Aerosol diffusion calculation model

The DPM model is a hybrid Eulerian–Lagrangian multiphase model. It treats the fluid and particles as the continuous phase and the discrete phase, respectively, determining the motion of the former by solving the Navier–Stokes equations and the motion of the latter by tracking particles in a Lagrangian coordinate system [20]. The DPM model is generally suitable for scenarios in which the volume fraction of the discrete phase is small. (1) (2)where denotes the gas volume fraction; ρ_g represents the gas density; p signifies the pressure; v_g and v_p are the gas and particle velocities, respectively; t is the time; corresponds to the viscous stress term; g refers to the gravity vector; K_{g s} is the momentum exchange coefficient between the air and the particles; and ε denotes the dissipation rate of turbulent kinetic energy for the turbulence model. (3) (4) (5)where m_s and ρ_s denote the particle mass and density, respectively; signifies the drag force on the particle; F is the additional force, such as Saffman lift, thermophoretic force and other external forces which are different from gravity and drag force; τ_r represents the particle relaxation time; Res is the relative Reynolds number; d_s is the particle diameter; μ is the continuum phase viscosity; and C_D is the drag function.

The particle analysis report of plutonium aerosol from low-yield nuclear explosions in the former Soviet Union indicates that these aerosols can be approximated as spherical particles. Consequently, the spherical drag model is employed to characterize the drag force on plutonium aerosol [21]. The drag coefficient of plutonium aerosol is expressed as: (6)where a₁, a₂, and a₃ denote constants that vary with Re.

Saffman lift refers to the lift force acting on solid particles in a flow field with a velocity gradient, where particles in low-velocity regions move toward high-velocity regions. This additional force is significant for particles with low Reynolds numbers, such as submicron particles. Because plutonium aerosol from chemical explosions includes submicron particles, the Saffman lift is negligible. This study uses the expression proposed by Li et al. [22]: (7)where K is a constant, dij denotes the deformation tensor, and v denotes the kinematic viscosity.

The thermophoretic force is the force exerted on particles in a gas due to a temperature gradient. During the aerosol diffusion phase, the substantial temperature difference between the smoke plume containing aerosol particles and the ambient air causes particle motion to be influenced by the thermophoretic force [23]. The thermophoretic force is defined as (8)where denotes the thermophoretic coefficient.

Plutonium aerosol source term after chemical explosion

The chemical explosion radio-aerosol dispersion studies focus on determining aerosol source-term parameters, including plutonium aerosol particle density, particle size distribution, nuclide composition, aerosolization rate, and respirable fraction, which are critical inputs for numerical simulations of aerosol dispersion. Few experiments have established the source terms of plutonium aerosol under chemical explosion conditions. Data from the nonnuclear explosion of a plutonium-bearing device during ORC in 1964 in Nevada, USA, are particularly informative [24]. ORC included four experiments: Double Tracks (DT), Clean Slate 1 (CS1), Clean Slate 2 (CS2), and Clean Slate 3 (CS3). The DT experiment, involving a single-point detonation of plutonium-containing materials under uncovered conditions, most closely resembles the conditions of this study.

Aerosol measurements from the ORC DT experiment indicate that the average density of aerosol particles formed after a chemical explosion of plutonium is 4.9 g cm^-3. Luna et al. analyzed air-sampling data from air samples used during the ORC experiment to determine the particle size distribution of plutonium aerosol [25]. Based on their findings, this study sets the particle size range for plutonium aerosol to 0.1 μm–120 μm, with the distribution following the Rosin–Rammler model.

The nuclide composition of weapons-grade plutonium materials was used to determine the nuclide composition and mass fraction of 1 kg of plutonium material [26]. Given the long half-lives of ²³⁹Pu, ²⁴⁰Pu, and ²⁴²Pu, their decay products can be disregarded. Our study focuses on ²³⁴U and ²⁴¹Am, produced by the decay of ²³⁸Pu and ²⁴¹Pu. The quantities of ²³⁴U and ²⁴¹Am are obtained from Eq. (9), and the radioactivity of different nuclides is determined from Eq. (10). Table 1 presents the nuclide composition, mass fraction, and radioactivity of nuclides released from 1 kg of plutonium material due to chemical explosions. (9) (10)

Plutonium aerosol primarily causes internal lung damage after inhalation [27]. In the ORC outfield experiments, plutonium materials exhibited a 100% aerosolization rate during chemical explosions, indicating that all of the plutonium was converted to aerosol. A respirable fraction (RF) of 16(%) was observed for DT, CS2, and CS3, and 20(%) for CS1 [28]. Based on these ORC experimental results, this study assumes a 100% aerosolization rate and a 20% respirable fraction for plutonium following a chemical explosion. The source-term parameters for the aerosol dispersion simulation are listed in Table 2.

