Click here
Click here
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
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 relates to the diffusion length in each energy group, determining the spatial curvature of the flux solution.

Legendre Expansion

The terms represent Legendre polynomials within the domain , 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 (), which is a key quantity in reactor calculations. This constraint primarily determines the coefficient due to the even nature of the hyperbolic cosine function.

Constraint 2-1: Flux Continuity

For the left node:

For the right node:

The flux continuity( ) condition determines the coefficient .

Constraint 2-2: Current Continuity

For the left node:

For the right node:

These current conditions() determine the coefficient .

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
  • The left-side and right-side surface fluxes are not equal.
  • The discontinuity factor is defined as follows to equalize the homogeneous surface fluxes.
  • The heterogeneous currents with discontinuity factors are defined by


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 using the average flux() and boundary surface current().
  • Using and , the A and B of can be determined, is determined as same to SENM method.
  • From , can be calculated and it has to be essentially same to due to nodal balance eq. (NBE)
  • However, the surface flux of nodal method is different from the reference surface flux .
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.
  • However, 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.
  • Finally, the homogeneous reflector xs is modified by the discontinuity factor to be applied for the core calculation.
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 () provide an accurate representation for core calculations.