-range forces are present. If the fluid you are modelling is non-polar and doesn't have things like van der Waals forces between molecules, then the ideal gas law is a great approximation.

It is also considered ideal because it assumes all collisions are equal -- so when two molecules hit one another, they react in the same way. This is because it assumes all molecules are the same size. So if you have a gas that has molecules that are roughly the same size (like air, since nitrogen and oxygen molecules are quite close), then the approximation is a good one. But if you are doing some complex combustion processes where you have heavy organic fuels burning and releasing things like hydrogen trace species, then there is a wide range of molecule sizes and the model doesn't work so well anymore.

And of course, whether you choose the thermally or calorically perfect version of the ideal gas law also depends on your conditions. Calorically perfect works just fine when the change in temperatures in the fluid is small (note, the temperature doesn't have to be small, just the variation in temperature -- you can choose the heat capacities to be accurate at any temperature you want, they are just fixed). Thermally perfect is good over the range of temperatures that one has valid data for the heat capacities.

So to summarize -- you can use it whenever:

1. Molecular interactions are due to collisions alone (no long range forces)


2. Molecules are close in size


3. If the gas has a wide range of temperatures, the heat capacities need to vary with temperature; otherwise, careful selection of constant values will work

Of course, it's quite common to use it anyway when the above assumptions are not all that great. Many simulations of combustion use it despite the wide range in molecular sizes. It all depends on how accurate you want to be for the cost and how much data is actually available to use better models.