Uncertainty Propagation of Fission Product Yields from Uranium and Plutonium in Pebble-Bed HTGR Burnup Calculation

Abstract

Quantifying fission product yield uncertainty contribution to reactor burnup calculation is an important aspect of pebble-bed High Temperature Gas-cooled Reactor (pebble-bed HTGR) uncertainty analysis. In this work, uncertainty propagation of fission product yield to pebble-bed HTGR burnup calculation is conducted. Uncertainty of fission product yields from four fissile isotopes, namely ²³³U, ²³⁵U, ²³⁹Pu and ²⁴¹Pu, are considered. The stochastic sampling-based uncertainty analysis method is adopted and fission product yield covariance matrices are estimated from ENDF/B-VII.1. The covariance matrix for each fissile actinide is estimated based on the Bayesian method and fission product yields are assigned with log-normal distribution in the sampling process with the Latin Hypercube Sampling (LHS) method. Since the fission fraction from ²³⁹Pu plays an important role in fissions of fuels with high burnup value in pebble-bed HTGR, the fission product yield uncertainty contribution from ²³⁹Pu is highlighted in this work. The result shows that, in the burnup equilibrium state of pebble-bed HTGR, fission product yield uncertainty contributions from ²³⁵U and ²³⁹Pu to relative uncertainty of k_eff are 0.027% and 0.026%, respectively. The overall uncertainty contribution from four fissile isotopes (²³³U, ²³⁵U, ²³⁹Pu and ²⁴¹Pu) to relative uncertainty of equilibrium core k_eff is 0.038%. Furthermore, fission product yield uncertainty has an important contribution to the nuclide density uncertainty of fission products. The most relative uncertainty, 10.82%, is observed in ¹⁰⁹Ag contributed from the fission product yield uncertainty of ²³⁹Pu at the burnup equilibrium state. This indicates the uncertainty contribution from the fission product yield of ²³⁹Pu cannot be neglected in pebble-bed HTGR burnup uncertainty analysis.

1. Introduction

Burnup uncertainty analysis is an important aspect in design and evaluation of inherent safety characteristics for pebble-bed High Temperature Gas-cooled Reactor (pebble-bed HTGR). Pebble-bed HTGR adopts online refueling fuel management during its operation. Such a fuel cycle process allows fuel pebbles to pass through the core multiple times before they are discharged [1]. Compared with Light Water Reactors (LWRs), this multi-pass refueling fuel management gives pebble-bed HTGR two features. One is that fuels could reach a higher burnup value, which makes fissions from ²³⁹Pu take higher fractions than LWR [2]. Furthermore, this pebble flow spatially couples fuels with different burnup values inside the core. Therefore, burnup characteristics in pebble-bed HTGR are different with that in LWRs. Considering the safety analysis of HTGR, an important aspect of the safety analysis in HTGR is the uncertainty quantification of peak temperature of fuel pebbles under an accidental scenario. One of the main contributors to the above uncertainty is related the burnup of HTGR. Fission product yield uncertainty is one of the main uncertainty contributors in the burnup process of HTGR, and the quantified uncertainties are an indispensable part of HTGR’s safety analysis. Therefore, contributions from different uncertainty sources in the reactor burnup calculation should be quantified properly to fully investigate the safety features of pebble-bed HTGR.

Previous studies from LWRs’ burnup uncertainty analysis have assured the necessity of quantifying fission product yield uncertainty contribution to reactor burnup calculation. Based on the XSUSA library, Martinez et al. [3] extended the propagation of fission product yield and decay data uncertainties to depletion simulation in the UAM6 PWR pin-cell burnup benchmark. Results showed that fission product yield uncertainty has a non-negligible impact on reactor criticality and nuclide inventory calculation. This finding is reported in Leray et. al. [4], which has assessed the contribution of fission product yield uncertainty to reactor criticality calculation and spent fuel behavior analysis. Quantifying the uncertainty of the infinite multiplication factor (k_inf) and nuclide density induced by cross sections, fission product yields, and the half-life of decay, the SUNDEW code was developed for burnup sensitivity and uncertainty analysis. Its results from TMI-1 PWR pin cell [5] indicate that fission product yields have obvious contributions for nuclide density uncertainty.

