## 1. Introduction

The cumulative plutonium production is proportional to the neutron fluence and so does the change of the impurity elements in the graphite moderator. So, if we know the change of the impurity elements in the graphite moderator, we can estimate the cumulative plutonium production. In many cases, however, we don’t know the change of impurity elements because the initial amount of impurities in the graphite varies from graphite to graphite and is usually unknown.

To overcome this difficulty, GIRM was developed at Pacific Northwest National Lab (PNNL) in 1990’s [1, 2]. Although the initial amount of impurity elements is unknown, the isotopic ratios of the impurity elements such as ^{10}B/^{11}B and ^{36}Cl/^{35}Cl are known initially and their change does not depend on the amount of impurity elements but depends only on the neutron fluence. Once the cumulative plutonium production is tabulated as a function of the isotopic ratio of an impurity element using a simple twodimensional (2-D) unit fuel pin cell model, the cumulative plutonium production can be estimated by measuring the isotopic ratio of the impurity elements in the graphite moderator of the reactor of interest under the assumption that the correlation between the cumulative plutonium production and the isotopic ratio of the impurity elements from the unit fuel pin cell model is applicable everywhere in the core.

In our previous work, a suitability study was done for many candidate indicator isotopes of impurity elements in the graphite moderator, and it was found that ^{10}B/^{11}B, ^{36}Cl/^{35}Cl, ^{48}Ti/^{49}Ti and ^{235}U/^{238}U have a consistent correlation with the cumulative plutonium production, regardless of the initial impurity concentration of the graphite. On the other hand, the correlation between ^{6}Li/^{7}Li and plutonium production depends on the initial concentration of the boron impurities in the graphite because ^{7}Li can be produced both by the neutron capture reaction of ^{6}Li and by the (n, α) reaction of ^{10}B [3].

When GIRM is applied to estimate the cumulative plutonium production of a graphite-moderated reactor, isotopic ratio measurements of impurity elements are not performed for all fuel channels for practical reasons but for some fuel channels. In this case, the cumulative plutonium production of the whole core should be estimated from the measured isotopic ratios. A 3-D cumulative plutonium production map can be produced by a 3-D regression based on the measured isotopic ratios [4].

In this work, the accuracy of a 3-D polynomial regression technique for estimating cumulative plutonium production in a graphite-moderated reactor is assessed. As the reference reactor, a Magnox-type reactor, British Calder Hall reactor [5] was selected. The 3-D depletion calculation for the reference reactor was performed using the MCS code [6], a continuous-energy neutron transport Monte- Carlo code, developed at the COmputational Reactor physics and Experiment laboratory (CORE) of Ulsan National Institute of Science and Technology (UNIST). From the 3-D depletion calculation of the reference reactor, the total cumulative plutonium production of the whole core was calculated for every burnup steps and the result are taken as the reference one. On the other hand, the cumulative plutonium production of the whole core was estimated using the 3-D polynomial regression technique and it was compared with the reference result. In the 3-D regression technique, the cumulative plutonium production was estimated using the ^{10}B/^{11}B ratio at some points from the reference 3-D depletion calculation and the tabulated cumulative plutonium production as a function of ^{10}B/^{11}B ratio using the unit fuel pin cell model of the reference reactor.

## 2. Method

In this section, the specifications of Calder Hall reactor used in this study and the depletion calculation results for the reactor using MCS are provided. Also, the process of GIRM combined with polynomial regression is explained.

### 2.1 Calder Hall Reactor

The detailed specifications of the reactor used to produce plutonium in Yongbyon nuclear scientific research center of North Korea are unknown. However, it is known to be a smaller version of British Calder Hall reactor [7]. Therefore, the Calder Hall reactor whose specifications are well known was taken as the reference reactor in this study. The Calder Hall reactor was used to produce plutonium as well as to produce electricity as the world’s first commercial reactor. The fuel, the moderator, and the coolant of the reactor are natural uranium metal, graphite, and carbon dioxide (CO_{2}) gas, respectively. The core has three zones, Zone A, Zone B, and Zone C and the radius of the coolant hole is different for each zone. The radial and axial layout of Calder Hall reactor are presented in Fig. 1, the geometry of fuel pin for Zone A is shown in Fig. 2. Also, Table 1 presents the design parameters.

The depletion simulation of the Calder Hall reactor was performed using MCS with a nuclear cross-section library based on ENDF/B-VII.1 and HELIOS kappa library. The effective multiplication factors (*k _{eff}*) for the depletion steps are shown in Fig. 3 and Table 2. The standard deviations of the multiplication factor are within 20 pcm.

Cumulative plutonium production at each depletion step calculated by MCS simulation is presented in Fig. 4. Table 3 shows the plutonium isotopes production calculated by MCS in Calder Hall reactor at burnup steps. Since the other isotopes except for ^{238}Pu, ^{239}Pu, ^{240}Pu, ^{241}Pu and ^{242}Pu are produced with less than 0.01 kg, the total plutonium production is assumed to be the sum of ^{238}Pu, ^{239}Pu, ^{240}Pu, ^{241}Pu and ^{242}Pu production. In this study, these values are assumed to be the actual plutonium production from the Calder Hall reactor and they are used as the reference values for the comparison with the values estimated by GIRM combined with polynomial regression in this study.

