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Methodology for Discontinuity Factors Generation for Simplified P_{3} Solver Based on Nodal Expansion Formulation

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## Abstract

**:**

_{N}) approximation was first introduced as a three-dimensional (3D) extension of the plane-geometry Spherical Harmonic (P

_{N}) equations. A third order SP

_{N}(SP

_{3}) solver, recently implemented in the Nodal Expansion Method (NEM), has shown promising performance in the reactor core neutronics simulations. This work is focused on the development and implementation of the transport-corrected interface and boundary conditions in an NEM SP

_{3}solver, following recent published work on the rigorous SP

_{N}theory for piecewise homogeneous regions. A streamlined procedure has been developed to generate the flux zero and second order/moment discontinuity factors (DFs) of the generalized equivalence theory to minimize the error introduced by pin-wise homogenization. Moreover, several colorset models with varying sizes and configurations are later explored for their capability of generating DFs that can produce results equivalent to that using the whole-core homogenization model for more practical implementations. The new developments are tested and demonstrated on the C5G7 benchmark. The results show that the transport-corrected SP

_{3}solver shows general improvements to power distribution prediction compared to the basic SP

_{3}solver with no DFs or with only the zeroth moment DF. The complete equivalent calculations using the DFs can almost reproduce transport solutions with high accuracy. The use of equivalent parameters from larger size colorset models show a slightly reduced prediction error than that using smaller colorset models in the whole-core calculations.

## 1. Introduction

_{N}) or spherical harmonics (P

_{N}), for the three-dimensional (3D) transport problem can be challenging, even with the rapid increase in computing power. Thus, low-order approximations to the transport equation that can be solved at a significantly reduced computational cost are of special interest. The coarse mesh diffusion equations are traditionally the first choice to the program developer; however, such methods may be inadequate for advanced reactor designs, which usually demonstrate a high level of heterogeneity, or in core regions near material boundaries and strong absorbers.

_{N}(SP

_{N}) equations were first proposed to introduce additional transport effects into the standard P

_{1}equations without introducing the complexities and undesired increase in the computational cost that a full transport theory solution would entail. The P

_{N}equations are obtained from inserting the truncated spherical harmonics expansion (to some order n) of the angular flux and differential scattering cross section into the transport equation.

_{N}equation in planar geometry with the 3D gradient and divergence operators, one arrives at a 3D generalization of the 1D P

_{N}equations [1]. Such a substitution was performed in an ad hoc manner, leaving the method to lack a theoretical foundation. As asymptotic and variational derivations later assisted to establish the theoretical basis for the SP

_{N}method [2,3,4], the application of the method became more popular. The numerical results show that the SP

_{N}approximation can yield accurate solutions of the transport problems, which outperforms the diffusion method, but with considerably less computational expense than those for the S

_{N}or the full P

_{N}methods [5,6]. For example, the computational time for SP

_{3}is only about twice that of the diffusion method. At locations in the core with a high flux gradient, such as material boundaries, the SP

_{3}has a higher precision due to the use of the higher Legendre moments P

_{N}scattering library and angular flux distribution. It is suggested that the SP

_{3}method should be utilized to a homogenized pin cell (other than a large assembly node) level calculation using a few groups (instead of two groups) to retain its superiority in accuracy over diffusion approximation [7].

_{3}calculation without DFs.

_{3}solver was established in the NEM by taking advantage of the original diffusion-based solver to achieve a pin cell resolution of highly heterogeneous reactor cores [13]. Further development and enhancement have since been conducted to improve its performance, such as incorporating the higher order scattering cross sections and discontinuity factors (DFs) [14,15]. However, this implementation adopted the ad hoc interface and boundary conditions based on the assumption of 1D behavior near a surface, which prevents the angular flux from being represented by the SP

_{N}flux components. Therefore, an SP

_{N}solver cannot provide the exact scalar flux solution as the P

_{N}solver.

_{3}solver, following recent published work on the rigorous SP

_{N}theory for piecewise homogeneous regions [16,17]. The new theory proves that the SP

_{N}solutions can represent a particular set of the angular flux solution by the P

_{N}theory. The resulting interface and boundary conditions now involve terms of higher order gradients of flux as well as tangential gradients, although the SP

_{N}equations are identical to the conventional ones. The transported corrected interface and boundary conditions are formulated in the nodal expansion expression of the flux components and the solution method involving the response matrix utilized in the NEM. Then, a side-dependent DF generation approach is devised based on the GET for homogenization with the intention to eliminate the error introduced by pin-wise homogenization. Both zeroth- and second-moment DFs can be generated using an in-house developed code using the solution from the transport solutions using Method of Characteristics (MOC). Several colorset models with varying sizes and configurations are later explored for their capability of generating DFs that can produce results equivalent to that using the whole-core model for more practical implementations. The new developments are tested and demonstrated on the C5G7 benchmark.

