Thank you for the reply! Really appreciate it, as I knew I was making some silly mistakes and wasn't even sure where to start.
This is dimensionally incorrect. How did you get it from Fick's first (not second) law?
I actually got this from some university material, but it seems I may have miscalculated it. The problem they showed was a tank of water exposed to air, therefore making the water evaporate slowly and diffuse as vapour outwards into the air. They demonstrated how to calculate the flowrate of water vapour from the surface of the water out into the air, which they did via Fick's law of diffusion. I figured maybe it would be possible to rearrange that into a spherical coordinate for a bubble, so I got the following:
Φm = D*A*(-dc/dr) = D*(4*pi*r^2)*(-dc/dr)
Φm/(D*4*pi*r^2)*dr = -dc
Integrating both sides with the following boundary conditions: c(r) = c, c(R) = c0 (e.g. water concentration is higher at the outside surface (r = R) of the bubble)
Φm/(D*4*pi)*(1/r - 1/R) = (c - c0)
Φm = (D*4*pi)*(∆c)/(1/r - 1/R)
I already knew something was off as the radii don't match up to the concentrations in order to make the flowrate positive, but I was hoping maybe something could be done to fix that, but I guess I was being too hopeful.
Where do you get this from? I find values of ca. 2e-9 m2/s for the self-diffusion coefficient. Have you found something different for diffusion across a surface?
I actually just got this off of wikipedia. The diffusion coefficient of water vapour in air is given as 0.282 cm2
Shouldn't that be something more like 103 kg/m3 from liquid to vapour?
I'm wondering if a diffusion approach is appropriate at all, as evaporation, unlike diffusion, is an energy-absorbing process, and its rate depends on how much energy (if any) you put into the liquid.
I see what you mean, and I should have actually clarified initially that the concentration c0 is defined as the partial pressure of water vapour times the molar mass of water. So it isn't actually kg/m3 of water in liquid water, but the concentration of water vapour immediately at the gas-liquid interface, so at r = R.
Overall I was also skeptical of the whole diffusion approach, especially since there are so many other factors involved (e.g. convection, eddies, etc), but I don't have the expertise in the slightest to try to solve the model with something like CFD. I found some theses online that included diffusion using a modified Fick's law ( Φm'' = (ka)*Vliquid*(Ca - Ca,eq) ), so I tried to potentially imitate those. Do you possibly have any other suggestions on how to model this kind of situation?