Calculation methods

After an accident, the whole-core fuel claddings damage and nuclides are released from the fuel cladding gap and the primary coolant into the containment and subsequently into the environment after a series of removal actions. The process of the accident radiological consequence analysis is shown in Fig. 1.

Fig. 1Full-screen View

Flowchart of accident radiological consequence analysis.

For calculation of the radioactivity released into the containment and environment, the nuclides were divided into two categories according to their decay chains: precursor and daughter nuclides. Figure 2 shows several scenarios of the nuclide decay process, where N₁ is the precursor nuclide and N₂ and N₃ are daughter nuclides.

Fig. 2Full-screen View

(Color online) The decay process of nuclides.

In the process of nuclide release from the reactor core into the containment and environment following an accident, the generation and depletion mechanisms of nuclides differ between time periods. The generation of nuclides in the containment atmosphere includes the release of the core fuel cladding gap and primary coolant as well as the decay of the precursor nuclides. The depletion factors to be considered vary between nuclides in the containment atmosphere, and include nuclide decay and containment leakage for inert gases, and nuclide decay, natural removal, and containment leakage for iodine and aerosols, since this would reduce their radioactivity in the containment atmosphere. Furthermore, the nuclides in the environment originate from leakage from the containment.

In the analysis of the accident source term, some nuclides, such as ¹³¹I, ¹³²I, ¹³³I, ¹³⁴I, ¹³⁵I, ⁸⁷Kr, ⁸⁸Kr, ¹³⁸Xe, and ¹³⁶Cs, were precursor nuclides, and others, such as ⁸⁵Kr, ¹³¹Xe^m, ¹³³Xe^m, ¹³³Xe, ¹³⁵Xe^m, ¹³⁵Xe, and ¹³⁸Cs, were daughter nuclides. Radioactivity models of the precursor and daughter nuclides were established to calculate their release amounts into the containment and environment. The calculation models of nuclide radioactivity in the containment atmosphere after the accident were as follows [24]:

I. The precursor nuclide radioactivity model: $\frac{\text{d}{A}_{c,i}(t_n)}{dt}=P_{A_{c,i}}{e}^{-{\lambda }_i{t}_n}-({\lambda }_i+{\lambda }_{\text{di}}\text{+}{\lambda }_0)A_{c,i}(t_n)$ (1)

II. The daughter nuclide radioactivity model: $\frac{\text{d}{A}_{c,j}(t_n)}{\text{d}t}=P_{A_{c,j}}{e}^{-{\lambda }_j{t}_n}+{\lambda }_j{\displaystyle \sum _{i=1}^{j-1}{\xi }_{ij}{A}_{c,i}(t_n)}-\left({\lambda }_j+{\lambda }_{\text{dj}}+{\lambda }_0\right)A_{c,j}(t_n),$ (2) where $A_{c,i}(t_n)$ and $A_{c,j}(t_n)$ are the radioactivity of nuclides i and j in the containment, respectively (Bq); $P_{A_{c,i}}$ and $P_{A_{c,j}}$ are the release rates of nuclides i and j in the reactor core and primary coolant, respectively (Bq/h); ${\lambda }_i$ and ${\lambda }_j$ are the decay constants of nuclides i and j, respectively (h^-1); ${\lambda }_0$ is the containment leakage rate (h^-1); ${\lambda }_{\text{di}}$ and ${\lambda }_{\text{dj}}$ are the removal constants of nuclides i and j in the containment (h^-1); ${\xi }_{ij}$ is the branching ratio of nuclide i decaying into nuclide j; and t_n is the time that has passed since the accident (h).

The nuclide radioactivity released from the containment into the environment was calculated using the following equations [24]: $A_{e,i}(t_{\text{n}})={\displaystyle {\int }_{t_{n-1}}^{t_n}{\lambda }_0{A}_{c,i}(t_{\text{n}})\text{d}t},$ (3) $A_{e,j}(t_{\text{n}})={\displaystyle {\int }_{t_{n-1}}^{t_n}{\lambda }_0{A}_{c,j}(t_{\text{n}})\text{d}t,}$ (4) where $A_{e,i}(t_{\text{n}})$ and $A_{e,j}(t_{\text{n}})$ are the radioactivity of nuclides i and j in the environment, respectively (Bq).