_{3}equations based on the nodal expansion formulation are derived and the solution method of SP

_{3}equations using the response matrix is introduced. Section 3 is focused on the derivation of the interface and boundary conditions in the new SP

_{N}theory and their implementation in the NEM SP

_{3}solver. Section 4 introduces the equivalent calculation scheme developed for the SP

_{3}solver with the emphasis placed on the generation of equivalence parameters using different lattice models to facilitate practical applications. The transport-corrected SP3 solver is tested on a series of problems from the C5G7 benchmark, and the results are shown in Section 5. Finally, conclusions and future perspectives are given in Section 6.

## 2. SP_{3} Method Based on Nodal Expansion Formulation

_{N}equations can reduce to a (n + 1)-th order differential equation for the lowest order scalar flux, which is equivalent to $\left(n+1\right)/2$ coupled second order differential equations. This indicates that being diffusion theory type equations, the SP

_{N}solvers have commonly been cast in such a way as to leverage existing diffusion machinery by iterating over the SP

_{N}moment equations. The steady-state SP

_{3}equations used in the NEM are the following [13]:

_{3}equations shown in Equation (4) by multiplying ${f}_{1}$ and ${f}_{2}$, and then performing the transverse integration along one direction as follows:

_{3}removal matrix. Outgoing partial currents are computed using the incoming partial currents and the node-dependent response functions. These outgoing partial currents become the incoming partial currents in the neighboring nodes. Outer fission source iterations are then performed around the inner iterations to calculate values for the problem multiplicative eigenvalue (${k}_{\mathrm{eff}}$) and the space- and energy-dependent fission neutron source distribution.

## 3. Transport-Corrected SP3 Method for Equivalent Calculations

#### 3.1. Derivations of Interface and Boundary Conditions

_{3}has long been problematic due to the ad hoc interface and boundary conditions adopted in the early developments of the SP

_{N}theory, which involves the diffusion theory type of first order derivatives in the surface normal direction. It prevents the angular flux from being represented by the SP

_{3}solution in the conventional SP

_{3}theory or being compared to the reference angular flux solution from the heterogeneous transport calculation. Simply put, it is not clear which higher order quantities should be continuous across the node surface, which makes it difficult to rigorously define and calculate the high order DF in SP

_{3}.

_{N}equations from the P

_{N}, proved that ${\varphi}_{0}$ solutions in 3D SP

_{3}are one set of solution to 3D P

_{3}theory, and provided an exact expression of the angular flux using the SP

_{3}flux solutions [16,17]. This method has been adopted, modified, and implemented in the NEM SP

_{3}solver. For the convenience of the readers, we provide a concise review summary of Chao’s work regarding the derivation of the transport-corrected interface and boundary conditions.

_{N}equations can be derived via the variational method by introducing the following trial function into the even parity transport equation (Equation (18)):

_{N}solution functions, as shown below.

#### 3.2. Implementation of Interface and Boundary Conditions

_{3}theory:

_{3}equations is not impacted during the derivation in Section 3.1 and, thus, still valid for the new SP

_{3}theory, the corrected second moment flux term and third moment net partial current become the following:

## 4. Equivalent Calculation Scheme

_{3}solver performs the finite volume integration, which requires material parameters to be constant in the node. Fuel pin homogenized cross sections are prepared using the flux-volume weighting procedure to preserve reaction rates between lattice calculations and SP

_{3}calculations. Here, we will only focus on the DF generation approach developed and implemented for the SP

_{3}solver aiming to preserve the leakage rate between nodes.

