How to estimate the settling time of atmospheric particulates as a function of aerodynamic size? Understanding the behavior of particulates in the atmosphere is important for modeling climate, weather and public health. They can be nucleation sites for rain, fog and smog, they can have thermal impact due to absorption of sunlight and possibly radiation in thermal infrared, and the smallest can deposit chemicals deep inside our lungs.
Their behavior can be characterized by their particulate aerodynamic diameter, one way to characterize their size according to their aerodynamic behavior.
The rate at which particulate matter deposited in the atmosphere settle to the ground is a strong function of size. Throw a handful of sand into the air and it returns to the ground within second, create soot with fire and it rises in the column of hot air produced and may take weeks, months or even years before it returns to Earth.
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Is there any way to estimate at least approximately as a function of size the time it takes airborne particulates found high in the atmosphere to return to Earth, and which ones return due to gravity and which due to formation of precipitation? I know it's a complex topic and it may depend strongly on the altitude at which they start. Perhaps some rules-of-thumb or examples might be sufficient to get an idea of what's involved in such estimates. That may be helpful in order to formulate more specific follow-up questions.
I took a class once, and we had an approximate equation for an particle (equation sheet is still up):
$$\frac{d\vec{v}}{dt}=\frac{\rho_{particle}-\rho_{air}}{\rho_{particle}}\vec{g}-\frac{3\rho_{air}C_D}{4\rho_{particle}CD_{particle}}\vec{v}|\vec{v}|$$ where $\rho$ is density, $\vec{g}$ is the gravity vector (usually $=g\hat{k}$ but can be changed if the particle has a charge), $C_D$ is the surface drag coefficient, $C$ is the Cunningham correction factor, $\vec{v}$ is the particle velocity, and $D_{particle}$ is the aerodynamic diameter of the particle.
Solving for $\frac{d\vec{v}}{dt}=0$ to get the terminal velocity is one option. I can't remember if the above equation considers environmental motion. You can work through the other equations in the equation sheet and see that even getting a terminal velocity is complicated and requires an iterative method since $C_D=C_D(\vec{v})$. This might work for engineering purposes, but may not practical for atmospheric modeling.
There is another take on this issue of deposition. One parameterization for dry deposition is the (see pgs. 6-9): $$v_{deposition}=\frac{1}{r_a+r_b+r_a r_b v_s}+v_s$$
$$r_a=\frac{1}{ku_*}\left[\ln\left(\frac{z-d}{z_0}\right)-\Psi_h\left(\frac{z}{L}\right)\right]$$
$$v_s=\frac{D_p\rho_{particle}g}{18C\mu}$$, where $v_s$ is the settling velocity, $\mu$ is the dynamic viscosity, $L$ is the Monin-Obukhov length, $\Psi_h$ is the integrated similarity/Businger-Dyer function for heat, $k$ is the von-Karman constant, $z_0$ is the roughness length, $d$ is the displacement height, $u_*$  is the friction velocity, and $r_b$ is the viscous  sub-layer resistance.