#
Verification and Validation of the SP_{L} Module of the Deterministic Code AZNHEX through the Neutronics Benchmark of the CEFR Start-Up Tests

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## Abstract

**:**

## 1. Introduction

## 2. Description of China Experimental Fast Reactor

## 3. Description of the Codes and Models

#### 3.1. AZNHEX Code Background

#### $S{P}_{L}$ Implementation in AZNHEX

#### 3.2. CEFR Core Modelling

#### 3.2.1. Model for Serpent

#### 3.2.2. Cross-Section (XS) Generation

## 4. Results

#### 4.1. Verification and Validation Exercise

#### Description of Fuel Loading and Criticality Experiment

#### 4.2. Verification and Validation of Serpent Model

#### 4.3. Verification and Validation of $S{P}_{L}$ Implementation in AZNHEX

#### Mesh Sensitivity Analysis with AZNHEX $S{P}_{L}$

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

AZNHEX | AZtlan Nodal HEXagonal |

AZKIND | AZtlan KInetics in Neutron Diffusion |

AZTHECA | AZtlan THErmohydraulics Core Analysis |

AZTRAN | AZtlan TRANsport |

AZTUSIA | AZtlan Tool for Uncertainty and SensItivity Analysis |

BROND | Library of Recommended Evaluated Neutron Data (in Russian abbreviations) |

CEFR | China Experimental Fast Reactor |

CENDL | Chinese Evaluated Nuclear Data Library |

CIAE | China Institute of Atomic Energy |

CONACYT | National Council for Science and Technology (in Spanish abbreviations) |

ENDF | Evaluated Nuclear Data File |

IAEA | International Atomic Energy Agency |

ININ | National Institute of Nuclear Research (in Spanish abbreviations) |

IPN | National Polytechnic Institute (in Spanish abbreviations) |

JEFF | Joint Evaluated Fission and Fusion File |

JENDL | Japanese Evaluated Nuclear Data Library |

NEA | Nuclear Energy Agency |

OECD | Organisation for Economic Co-operation and Development |

SENER | Secretariat of Energy (in Spanish abbreviations) |

## Nomenclature

Symbols | |

${D}_{g}$ | Diffusion coefficient for energy group g, m |

${\mathtt{D}}_{i}$ | Diffusion coefficient for artificial energy-group i, m |

g | Energy group g |

${g}^{\prime}$ | Energy group ${g}^{\prime}$ |

G | Total number of energy groups considered in a given problem |

i | Integer number that describes the array of the diffusion coefficients and |

cross-sections depending on the energy group g and the order L implemented | |

j | Integer number that describes the array of the diffusion coefficients and |

cross-sections depending on the energy group g and the order L implemented | |

k | Multiplication factor |

${k}_{eff}$ | Effective multiplication factor |

L | Implemented order of the $S{P}_{L}$ approximation |

m | Integer number that identifies the implemented $(L+1)/2$ order of the $S{P}_{L}$ |

approximation | |

N | Order of discrete ordinate |

$NC$ | Number of unknowns per discrete ordinate |

n | Integer number that identifies the implemented $(L+1)/2$ order of the $S{P}_{L}$ |

approximation | |

pcm | percent mili-rho |

$\overrightarrow{r}$ | A vector that indicates the spatial position of a neutron in a three-dimensional space |

${S}_{l}^{{g}^{\prime}}$ | l-th angular moment of the neutron source term for energy group ${g}^{\prime}$, 1/m${}^{3}\xb7$s |

$TNU$ | Total number of unknowns |

${\mu}_{m}$, ${\mu}_{n}$ | $S{P}_{L}$-order dependent constant |

${\nu}_{g}$ | Average number of neutrons released by fissions which are induced by neutrons |

with energies in the energy group g | |

${\nu}_{{g}^{\prime}}$ | Average number of neutrons released by fissions which are induced by |

neutrons with energies in the energy group ${g}^{\prime}$ | |

$\nu {\Sigma}_{\mathtt{f}i}$ | Macroscopic fission cross-section multiplied by the average number of |

neutrons produced per fission, for artificial energy-group i, 1/m | |

$\nu {\Sigma}_{\mathtt{f}j}$ | Macroscopic fission cross-section multiplied by the average number of |

neutrons produced per fission, for artificial energy-group j, 1/m | |

$\rho $ | Reactivity |

${\Sigma}_{{f}_{g}}$ | Macroscopic fission cross-section for energy group g, 1/m |

