Calculation and Analysis of the Source Term of the Reactor Core Based on Multivariate Analysis of Variance

School of Nuclear Energy and Environment, Southeast University, Nanjing 210096, China Institute of Nuclear "ermal-hydraulic Safety and Standardization, Beijing, China School of Nuclear Science and Engineering, North China Electric Power University, Beijing 102206, China Beijing Key Laboratory of Passive Safety Technology for Nuclear Energy, North China Electric Power University, Beijing 102206, China Key Laboratory of Condition Monitoring and Control for Power Plant Equipment, North China Electric Power University, Beijing 102206, China


Introduction
When a serious accident occurs in a nuclear power plant, a large number of radioactive fission products are released from the core due to meltdown, which will cause environmental pollution and casualties. e amount of core source terms directly affects the quantity released. Jia [1] analyzed the important actinides nuclides, fission product nuclides, and activation product nuclides and proposed a reasonable disposal plan. Liu and Zhu [2] proposed an analysis model of the minimum nuclear criticality accident source term and provided a related calculation method. Wheeler et al. [3] used Serpent 2 to calculate the fuel consumption of molten salt reactors and found that gas removal can affect many different fission products beyond the gaseous chemical group. Actinides were also affected, although not in any gaseous fission product decay chain. Sun et al. [4] used MELCOR to study the source term estimation of an AP1000 nuclear power plant under severe accident conditions. Lee and Ko [5] developed a method to quantify the source term in each source term category identified in the level II probabilistic safety assessment analysis of nuclear power plants; then, the obtained source term characteristics were compared with other source terms. Liu et al. [6] conducted a series of experiments on the primary coolant and in various deduction methods was clarified. Fang et al. [7] carried out ab initio calculations of the antineutrino flux of a new isotope reactor and provided accurate numerical calculations of the lepton wave function.
eir results showed that the accumulated antineutrinos were within the range of high-energy antineutrinos, and the electron and electron energy spectra have significant but opposite spectral deviations in the range of 2%-4%. Yang et al. [8] introduced the analysis method, scope, and main operating procedures of the source items following the Qinshan Nuclear Power Plant accident and calculated the release, migration, and distribution of nuclides under the design-basis accident through the accident analysis program. Liu et al. [9] analyzed the problems in the calculation of the fission product source term of M310/CPR1000, EPR, and AP1000 and modified the calculation process of the fission product source term. Jin et al. [10] used the modular accident analysis program-integrated severe accident analysis code (MAAP-ISAAC) to quantitatively evaluate the amount of radionuclides released into the environment following an interfacing system loss-of-coolant accident. e analysis results showed that the decontamination factor for nuclide groups, excluding inert gas, Sb, and Te 2 nuclide groups, was approximately 2.5. Bahadir and Lindahl [11] used the nodal code SIMULATE to simulate the reactor core to calculate the node-wise burnup and the power of the target assembly (boron concentration, fuel temperature, moderator temperature, and power level) under different burnups. ey found that SIMULATE-5 can accurately describe the neutronic and thermal-hydraulic behavior of boiler water reactor and pressured water reactor (PWR) cores. Gera et al. [12] estimated the source terms of some postulated severe accident scenarios in the 220 MW Indian pressurized heavy-water reactor. ey found that the estimated source term and corresponding consequences were higher in reactor inlet header break cases than in reactor outlet header break cases. Ahn et al. [13] used MELCOR 2.2 and MAAP 5.04 to calculate the source terms of a PWR after a serious accident. ey found that the dedicated mitigation strategy greatly decreased the environmental release of the fission product cesium. With regard to the evolution of severe accident and plant responses, both codes predicted the general trend of each base and mitigation scenarios. ere are relatively few studies on the calculation of the weight of factors affecting the core source term in this type of literature. By calculating the radioactivity and photon source strength of actinides and fission products, the weight proportion of each factor was calculated by the multifactor variance analysis method [14]. It is of great significance to understand the importance of each influencing factor of the source term of the reactor core and to study the source term of the reactor under normal operations and serious accidents.

