OECD/NEA has initiated an international Uncertainty Analysis in Modeling (UAM) benchmark focused on uncertainties in modeling of Light Water Reactor (LWR). The first step of uncertainty propagation is to perform sensitivity to the input data affected by the numerical errors and physical models. The objective of the present paper is to study the effect of the numerical discretization error and the manufacturing tolerances on fuel pin lattice integral parameters (multiplication factor and macroscopic cross-sections) through sensitivity calculations. The two-dimensional deterministic codes NEWT and HELIOS were selected for this work. The NEWT code was used for analysis of the TMI-1, PB-2, and Kozloduy-6 test cases; the TMI-1 test case was investigated using the HELIOS code. The work has been performed within the framework of UAM Exercise I-1 “Cell Physics.”
OECD/NEA has initiated an international Uncertainty Analysis in Modeling (UAM) benchmark focused on propagation of uncertainties in the entire modeling chain of Light Water Reactor (LWR) in steady-state and transient conditions. The final objective is to benchmark uncertainty and sensitivity analysis methods in coupled multiphysics and multiscale LWR calculations.
The present paper is concerned only with cell and lattice physics. In reactor analysis, the lattice physics calculations are used to generate nodal (lattice-averaged) parameters, used for the full-core simulation. Similarly to other numerical simulations, the lattice-averaged parameters are affected by uncertainties. In lattice physics, these uncertainties can be divided into 3 types: Multigroup cross sections uncertainties, Uncertainties associated with methods and modeling approximations used in lattice physics codes, and Fuel/assembly manufacturing tolerances.
The objective of the present paper is to study the effect of the last two uncertainty sources, within the framework of UAM Exercise I-1 “Cell Physics.” This exercise is focused on derivation of the multigroup microscopic cross-section libraries. Even if the intention for Exercise I-1 is to propagate the uncertainties in evaluated Nuclear Data Libraries—NDL—(microscopic point-wise cross sections) into multigroup microscopic cross-sections, here the NDL data have been used directly to perform lattice physics calculations (fuel pin lattices) in order to evaluate neutronics-related parameters.
For uncertainty propagation, the first step is to perform sensitivity to the input data affected by the errors or uncertainties. In this paper, the effect of numerical discretization errors and manufacturing tolerances on fuel pin lattice integral parameters (multiplication factor and cross-sections) has been analyzed through sensitivity calculations.
The two-dimensional deterministic codes NEWT and HELIOS were selected for this work. The NEWT code was used for analysis of the TMI-1, PB-2, and Kozloduy-6 test cases. Then, the TMI-1 test case was investigated using the HELIOS code. Finally, a comparison has been made between the two lattice codes.
Two deterministic lattice codes have been used to perform the uncertainties studies: NEWT and HELIOS.
NEWT (New ESC-based Weighting Transport code) is a two-dimensional (2D) discrete-ordinates transport code developed at Oak Ridge National Laboratory [
HELIOS is a generalized-geometry 2D lattice physics code developed by Studsvik-Scandpower [
The two-dimensional fuel pin-cell test problems representative of BWR PB-2, PWR TMI-1, and Kozloduy-6 VVER-1000 have been analyzed. The UAM specifications were used to define these three test problems, the details are shown in Figures
Configuration of PB-2 BWR unit cell.
Configuration of TMI-1 PWR unit cell.
Configuration of Kozloduy-6 VVER-1000 unit cell.
In addition to the cell geometry, material compositions, and material temperatures, each code requires code-specific numerical parameters. Table
Reference simulation parameters.
Parameter | NEWT | HELIOS |
---|---|---|
Cross-sections library | ENDF/B-VII.0 | HELIOS master library |
Number of energy group | 238 | 190 |
Grid structure |
|
|
Number of sides per cylinder | 12 | — |
Number of fuel pin azimuthal regions | — | 4 |
Convergence Criteria | ||
Inner iterations | 10−4 | 10−4 |
Outer iterations | 10−6 | — |
|
10−6 | 10−6 |
The focus of the paper is on the numerical discretization error and the manufacturing tolerance sensitivities. The specific description of each type of calculation is described in the section below.