${\Sigma}_{{f}_{{g}^{\prime}}}$ | Macroscopic fission cross-section for energy group ${g}^{\prime}$, 1/m |

${\Sigma}_{\mathtt{R}i}$ | Macroscopic removal cross-section for artificial energy-group i, 1/m |

${\Sigma}_{{s}_{0}}^{g\to g}$ | 0-th angular moment of the macroscopic scattering cross-section from energy |

group g to energy group g, 1/m | |

${\Sigma}_{{s}_{0}}^{{g}^{\prime}\to g}$ | 0-th angular moment of the macroscopic scattering cross-section from energy |

group ${g}^{\prime}$ to energy group g, 1/m | |

${\Sigma}_{\mathtt{s}j\to i}$ | Macroscopic scattering cross-section from artificial energy-group j to artificial |

energy-group i, 1/m | |

${\Sigma}_{{t}_{g}}$ | Macroscopic total cross-section for energy group g, 1/m |

${\varphi}_{0}^{g}$ | Scalar neutron flux for energy group g, 1/m${}^{2}\xb7$J·s |

${\varphi}_{0}^{{g}^{\prime}}$ | Scalar neutron flux for energy group ${g}^{\prime}$, 1/m${}^{2}\xb7$J·s |

${\mathsf{\Phi}}_{i}$ | Array of neutron flux moments for artificial energy-group i, 1/m${}^{2}\xb7$s |

${\mathsf{\Phi}}_{j}$ | Array of neutron flux moments for artificial energy-group j, 1/m${}^{2}\xb7$s |

${\mathsf{\Phi}}_{m}^{{g}^{\prime}}$ | m-th array of neutron flux moments for energy group ${g}^{\prime}$, 1/m${}^{2}\xb7$s |

${\mathsf{\Phi}}_{n}^{g}$ | n-th array of neutron flux moments for energy group g, 1/m${}^{2}\xb7$s |

${\chi}_{g}$ | Probability that a neutron is born in the energy group g |

${\chi}_{i}$ | Neutron fission spectrum for artificial energy-group i |

${\omega}_{m}$, ${\omega}_{n}$ | $S{P}_{L}$-order dependent constant |

∇ | Gradient |

$\nabla \xb7$ | Divergence |

${\nabla}^{2}$ | Laplace operator |

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**Figure 5.**${k}_{eff}$ values obtained with AZNHEX-$S{P}_{L}$ and their comparison with the experimental data.

**Figure 7.**Radial (

**a**) $1\times 1$, (

**b**) $2\times 2$, (

**c**) $3\times 3$, and (

**d**) $4\times 4$ refinements made in the AZNHEX code.

**Figure 8.**Axial (

**a**) $\times 1$, (

**b**) $\times 2$, (

**c**) $\times 3$, and (

**d**) $\times 4$ refinements made in the AZNHEX code.

n = 1,⋯,(L+1)/2 | μ_{n} | ω_{n} |
---|---|---|

$L=1$ | ||

1 | $1/\sqrt{3}$ | 1 |

$L=3$ | ||

1 | 0.339981043584856 | 0.652145154862546 |

2 | 0.861136311594053 | 0.347854845137454 |

$L=5$ | ||

1 | 0.238619186083197 | 0.467913934572691 |

2 | 0.661209386466265 | 0.360761573048139 |

3 | 0.932469514203152 | 0.171324492379170 |

$L=7$ | ||

1 | 0.183434642495650 | 0.362683783378360 |

2 | 0.525532409916329 | 0.313706645877890 |

3 | 0.796666477413627 | 0.222381034453374 |

4 | 0.960289856497536 | 0.101228536290376 |

Group | Upper Limit (MeV) |
---|---|

1 | 2.000000 × 10${}^{1}$ |

2 | 1.353400 × 10${}^{0}$ |

3 | 5.23400 × 10${}^{-1}$ |

4 | 6.73790 × 10${}^{-2}$ |

5 | 3.35460 × 10${}^{-3}$ |

6 | 7.48520 × 10${}^{-4}$ |

Material | Energy Group | Group Constants | |||||||
---|---|---|---|---|---|---|---|---|---|