Core Description
e AP1000 core was selected as the research object. e basic parameters of the AP1000 fuel assembly [15] are listed in Table 1.
e influence of four factors: burnup, fuel enrichment, specific power of the reactor, and operation mode on the source term of the reactor core was investigated. e values for each factor are listed in Table 2.

Point Reactor Kinetics Model.
In this study, the Oak Ridge Isotope Generation and Depletion Code (ORIGEN) was used for burnup calculations [16]. ORIGEN was developed by the Oak Ridge National Laboratory for nuclide ignition consumption, decay, and radioactive material treatment [17]. e input file specifications include a database of over 700 nuclides that are widely used in various types of reactors. e ORIGEN program uses the point reactor kinetics model. According to the average neutron flux or the power of space and energy within a certain range and read in terms of various cross sections, decay, and other data from the database, we can calculate the accumulation and change of any nuclide in a given homogeneous material. In the program, a nonhomogeneous first-order ordinary differential equation [18] is used in the following equation: where i (1, 2, 3, . . .. . ., N) is the number of nuclides; X i is the atom density of nuclide i; l ij is the fraction of radioactive disintegration by nuclide j which leads to formation of nuclide i; λ i is the radioactive decay constant of nuclide i (1/ s); ϕ is position-and energy-averaged neutron flux, n/ (cm 2 ·s); f ik is the fraction of neutron absorption by nuclide k which leads to formation of nuclide i; σ k is the spectrumaveraged neutron absorption cross section of nuclide k; and F i is the continuous feed rate of nuclide i.

Neutron Fluence Calculation Model.
In the ORIGEN2 program, the neutron flux is calculated from the power. For the sake of clarity, we assume that the power to be generated from the fuel is specified and that the flux must be calculated. e first approximation of instantaneous neutron flux at the beginning of the irradiation time step is shown in the following equation: where ϕ is the instantaneous neutron flux (n·cm −2 ·s −1 ); P is the power (MW); X i f is the amount of fissile nuclide i in fuel (g·atom); σ i f is the microscopic fission cross section for nuclide i, (barn); andR i is the recoverable energy per fission for nuclide i (MeV/fission).

Multivariate Analysis of Variance.
e main function of multivariate analysis of variance [19] is to determine whether multiple factors have a significant influence on dependent variables through the process of hypothesis testing. e calculation model of the multivariate analysis of variance [20] is used in the following equation: where F is the statistic; MS B is the variance between groups; MS W is the variance within groups; SS B is the sum of squares between groups; SS W is the sum of squares within groups; df B is the degree of freedom between groups, df B is K −1 ; K is the number of groups; df W is the degree of freedom within groups, df W is K(n-1); and n is the number of levels in each group. (1) Calculation of radioactivity: the radioactivity of actinides and fission products under different fuel consumptions is shown in Figure 1.

Analysis of the
It can be seen that the radioactivity of actinides and fission products generally increased with the increase in fuel consumption. However, after reaching 10000 MWd/tU, the rising trend of fission product radioactivity slowed down, while that of actinides increased. e activity of fission products is one order of magnitude higher than that of actinides. Evidence has shown [21] that owing to the different half-lives, the radioactive decay law of various nuclides that change with fuel consumption is similar. e total amount of long-lived radionuclides increased with fuel consumption. In contrast, short-lived radionuclides were not very sensitive to changes in fuel consumption except in the initial accumulation stage. is is one of the reasons behind the different trends in the radioactivity of actinides and fission products.
(2) Calculation of photon source strength: the photon source strengths under different fuel consumptions are shown in Figure 2.
It can be seen that the photon source strength of each energy group increased with the increase in fuel consumption. Because of the variation of fission products and actinides with fuel consumption, the photon source strength of groups with energies of 1.36-1.80 MeV, 2.2-2.6 MeV, and 3.0-3.5 MeV tend to decrease when the fuel consumption exceeds 10000 MWd/tU. For the same fuel consumption, the photon source strength with low energy is more than 10 orders of magnitude greater than that with high energy.