Calculation method of internal irradiation dose caused by aerosol

Inhalation is the primary route for plutonium deposition in the human respiratory tract, with the extensive surface area of the alveoli facilitating the deposition of plutonium aerosol. Additionally, the plutonium aerosol produced by the explosion has limited solubility in the human body. Because respirable aerosol particles are less than 10 μm, only the effective dose from inhalation of plutonium aerosol smaller than 10 μm is considered [29]. According to GB/T 16148-2009, the committed dose from inhaling a given radioactive aerosol is calculated as follows [30], (11)where A₀ denotes the radionuclide inhaled by the personnel; and e(τ) is the effective dose coefficient corresponding to the radionuclide in Sv Bq^-1, which depends on the radionuclide’s entry route into the human body, aerosol particle size, and the radionuclide form, as shown in Table 3. (12) (13)where IR represents the respiration rate of the personnel; the nasal respiration rate during normal activity is 2.67 × 10^-4 m³ s^-1, and the respiration rate during exercise is 6.67 × 10^-4 [31]; IP signifies the duration of inhalation in seconds; C_r is the activity concentration at a specific location in; C_p refers to the plutonium aerosol particle concentration in the region in; mj is the mass of the j-th plutonium isotope in the region in μg; Aj is the total activity of the j-th plutonium isotope in Bq.

Simulation analysis of chemical explosion

The diffusion of aerosol following a chemical explosion in a confined space can be divided into two main stages. The first stage involves the chemical explosion and aerosol formation. The explosive material containing the plutonium is subjected to external forces, resulting in a single-point detonation. The resulting shock wave atomizes the plutonium metal under extremely high pressure and temperature. The energy released from the explosion is converted into kinetic and thermal energy, generating a high-pressure wave that violently disperses the radioactive aerosol radially. This process occurs on a microsecond timescale.

The second stage involves the spatial diffusion of the aerosol. The plutonium aerosol is carried by buoyant smoke and continues to diffuse until reaching a stable state. Due to the difficulty of experimentally observing the initial stage and the complexity of the associated physical and chemical processes, this study excludes the potential influence of plutonium materials on the chemical explosion process, and assumes complete aerosolization of plutonium materials. Simulation calculations provide the spatial distributions of pressure, velocity, density, and temperature of the gas phase following the single-point detonation, forming the basis for constructing the aerosol diffusion source.

Assuming a TNT mass of 20 kg in a chemical explosion accident, the shock wave propagation following the chemical explosion is numerically simulated using Euler’s finite difference method. The state of the explosion products is described using the Jones–Wilkins–Lee (JWL) equation of state and the ideal gas equation of state, with the JWL equation expressed as follows [32]. (14)where p, V, and e denote the explosion product pressure, specific volume, and internal volumetric energy, respectively; A, R₁, R₂, and ω represent the JWL equation of state parameters.

Upon analyzing the spatial distribution of various explosion-product parameters at different times, it is observed that at t = 0.51 ms, the shape of the shock wave transitions from spherical to hemispherical. The fireball then continues to expand in this form. The plutonium metal undergoes aerosolization and begins diffusing into the surrounding space. Figure 2 illustrates the spatial distribution of the explosion-product parameters at 0.51 ms, revealing a distinct double-layer structure. This structure consists of an inner layer characterized by a negative-pressure environment and an outer layer exhibiting overpressure conditions.

