In atmospheric dispersion modelling the deposition velocity of particulate matter is often described by: $$ v_d = \frac{1}{r_A+r_B+r_A r_B v_g}+v_g \hspace{10mm}(1) $$ where $r_A$ is the aerodynamic resistance (describing turbulent fluxes in the surface layer), $r_B$ is the viscous resistance (molecular diffusion in the quasi-laminar sublayer) and $v_g$ is the terminal fall velocity of the particles.
How is $(1)$ derived?
It is used in numerous models but I haven't found any references explaining how the expression is derived. The closest I've found is in this book by Seinfeld and Pandis (ISBN: 9781118947401), chapter 19.
There, it is explained that $(1)$ is derived from the following system of equations describing the total flux through an assumed constant flux layer: $$ F = \frac{C_3-C_2}{r_A} + v_g C_3 = \frac{C_2+C_1}{r_B} + v_g C_2 = \frac{C_3-C_1}{r_t} \hspace{10mm} (2) $$ where $C_3$ is the particle concentration at the top of the surface layer, $C_2$ is the concentration between the surface layer and the viscous sublayer and $C_1$ is the concentration of still airborne particles at the surface.
At this stage it is assumed all particles close to the surface will adhere to the surface so that $C_1$ is practically 0. Then $(2)$ can be simplified to: $$ \frac{C_3}{r_t} = \frac{C_3-C_2}{r_A} + v_g C_3 = \frac{C_2}{r_B} + v_g C_2 \hspace{10mm} (3) $$
By elimination of $C_3$ and $C_2$ from $(3)$ one is supposed to end up with $(1)$, since $v_d=r_t^{-1}$. I've tried doing this but end up with this instead: $$ v_d = \frac{1}{r_t} = \frac{1}{r_A+r_B+r_A r_B v_g} + \frac{v_g}{r_A(\frac{1}{r_B}+v_g) +1} $$
So either my algebra is a bit rusty or there are additional assumption involved...