Analysis of the Influence of Enrichment.
e influence of enrichment was analyzed under the following conditions: fuel consumption of 30000 MWd/tU; a specific power of 4040 MW/tU; continuous operation mode; and enrichment degrees of 2%, 3%, 4%, and 5%.
(1) Calculation of radioactivity: the radioactivity of actinides and fission products with different enrichments is shown in Figure 3.
It can be seen that the radioactivity of fission products and actinides decreased with the increase in enrichment. When the enrichment increased from 2% to 3%, the radioactivity decreased significantly. When the enrichment increased from 3% to 5%, the radioactivity did not decrease. e main reason for this situation is that, under the same total fuel installed capacity, the 235 U content at low enrichment was less, while at high enrichment, it was greater, which leads to changes in the radioactivity of actinides and fission products.
(2) Calculation of photon source strength: the photon source strengths under different enrichment are shown in Figure 4.
It can be seen that the photon source strength of each group of photons tended to decrease depending on the enrichment. Among them, the photon source strength of the group with energy greater than 2 MeV decreased significantly. For the same enrichment, the lower the photon energy, the greater the photon source strength; the higher the photon energy, the smaller the photon source strength.
e maximum values of both can reach 10 orders of magnitude.

Analysis of the Influence of Specific Power.
e influence of specific power was analyzed under the following conditions: fuel consumption of 30000 MWd/tU; enrichment of 4%; continuous operation mode; and specific powers of 20 MW/tU, 30 MW/tU, 40 MW/tU, and 50 MW/tU.
(1) Calculation of radioactivity: the radioactivity of actinides and fission products at different specific powers is shown in Figure 5.
It can be seen that as the specific power increased, the activity of actinides and fission products increased gradually. However, the activity of fission products is several orders of magnitude higher than that of actinides. With the increase in specific power, the magnitude of change of fission products with specific power is higher than that of actinides. e radioactivity of fission products and actinides varied linearly with specific power, with a correlation coefficient of 1.
(2) Calculation of photon source strength: the photon source strengths at different specific powers are shown in Figure 6.
It can be seen that, in the low energy region, the higher the specific power, the stronger the photon source strength.
In the high-energy region, the photon source strength was not affected by the specific power. At the same specific            (1) Calculation of radioactivity: the radioactivity of actinides and fission products under different operating modes is shown in Figure 7.
It can be seen that the radioactivity of fission products and actinides did not change in both continuous operation and interval operation modes. is indicates that the operation mode has little effect on the activity.
(2) Calculation of photon source strength: the photon source strengths under different operation modes are shown in Figure 8.
It can be seen that the two operation modes have little effect on the photon source strength. us, under the same burnup and long operation time, the change in the operation mode will not have a significant impact on the photon source strength in the reactor.

Weight Calculation and Analysis.
e weight of each impression factor was calculated and analyzed by multivariate analysis of variance. Section 4.4 shows that the operation mode does not affect the radioactivity and photon source strength of actinide nuclides and fission products; hence, the three factors of burnup, enrichment, and specific power were selected. Based on the data in Table 2, an orthogonal table of fuel consumption, specific power, and enrichment was established, as shown in Table 3.
According to Table 3, the multifactor analysis of variance program was completed, and Table 4 was obtained after calculation. e values of the influence factor F of the operation results are shown in Figure 9. Figure 9 shows that the accumulation of specific power to the core source term is the largest, far more than the other two factors. e enrichment degree is a negative correlation. erefore, the type of reactor should be the main consideration in the selection of specific power to minimize the generation of the core source term in the reactor.

Conclusions
In this study, the key parameters for the calculation of the source terms for supercritical water-cooled reactors were selected. e ORIGEN2 code was used to simulate the radioactive activity and photon source strength under different fuel consumptions, enrichments, specific powers, and operation modes.
e results show that the radioactivity of actinides and fission products increased with the increase in fuel consumption and decreased with the increase in enrichment. e radioactivity of fission products and actinides varied linearly with the specific power, with a correlation coefficient of 1. Simultaneously, a multifactor analysis program was established to calculate the influence of various factors on the activity of fission products. It was found that the specific power was the most important factor, followed by the enrichment degree; fuel consumption was the least important.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare no conflicts of interest in the publication of this paper.