Fewer studies have been conducted to address fission product yield uncertainty contribution in pebble-bed HTGR burnup uncertainty analysis. Wang et al. [6] examined the uncertainty contribution of fission product yield in criticality calculation for the burnup equilibrium state of HTGR. However, only fission product yield uncertainty from ²³⁵U is considered in this work. The uncertainty propagation of fission product yield from other fissile nuclides is still absent, specifically in pebble-bed HTGRs, given that fission fractions from ²³⁹Pu in fuels with higher burnup value is comparable with that from ²³⁵U. The fission product yield uncertainty contribution from ²³⁹Pu and other fissile nuclides should be taken into account properly in pebble-bed HTGR burnup uncertainty analysis.

Based on HTGR physics design code VSOP [7], an uncertainty analysis code VSOP–UAM is developed to analyze the uncertainty contribution from nuclear data to pebble-bed HTGR reactor calculation [1]. Currently, this code has been developed with fission product yield uncertainty analysis capacity [8,9]. By using Bayesian based fission product yield covariance estimation procedures [10] and lognormal-based sampling techniques in VSOP–UAM [6], fission product yield uncertainty from ²³³U, ²³⁹Pu and ²⁴¹Pu is studied in this work. Their contributions to effective multiplication factor (k_eff) and nuclide density uncertainty are quantified at the burnup equilibrium state of pebble-bed HTGR. The present work is highlighted in the following aspects: covariance matrices of fission product yield from ²³³U, ²³⁹Pu and ²⁴¹Pu are estimated based on the Bayesian method. There is further emphasis on the comparison of fission product yield uncertainty contributions among these actinides.

The rest of this paper is organized as follows: In Section 2, we briefly introduce the normal distribution-based Bayesian method for estimating the fission product yield covariance matrix and the log-normal distribution based sampling procedures for generating fission product yield perturbed samples. Section 3 introduces the proposed framework for HTGR burnup uncertainty analysis. Section 4 presents the uncertainty quantification (UQ) results regarding k_eff and nuclide density. Finally, conclusions are summarized in Section 5.

2. Methodology

2.1. Fission Product Yield Covariance Matrix Estimation

Fission product yield is an important nuclear metric relating to the production of fission product isotopes in reactor burnup calculation. In the release of ENDF/B-VII.1, the covariance matrix of fission product yield is still absent [4]. Uncertainty propagation with properly estimated fission product yield covariance matrix could guarantee the consistency between independent and cumulative fission product yield as well as the physical constraints imposed on fission product yield data, which is crucial for guaranteeing unbiased uncertainty quantification results. Based on the Bayesian method provided in VSOP–UAM, the covariance matrix for fission product yield from thermal neutron induced fission of ²³³U, ²³⁹Pu and ²⁴¹Pu is estimated. A brief illustration about this estimation is recaptured in Figure 1. For a detailed derivation of this estimation, please refer to Wang et at. [9], which focuses on estimating the covariance matrix of independent fission product yield from thermal neutron induced ²³⁵U fission based on the Bayesian updating method. Under the Bayesian method, multivariate normal distributions are assigned as fission product yields' prior distribution and its mean vector as well as covariance matrix (which is a diagonal matrix) is taken from the ENDF/B-Ⅶ.1 fission product yield sub-library. There are four constraints regarding to the independent fission product yield that are used to construct the likelihood function. The prior probability distribution of fission product yield is updated with the weights from likelihood function, and the posterior probability distribution is obtained. Compared with fission product yield covariance matrix under the prior probability distribution, the covariance matrix under the posterior probability distribution pertains to the above physical constraints. The mentioned four physical constraints considered in this work are: cumulative and independent fission product yield consistency; binary fission constraint (total independent fission product yield conservation); fission system mass number conservation constraint and fission system charge number conservation constraint. This covariance matrix estimation process is schematically illustrated in Figure 1.

