I know that forced convection requires some external bulk flow, but is there an equation to determine which form of Convection is dominant? I looked around and found the Richardson number $\mathrm{Ri}=\frac{\mathrm{Gr}}{\mathrm{Re}^2}$ but I don't know how to interpret the number or if it even applies to mass transfer.
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The Richardson number $\mathrm{Ri} = \frac{\mathrm{Gr}}{\mathrm{Re}^2} = \frac{\text{buoyancy term}}{\text{flow gradient term}}$ (1) can be used to determine if forced or natural convenction dominate in thermal convection.
$\mathrm{Re}$ = Reynolds number
$\mathrm{Gr}$ = Grashof number
You can neglect natural convection for $\mathrm{Ri} < 0.1$ and forced convection for $\mathrm{Ri} > 10$.
I won't get into more detail about the number as you can learn more about it in the linked Wikipedia article.
The interesting question is: is this applicable to mass convection too?
I would lean towards saying yes, but I can't tell you for sure; it should be applicable in order to get at least a ballpark estimate.
Dimensional analysis of convective mass transfer yields a number of pertinent dimensionless groups that are, in general, analogous to dimensionless groups for convective heat transfer.
[...]
A close analogy exists between convective heat and convective mass transfer owing to the fact that conduction and diffusion in a fluid are governed by physical laws of identical form, that is, Fourier’s and Fick’s laws, respectively.
Kreith, F.; Boehm, R.F.; et. al. “Heat and Mass Transfer” Mechanical Engineering Handbook p. 266 ff.(2)
Give that chapter a read and decide for yourself what you can apply from thermal convection.