The time that passed since the accident was divided into several time periods (t_n-1, t_n), and the nuclide radioactivity in each was calculated iteratively. The division of the time steps comprehensively considered factors such as changes in the aerosol removal constant with time, the removal time of iodine, and the time node of nuclide radioactivity output results.

The natural removal effects of iodine and aerosols in the containment atmosphere are restricted by the natural removal coefficient and decontamination factor. The calculation model for the nuclide removal time in the containment atmosphere is therefore as follows [24]: $\begin{array}{ll}t=t_0-\frac{\ln \frac{1}{K\cdot DF}}{{\lambda }_{\text{d}}},\hfill & K=\frac{1}{0.5{\lambda }_{\text{d}}}\left(1-{\text{e}}^{-0.5{\lambda }_{\text{d}}}\right)\hfill \end{array},$ (5) where DF is the maximum decontamination factor of a nuclide; ${\lambda }_{\text{d}}$ is the removal constant of a nuclide (h^-1); and t₀ is the release time of a nuclide from the reactor core and primary coolant into the containment (h).

The 95 % χ/Q values for 16 downwind directions and the overall site were calculated using ARCON96 [24-25]. However, ACRON96 did not calculate the maximum sector at the 99.5th percentile. Instead, the 99.5th-percentile χ/Q value for each sector was obtained according to the calculation method described in RG 4.28 [21].

First, the percentage of time at which each χ/Q threshold was exceeded was calculated using the following formula: $P=\left(1-\frac{H_{\text{avg}}}{H_{\text{total}}}\right)\times 100,$ (6) where P is the percentage of time at which each χ/Q threshold is exceeded (%); H_totalis the total number of hours in the. LOG file for a given averaging interval; and H_avg is the number of hours exceeded for each χ/Q threshold.

A simple linear interpolation could then be used to determine the 99.5th-percentile χ/Q value using the following equation: $y=y_1+\frac{x-x_1}{x_2-x_1}\cdot \left(y_2-y_1\right)$ (7) where y is the 99.5th-percentile χ/Q value (s/m³); y₁ and y₂ are the smaller and greater of the two bounding χ/Q values, respectively (s/m³); x = 99.5; and x₁ and x₂ are the smaller and greater of the two exceedance frequencies, respectively.

The larger of the two χ/Q values, either the 99.5 % maximum sector value or the 95 % overall site value, were selected to represent the χ/Q value for the 0–2 h, 2–8 h, 8–24 h, 24–96 h, and 96–720 h time intervals.

The individual total effective dose included the inhalation dose from internal exposure, the cloud immersion dose, and the ground deposition dose from external exposure. The following equations were used to calculate the individual total effective doses [19]: $D_a={\displaystyle \sum _i{\text{DCF}}_i{}^a{\displaystyle \sum _j{Q}_{ij}\cdot (\chi /Q{)}_{xj}}},$ (8) $D_g={\displaystyle \sum _i{\text{DCF}}_i{}^g{\displaystyle \sum _j{Q}_{ij}\cdot {\omega }_{ij}\cdot \left(1-e^{-{\lambda }_i\cdot t_j}\right)/{\lambda }_i}}$ (9) $D_{\text{h}}={\displaystyle \sum _i{\text{DCF}}_i{}^h{\displaystyle \sum _j{Q}_{ij}\cdot {\left(\chi /Q\right)}_{xj}\cdot \text{B}{r}_j}},$ (10) $D_{\text{t}}=D_{\text{a}}+D_{\text{g}}+D_{\text{h}},$ (11) where D_a and D_g are cloud immersion dose and ground deposition dose from external exposure, respectively (Sv); D_h is the inhalation dose from internal exposure (Sv); D_t is the individual total effective dose (Sv); Q_ij is the amount of nuclide i released into the environment during time interval j (Bq); (χ/Q)_xj is the offsite atmospheric dispersion factor at x m from the release point during the time interval j (s/m³); DCF_i^a is the effective dose conversion factor via cloud immersion of nuclide i (Sv/s)/(Bq/m³); DCF_i^g is the effective dose conversion factor via ground deposition of nuclide i (Sv/s)/(Bq/m³); DCF_i^h is the effective dose conversion factor via inhalation of nuclide i (Sv/Bq); ω_ij is the ground deposition factor for nuclide i during time interval j (1/m²); λ_i is the decay constant for nuclide i (1/s); t_j is the release time of the time interval j (s); and Br_j is the adult breathing rate during the time interval j (m³/s).