In Figure 2, each element is the correlation coefficient between independent fission product yields from the ²³⁹Pu thermal neutron induced fission system. The correlation coefficient has its absolute value larger than 0.01, as shown in Figure 2. The blue plots are negative correlation coefficients between the fission product yields and the red plots represent the positive correlations. The correlation coefficient ρ(y_i, y_j) between independent fission product yield y_i and y_j is calculated as in Equation (1). Here, cov(y_i, y_j) is the covariance between y_i and y_j, while σ(y_i) and σ(y_j) are their standard deviations, respectively.

$$\rho \left(y_i,y_j\right)=\frac{cov\left(y_i,y_j\right)}{\sigma \left(y_i\right)\sigma \left(y_j\right)}$$

(1)

When adopting the stochastic sampling based method to assess independent fission product yield uncertainty propagation, these estimated correlation coefficients here are used to comply with the inherent constraints within each generated independent fission product yield sample.

2.2. Fission Product Yield Stochastic Sampling Method Based on Log-Normal Distribution

Stochastic sampling-based uncertainty analysis is used in this work to conduct fission product yield uncertainty analysis. A brief description of sampling theory for multivariate distribution can be found in Hao et al. [11]. Since most independent fission product yields have large relative uncertainties [6], conventional normal distribution-based sampling procedures would generally produce independent fission product yield samples with negative values. These negative values violate the non-negativity property of fission product yield, and it would bring biases into the uncertainty analysis results. In VSOP–UAM, a log-normal distribution based sampling procedure is used to overcome this issue [12,13]. This sampling procedures are briefly recalled as follows in Figure 3, and for a detailed description regarding this, please refer to Wang et al. [6].

When given mean vector ${\overrightarrow{\mu }}_i$ and covariance matrix V_i,4 of the independent fission product yield, perturbed independent fission product yield samples could be generated by adopting log-normal based sampling procedures provided in VSOP–UAM. First, the original independent fission product yield mean vector and covariance matrix are converted into those of logarithmic independent fission product yields by Equations (2) and (3).

$${\mu }^{\ln }\left(\ln y_i\right)=\ln {\mu }_i\left(y_i\right)-\frac{1}{2}{V}_{i,4}^{\ln }\left(\ln y_i,\ln y_i\right)$$

(2)

$$V_{i,4}^{\ln }\left(\ln y_i,\ln y_j\right)=\ln \left[\frac{V_{i,4}\left(y_i,y_j\right)}{{\mu }_i\left(y_i\right){\mu }_j\left(y_j\right)}+1\right]$$

(3)

The Latin Hypercube Sampling (LHS) technique [14] is used to generate logarithmic independent fission product yield samples with ${\overrightarrow{\mu }}_i^{ln}$ and V_i,4. Then, the exponential transformation regarding to each obtained logarithmic independent fission product yield sample is taken to obtain the non-negative valued independent fission product yield perturbed samples.

3. Burnup Uncertainty Analysis Based on VSOP–UAM

A burnup uncertainty analysis should be studied after generating independent fission product yield samples through the log-normal distribution based sampling procedures. A preliminary study should be conducted to examine the consistency between the VSOP fission product yield database and independent fission product yield perturbed samples generated based on the ENDF/B-VII.1 fission product yield sub library. The VSOP fission product yield database differs from the sampled perturbation database in the following aspects:

(1) The two databases have different data sources: the VSOP fission product yield database is based on the ENDF/B-IV and ENDF/B-V databases [7]. They are different from the ENDF/B-VII.1 database. Hence, it is necessary to examine the consistency between these two databases.