#### 4.1. SP_{3} Discontinuity Factors

_{N}angular flux solution. The work presented below is similar to a previous effort [11] in the sense that the DFs are generated for both zeroth and second order flux moments; however, angular moments in the SP

_{3}method are different and so is the application of DFs. It should be emphasized that the DF is defined for and applied to ${\mathsf{\Psi}}_{0}$ and ${\mathsf{\Psi}}_{2}$ in the following way:

_{3}equations. In other words, given a surface limiting two adjacent homogenized regions, the DFs enforce the continuity for the heterogeneous reconstructed flux $D{F}_{0}^{-}{\mathsf{\Psi}}_{0}^{\mathrm{hom},-}=D{F}_{0}^{+}{\mathsf{\Psi}}_{0}^{\mathrm{hom},+}$ and $D{F}_{2}^{-}{\mathsf{\Psi}}_{2}^{\mathrm{hom},-}=D{F}_{2}^{+}{\mathsf{\Psi}}_{2}^{\mathrm{hom},+}$.

_{3}solution (${\mathsf{\Psi}}_{0}^{\mathrm{hom}}$ and ${\mathsf{\Psi}}_{2}^{\mathrm{hom}}$) inside the homogenized cell.

_{3}solver can be carried out in the following steps:

- Perform the transport calculations and generate quantities including ${k}_{\mathrm{eff}}$, cell homogenized cross sections, cell averaged scalar flux, and side-dependent surface fluxes and currents.
- Solve the fixed interface problem for each cell sequentially by taking the reference values obtained in step one, generate the SP
_{3}surface fluxes and currents, then compute the DFs according to their definitions. - Execute a normal SP3 calculation on a pin-by-pin level using the homogenized cross sections from step one and DFs from step two to obtain the SP
_{3}solution.

#### 4.2. Practical Approach to Generate SP_{3} Discontinuity Factors

_{3}. Therefore, efforts are also made to explore the feasibility of colorset lattice models to generate equivalent parameters for the core simulation. The three colorset models under consideration are as follows:

- Single-pin model: cross sections are homogenized over the pin cell and DFs are calculated for each of the four surfaces of a pin cell.
- Double-pin model: two pin cells of different type (i.e., material) are placed next to each other and DFs are calculated for each of the four surfaces in each pin cell. The cross sections are taken from the first model.
- Assembly model: both homogenized cross sections and DFs are location dependent, i.e., they are generated for each of the pin cells.

## 5. Verification of Transported Corrected SP_{3} Solver

_{3}solver and the corresponding equivalent calculation scheme are verified using the C5G7 benchmark [22]. It is a miniature light water reactor (LWR) with sixteen fuel assemblies (mini core): eight uranium oxide (UO

_{2}) assemblies and eight mixed oxide (MOX) assemblies, surrounded by a water reflector. It features a quarter-core radial symmetry in the 2D configuration, as shown in Figure 2. On the fuel pin level, there are one UO

_{2}pin, three MOX pins, one guide tube, and one fission chamber. The three MOX fuels pins are 4.3, 7.0, and 8.7% plutonium weight enriched.

_{3}solver [15].

#### 5.1. Transport-Corrected SP_{3} Method

_{3}solver without applying DFs to reveal the impact of the updated interface and boundary conditions. Three cases are selected from the C5G7 benchmark for this purpose, including the UO

_{2}assembly, MOX assembly, and C5G7 core. In the single assembly cases, pin-wise cross sections are generated using the assembly model with the infinite lattice approximation. For the last case, they are prepared in a whole-core transport calculation, including the cross sections of the water reflector.

_{3}solver yields slightly better pin power distributions, especially at the center of the UO

_{2}assembly (upper left assembly), regions close to the water reflector (right and bottom surfaces), as well as pin cells next to the fission chamber and guide tubes. As expected, the transport-corrected solver cannot eliminate the prediction error because the spatial homogenization errors still exist.

#### 5.2. Test and Comparison of DFs

_{3}calculation.

_{3}solver in terms of its prediction of eigenvalue and pin power distribution. It can be seen that the DFs of the GET help reduce the prediction error significantly in all four test cases. The largest improvement is observed in the C5G7 core case, where the deviation from the reference value is reduced by ~200 pcm in eigenvalue and over halved in the root-mean-square (RMS) error of pin power distribution. The prediction accuracy is also drastically improved in the MOX assembly case where the local flux gradient is more profound than that in the UO

_{2}assembly.

- Single-pin model: There are seven models each corresponding to one type of pins. The non-fissionable node, such as guide tubes and water reflector cells, is placed in the center of a 3 × 3 configuration surrounded by UO
_{2}fuel pins. - Double-pin model: Eight sets of 2 × 1 pin colorset models are developed for different combination of pin cells. These DFs will be used in the whole core on the interface between neighboring different cells.
- Assembly model: Single UO
_{2}and MOX assembly models.