Main parameters

The main assumptions and parameters for the radioactivity and radiation dose calculations after the accident were as follows [26, 27, 28, 29, 30]:

· It was assumed that the core was uncovered by a pipe break, all fuel claddings in the reactor core were damaged, and that the containment maintained its integrity. After the accident, nuclides were released from the fuel cladding gap and primary coolant into the containment. The core inventory used to calculate the accident source term was obtained from the end of cycle data (Table 1).

· In the analysis of the radiological consequences of the accident, inert gas, iodine, and Cs released into the environment were considered the key nuclides. The production term of the precursor nuclides was taken into account.

· After the accident, the release flow of the primary coolant was constant and the release time was 30 s. The elemental and organic iodine contents in the primary coolant were 97 % and 3 %, respectively.

· The radionuclides in the cladding gap of the damaged fuel rods were linearly released into the containment within 30 min after the accident. The radionuclides released from the cladding gap mainly included inert gas, iodine, and alkali metals, and the release proportion was 5 % of the core inventory. The compositions of the particulate, elemental, and organic iodine released from the cladding gap into the containment were 95 %, 4.85 %, and 0.15 %, respectively.

· Considering the natural removal of elemental iodine and aerosols in the containment, the decontamination factor limits were 200 and 1000, respectively. The natural removal rates of elemental iodine and aerosols were 1.3 h^–1 and 0.1 h^–1, and the corresponding removal times were 4.35 h and 69.34 h, respectively. No deposition, resuspension, or filtration of inert gas was considered due to their inert nature.

· Within 24 h after the accident, the leakage rate of the containment was 0.2 % of the containment free volume per day, and the leakage rate was halved after 24 h.

· The meteorological data ARCON96 required for χ/Q calculations, including precipitation, wind direction, wind speed, and cloud cover, were obtained from the annual observation statistics of a site in Heilongjiang, and covered more than 8700 h of data. Atmospheric stability was obtained using meteorological data. The release type was a ground release and the building aera was 0.01 m².

· The dose conversion factors were obtained from the GB18871, ICRP71, and federal guidance reports (FGR) No.11 and No.12, and the breath rate from the RG 1.4.

The site boundary distance of the SMR was 160 m. According to the safety target of the small PWR, for the important sequence of the beyond design basis accident the individual total effective dose at the site boundary should be less than 10 mSv throughout the entire duration of the accident [22].

Radioactivity released into the environment

The nuclide radioactivity released from the primary coolant after the accident was several orders of magnitude smaller than that released from the reactor core fuel cladding gap; therefore, it was negligible. The total radioactivity released into the environment after the accident was 10¹⁴ Bq. Figure 3 shows the change in iodine radioactivity in the environment 30 days after the accident. The radioactivity of various forms of iodine in the environment was closely related to the iodine radioactivity in the cladding gap and containment. Although the particulate iodine in the containment undergoes natural removal over a long period of time, the radioactivity of the particulate iodine released into the environment reached 10¹³ Bq after 30 days due to its high proportion in the cladding gap, which was much higher than that of elemental and organic iodine. While the proportion of elemental iodine in the cladding gap was higher than that of organic iodine, elemental iodine is affected by natural removal, such as sedimentation, heat conduction, and water vapor condensation, resulting in a similar release amount of both.

Fig. 3Full-screen View

(Color online) Radioactivity of iodine released into the environment after the accident.