The numerical discretization error was quantified for The spatial discretization of the cell grid: users can define a computational grid in which the NEWT ESC solution algorithm is applied. Convergence studies have been performed on The approximation of a circle with an equilateral polygon with a certain number of sides. The default number is 12. The influence of this approximation on the
The study has been carried out for the three test cases in HZP condition only.
Sensitivity of lattice-averaged parameters to manufacturing tolerances has been studied using data provided by the UAM specifications for Phase I. For TMI-1 and PB-2 test cases, the manufacturing uncertainties are shown in Table
Manufacturing tolerances for TMI-1 test case.
Parameter | Reference value | Variation | |
---|---|---|---|
TMI-1 | PB-2 | ||
Fuel density | 10.283 g/cm3 | ±0.17 g/cm3 | ±0.91% |
Fuel pellet diameter | 9.391 mm | ±0.013 mm | ±0.013 mm |
Gap thickness | 0.0955 mm | ±0.024 mm | — |
Clad thickness | 0.673 mm | ±0.025 mm | ±0.04 mm |
235U concentration | 4.85 w/o | ±0.00224 w/o | — |
For VVER-1000 test case, the manufacturing tolerances are shown in Table
Manufacturing tolerances for Kozloduy-6 test case.
Parameter | Reference value | Lower limit | Upper limit |
---|---|---|---|
Inner hole diameter | 1.4 mm | 1.4 mm | 1.7 mm |
Fuel density | 10.4 g/cm3 | 10.4 g/cm3 | 10.7 g/cm3 |
Fuel pellet diameter | 7.56 mm | 7.53 mm | 7.56 mm |
Clad inner diameter | 7.72 mm | 7.72 mm | 7.78 mm |
Clad outer diameter | 9.1 mm | 9.05 mm | 9.15 mm |
235U concentration | 3.3 w/o | 3.25 w/o | 3.35 w/o |
It should be noted that these uncertainties were specified for the fuel assembly, but in this work they have been applied for the single pin-cell. The sensitivities have been performed by changing the listed parameters affected by manufacturing tolerances. For TMI-1 and PB-2 cases all the listed parameters have been increased by the quantity indicated in Table
It should be noted that in this work the term “sensitivity” has not been used in the usual way but as reactivity differences (
The reference (base case) results for the three test models are presented in this section. The lattice
The results from HELIOS and NEWT are compared for TMI-1 pin-cell on Table
TMI-1 fuel pin—NEWT and HELIOS results for reference model.
HZP | HFP | |||
---|---|---|---|---|
NEWT | HELIOS | NEWT | HELIOS | |
|
1.41481 | 1.42595 | 1.39138 | 1.40788 |
Reactivity |
29319 | 29871 | 28129 | 28971 |
The difference between HZP and HFP values is of about 1200 pcm in NEWT evaluations and 900 pcm in HELIOS ones. Criticality results for PB-2 and Kozloduy-6 test cases are provided in Tables
PB-2 fuel pin—NEWT results for reference model.
HZP | HFP | |
---|---|---|
NEWT | NEWT | |
|
1.33869 | 1.21906 |
Reactivity |
25300 | 17932 |
Kozloduy-6 fuel pin—NEWT results for reference model.