(2) The fission product yield types are different. VSOP contains a shortened fission product isotope decay chain with 44 fission products [7]. These fission products have been assigned a mixing fission product yield data with both independent and cumulative fission product yield. Among them, 14 fission products are provided as independent fission product yields, and the remaining 30 fission products are provided as cumulative fission product yields. Hence, sampled fission product yields should be accorded between these two types. Furthermore, there is an additional pseudo fission product FP-44, accounting for the sum of fission product yields which are not explicitly included in the chain [7,15].

Substituting the VSOP fission product yield database directly with independent fission product yield perturbed samples would cause inconsistency [8]. A method of mapping these two types of fission product yield is established in VSOP–UAM. The cumulative fission product yield sampled data are produced through decay chain mapping based on independent fission product yield sampled data. The fission product yield perturbing samples that are suitable for VSOP burnup calculation are generated after mixing the two types of fission product yield sampled data above.

The fission product yield uncertainty analysis flowchart in VSOP-UAM is shown in Figure 4. Using this program, the contribution of the fission product yield uncertainties from ²³³U, ²³⁵U, ²³⁹Pu and ²⁴¹Pu to burnup calculation could be analyzed.

As stated in Figure 4, independent fission product yield samples are generated by log-normal distribution-based sampling procedures. Corresponding cumulative fission product yield samples are generated for each independent fission product yield sample. Finally, VSOP fission product yield perturbed databases are prepared based on both independent and cumulative fission product yields. After producing N perturbed samples, the VSOP burnup calculation can be conducted repeatedly with these samples. For uncertainty qualification (UQ) analysis, the uncertainty of k_eff and nuclide densities are the main consideration.

4. Uncertainty Analysis Results

Based on the fission product yield uncertainty analysis capability provided in VSOP-UAM, uncertainty contributions to k_eff and nuclide density in the burnup equilibrium state of HTR-PM (High-Temperature Reactor—Pebble-bed Module) are analyzed. Parameters regarding to HTR-PM burnup model used in this work are listed in

Table 1. HTR-PM main burnup model parameters.

Parameter	Unit	Value
Fresh fuel enrichment of 235U	wt%	8.5
Heavy metal mass per fuel pebble	g	7
Total irradiation time in core	d	1056
Refueling times	-	15
Reactor thermal power	MWt	250
Designed average discharge burnup value	MW·d/tU	90,000

.

Fission product yields from ²³³U, ²³⁵U, ²³⁹Pu, and ²⁴¹Pu are used in the HTR-PM burnup model. In order to systematically investigate their uncertainty contributions, a stochastic sampling based uncertainty analysis is conducted in the following three phases.

(1) Individual uncertainty analysis: In the first phase, fission product yield uncertainties from ²³³U, ²³⁵U, ²³⁹Pu, and ²⁴¹Pu are propagated into HTR-PM burnup calculation individually. This study is expected to highlight the importance of fission product yields from these four actinides separately in the HTR-PM burnup calculation.

(2) Comparative uncertainty analysis: In the second phase, fission product yield from Uranium (involving ²³³U and ²³⁵U) and Plutonium (involving ²³⁹Pu and ²⁴¹Pu) are perturbed separately. This study is conducted to address the importance of fission product yield uncertainties from Plutonium fissions in HTR-PM burnup calculations.

(3) Total uncertainty analysis: In the last phase, fission product yield uncertainties from these four actinides are propagated into HTR-PM burnup calculation simultaneously. This study would indicate the total uncertainty contributions from fission product yield in HTR-PM burnup calculations.

In this analysis, the fission product yield perturbed sample size is 1000 to ensure a reliable uncertainty analysis result. k_eff and nuclide density in the equilibrium core of HTR-PM are quantified in each analysis phase.

Under large burnup values, fission fractions from ²³⁹Pu become comparable with that from ²³⁵U, as shown in Figure 5 using ENDF/B-Ⅳ and Ⅴ cross section data from VSOP, which plotted the fission fraction curve of ²³⁵U and ²³⁹Pu against the burnup value. Data in Figure 5 are calculated in VSOP, whose database originated from ENDF/B-Ⅳ and Ⅴ cross section data. Therefore, the fission product yield uncertainty contribution from ²³⁹Pu is of major concern in this work.

