# Solution of diffusion equation for 2 infinite slabs reactor

I have exercise where I have to calculate flux for reactor with 2 same infinite slabs of multiplying medium in vacuum. I have to calculate flux inside slabs and in the slot between them. I am following this calculation as my guideline. I have problems with finding boundary conditions and I also want to know if my calculation makes sense.

I am solving diffusion equation for 1 D system

$$D \frac{d^2 \phi(x)}{dx^2}+B_g^2\phi(x)=0,$$ where $$B_g^2=\frac{v\Sigma_f-\Sigma_a}{D}$$, $$\Sigma_f$$ is fission cross section and $$\Sigma_a$$ absorption cross section.

The general solution is:

$$\phi(x)=A\cdot sin(B_g x)+C \cdot cos(B_gx).$$

For external boundary condition I am assuming vacuum boundary condition, eg. $$\phi(\pm|3/2a+d|)=0$$. I know, this is only mathematical tool, and the flux in vacuum just has some constant value, which equals the flux on the outer edge of slab. But I don't know, what are boundary conditions on internal borders of slabs ($$x=\pm a/2$$ in sketch).

So my idea is that flux in the slot between slabs $$\phi_{i}$$ is just sum of fluxes, which both slabs will have on outside ($$\phi_{out}$$), if there would be only 1 slab. In that case I would have relation $$\phi_i=2\cdot \phi_{out}$$. This leads to the condition $$\phi(1/2 a)=2 \cdot \phi(3/2 a)$$.

This is the point, where I cant, continue the calculation. I this the right choice of boundary conditions, and what is the solution for them?

• The problem with your idea is that you are considering the leakage out of the two source regions into the center area, but you also need to consider that there will be leakage into the source regions from the other source. This is not an easy problem to solve. – NuclearFission May 1 '20 at 11:46
• In your opinion, what will happens if you increase or decrease the width of the central vacuum region ? Will it change the result ? – manu190466 May 1 '20 at 20:47