Trying to figure out the mathematical model that might correlate to laboratory results.
I have a cylindrical pressure vessel (picture a can) with height, h, and radius, r. It is filled with distilled water and held at a high ambient hydro-static pressure, P. From the bottom of the can (h=0), I release N2 gas into the vessel at flowrate, f. Using Henry's Law, the gas diffuses into the liquid very quickly because of pressure P. If I have an N2 sensor on the top of the vessel....how long will it take for it to start seeing concentrations of N2 resulting from the input at the bottom of the vessel?
I know this has to do with Fick's law but I'm having a lot of trouble wrapping my head around it.
One possible solution is as follows.
The model with one-dimensional fick diffusion
where C(x,t) is the concentration of the N2; n is the diffusivity of the N2 in water.
The solution is
An approximated time for the sensor response is