4.1. k_eff Uncertainty Quantification

At the burnup equilibrium core state of HTR-PM, the k_eff. value remains unchanged with time.

Table 2. k_eff UQ results of equilibrium core.

Phases	Perturbed Fission Product Yield	Relative Uncertainty (95% Confidence Interval) (%)
Individual uncertainty analysis (I)	235U	0.027 (0.026–0.029)
	239Pu	0.025 (0.024–0.027)
Comparative uncertainty analysis (II)	233U + 235U	0.027 (0.026–0.029)
	239Pu + 241Pu	0.026 (0.025–0.028)
Total uncertainty analysis (III)	233U+ 235U+ 239Pu+241Pu	0.038 (0.036–0.040)

presents the contribution of different fission product yield uncertainties to k_eff uncertainties and the corresponding 95% confidence interval. The fission fraction and nuclide amount for each fissile actinide is listed in

Table 3. Fission fraction and nuclide amount for each fissile at burnup equilibrium core.

	233U	235U	239Pu	241Pu
Fission fraction	3.15 × 109	6.60 × 10−1	2.68 × 10−1	6.86 × 10−2
Nuclide amount (g)	4.65 × 10−4	1.07 × 105	1.40 × 104	3.76 × 103
Nuclide amount ratio(%)	233U/235U: 4.34 × 10−7		241Pu/239Pu: 26.86

.

Table 2 states several burnup uncertainty calculation results. The result from perturbing ²³⁹Pu fission product yield alone a has similar contribution to the relative uncertainty of k_eff comparted with ²³⁵U, which indicated that it is necessary to consider the ²³⁹Pu fission product yield uncertainty contribution in the pebble-bed HTGR burnup uncertainty analysis. Secondly, the square of perturbing all fissile nuclide fission product yields’ value equals to the sum of squares of results in the second phase values; therefore it could be assured that each fission product yield uncertainty propagation is independent from each other. From Table 3, the fission fraction and nuclide density value of ²³³U are much less than other actinides. As a result, the contribution of its fission product yield can be ignored

4.2. Quantification of Average Nuclide Density Uncertainties in Burnup Equilibrium Core

Relative uncertainties of average nuclide densities of actinides and fission products in the core are obtained after perturbing VSOP fission product yield databases. Four actinides’ fission product yields are perturbed simultaneously, and relative nuclide density uncertainties of isotopes included in VSOP are shown in Figure 6.

It is obvious that fission product yield contributions to the relative uncertainties of actinide densities are much smaller than those of fission product isotopes. The main reason is that the effect of fission product yields on actinide densities is indirect. With fission products’ accumulating, it affects actinides’ concentration by influencing neutron energy spectrum calculations and diffusion calculations. Compared with that, fission product yields affect the production of fission products directly. As shown in Figure 6, the contribution of fission product yield uncertainty to the uncertainty of the fission product densities is bounded between 0.2% and 10.8%. This range is much larger than that of actinides.

Figure 7 shows relative uncertainties of nuclide densities when perturbing fission product yields from individual actinide. Results from Figure 7a,b show that individual fission product yield uncertainties from ²³⁵U and ²³⁹Pu are the major contributors to the uncertainties of nuclide densities, compared with ²³³U and ²⁴¹Pu. This is due to the fact that ²³⁵U and ²³⁹Pu are the major fissile actinides in the equilibrium core of HTR-PM. Specially, for the case of ²⁴⁰Pu nuclide density uncertainty results in Figure 7a, the contribution from ²³⁹Pu is slightly larger than that from ²³⁵U. An additional contribution from neutron absorption of ²³⁹Pu is responsible for this difference.