There were differences in the release curves of each iodine form. For elemental and particulate iodine, ¹³³I released the largest amount, reaching 7.42×10¹⁰ Bq and 1.26×10¹³ Bq 30 days after the accident, respectively, which resulted from the combined effects of nuclide half-life, initial radioactivity, and natural removal. Regarding organic iodine, ¹³¹I released the largest amount, with a radioactivity of 1.84×10¹¹ Bq. This was because organic iodine has no natural removal effect and is only affected by self-attenuation and the initial radioactivity. The longer the time after the accident, the more significant the effect of self-attenuation. Among organic iodine nuclides, ¹³¹I has the longest half-life, resulting in a larger release amount compared to other iodine nuclides 30 days after the accident.

The radioactivity of Cs released into the environment after the accident is shown in Figure 4. ¹³⁸Cs had the largest initial radioactivity in the fuel cladding gap; therefore, the release amount of ¹³⁸Cs was the largest at the initial stage of the accident but, owing to its very short half-life (approximately 32 min) combined with natural removal effects, such as gravity settlement, thermal electrophoresis, and diffusion electrophoresis, the ¹³⁸Cs in the containment decayed quickly and was no longer released into the environment a few hours after the accident. In turn, the radioactivity of ¹³⁴Cs, ¹³⁶Cs, and ¹³⁷Cs in the environment gradually increased with time, but the rate of increase gradually slowed. After 30 days, the radioactivity of ¹³⁴Cs, ¹³⁶Cs, and ¹³⁷Cs in the environment reached equilibrium, where the released amounts of ¹³⁴Cs and ¹³⁷Cs were almost identical, reaching a radioactivity level of 10¹² Bq.

Fig. 4Full-screen View

(Color online) Radioactivity of cesium released into the environment after the accident.

The depletion factor of an inert gas is its own decay. At the beginning of the accident, the release amount of ⁸⁸Kr was the largest, and the release amounts of ⁸⁵Kr^m, ⁸⁷Kr, and ⁸⁸Kr in the environment gradually reached equilibrium with time, whereas the radioactivity of ⁸⁵Kr continuously increased and did not reach equilibrium after 30 days (Fig. 5). ⁸⁵Kr released from the fuel cladding gap decayed slowly in the containment atmosphere owing to its long half-life (10.72 years) and was gradually released into the environment; therefore, its release amount was the largest among krypton nuclides.

Fig. 5Full-screen View

(Color online) Radioactivity of krypton released into the environment after the accident.

Regarding xenon nuclides, ¹³³Xe released the highest amount, with a radioactivity of 10¹⁴ Bq in the environment (Fig. 6). The released amounts of ¹³⁵Xe^m, ¹³⁵Xe, and ¹³⁸Xe reached a steady state and ¹³³Xe^m tended to be balanced, while those of ¹³¹Xe^m and ¹³³Xe continued to increase after the accident. This was mainly because the half-life of ¹³¹Xe^m and ¹³³Xe is 11.84 days and 5.25 days, respectively, which is longer than that of other xenon nuclides. The time required for inert gases released into the environment to reach equilibrium is proportional to their half-life; the longer the half-life of the nuclide, the longer it takes for radioactivity to reach equilibrium.

Fig. 6Full-screen View

(Color online) Radioactivity of xenon released into the environment after the accident.

Radiation dose at the site boundary

The 99.5th-percentile χ/Q values for the 16 downwind directions and the 95th-percentile χ/Q values for the overall site at different time periods after the accident were obtained (Fig. 7). The maximum value of the 99.5th-percentile χ/Q for the 16 downwind directions appeared in the east-north-east (ENE) and northeast (NE) directions, and the χ/Q values decreased with time. In the 0–2 h and 2–8 h time intervals, the 95th-percentile χ/Q values for the overall site were less than the maximum values of the 99.5th-percentile χ/Q for the 16 downwind directions, while the 95th-percentile χ/Q values for the overall site in the 8–24 h, 24–96 h, and 96–720 h time intervals were significantly greater than the maximum values of the 99.5th-percentile χ/Q for the 16 downwind directions.

Fig. 7Full-screen View

(Color online) Atmospheric dispersion factors at different time intervals after the accident.