${\mathit{D}}_{\mathit{g}}$ | ${\mathbf{\Sigma}}_{{\mathit{r}}_{\mathit{g}}}$ | $\mathit{\nu}$∗${\mathbf{\Sigma}}_{{\mathit{f}}_{\mathit{g}}}$ | $\mathit{\kappa}$∗${\mathbf{\Sigma}}_{{\mathit{f}}_{\mathit{g}}}$ | ${\mathit{\chi}}_{\mathit{g}}$ | ${\mathbf{\Sigma}}_{\mathit{s}{\mathit{g}}^{\prime}\to \mathit{g}}$ | ||||

Fuel SA’s (Ring 1) | 1 | 2.968 | 4.290 × 10${}^{-2}$ | 2.205 × 10${}^{-2}$ | 2.031 × 10${}^{2}$ | 5.794 × 10${}^{-1}$ | 1.220 × 10${}^{-1}$ | 2.474 × 10${}^{-2}$ | 8.885 × 10${}^{-3}$ |

3.425 × 10${}^{-4}$ | 2.323 × 10${}^{-6}$ | 4.666 × 10${}^{-7}$ | |||||||

2 | 2.150 | 2.978 × 10${}^{-2}$ | 1.529 × 10${}^{-2}$ | 2.023 × 10${}^{2}$ | 2.825 × 10${}^{-1}$ | 0.00 | 1.739 × 10${}^{-1}$ | 2.223 × 10${}^{-2}$ | |

4.088 × 10${}^{-4}$ | 3.151 × 10${}^{-6}$ | 1.542 × 10${}^{-7}$ | |||||||

3 | 1.527 | 1.407 × 10${}^{-2}$ | 1.747 × 10${}^{-2}$ | 2.022 × 10${}^{2}$ | 1.303 × 10${}^{-1}$ | 0.00 | 0.00 | 2.352 × 10${}^{-1}$ | |

4.892 × 10${}^{-3}$ | 1.206 × 10${}^{-6}$ | 6.873 × 10${}^{-8}$ | |||||||

4 | 1.066 | 1.811 × 10${}^{-2}$ | 2.767 × 10${}^{-2}$ | 2.022 × 10${}^{2}$ | 7.495 × 10${}^{-3}$ | 0.00 | 0.00 | 0.00 | |

3.248 × 10${}^{-1}$ | 1.253 × 10${}^{-3}$ | 8.235 × 10${}^{-7}$ | |||||||

5 | 6.931 × 10${}^{-1}$ | 5.512 × 10${}^{-2}$ | 7.978 × 10${}^{-2}$ | 2.022 × 10${}^{2}$ | 8.053 × 10${}^{-5}$ | 0.00 | 0.00 | 0.00 | |

0.00 | 5.400 × 10${}^{-1}$ | 3.866 × 10${}^{-3}$ | |||||||

6 | 7.796 × 10${}^{-1}$ | 9.172 × 10${}^{-2}$ | 1.488 × 10${}^{-1}$ | 2.022 × 10${}^{2}$ | 9.843 × 10${}^{-6}$ | 0.00 | 0.00 | 0.00 | |

0.00 | 0.00 | 3.634 × 10${}^{-1}$ |

RE2 Position | Exp. Measurement | Serpent | Absolute Dev. |
---|---|---|---|

${\mathit{\rho}}_{\mathit{exp}}$ (pcm) | ${\mathit{\rho}}_{\mathit{Serpent}}$ (pcm) | ${\mathit{\rho}}_{\mathit{exp}}-{\mathit{\rho}}_{\mathit{Serpent}}$ | |

