The chemical explosion of explosives wrapped with radioactive materials is a typical nuclear accident scenario. This study simulates and analyzes the process of a radioactive smoke cloud rising after the chemical explosion of an explosive device wrapped with plutonium materials. The state of the explosive products after the single-point detonation of the explosives was simulated. The numerical simulation of plutonium-aerosol rise was carried out based on the discrete phase model according to the simulation results of the chemical explosion. Based on the Chapman model and Levenberg–Marquardt algorithm, the simulation results of the plutonium aerosol-cloud height were fitted to obtain it as a function of time. The spatial distribution and velocity change of plutonium aerosol particles are analyzed. The reasons for the formation of vortex rings in the smoke cloud are discussed, which are significant in the emergency response to nuclear accidents.

Topics
Nonequilibrium thermodynamics, Computer simulation, Nuclear explosions, Aerosols, Navier Stokes equations, Viscosity, Vortex dynamics, Regression analysis, Particle distributions

The chemical explosion of plutonium material encased in explosives causes its aerosolization under high temperatures and pressure, forming a radioactive smoke cloud. Inhaling plutonium aerosols can lead to fatal alpha internal irradiation.1 The process from the formation to the diffusion of plutonium aerosol can be divided into three parts: blast conditions for the plutonium-aerosol formation stage, plutonium aerosol-smoke cloud formation and rising stage, and plutonium aerosol-atmospheric diffusion stage.2 At present, the third stage can be modeled using the Gaussian diffusion model,3 the Lagrangian diffusion model,4 the meteorological prediction model,5 and other means of simulation studies. Currently, the typical experimental research on chemical explosions utilizes the United States Operation Roller Coaster (ORC) experiment6 and Israel’s Green Field (GF) experiment.7 The main research content of the ORC experiment is the plutonium-aerosol diffusion. The GF experiment is mainly used to study the diffusion process of the explosion smoke cloud. As the data from the ORC experiment on smoke cloud rise are incomplete, GF experiments are used to establish the model to verify the smoke cloud burst height data, citing the ORC experiment for the particle size, mass distribution, and other data of the plutonium aerosol.

Church analyzed the height of the smoke cloud of the explosion of more than 53 kg of trinitrotoluene (TNT) and obtained the explosion height formula by fitting.8 However, this formula is unsuitable for low-yield conditions and cannot describe the change rule of smoke cloud height with time. Makhviladze et al.,9 Kansa,10 and Kanarska et al.11 calculated the vortex ring motion of the explosion cloud under different conditions but ignored the motion of the generated aerosol particles. Duan simulated the rising process of the explosion smoke cloud under various conditions and verified it experimentally.12 Hao simulated the explosion cloud flow field using OpenFOAM.13 

Most of the above-mentioned studies have focused on smoke clouds generated by explosives or nuclear explosions. However, there is a lack of studies on the rise of radioactive smoke clouds following chemical explosions of explosive devices containing radioactive materials. This study takes a single-point initiation event of an explosive device wrapped with plutonium material as the research object to investigate the rising process of radioactive smoke cloud under chemical explosion conditions. First, the chemical explosion process of the explosive is simulated, and the plutonium-aerosol diffusion source is constructed by combining the plutonium-aerosol parameters with the chemical explosion simulation results. Then we simulated the rising process of the plutonium aerosol-smoke cloud based on the discrete phase model (DPM), fitted the height of the plutonium aerosol-smoke cloud with time, and analyzed the spatial distribution and particle velocity of plutonium aerosol.

DPM is a hybrid Eulerian–Lagrangian multiphase model. The model treats the fluid as a continuous phase and calculates its motion by solving the Navier–Stokes equations. It treats the particles as a discrete phase and solves for their motion by tracing them in a Lagrangian coordinate system. The DPM model is generally suitable for cases with a small volume fraction of the discrete phase.

The continuous phase in aerosol diffusion is air, and its motion can be described by the Navier–Stokes equation
(1)
(2)
where αg is the gas volume fraction, ρg is the gas density, p is the pressure, vg and vs are the gas and particle velocities, t is the time, τ̄ is the viscous stress term, g is the gravity vector, Kgs is the momentum exchange coefficient between the air and the particles, and ɛ is the rate of dissipation of turbulent kinetic energy in the turbulence model.
The discrete phase in this study is plutonium-aerosol particles, and the particle motion is obtained by solving the control equations based on Newton's second law in the Lagrangian coordinate system,
(3)
(4)
(5)
where ms is the particle mass, ρs is the particle density, msvg−vsτr is the particle trailing force, F is the additional force, τr is the particle relaxation time, Res is the relative Reynolds number, ds is the particle diameter, μ is the continuum phase viscosity, and CD is the drag function.
According to the particle analysis report on plutonium aerosols from low-yield nuclear explosions, plutonium aerosols can be approximated as spherical particles.14 Consequently, the spherical drag model is employed to characterize the drag force on plutonium aerosols. The drag coefficient of plutonium aerosols can be described as
(6)
where a1, a2, and a3 are constants that vary with the range of Re.
The Saffman lift is defined as the lift force acting on solid particles moving in a flow field with a velocity gradient, where the low-velocity region points in the direction of the high-velocity region. This additional force applies only to particles with small Reynolds numbers, specifically submicrometer particles. Since plutonium aerosols from chemical explosions contain submicrometer particles, the role of the Saffman lift must be considered. The expression proposed by Li and Ahmadi is used in this study,15 
(7)
where K is a constant and dij is the deformation tensor.
The thermophoretic force is the force opposing the temperature gradient to which particles are subjected in a gas with a temperature gradient. In the aerosol diffusion phase, there is a large difference in temperature between the smoke plume holding the aerosol particles and the outside air, so the particle movement is affected by the thermophoretic force. The expression for the thermophoretic force is as follows:
(8)
where DT,p is the thermophoretic coefficient.

