Time-dependent thermal expansion of air?

Every solid, gas, liquid has a coefficient of thermal expansion (measured in e.g. %/K). This assumes an equilibrium state at the new temperature

My question is: what is the time rate of change thermal expansion of air (i.e. the units would be %/K/s)? Therefore enabling you to answer not only the question “what is the new volume?” but also “how long did it take to reach that new volume?”

I’m not expecting a simple relationship as there are spatial and physical non linearities in the problem. But are there e.g. empirical estimates?

Application: I have hot gas flowing into a duct in which a fan is installed. The gas is not at equilibrium (i.e. it is still expanding). The velocity in the duct will increase above the fan-induced velocity due to the expansion of the gas. I want to quantify this increase in velocity.

Is the answer: solve the compressible Navier Stokes equations?