Copy of Two-Node Nodal Kernel With Equivalence Theory

Two-Node Nodal Kernel With Equivalence Theory
Source Expansion Nodal Method
Introduction and implementation approach
Equivalence Theory
For the reflector cross-section generation
General Solution Structure
\phi_g(\xi) = A_g \cosh(k_g \xi) + B_g \sinh(k_g \xi) + \sum_{i=0}^{4} c_{g,i}  P(\xi)
Flux Function Components
The general solution combines hyperbolic functions with Legendre polynomial expansions, providing flexibility in representing spatial flux variations across node boundaries.
Wave Number Definition
The parameter k_g = \frac{h}{2}\sqrt{\frac{\sigma_{r,g}}{D_g}} relates to the diffusion length in each energy group, determining the spatial curvature of the flux solution.
Legendre Expansion
The P(\xi) terms represent Legendre polynomials within the domain [-1,1], enabling accurate representation of complex flux shapes with a small number of terms.
Constraints for the Two-node Kernel
Constraint 1: Average Flux
The first constraint utilizes the node-average flux (\bar{\phi}_g), which is a key quantity in reactor calculations. This constraint primarily determines the coefficient A_g due to the even nature of the hyperbolic cosine function.
Constraint 2-1: Flux Continuity 
For the left node:  \phi_{l,g}(1)
For the right node: \phi_{r,g}(-1)
The flux continuity(  \phi_{l,g}(1)=\phi_{r,g}(-1) ) condition determines the coefficient B_g.
Constraint 2-2: Current Continuity
For the left node: J_l = -D_{l,g} \cdot \frac{d\phi_{l,g}}{d\xi}(1)
For the right node: J_r = -D_{r,g} \cdot \frac{d\phi_{r,g}}{d\xi}(-1)
These current conditions(J_l=J_r) determine the coefficient B_g.
Generalized Equivalence Theory
Flux Discontinuity in the Equivalence Theory (T. Kozlowski, 2017)
(Original reference : Assembly homogenization techniques for light water reactor analysis (Smith, 1984) )
The homogeneous surface flux to preserve heterogeneous solutions is
\phi_s^{hom}=\phi_s^{(L\ or\ R)}=\bar{\phi}^{het} - {1 \over 2} \beta\ J^{het}The left-side and right-side surface fluxes are not equal.   
The discontinuity factor is defined as follows to equalize the homogeneous surface fluxes.
f_L\ \phi_s^L=\phi_s^{het}=f_R\ \phi_s^R
f=\frac{\phi_s^{het}}{\phi_s^{hom}}=\frac{\phi_s^{het}}{{\bar{\phi}}^{het}-\frac{1}{2\beta}J^{het}}
The heterogeneous currents with discontinuity factors are defined by
J^{het}=-2\frac{\left(\frac{\beta_{i+1}}{f_R}\right)\left(\frac{\beta_i}{f_L}\right)}{\left(\frac{\beta_{i+1}}{f_R}\right)+\left(\frac{\beta_i}{f_L}\right)}\left(\phi_{i+1}f_R-\phi_if_L\right)
Equivalence Theory for Relector XS
1. Reference Solution is given as :
Equivalence Theory for Relector XS
2. Nodal calculations for each of nodes, independently
For the left node (or right node)
Determine higher order flux function \phi(\xi) using the average flux(\overline{\phi_l^*}) and boundary surface current(J_l^*).
Using \overline{\phi_l^{\ast}} and J_l^{\ast}, the A and B of \phi(\xi) can be determined, c_i is determined as same to SENM method.
From \phi(\xi), J_c^n can be calculated and it has to be essentially same to J_c^{\ast} due to nodal balance eq. (NBE)
However, the surface flux of nodal method (\phi_s^n) is different from the reference surface flux (\phi_s^{\ast}).
\phi_g(\xi) = A_g \cosh(k_g \xi) + B_g \sinh(k_g \xi) + \sum_{g'=1}^{G} c_{g,i}  P(\xi)NBE : {J_{r,g}-J_{l,g} \over h} + \Sigma_{t,g} \phi_g={\chi_g \over k_{eff}}\Psi + \sum_{g=1}^{G}{\Sigma_{g'g} \phi_{g'}}
Equivalence Theory for Relector XS
3. Calculate the discontinuity factor
The two surface fluxes calculated by the nodal calculations are not equal to each other.
As introducing the discontinuity factors on the left and right-side node, the flux continuity condition can be established.
f_l \phi_s^l = f_r \phi_s^rHowever, the left discontinuity factor can not be applied for the fuel node since the fuel xs is independently generated.
Therefore, the whole discontinuity effect is regarded as the discontinuity of reflector as the following equations.
\phi_s^l = f_{refl} \phi_s^r\,\,\, \text{where} \,\,\, f_{refl}={f_r \over f_l}Finally, the homogeneous reflector xs is modified by the discontinuity factor to be applied for the core calculation.
\Sigma_{refl,x}' = {\Sigma_{refl,x} \over f_{refl}}
Conclusion
General Solution Structure
Combines analytic functions with polynomial expansions to accurately represent neutron flux distributions.
Two-Node Kernel Constraints
Ensure proper nodal balance while preserving computational efficiency.
Equivalent Theory for Reflectors
Introduces discontinuity factors that effectively handle the fuel-reflector interface.
Modified Cross-Sections
The modified homogeneous reflector cross-sections (\Sigma_{refl,x}' = {\Sigma_{refl,x} \over f_{refl}}) provide an accurate representation for core calculations.