190 mm | 40 | 48.0 | −8.0 |

170 mm | 34 | 41.0 | −7.0 |

151 mm | 25 | 28.0 | −3.0 |

70 mm | 0 | 4.0 | −4.0 |

RE2 Position | Exp. Measurement | AZNHEX (pcm) | |||||||
---|---|---|---|---|---|---|---|---|---|

${\mathit{\rho}}_{\mathit{exp}}$ | ${\mathit{\rho}}_{{\mathit{SP}}_{1}}$ | Dev | ${\mathit{\rho}}_{{\mathit{SP}}_{3}}$ | Dev | ${\mathit{\rho}}_{{\mathit{SP}}_{5}}$ | Dev | ${\mathit{\rho}}_{{\mathit{SP}}_{7}}$ | Dev | |

190 mm | 40 | −2031.34 | 2071.34 | 356.52 | −316.52 | −143.08 | 183.08 | −123.57 | 163.57 |

170 mm | 34 | −2032.81 | 2066.81 | 355.13 | −321.13 | −144.24 | 178.24 | −124.73 | 158.73 |

151 mm | 25 | −2038.18 | 2063.18 | 349.97 | −324.97 | −149.21 | 174.21 | −129.70 | 154.70 |

70 mm | 0 | −2065.01 | 2065.01 | 324.15 | −324.15 | −174.29 | 174.29 | −154.75 | 154.75 |

${\mathit{SP}}_{\mathbf{1}}$ | ${\mathit{SP}}_{\mathbf{3}}$ | ${\mathit{SP}}_{\mathbf{5}}$ | ${\mathit{SP}}_{\mathbf{7}}$ | |
---|---|---|---|---|

Time [s] | 237.7 | 2260.4 | 4244.9 | 6546.5 |

Time factor vs. $S{P}_{1}$ | 1 | 9.5 | 17.8 | 27.5 |

Time factor vs. $S{P}_{3}$ | – | 1 | 1.8 | 2.9 |

**Table 7.**Reactivity values and deviation in pcm from AZNHEX calculations with radial refinement and experimental data.

RE2 Position | 190 mm | Dev | 170 mm | Dev | 151 mm | Dev | 70 mm | Dev | |
---|---|---|---|---|---|---|---|---|---|

Exp. | |||||||||

Measurement | 40 (pcm) | 34 (pcm) | 25 (pcm) | 0 (pcm) | |||||

$S{P}_{1}$_$1\times 1$ | −2031.34 | 2071.34 | −2032.81 | 2066.81 | −2038.18 | 2063.18 | −2065.01 | 2065.01 | |

$S{P}_{1}$_$2\times 1$ | −1649.18 | 1689.18 | −1650.81 | 1684.81 | −1656.31 | 1681.31 | −1683.73 | 1683.73 | |

$S{P}_{1}$_$4\times 1$ | −1508.35 | 1548.35 | −1510.34 | 1544.34 | −1515.89 | 1540.89 | −1543.48 | 1543.48 | |

$S{P}_{3}$_$1\times 1$ | 356.72 | −316.72 | 354.74 | −320.74 | 349.77 | −324.77 | 323.95 | −323.95 | |

$S{P}_{3}$_$2\times 1$ | 702.06 | −662.06 | 700.42 | −666.42 | 695.17 | −670.17 | 668.81 | −668.81 | |

AZNHEX | $S{P}_{3}$_$4\times 1$ | 827.93 | −787.93 | 826.24 | −792.24 | 821.08 | −796.08 | 794.40 | −794.40 |

$S{P}_{5}$_$1\times 1$ | −143.20 | 183.20 | −144.21 | 178.21 | −149.22 | 174.22 | −174.30 | 174.30 | |

$S{P}_{5}$_$2\times 1$ | 202.30 | −162.30 | 201.03 | −167.03 | 195.93 | −170.93 | 170.28 | −170.28 | |

$S{P}_{5}$_$4\times 1$ | 327.31 | −287.31 | 326.01 | −292.01 | 320.88 | −295.88 | 295.01 | −295.01 | |