Fig. 2Full-screen View

(Color online) Spatial distribution of explosion product parameters at 0.51ms after the chemical explosion: (a) Spatial distribution of density, (b) Spatial distribution of pressure, (c) Spatial distribution of velocity magnitude, and (d) Spatial distribution of temperature

Simulation scene construction

A typical underground facility structure with an operation room and a tunnel is selected for analysis, as illustrated in Fig. 3. The ground center of the Room is assumed to be the location of the chemical explosion accident. The Room measures 10 m × 10 m × 5 m, and the Channel connecting the Room to the tunnel is 3 m in length with a cross-section of 2 m × 3 m. The tunnel is 200 m in length and has a semicircular cross-sectional area with a diameter of 20 m. The wall-jet model is employed as the ground boundary condition for the underground facility, and the reflect model is applied to the walls, considering the inviscid nature of plutonium aerosol particles. It is assumed that the explosion results in the Channel becoming open after the accident, and that no directional ventilation is performed in the underground facility. The ambient pressure before the accident is taken as the standard atmospheric pressure.

Fig. 3Full-screen View

(Color online) Typical underground facility model

The coordinate origin is placed at the center of the Room’s floor. Seven monitoring areas are established to support subsequent monitoring of aerosol parameters. The monitoring area is defined based on the hemispherical breathing zone specified by the US Department of Labor, which encompasses the area in front of the shoulders of personnel [33]. This breathing zone is approximately 6–9 in. As the orientation of personnel is unknown, the monitoring zones are defined as spheres with a radius of 1.2 m. The coordinates of the spherical center of each monitoring area are listed in Table 4.

Simulation of plutonium aerosol diffusion of chemical explosion

After the chemical explosion, plutonium aerosol initially diffuses in a hemispherical radial pattern within the Room due to the strong shock wave. The particles then rapidly enter the tunnel through the Channel, and because of their high initial velocity, continue to diffuse and spread toward both ends of the Channel after colliding with the tunnel wall. The simulation results analyze the particle size, particle velocity, air density, particle number, mass fraction in space, and particle concentration at different positions.

3.3.1

Spatial distribution of plutonium aerosol

Figure 4(a) shows the radial distribution of aerosol particles in the Room at 0.01 s following the chemical explosion, with larger particles diffusing at a faster rate. Figure 4(b) illustrates the rapid influx of aerosol particles into the tunnel, driven by the substantial pressure and density differential between the puff interior and the surrounding environment. Figure 4(c) and (d) depict the diffusion behavior of aerosol particles upon collision with the tunnel wall, leading to their distribution toward both ends. As the particles decelerate due to the wall impact, their velocity decreases during the diffusion process until the aerosol state in the environment reaches equilibrium.

Fig. 4Full-screen View

(Color online) Spatial distribution of plutonium aerosol of various particle sizes at different points in time

Figure 5(a) and (b) show that after 300 s, most aerosol particles are concentrated at both ends of the tunnel, with particle velocity reduced to below 1 m s^-1. In Fig. 5(c), significant layering of particles within the tunnel is evident at 600 s. Particles in the upper layer diffuse slowly to both ends at a velocity of less than 0.15 m s^-1, while a small number of particles continue to migrate into the tunnel through the Channel.

Fig. 5Full-screen View

(Color online) Plutonium aerosol velocity distribution at different points in time

3.3.2

Plutonium aerosol concentration distribution

By establishing monitoring zones within the reference person’s breathing area at various locations (Table 4), this study observed fluctuations in particle concentration within these zones 10–600 s after the accident. These observations were used to determine the internal irradiation dose received by subsequent personnel from inhaling plutonium aerosol.

As shown in Fig. 6, in Area 1, particle concentrations remained above 100 μg m^-3 before 200 s, gradually declining thereafter. A sharp decrease below 10 μg m^-3 occurred between 200 s and 250 s, followed by a slow decline to approximately 1 μg m^-3 after 250 s. Area 2 exhibited a decrease to 100 μg m^-3 at approximately 150 s, followed by a rapid increase peaking at about 300 s, and subsequently decreasing rapidly to 1 μg m^-3 after 400 s. As illustrated in Fig. 7, the concentration trends in Areas 3 and 7 mirrored each other, starting near zero before 180 s, rising to approximately 5 μg m^-3 at 200 s, and subsequently decreasing. A secondary rise occurred around 250 s, peaking at about 11 μg m^-3 at 300 s, and then decreasing between 300 s and 600 s, with minor fluctuations observed between 350 s and 450 s. Areas 4 and 6 show similar trends, peaking initially at 70 s, dropping to 0, and subsequently rising again to a maximum of 10 μg m^-3 at 300 s, gradually decreasing to 0. Area 5 peaked at 13.5 μg m^-3 at 30 s, declining to near zero between 30 s and 190 s, with a second peak reaching 9 μg at 300 s before gradually declining again to near zero between 300 s and 600 s. The internal irradiation dose received by personnel in an accident area is directly linked to the mass of plutonium aerosol inhaled in that area. This mass can be determined by computing the plutonium aerosol concentration at different inhalation times.

