What formula can serve as an approximate estimate of the time taken for the smell of a perfume to reach somebody?

Question

I am in an attempt to calculate the time required for the smell of a bottle of perfume to reach a person's nose $10$m away. Real life experience tells me that it takes several seconds. I tried to work out this time theoretically in two different ways, but none of them gives me the right amount of time.

1. I tried to find the average speed of air particles, but in room temperature, this is something near $300$ $m/s$. Clearly, this cannot be used to calculate the time taken by the perfume molecules to travel a certain distance. Their apparent speed is much slower, in that collision and the random walk takes place.
2. So I tried to use the random walk formula here instead: $\langle r^2\rangle=6Dt$. But from my calculation, it takes the particles more than half a year to travel 5 meters!

I know why both method 1 and 2 are problematic. Air is not static. It is flowing all the time. I believe that it is the air's flow that allows the particles to travel 5 meters in just a few seconds. But I struggle to find a formula and a theory to explain this kind of random flow of gas. Can anyone give a model for this?

Answer 1

Here is a way to estimate. From this Wiki article diffusivity of CO$_2$ in air is 16 mm$^2$/s, and I assume the value is similar for perfume. Time scale for molecular diffusion over a distance $L$ is $\sim L^2/\nu=5^2/16\times10^{-6}\approx 18$ days.

But as you guessed correctly the actual time is much less because there is motion of air inside the room, either due to winds or due to convection currents. If there is a wind blowing with speed $U$ then the time for perfume to travel distance $L$ is $\sim L/U$; for $U\sim 1$ ms$^{-1}$ the time is 5 second.

The more complicated case is when there is convection, say due to temperature difference between room-air and your body (or any other factor that makes room air unstably stratified). Estimate for a similar scenario is provided in the first chapter of A first course in Turbulence by Tennekes and Lumley. If the convection is driven by temperature difference $\Delta T$ then typical acceleration of air parcels is $\sim g\beta\Delta T$, where $\beta$ is volumetric expansion coefficient of air. If $S$ is the typical size of eddies or convection cells that are set up, then typical convection velocity is $\sim\sqrt{Sg\beta\Delta T}$ (dimensional reasoning). Perhaps it is reasonable to assume that eddy size is of the order of dimensions of human body, i.e. $S\sim 1$ m. Then for $\Delta T\sim 10~^\circ$C, $\beta=0.0034~/^\circ$C, convection velocity $\sim 0.5$ m/s. Convection velocity is not like a wind which blows in one direction; nevertheless if convection exists in the entire room (which it does in usual circumstances) convection currents carry the perfume farther more quickly than would molecular diffusion (see also eddy diffusivity). If convection is occurring throughout the room then the time for diffusion of perfume may be estimated as $\sim$5 (m)/0.5 (ms$^{-1}$)=10 second.