How to estimate the settling time of atmospheric particulates as a function of aerodynamic size?
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.