The basis of chemical explosion radioaerosol dispersion studies is the determination of aerosol source term parameters, including plutonium aerosol-particle density, particle-size distribution, nuclide composition, aerosolization rate, and respirable share, which are important inputs for subsequent numerical simulations of aerosol dispersion. Among the few experiments conducted to determine the source terms of plutonium aerosols under chemical explosion conditions, data from the chemical explosion experiments on plutonium materials in the ORC conducted in 1964 in NV, USA, are more informative.16 Four experiments were conducted in the ORC: Double Tracks (DT), Clean Slate 1 (CS1), Clean Slate 2 (CS2), and Clean Slate 3 (CS3), of which the DT experiment was a single-point detonation experiment of plutonium-containing materials under uncovered conditions, similar to the conditions of this study.

According to the results of aerosol measurements in the ORC DT experiment, the average density of aerosol particles formed after the chemical detonation of plutonium material was 4.9 g/cm3. Luna et al. statistically analyzed the aerosol data collected by the air sampler in the ORC experiment and obtained the particle-size distribution of plutonium aerosol after chemical detonation. The present study refers to these data and sets the range of the particle size of plutonium aerosol as 0.1–120 μm.17 The particle-size distribution obeyed the Rosin-Rammler distribution.

The plutonium aerosols cause damage to the human body primarily through internal irradiation in the lungs after inhalation. All the plutonium materials of the ORC outfield experiments had an aerosolization rate of 100% upon chemical explosion, i.e., all plutonium metals existed in the form of aerosols after a chemical explosion occurred, with a respiratory rate of 16% for DT, CS2, and CS3, and 20% for CS1.18 Referring to the results of the ORC experiment, the aerosolization rate of plutonium material after the chemical explosion is assumed to be 100%, and the respirable share is 20%. The source term parameters of the chemically exploded plutonium aerosol setup for aerosol dispersion simulation in this study are listed in Table I.

TABLE I.

Parameters of aerosol source items for chemical explosive plutonium.

Plutonium aerosol source parametersThis study set
Aerosol particle density 4.9 g/cm3 
Particle-size distribution 0.1–120 μm (Rosin–Rammler) 
 distribution 
Aerosol rate 100% 
Respirable share 20% 
Plutonium aerosol source parametersThis study set
Aerosol particle density 4.9 g/cm3 
Particle-size distribution 0.1–120 μm (Rosin–Rammler) 
 distribution 
Aerosol rate 100% 
Respirable share 20% 
View Large

The dispersion of aerosols following the explosion of explosives encasing plutonium material can be simplified into two main phases: the chemical explosion and aerosol formation. The explosive material containing plutonium is subjected to an external force that produces a single-point explosion. The plutonium metal is subjected to a shock wave and aerosolizes at very high pressure and temperature. The energy released by the chemical explosion is converted into kinetic and thermal energy, generating a high-pressure wave. The radioactive aerosols then spread radially in an erratic manner. The time scale of this process is on the order of microseconds. The second stage is the diffusion of the aerosol in the atmosphere. The plutonium aerosol is carried by buoyant smoke and diffuses in the atmosphere until it reaches a steady state. Since the initial phase is difficult to observe in an experimental setting and contains intricate physical and chemical processes, this study assumes complete aerosolization of plutonium material, excluding the potential influence of plutonium material on the chemical explosion process. The spatial distribution of pressure, velocity, density, and temperature of the gas phase after single-point initiation of the explosives was obtained through simulations, thus providing a basis for the subsequent construction of an aerosol diffusion source. By comparing the spatial distribution of various parameters of the explosion products at different times, 1 ms was selected as the moment the explosion products stably diffuse outward. After this time, the fireball expanded continuously in this shape, and the fully aerosolized plutonium metal began to diffuse into the atmosphere. The state distribution of the explosion products at this time is the state distribution of the aerosol diffusion source gas. Figure 1 shows the spatial distribution of the parameters of the explosion products at 1 ms, with a clear double-layer structure, in which the inner layer is a negative-pressure environment, and the outer layer is an overpressure environment.