The individual total effective doses (adults) at the site boundaries are shown in Fig. 8. Of the 16 downwind directions, the ENE direction received the largest radiation dose, followed by the NE direction. The individual total effective doses calculated using the 99.5th-percentile χ/Q values for the 16 downwind directions were lower than those calculated using the 95th-percentile χ/Q values for the overall site. According to the larger χ/Q value of the 99.5-percent maximum sector value and the 95-percent overall site value for each time interval, the total effective dose at the site boundary was 8.65 mSv. The inhalation dose from internal exposure was the main contributor to the individual total effective dose, accounting for 88.5 %, followed by the cloud immersion dose from external exposure, which accounted for 8.7 %, while the contribution of the ground deposition dose from external exposure was small, accounting for only 2.8 %. The dose resulting from ground deposition had a long-term effect. After accidents where nuclides are released into the environment within a short time span, the ground deposition dose from external exposure can be ignored. For example, the NuScale SMR does not consider the ground deposition dose for the analysis of radiological consequences in fuel handling accident.

Fig. 8Full-screen View

Total effective doses at the site boundary 30 days after the accident.

The χ/Q value and nuclide radioactivity released into the environment at different time periods after the accident jointly affected the individual total effective dose. At the 0–2 h time point, the cumulative release amount of nuclides in the environment was small, even though the χ/Q value of the ENE direction was higher than that of the overall site. At the 2–8 h time point, the χ/Q values of the ENE and NE direction were similar to that of the overall site. After 8 h, the χ/Q value of the ENE direction was lower than that of the overall site, the nuclides in the containment were continuously released into the environment, especially those with a long half-life, and the release amount gradually increased with time. Therefore, after the accident the individual total effective dose calculated by the 95th-percentile χ/Q value for the overall site was higher than that calculated by the 99.5th-percentile χ/Q values for the 16 downwind directions.

Uncertainty in the calculation results

Uncertainty in calculation results is an unavoidable problem in nuclear and radiation safety analyses [31, 32, 33, 34]. In general, the sources of uncertainty in accident radiological consequences include: 1) uncertainties in the models of radiological consequences, 2) uncertainties from parameters such as experimental and empirical data, and 3) uncertainties caused by human factors.

The uncertainties caused by models mainly refer to the radioactivity model of nuclides in the containment and environment, the atmospheric dispersion factor, and the radiation dose models. Herein, the accident source term model was verified using the recognized TACTIII and TITAN 5 computer codes. The results showed that the relative deviation of nuclide radioactivity in the environment calculated by the established model and TACT III was within ± 5%, and that of the nuclide effective removal time in containment calculated by the established model and TITAN 5 was less than 1 % [23]. The main reason for the differences in the nuclide radioactivity released into the environment was the fact that the nuclide decay constants used in the established calculation model were obtained from the radioisotope manual, which differs from the nuclide data of the TACT III database, such as the decay constant of ¹³⁸Xe. The relative deviation of the calculation results was within ± 0.05 % when using the same nuclide decay constants. The atmospheric dispersion factor was calculated using ARCON96, which was based on field measurements obtained at seven reactor sites. The diffusion coefficients were revised and the dispersion algorithms improved the model performance, resulting in higher prediction accuracy [21, 25]. The individual total effective dose calculation model was verified through analysis of radiological consequences of fuel handling accident at the NuScale SMR. The individual total effective dose calculated by the established model was 5.4 mSv and that in NuScale final safety analysis report was 5.5 mSv, which proves the accuracy of the radiation dose model.

The parameter values used in the accident radiological consequence analysis were collected from internationally recognized and widely used literature, such as the International Commission on Radiological Protection reports, federal guidance reports from the Environmental Protection Agency, and Nuclear Regulatory Commission regulatory guides. To analyze the changes in nuclide radioactivity in the reactor core, the uncertainties in power (1 %) and fuel management plan changes (4 %) were considered for the core inventory based on existing nuclide presence data. Meteorological data were provided by a small modular reactor designer. Uncertainties caused by human factors usually result from the user's inexperience in model and calculation processes, inaccurate division of time nodes, and calculation input data errors. These uncertainties were controlled for through independent recalculation conducted by different professionals.