HZP | HFP | |
---|---|---|
NEWT | NEWT | |
|
1.34311 | 1.32530 |
Reactivity |
25546 | 24546 |
One of the user-defined values is a computational grid in which the NEWT ESC solution algorithm is applied. Convergence study has been performed on
Grid influence on
For all the three test cases increasing the number of computational points the
It is important to note that the value corresponding to a
For subsequent calculations a
Another important NEWT user-defined value is the approximation of a circle with an equilateral polygon with a certain number of sides. Convergence study has been performed on
Number of polygon sides’ influence on
Nonmonotonic convergence is observed when varying the number of polygon sides, in particular for TMI-1 and PB-2 cases, while for Kozloduy-6 trend is much more monotonic. A possible reason for such behavior is that the use of the
Number of polygon sides’ influence on
With the
An important effect is that the spatial discretization error (Figure
Sensitivity of lattice-averaged parameters to manufacturing tolerances has been studied for the 3 test cases. For TMI-1 and PB-2 cases the sensitivities were calculated by increasing the parameters listed in Table
Table
Manufacturing sensitivities for TMI-1 and PB-2 test cases.
Parameter | TMI-1 | PB-2 | ||||
---|---|---|---|---|---|---|
Variation |
|
Variation |
| |||
HZP | HFP | HZP | HFP | |||
Fuel density | +0.17 g/cm3 |
|
|
+0.91% |
|
|
Fuel pellet diameter | +0.013 mm |
|
|
+0.013 mm |
|
|
Gap thickness | ||||||
from outside | +0.024 mm |
|
|
— | — | — |
from inside | +0.024 mm |
|
|
— | — | — |
Clad thickness | ||||||
from outside | +0.025 mm |
|
|
+0.04 mm |
|
|
from inside | +0.025 mm |
|
|
+0.04 mm |
|
|
235U concentration | +0.00224 w/o |
|
|
— | — | — |
Manufacturing sensitivities for TMI-1 test case.
Manufacturing sensitivities for PB-2 test case.
It should be noted that errors in gap and clad thickness can be considered in two ways. The gap thickness can increase because the fuel pellet diameter decreases or the clad thickness decreases (increase of the internal clad diameter); The clad thickness can increase because the gap thickness decrease (while outer clad diameter remains the same) or the outer clad diameter increases (while the inner clad diameter remains the same).
Each of the gap and clad thickness variations have been analyzed and are included in Table
The largest
Kozloduy-6
Manufacturing sensitivities for Kozloduy-6 test case.
Parameter | Variation |
| |
---|---|---|---|
Kozloduy-6 | |||
HZP | HFP | ||
Inner hole diameter | |||
Upper limit (0.17 cm) | +0.03 cm | 80 | 87 |
Fuel density | |||
Upper limit (10.7 g/cm3) | +0.3 g/cm3 | −151 | −163 |
Fuel pellet diameter | |||
Lower limit (0.753 cm) | −0.003 cm | 50 | 54 |
Clad inner diameter | |||
Upper limit (0.778 cm) | +0.006 cm | 31 | 31 |
Clad outer diameter | |||
Upper limit (0.915 cm) | +0.005 cm | −135 | −144 |
Lower limit (0.905 cm) | −0.005 cm | 132 | 141 |
235U concentration | |||
Upper limit (3.35%) | −0.05 w/o | 168 | 170 |
Lower limit (3.25%) | +0.05 w/o | −173 | −175 |
Manufacturing sensitivities for Kozloduy-6 test case.
The same sensitivity calculations have been performed using a
Table
Manufacturing sensitivities on macroscopic cross-sections for TMI-1 fuel pin lattice.
Parameter |
Variation |
|
|
|
---|---|---|---|---|
% | % | % | ||
Fuel density | +0.17 g/cm3 | 1.16% | −1.22% | −1.16% |
Fuel pellet diameter | +0.013 mm | 0.22% | −0.24% | −0.22% |
Gap thickness | ||||
from outside | +0.024 mm | −0.08% | 0.07% | 0.08% |
from inside | +0.024 mm | −0.81% | 0.87% | 0.81% |
Clad thickness | ||||
from outside | +0.025 mm | −0.10% | 0.10% | 0.10% |
from inside | +0.025 mm | −0.02% | 0.02% | 0.02% |
235U concentration | +0.00224 w/o | 0.03% | −0.03% | −0.03% |
The geometric data used so far for our analysis corresponds to the atmospheric temperature, even for HFP calculations. At high temperatures, corresponding to the HFP conditions, both the fuel pellet and the clad undergo thermal expansion. To analyze this effect, we calculated the actual fuel pellet diameter and clad dimensions following the thermal expansion (at HFP temperatures) [
The “hot” pin dimensions have been calculated using a thermal expansion coefficient for the fuel of
After the calculation of “hot” pin dimension, the manufacturing tolerances sensitivities have been recalculated, at HFP conditions, as before (cf. Section 4.2).
