Understanding the Adjoint Multigroup Diffusion Equation

[DimZero]

Hello everyone,

I wish to know if someone could help me with the adjoint multigroup diffusion equation. In particular with the terms that make up the macroscopic removal cross section. Below, both the multigroup diffusion equation and its adjoint are shown, but I'm not sure about the latter. I would be extremely grateful to anyone who could help me out with it.

Thank you very much.

Multigroup Diffusion Equation: $$ -D_i\nabla^2 \phi_i + \left( \Sigma_{c,i} + \Sigma_{f,i} \right) \phi_i + \left( \Sigma_{el,i\rightarrow \left( i + 1 \right)} \right) \phi_i + \sum_{j = i + 1}^{Ng} \left( \Sigma_{el,i\rightarrow j} \right) \phi_i = \left( \Sigma_{el,{\left(i - 1 \right)}\rightarrow i} \right) \phi_{\left(i - 1 \right)} + \sum_{j = 1}^{i - 1} \left( \Sigma_{inel,j\rightarrow i} \right) \phi_j + \chi_i \sum_{j = 1}^{Ng} \left( \nu \Sigma_{f,j} \right) \phi_j $$

Adjoint Diffusion Equation: $$ -D_i\nabla^2 \phi^*_i + \left( \Sigma_{c,i} + \Sigma_{f,i} \right) \phi^*_i + \left( \Sigma_{el,i\rightarrow \left( i + 1 \right)} \right) \phi^*_i + \sum_{j = i + 1}^{Ng} \left( \Sigma_{el,i\rightarrow j} \right) \phi^*_i = \left( \Sigma_{el,i \rightarrow \left( i + 1 \right)} \right) \phi^*_{ \left(i +1 \right)} + \sum_{j = i + 1}^{Ng} \left( \Sigma_{inel,i\rightarrow j} \right) \phi^*_j + \nu \Sigma_{f,i} \sum_{j = 1}^{Ng} \left( \chi_j \phi^*_j \right) $$

 

[Astronuc]

I'm looking at this. Has one tried to write the equations for 2 groups and compared it to examples from textbooks?

What text is one using?

 

[DimZero]

Hello,

thanks for replying.
No, I haven't tried this but I will and see what I can get from it. Anyways, I'm not using any books (regarding the adjoint equation). I've derived the diffusion equation for the real flux using several notes, but I cannot find anything on the adjoint diffusion equation. I derived that form of the adjoint equation myself and I'm not sure whether it's right or wrong, this is why I was seeking for help.

 

[DimZero]

In order to derive the diffusion equation for the real flux I used the multigroup diffusion equation in its compact form, thus containing the leakage, the total macroscopic cross section, the scattering source and the fission source terms. Then I wrote the total macroscopic cross section term and the scattering one in their explicit form, showing all the parameters. Furthermore, putting them together and using the "directly coupled" approximation for the elastic scattering and the "only down scattering" for the inelastic scattering, I've obtained the equation presented above.
The same approximations have been used to deduce the adjoint diffusion equation, considering that the quantity there considered is not a real flux.

 

[rpp]

It is a little harder for me to dig through because you've separated the inelastic and elastic, and assume downscatter can only occur over one group.
Many of the terms on the LHS are commonly grouped together as a "removal" cross section.

From Hebert, "Applied Reactor Physics":

Multigroup Diffusion Equation: $$ -D_i\nabla^2 \phi_i + \Sigma_{ri} \phi_i =
\sum_{j \ne i}^{Ng} \left( \Sigma_{j\rightarrow i} \right) \phi_j + \chi_i / \lambda \sum_{j = 1}^{Ng} \left( \nu \Sigma_{f,j} \right) \phi_j $$

Adjoint Diffusion Equation: $$ -D_i\nabla^2 \phi^*_i + \Sigma_{ri} \phi^*_i =
\sum_{j \ne i}^{Ng} \left( \Sigma_{i\rightarrow j} \right) \phi^*_j + \nu \Sigma_{f,i} /\lambda \sum_{j = 1}^{Ng} \left( \chi_j \phi^*_j \right) $$

You basically switch the indices on the scattering matrix, and switch the indices on the $\chi$ and $\Sigma_f$ terms.

 

[DimZero]

It is a little harder for me to dig through because you've separated the inelastic and elastic, and assume downscatter can only occur over one group.
Many of the terms on the LHS are commonly grouped together as a "removal" cross section.

From Hebert, "Applied Reactor Physics":

Multigroup Diffusion Equation: $$ -D_i\nabla^2 \phi_i + \Sigma_{ri} \phi_i =
\sum_{j \ne i}^{Ng} \left( \Sigma_{j\rightarrow i} \right) \phi_j + \chi_i / \lambda \sum_{j = 1}^{Ng} \left( \nu \Sigma_{f,j} \right) \phi_j $$

Adjoint Diffusion Equation: $$ -D_i\nabla^2 \phi^*_i + \Sigma_{ri} \phi^*_i =
\sum_{j \ne i}^{Ng} \left( \Sigma_{i\rightarrow j} \right) \phi^*_j + \nu \Sigma_{f,i} /\lambda \sum_{j = 1}^{Ng} \left( \chi_j \phi^*_j \right) $$

You basically switch the indices on the scattering matrix, and switch the indices on the $\chi$ and $\Sigma_f$ terms.

Unfortunately I was forced to break down each term, in order to proceed with the iterative process needed to calculate the adjoint flux. Anyways, starting out from the equation that I wrote and proceeding backwards, I end up with your equation, thus mine should be correct. I only have some notes and no book to look up to, and it can be quite tricky to derive some equations.
Thanks for the reply, it was really helpful.