Fig. 6Full-screen View

(Color online) Plutonium aerosol concentrations in Areas 1 and 2 as a function of time

Fig. 7Full-screen View

(Color online) Plutonium aerosol concentrations in Areas 3–7 as a function of time

Assessment of internal irradiation dose to personnel

The internal irradiation dose received by personnel near the chemical explosion accident site correlates with the mass of plutonium aerosol inhaled in that area. The mass of plutonium aerosol inhaled at different times can be determined by computing the concentration of plutonium aerosol during the initial stages of the incident. Because personnel are most affected shortly after the explosion, only the internal irradiation of individuals at various locations after 5 s of the chemical explosion is considered. The effective doses received by personnel in each monitoring area over different time periods are provided in Tables 5 and 6.

The variation in effective doses across regions over different time periods closely corresponds to changes in particle concentration in those regions. The decision to remain in Area 2 is likely influenced by the considerable internal exposure risk. First, because the Channel serves as the passage between the Room and the Tunnel, any outward diffusion of plutonium aerosol must pass through this area. Second, the low channel height restricts the movement of smaller particle-size plutonium aerosol particles, which tend to accumulate at higher positions in space. This constraint increases the likelihood that these particles will be present at a person’s breathing height.

According to data reported by Giardina, the settlement velocity of 5 μg m^-3 aerosol particles within a confined space is 0.002, ensuring that small particles at breathing height settle within one hour after a chemical explosion [34]. Simulation results (Table 5 and 6) for plutonium aerosol diffusion show clear stratification at 600 s, with the horizontal velocity becoming negligible. Therefore, gravity deposition is the primary factor to consider [35]. It can be assumed that the internal radiation dose received by subsequent personnel is consistent with the computed results between 540 s and 600 s. The effective dose at each monitoring point within one hour can thus be estimated, as shown in Table 7 and 8. As indicated by the estimates presented in these tables, the likelihood of inhalation of plutonium aerosols and internal exposure of personnel in the various areas 10 min after the accident is markedly reduced. In particular, the cumulative dose increase in Areas 4 and 6 is nearly negligible, at less than 0.01%.

According to ICRP recommendations, a brief exposure to an internal radiation dose exceeding 250 mSv surpasses the emergency dose threshold, potentially causing irreversible damage and significantly increasing the incidence of lung cancer [36]. Newman et al. showed that the risk of pulmonary fibrosis increased sharply when the lung dose exceeded 10 Sv [18].

The estimated internal irradiation dose to personnel may be higher in the rooms and channels than that in other regions. If personnel enter the room within 240 s after an accident and stay there for more than a minute, they are likely to receive a dose exceeding 250 mSv. Even after 540 s, traversing the channel can cause lung damage. While the estimated cumulative internal irradiation dose to personnel in other areas may not reach the threshold, increased respiratory rates and the resuspension of ground-deposited aerosols owing to the movement of personnel could increase the actual internal irradiation dose beyond the estimated value presented in this paper.

Based on the results of the previous simulation, the following emergency response recommendations were proposed for this type of accident involving a plutonium-containing explosion in a confined space. Respondents should avoid entering the area within 10 min of the accident because of the high concentration of plutonium aerosols at breathing height. If entering an area is necessary, appropriate respiratory protection should be provided to prevent the inhalation of plutonium aerosols. Avoid Areas 1 and 2, where the concentration of plutonium aerosols at breathing height is significantly higher than in other areas. For people in the danger zone at the time of the accident, the following evacuation recommendations may help reduce the risk of inhaling plutonium aerosols: move as low as possible, as simulation results show that inhalable plutonium aerosols tend to rise and concentrate above the room during the early stages of the accident; cover the nose and mouth with clothing (such as a T-shirt or cloth) to filter some of the aerosol; although this is not a substitute for professional respiratory protection, it may reduce the amount of plutonium inhaled to some extent; move to the nearest emergency exit as quickly as possible, but calmly, to avoid panic-induced increases in respiratory rate. In similarly confined spaces, the installation of an air suction system can also rapidly reduce the aerosol concentration in a facility following such incidents.