The new results have been compared with the original sensitivities obtained with “cold” dimensions and they are shown in Table
Manufacturing sensitivities for TMI-1 at HFP conditions considering thermal expansion.
Parameter | Variation | Δ | |
---|---|---|---|
HFP | HFP with thermal expansion | ||
Fuel density | +0.17 g/cm3 | −104 | −105 |
Fuel pellet diameter | +0.013 mm | −20 | −20 |
Gap thickness | |||
from outside | +0.024 mm | −91 | −92 |
from inside | +0.024 mm | 74 | 74 |
Clad thickness | |||
from outside | +0.025 mm | −110 | −111 |
from inside | +0.025 mm | −15 | −15 |
235U concentration | +0.00224 w/o | 4 | 4 |
The reactivity sensitivities are the same as those calculated with the “cold” geometry.
The TMI-1 sensitivities have been recalculated with lattice code HELIOS and compared with the NEWT results. The results are summarized in Table
Comparison between HELIOS and NEWT sensitivities for TMI-1.
Parameter | Variation | Δ | |||
---|---|---|---|---|---|
NEWT | HELIOS | ||||
HZP | HFP | HZP | HFP | ||
Fuel density | +0.17 g/cm3 | −97 | −104 | −95 | −101 |
Fuel pellet diameter | +0.013 mm | −19 | −20 | −21 | −23 |
Gap thickness | |||||
from outside | +0.024 mm | −84 | −91 | −68 | −73 |
from inside | +0.024 mm | 69 | 74 | 78 | 84 |
Clad thickness | |||||
from outside | +0.025 mm | −104 | −110 | −100 | −106 |
from inside | +0.025 mm | −16 | −15 | −29 | −30 |
235U concentration | +0.00224 w/o | 4 | 4 | 3 | 4 |
Comparison between HELIOS and NEWT sensitivities for TMI-1.
This work has been carried out in the framework of UAM Exercise I-1 “Cell Physics.” Three test cases (TMI-1, PB-2, and Kozloduy-6) have been analyzed with the deterministic code NEWT. In addition, the TMI-1 fuel pin has also been modeled with the HELIOS code in order to compare the results of the two codes.
The infinite multiplication factor has been calculated for each of the lattice configuration. A significant discrepancy was found in the multiplication factor between NEWT and HELIOS for the TMI-1 case. The difference was about 600 pcm at HZP conditions and about 900 pcm at HFP conditions.
Sensitivity calculations have been performed in order to study the influence of numerical approximations and manufacturing tolerances on
The following important conclusions related to the NEWT numerical approximation can be highlighted. The spatial discretization error for The spatial discretization error is very large with the default discretization ( The equilateral polygon approximation of a circle has relatively small influence on The spatial discretization error and circle polygon approximation error are in opposite direction, causing a fortunate cancelation of error.
The following important conclusions related to the manufacturing sensitivities can be highlighted. Sensitivities change linearly with manufacturing tolerances for all the parameters considered. HFP sensitivities are larger than HZP ones, especially for PB-2 case. The manufacturing tolerance that has the largest influence on the The influence of manufacturing tolerances on two group macroscopic cross-sections has been analyzed and maximum variation is about 1.2%. Manufacturing tolerances sensitivities with “cold” and “hot” dimensions are the same. The spatial discretization has a significant effect on
The propagation of manufacturing tolerances for reactivity and few group nodal homogenized data at the fuel assembly level will be performed in the future.