In Figure 7b, the relative uncertainties of nuclide density for 18 fission products from ²³⁹Pu fission product yield uncertainty are larger than that of ²³⁵U. Some typical nuclides are listed in Figure 8. For ¹⁰⁹Ag, the contribution of ²³⁹Pu fission product yield uncertainty to its nuclide density relative uncertainty (up to 10.53%) is much larger than that of ²³⁵U. This is mainly because the β⁻ decay of ¹⁰⁹Pd to ¹⁰⁹Ag has a very short decay half-life of 13.7h [5]. In addition, fission product yield as well as its uncertainty of ¹⁰⁹Pd from fission of ²³⁹Pu is larger than that of ²³⁵U.

Moreover, the fission product uncertainty of ²³⁹Pu contributes significantly to the nuclide density relative uncertainties of ¹¹³Cd and ¹⁵⁷Gd. From Figure 8, ²³⁹Pu fission does not produce ¹¹³Cd directly, whereas it is generated by the decay of ¹¹³Ag, whose half-life is around 5.4h and is produced by ²³⁹Pu. Considering the relative uncertainty of ¹¹³Ag fission product yield in ²³⁹Pu is large (around 64%), and it could be responsible for the considerable nuclide density relative uncertainty of ¹¹³Cd.

Fission product yield and the fission product yield uncertainty data of ¹⁵⁷Gd (y_i = 2.90 × 10⁻⁷, σ_i = 1.86 × 10⁻⁷) (σ_i: standard deviation of each fission product yield) generated by ²³⁹Pu fission are larger than that of ²³⁵U (y_i = 1.48 × 10⁻⁹, σ_i = 9.47 × 10⁻¹⁰). Therefore, perturbing ²³⁹Pu fission product yield leads to a larger nuclide density relative uncertainty of ¹⁵⁷Gd.

Figure 9 shows the results of comparative uncertainty analysis and total uncertainty analysis. It can be seen that the contributions of both ²³³U and ²³⁵U fission product yields to the relative uncertainties of the actinide concentrations are a little larger than that of ²³⁵U alone. A similar phenomenon can be seen in that of ²³⁹Pu and ²⁴¹Pu.

The contributions of fission product yield uncertainties from U and Pu to the nuclide density relative uncertainties have the same magnitude. Thus, the contribution of Pu fission product yield uncertainty to burnup uncertainty cannot be neglected. It should also be noticed, in total uncertainty analysis, that the nuclide density uncertainty of ¹⁰⁹Ag reaches 10.82%, which is the largest relative uncertainty among the nuclide densities.

The above results also demonstrate that it is necessary to pay more attention to the impact of ²³⁹Pu fission product yield uncertainty on burnup uncertainty in HTGR.

5. Conclusions

This work investigates uncertainty contributions of fission product yield to pebble-bed HTGR burnup calculation by using VSOP-UAM. Fission product yields from ²³³U, ²³⁵U, ²³⁹Pu, and ²⁴¹Pu are perturbed to investigate their uncertainty contribution to k_eff and nuclide density uncertainties in the equilibrium core of HTR-PM. Several conclusions are listed, as follows:

(1) Comparisons between fission product yield uncertainties from Uranium and Plutonium indicates that uncertainty contribution from ²³⁹Pu fission product yield cannot be neglected in the pebble-bed HTGR burnup uncertainty analysis due to its high burnup value.

(2) Compared to k_eff, fission product yield uncertainties contribute significantly to the uncertainties of nuclide atomic densities, especially in the case of fission product nuclides.

(3) The contribution of ²³⁹Pu fission product yield uncertainty to ¹⁰⁹Ag nuclide density relative uncertainty exceeds 10%, which is the greatest uncertainty contribution among the others.

The results of the contribution of all fission product yield uncertainty to k_eff uncertainty is less than 0.038% compared with cross-section uncertainty, which reached 0.53% [5]. The next step can be focused on the comparison of contributions of cross section and fission product yield uncertainty for pebble-bed HTGR. The contribution of fission product yield uncertainty to the decay heat uncertainty of pebble-bed HTGRs can be studied further based on this work.