### 2.2 Graphite Isotope Ratio Method (GIRM)

The procedure of GIRM combined with polynomial regression applied in this study is as follow. First, through 2-D unit fuel pin cell depletion calculation, the cumulative plutonium mass density is tabulated as a function of the ^{10}B/^{11}B ratio. Fig. 5 shows the plutonium mass density as a function of the ^{10}B/^{11}B ratio in 2-D fuel pin. Then, the cumulative plutonium mass density at a sampling point in the reactor is estimated using the ^{10}B/^{11}B ratio sampled from the 3-D simulation and the tabulated cumulative plutonium mass density function. Alternatively, ^{10}B/^{11}B ratio from the 3-D simulation can be replaced by a measured data in actual application of GIRM. The 3-D spatial distribution of plutonium mass density over the entire core is derived through least-squares regression using the estimated plutonium mass density for each sampling point. Finally, the total plutonium production in the core can be estimated by integrating the 3-D spatial distribution of plutonium mass density over the entire core. The accuracy of GIRM combined with polynomial regression can be evaluated by comparing the total plutonium production estimated by GIRM combined with polynomial regression and that from the 3-D core depletion calculation using MCS.

The radial and axial sampling points in the Calder Hall reactor are given in Fig. 6. Since the configuration of fuel pins has a quarter core symmetry in a whole core, the sampling points were selected within a quarter core. The number of radial and axial sampling points are 28 and 5, respectively. Therefore, a total of 140 sampling data were used.

A 3-D space-dependent least-squares regression function based on the triangular basis was used to calculate the plutonium mass density for the whole core as shown in Eq. (1).

where *ƒ*(*x*, *y*, *z*) is the plutonium mass density for the (*x*, *y*, *z*) location in the core, *K* and *N* are the regression orders for the axial and the radial directions, respectively. In this study, several orders for the axial and the radial directions were tested as shown in Fig. 7 in order to find an optimized result. The results are similar to each other and quite good accuracy was observed regardless of the polynomial order. It is ascribed to the fact that the core has no control rod and therefore the flux shape is very smooth throughout the entire core. Cubic order for radial (*K* = 3) and quartic order for axial direction (*N* = 4) was chosen for the rest part of this study.

## 3. Results

### 3.1 Whole Core Plutonium Production

The total cumulative plutonium production calculated by MCS and that estimated by GIRM combined with polynomial regression were compared at each burnup steps as shown in Table 4 and Fig. 8. The maximum error of 3.1% is observed at the first burnup step and the error at the last burnup step is −4.5%. The root mean squares (RMS) of the errors throughout the burnup steps is 2.2%. The coefficients of determination (R2) for polynomial regression are also given for all depletion steps in Table 4. They are close enough to 1.0 (between 0.9879 and 0.9965), which indicates that the polynomial regression function was derived properly.

### 3.2 Axial and Pin-wise Plutonium Production

To verify the accuracy of polynomial regression, the plutonium productions at various axial points and those at various fuel pins were compared at the burnup steps of 1250, 2250, and 3250 days. Table 5–7 compare the axial cumulative plutonium production calculated by MCS and estimated by GIRM combined with polynomial regression at each burnup step. Although relatively large errors are observed at the top and bottom regions of active core, the RMS errors are 3.1%, 3.2%, and 3.5% at depletion steps of 1250, 2250, and 3250 days respectively. The correlation between the cumulative plutonium mass density and ^{10}B/^{11}B ratio is calculated based on a 2-D unit fuel pin cell. It has somewhat similar conditions along the central region of the core, but not in the top and bottom regions, where the spectrum is different because of the axial reflectors. Therefore, the top and bottom active core regions show a relatively larger error.

As for the pin-wise cumulative plutonium production, several fuel pin locations in different regions of the quarter core were selected. The chosen locations are shown in Fig. 9. The comparison of cumulative plutonium production calculated using both MCS and GIRM for each chosen fuel pin is presented in Table 8–10. The relative errors were found within acceptable range.

## 4. Conclusion

In this study, the accuracy of the GIRM combined with polynomial regression to estimate the total cumulative plutonium production was verified. The cumulative plutonium production in the Calder Hall reactor was estimated by the GIRM combined with polynomial regression and the results were compared with those calculated by a 3-D Monte- Carlo depletion calculation using the MCS code. With cubic and quartic order regression in axial and radial direction, respectively, the RMS error throughout the burnup steps is about 2.2%. Although the errors at top and bottom of the active core are relatively large, the error of the regression with cubic and quartic polynomial in axial and radial direction was acceptable. The RMS error was around 3.3%. The accuracy of the cumulative plutonium production estimated by the GIRM combined with polynomial regression at 12 fuel pins also assessed. The RMS error was about 2.4–2.8% depending on the burnup steps.

It was found that the accuracy of the regression with cubic and quartic polynomial was satisfactory for the estimation of cumulative plutonium production in Calder Hall reactor. However, no control rod was considered during the reactor operation in this study. The use of control rod will cause a distortion of flux shape. In that case, the accuracy of the cubic and quartic polynomial regression could be doubtful. The effect of the control rod on the accuracy of the polynomial regression are expected to be assessed in further works.