$S{P}_{7}$_$1\times 1$ | −124.15 | 164.15 | −125.16 | 159.16 | −130.17 | 155.17 | −155.24 | 155.24 | |

$S{P}_{7}$_$2\times 1$ | 221.92 | −181.92 | 220.65 | −186.65 | 215.56 | −190.56 | 189.99 | −189.99 | |

$S{P}_{7}$_$4\times 1$ | 346.73 | −306.73 | 345.43 | −311.43 | 340.30 | −315.30 | 314.57 | −314.57 |

**Table 8.**Reactivity values and deviation in pcm from AZNHEX calculations with axial refinement and experimental data.

RE2 Position | 190 mm | Dev | 170 mm | Dev | 151 mm | Dev | 70 mm | Dev | |
---|---|---|---|---|---|---|---|---|---|

Exp. | |||||||||

Measurement | 40 (pcm) | 34 (pcm) | 25 (pcm) | 0 (pcm) | |||||

$S{P}_{1}$_$1\times 1$ | −2031.34 | 2071.34 | −2032.81 | 2066.81 | −2038.18 | 2063.18 | −2065.01 | 2065.01 | |

$S{P}_{1}$_$1\times 2$ | −2302.14 | 2342.14 | −2303.70 | 2337.70 | −2309.08 | 2334.08 | −2336.42 | 2336.42 | |

$S{P}_{1}$_$1\times 4$ | −2374.52 | 2414.52 | −2376.13 | 2410.13 | −2381.54 | 2406.54 | −2408.99 | 2408.99 | |

$S{P}_{3}$_$1\times 1$ | 356.72 | −316.72 | 354.74 | −320.74 | 349.77 | −324.77 | 323.95 | −323.95 | |

$S{P}_{3}$_$1\times 2$ | 99.90 | −59.90 | 98.90 | −64.90 | 92.91 | −67.91 | 66.96 | −66.96 | |

AZNHEX | $S{P}_{3}$_$1\times 4$ | 28.99 | 11.01 | 27.99 | 6.01 | 22.00 | 3.00 | −4.00 | 4.00 |

$S{P}_{5}$_$1\times 1$ | −143.20 | 183.20 | −144.21 | 178.21 | −149.22 | 174.22 | −174.30 | 174.30 | |

$S{P}_{5}$_$1\times 2$ | −398.58 | 438.58 | −399.59 | 433.59 | −404.63 | 429.63 | −430.85 | 430.85 | |

$S{P}_{5}$_$1\times 4$ | −470.20 | 510.20 | −471.21 | 505.21 | −476.26 | 501.26 | −502.51 | 502.51 | |

$S{P}_{7}$_$1\times 1$ | −124.15 | 164.15 | −125.16 | 159.16 | −130.17 | 155.17 | −155.24 | 155.24 | |

$S{P}_{7}$_$1\times 2$ | −378.43 | 418.43 | −379.43 | 413.43 | −384.47 | 409.47 | −410.68 | 410.68 | |

$S{P}_{7}$_$1\times 4$ | −449.01 | 489.01 | −451.03 | 485.03 | −456.07 | 481.07 | −481.31 | 481.31 |

**Table 9.**Reactivity values and deviation in pcm from AZNHEX calculations with radial and axial refinements and experimental data.

RE2 Position | 190 mm | Dev | 170 mm | Dev | 151 mm | Dev | 70 mm | Dev | |
---|---|---|---|---|---|---|---|---|---|

Exp. | |||||||||

Measurement | 40 (pcm) | 34 (pcm) | 25 (pcm) | 0 (pcm) | |||||

$S{P}_{1}$_$1\times 1$ | −2031.34 | 2071.34 | −2032.81 | 2066.81 | −2038.18 | 2063.18 | −2065.01 | 2065.01 | |

$S{P}_{1}$_$2\times 2$ | −1916.62 | 1956.62 | −1918.37 | 1952.37 | −1923.88 | 1948.88 | −1951.81 | 1951.81 | |

