Can one prescribe flux boundary conditions on internal boundaries?

For space-dependent boundary conditions, I want to test a scenario as shown in the image. I want to set up the boundary condition for the interior boundary (7) along with other boundaries. Note that concentration in R2 is always greater than R1 and there is a known measured diffusion flux profile through the interior boundary (7).

From diffusion-advection module, it seems the option I can choose is the "flux discontinuity". For the test example, I just used the fixed diffusion flux as the boundary condition for (7).  I will also use the space-dependent diffusion flux along X as the interior boundary condition later.

However, I always found such boundary setup was not right for simulations, with materials unbalanced. My questions are:

(1) Is it the right setup for the interior boundary? What is the rule of setting positive or negative values?
(2) if boundary (2) and (4) are for flow-in and boundary (3) and (6) are for flow-out, will the diffusion flux from the top directly go out through these boundaries? I found the materials cannot be balanced, so I am wondering if there is some leaking from the test domain of R1 and R2.

I want to set up the boundary condition for the interior boundary (7) along with other boundaries. Note that concentration in R2 is always greater than R1 and there is a known measured diffusion flux profile through the interior boundary (7).

It is not really clear to me what specific boundary condition you want to apply to boundary (7), but it sounds like you want to set or prescribe a specific value of the flux on the boundary?. If this is what you are trying to do then this is not possible on internal boundaries. Specifically, the "flux discontinuity" boundary condition prescribes the difference in the flux/gradient between the two sides. See the image below for an explanation of the boundary conditions available to set on internal boundaries (the normal on internal boundaries is defined as pointing from the subdomain with higher number to lower).

Thanks for this explanation.
If I do not use the internal boundary, does it mean the featool will use the default for the interface (i.e. continuity)? If I choose discontinuity, how to set up the value of N?

James wrote

If I do not use the internal boundary, does it mean the featool will use the default for the interface (i.e. continuity)?

Yes, the default boundary condition for internal boundaries is always "continuity".

James wrote

If I choose discontinuity, how to set up the value of N?

It is hard to give a general recommendation as it is a very specific and uncommon boundary condition. Typically, if you really need this type of condition your application and problem would dictate an appropriate value.