Help with solving diffusion problem involving second order differential equations

I’m looking for help in understanding and implementing the formula to compute the saturation level of air with Butanol vapour delivered from a cylindrical wick. I think the best way of modelling this is based on finite diffusion into a solid cylinder and I’ve found a paper and YouTube video on it. My problem is I don’t understand what’s being said enough to get formula working in Mathematica. I would like to talk this through with someone who can confirm the modelling approach being taken and explain or provide the required formula. The equations to solve will involve Laplace transforms, partial second order differentials, or both.