$S{P}_{1}$_$4\times 4$ | −1846.55 | 1886.55 | −1848.37 | 1882.37 | −1853.94 | 1878.94 | −1882.17 | 1882.17 | |

$S{P}_{3}$_$1\times 1$ | 356.72 | −316.72 | 354.74 | −320.74 | 349.77 | −324.77 | 323.95 | −323.95 | |

$S{P}_{3}$_$2\times 2$ | 450.05 | −410.05 | 448.43 | −414.43 | 443.18 | −418.18 | 416.24 | −416.24 | |

AZNHEX | $S{P}_{3}$_$4\times 4$ | 509.14 | −469.14 | 507.44 | −473.44 | 502.13 | −477.13 | 475.00 | −475.00 |

$S{P}_{5}$_$1\times 1$ | −143.20 | 183.20 | −144.21 | 178.21 | −149.22 | 174.22 | −174.30 | 174.30 | |

$S{P}_{5}$_$2\times 2$ | −49.11 | 89.11 | −50.67 | 84.67 | −55.78 | 80.78 | −81.86 | 81.86 | |

$S{P}_{5}$_$4\times 4$ | 9.11 | 30.89 | 7.63 | 26.37 | 2.46 | 22.54 | −23.91 | 23.91 | |

$S{P}_{7}$_$1\times 1$ | −124.15 | 164.15 | −125.16 | 159.16 | −130.17 | 155.17 | −155.24 | 155.24 | |

$S{P}_{7}$_$2\times 2$ | −28.41 | 68.41 | −29.79 | 63.79 | −34.89 | 59.89 | −60.94 | 60.94 | |

$S{P}_{7}$_$4\times 4$ | 29.99 | 10.01 | 28.54 | 5.46 | 23.34 | 1.66 | −3.01 | 3.01 |

Refinement | ${\mathit{SP}}_{\mathbf{1}}$ | ${\mathit{SP}}_{\mathbf{3}}$ | ${\mathit{SP}}_{\mathbf{5}}$ | ${\mathit{SP}}_{\mathbf{7}}$ | ||||
---|---|---|---|---|---|---|---|---|

Time [s] | Factor | Time [s] | Factor | Time [s] | Factor | Time [s] | Factor | |

$1\times 1$ | 237.7 | 1 | 2260.4 | 1 | 4244.9 | 1 | 6546.5 | 1 |

$2\times 2$ | 1764.95 | 7.4 | 18,009.6 | 7.9 | 33,045.0 | 7.8 | 49,684.02 | 7.6 |

$4\times 4$ | 13,454.95 | 56.6 | 138,142.54 | 61.1 | 245,042.81 | 57.7 | 355,767.57 | 54.3 |

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## Share and Cite

**MDPI and ACS Style**

Muñoz-Peña, G.; Galicia-Aragon, J.; Lopez-Solis, R.; Gomez-Torres, A.; del Valle-Gallegos, E.
Verification and Validation of the *SP*_{L} Module of the Deterministic Code AZNHEX through the Neutronics Benchmark of the CEFR Start-Up Tests. *J. Nucl. Eng.* **2023**, *4*, 59-76.
https://doi.org/10.3390/jne4010005

**AMA Style**

Muñoz-Peña G, Galicia-Aragon J, Lopez-Solis R, Gomez-Torres A, del Valle-Gallegos E.
Verification and Validation of the *SP*_{L} Module of the Deterministic Code AZNHEX through the Neutronics Benchmark of the CEFR Start-Up Tests. *Journal of Nuclear Engineering*. 2023; 4(1):59-76.
https://doi.org/10.3390/jne4010005

**Chicago/Turabian Style**

Muñoz-Peña, Guillermo, Juan Galicia-Aragon, Roberto Lopez-Solis, Armando Gomez-Torres, and Edmundo del Valle-Gallegos.
2023. "Verification and Validation of the *SP*_{L} Module of the Deterministic Code AZNHEX through the Neutronics Benchmark of the CEFR Start-Up Tests" *Journal of Nuclear Engineering* 4, no. 1: 59-76.
https://doi.org/10.3390/jne4010005