Question

How to account for the interface between two different phases in a discretized diffusion model?

Answer 1

The two media are next to each other and the geometry is spherical because the system should simulate the diffusion out of a gas bubble. I have made a diffusion model for the gas phase (the left compartment) and a corresponding diffusion model for the liquid phase (the right compartment). The two diffusion models are as follows:

∂cleft∂t∂cright∂t=Dleft⋅1r2∂∂r(r2∂cleft∂r)=Dright⋅1r2∂∂r(r2∂cright∂r)∂cleft∂t=Dleft⋅1r2∂∂r(r2∂cleft∂r)∂cright∂t=Dright⋅1r2∂∂r(r2∂cright∂r)

Then I have discretized the volume. So the PDEs are turned into ODEs of the form:

dci,leftdtdci,rightdt=D(ci+1,left−2⋅ci+ci−1,leftΔr2+2ri,left⋅ci+1,left−ci−1,left2⋅Δr)=D(ci+1,right−2⋅ci+ci−1,rightΔr2+2ri,right⋅ci+1,right−ci−1,right2⋅Δr)dci,leftdt=D(ci+1,left−2⋅ci+ci−1,leftΔr2+2ri,left⋅ci+1,left−ci−1,left2⋅Δr)dci,rightdt=D(ci+1,right−2⋅ci+ci−1,rightΔr2+2ri,right⋅ci+1,right−ci−1,right2⋅Δr)

When discretizing I got the following expressions:

d2cidr2dcidr=ci+1−2⋅ci+ci−1Δr2=ci+1−ci−12⋅Δrd2cidr2=ci+1−2⋅ci+ci−1Δr2dcidr=ci+1−ci−12⋅Δr

But how does one incorporate the following two boundary conditions at the gas-liquid-interface, where HH is Henry's constant?

cleft,interfaceJleft,interface=H⋅cright,interface=Jright,interface(1)(2)(1)cleft,interface=H⋅cright,interface(2)Jleft,interface=Jright,interface

Meaning that there is equal flux across the interface between the two compartments.

The concentration in the last compartment of the gas phase (before the interface) is approximated by linear extrapolation. This means that the concentration just before the interface is given by:

c3=c2−c1r2−r1⋅r3+c1−r1c2−c1r2−r1c3=c2−c1r2−r1⋅r3+c1−r1c2−c1r2−r1

In this example it is assumed that only three compartments are present on each side of the interface.

took the help from Google..
hope it helps you