Diffusion equation from neutron flux¶

The diffusion equation, based on Fick’s law, provides an analytical solution of spatial neutron flux distribution in the multiplying system.

Links:

NuclearPower, possible similar formula.

diffusion_coefficient¶: neutron_diffusion_coefficient.

Symbol:: D
Latex:: \(D\)
Dimension:: length

macroscopic_fission_cross_section¶: macroscopic_cross_section of fission.

Symbol:: Sigma_f
Latex:: \(\Sigma_text{f}\)
Dimension:: 1/length

macroscopic_absorption_cross_section¶: macroscopic_cross_section of absorption.

Symbol:: Sigma_a
Latex:: \(\Sigma_\text{a}\)
Dimension:: 1/length

effective_multiplication_factor¶: effective_multiplication_factor.

Symbol:: k_eff
Latex:: \(k_\text{eff}\)
Dimension:: dimensionless

position¶: position.

Symbol:: x
Latex:: \(x\)
Dimension:: length

neutron_flux¶: neutron_flux as a function of position.

Symbol:: Phi(x)
Latex:: \(\Phi{\left(x \right)}\)
Dimension:: 1/(area*time)

neutron_flux_laplacian¶: Laplacian of the neutron_flux as a function of position.

Symbol:: Laplace(Phi)(x)
Latex:: \(\nabla^{2} \Phi{\left(x \right)}\)
Dimension:: 1/(length**4*time)

law¶

-D * Laplace(Phi)(x) + Sigma_a * Phi(x) = nu / k_eff * Sigma_f * Phi(x)

Latex:: \[- D \nabla^{2} \Phi{\left(x \right)} + \Sigma_\text{a} \Phi{\left(x \right)} = \frac{\nu}{k_\text{eff}} \Sigma_text{f} \Phi{\left(x \right)}\]