_{3}solution largely underestimates the fission power and neutron flux, while a uniform overestimation is observed in the central UO

_{2}assembly. An almost identical trend is also found in the comparison of the SP3 solution with option four against the transport results, which is expected because the DFs for nodes along the core/reflector interface are missing in both options three and four.

_{3}solver.

## 6. Conclusions and Outlook

_{3}solver for the equivalent reactor core calculation. The response matrix in the NEM SP

_{3}solver has been reformulated based on the recently published new SP

_{3}theory with rigorously derived interface and boundary conditions, while the computation scheme in the NEM is kept intact. A streamlined process of generating DFs of the GET for the correction of pin-by-pin homogenization error has been implemented, which incorporates the high-fidelity transport code and an in-house developed DF generation program. We also propose a few models with varying sizes for the generation of equivalent parameters aiming to further reduce the computational burden and explore their feasibility to practical applications. The transport-corrected SP

_{3}solver and various colorset models are tested on the mini-core C5G7 benchmark problem.

_{3}solver can compute the inter-node leakage rate correctly due to the updated interface and boundary conditions and, thus, improve the prediction of the pin power distribution. In the whole-core calculation, the regions benefiting the largest improvement are located at the center of the core, close to the water reflector, and next to the fission chamber and guide tubes.

_{3}methodology. For example, it has experienced numerical instabilities that cause an issue to achieve a converged solution when the DFs are imposed. Our preliminary investigation points to the use of the 1D forth order polynomial nodal expansion method implemented in the NEM solution method. We plan to implement and test a semi-analytical nodal expansion method in the future work to resolve this issue.

_{3}results from the reference transport solution. Thus, the DF generation program will be revisited and updated to solve the fixed interface problem for the reflector region. At last, effort will continue to investigate colorset models that can maximize the benefit of the equivalent calculations while maintaining the computational cost to a reasonable level for practical whole-core simulations.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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**Figure 3.**Differences in relative pin power using the base and transport-corrected SP

_{3}solvers. Blue regions indicate improved predictions by the transport-corrected SP

_{3}solver, while red regions mean the opposite.

**Figure 4.**Relative differences in pin power distribution predicted by a transport-corrected SP

_{3}solver from the reference transport solution. The cross sections and DFs are generated using the assembly model (option 3).

**Table 1.**Performance of the transport-corrected NEM SP

_{3}solver on various benchmark problems. The deviation from the reference results is given in pcm for the eigenvalue and RMS error for the pin power distribution.

Test Case | Solver | Δk_{eff} (pcm) | RMS Error of Pin Power |
---|---|---|---|

UO_{2} assembly | NEM SP_{3} | −125 | 0.010 |

NEM SP_{3} w/DFs | −5 | 0.004 | |

MOX assembly | NEM SP_{3} | −38 | 0.008 |

NEM SP_{3} w/DFs | 10 | 0.001 | |

C5 core | NEM SP_{3} | −171 | 0.048 |

NEM SP_{3} w/DFs | −75 | 0.023 | |

C5G7 core | NEM SP_{3} | 269 | 0.049 |

NEM SP_{3} w/DFs | −59 | 0.023 |

**Table 2.**Performance of the transport-corrected SP

_{3}solver using different DF generation models. The deviation from the reference results is given in pcm for the eigenvalue and RMS error for the pin power distribution.

Colorset Model | Δk_{eff} (pcm) | RMS Error of Pin Power |
---|---|---|

Single pin | 273 | 0.056 |

Double pin | 336 | 0.056 |

Assembly | 288 | 0.048 |

Assembly + double pin | 272 | 0.047 |

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**MDPI and ACS Style**

Xu, Y.; Hou, J.; Ivanov, K.
Methodology for Discontinuity Factors Generation for Simplified P_{3} Solver Based on Nodal Expansion Formulation. *Energies* **2021**, *14*, 6478.
https://doi.org/10.3390/en14206478

**AMA Style**

Xu Y, Hou J, Ivanov K.
Methodology for Discontinuity Factors Generation for Simplified P_{3} Solver Based on Nodal Expansion Formulation. *Energies*. 2021; 14(20):6478.
https://doi.org/10.3390/en14206478

**Chicago/Turabian Style**

Xu, Yuchao, Jason Hou, and Kostadin Ivanov.
2021. "Methodology for Discontinuity Factors Generation for Simplified P_{3} Solver Based on Nodal Expansion Formulation" *Energies* 14, no. 20: 6478.
https://doi.org/10.3390/en14206478