FIG. 1.
Spatial distribution of explosion product parameters at t = 1 ms. (a) Pressure, (b) density, (c) temperature, and (d) velocity.
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Spatial distribution of explosion product parameters at t = 1 ms. (a) Pressure, (b) density, (c) temperature, and (d) velocity.

FIG. 1.
Spatial distribution of explosion product parameters at t = 1 ms. (a) Pressure, (b) density, (c) temperature, and (d) velocity.
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Spatial distribution of explosion product parameters at t = 1 ms. (a) Pressure, (b) density, (c) temperature, and (d) velocity.

Close modal

The space fraction of explosion parameters obtained from the chemical explosion simulation of explosives in the previous section is used as the parameter input for the plutonium aerosol-diffusion source. The plutonium aerosol is assumed to be emitted radially from the overpressurized environment of the outer layer. The parameters of the plutonium aerosol are the values assumed in Table I. The motion of the plutonium aerosol is modeled using DPM.

The GF experiment was an explosion smoke cloud experiment conducted in Israel from 2006 to 2009. This study compares the explosion cloud height data of 10 kg TNT obtained from the experiment to verify the accuracy of the present simulation and to investigate the pattern of plutonium aerosol-smoke cloud height. The simulated and GF experiment smoke cloud height vs time and the fitted curve of smoke cloud height in this study are shown in Fig. 2.

FIG. 2.
Plutonium aerosol simulation results vs GF smoke cloud heights.
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Plutonium aerosol simulation results vs GF smoke cloud heights.

FIG. 2.
Plutonium aerosol simulation results vs GF smoke cloud heights.
View largeDownload slide

Plutonium aerosol simulation results vs GF smoke cloud heights.

Close modal
Comparing the smoke-cloud heights in this study with the experimental values in GF, the differences are mainly reflected in the pre-25 s period after explosion, and the experimental values after 25 s are the same as the simulated values. Based on the Chapman-Richards growth model19 and using the Levenberg–Marquardt algorithm20 to fit the simulated smoke-cloud height curves, the following relationship was obtained between the smoke cloud heights of plutonium aerosol and time, at 10 kg TNT equivalent:
(9)
where H is the height of the plutonium aerosol-smoke cloud (m), w is the TNT equivalent (kg), and t is the time of the chemical explosion (s). It can be seen from Fig. 2 that the fitting results of smoke cloud height with time show logarithmic distribution characteristics. The difference between the simulated value and the experimental value mainly occurs before 20 s. At this stage, the simulated cloud height is higher than the GF experimental value. This difference may be due to the deviation of the driving force of the cloud in the early stage of the rising stage, which will be further studied in the follow-up study.

The spatial distribution of plutonium aerosol particles of different sizes at different moments is shown in Fig. 3. The plutonium material first spreads radioactively after aerosolization due to the chemical explosion. Then, the plutonium aerosols with smaller internal particle sizes rise rapidly under the action of the flow field, forming the top of the ellipsoidal smoke cloud, and continue to rise gradually. Figure 3 shows that the plutonium aerosols with smaller particle size (<50 μm) are mainly concentrated in the upper part of the smoke cloud, forming the top of the cloud. When the aerosol diffuses radially, the plutonium-aerosol particles with larger particle size (>50 μm) have a greater velocity and are located at the edge of the diffusion. However, large-size particles have more mass and tend to be more dragged than the smaller aerosol particles, so they decelerate faster and are more difficult to accelerate when subjected to the same forces in the flow field. Some of the large-size aerosols will diffuse outward along the direction of the ground, and the rest of the large-size aerosols will move upward under the action of the flow field. However, owing to the large mass, the rising speed is not as fast as that of the small-size aerosols, mainly concentrated at the bottom of the smoke cloud. With time, the flow field weakens gradually, and the large-size aerosols will settle before the small-size aerosols under gravity.

FIG. 3.
Spatial distribution of plutonium aerosol with different particle sizes at different moments in time. (a) t = 1 s, (b) t = 5 s, (c) t = 30 s, and (d) t = 60 s.
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Spatial distribution of plutonium aerosol with different particle sizes at different moments in time. (a) t = 1 s, (b) t = 5 s, (c) t = 30 s, and (d) t = 60 s.

FIG. 3.
Spatial distribution of plutonium aerosol with different particle sizes at different moments in time. (a) t = 1 s, (b) t = 5 s, (c) t = 30 s, and (d) t = 60 s.
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Spatial distribution of plutonium aerosol with different particle sizes at different moments in time. (a) t = 1 s, (b) t = 5 s, (c) t = 30 s, and (d) t = 60 s.

Close modal

Plutonium-aerosol diffusion in the atmosphere is mainly due to the effect of the ambient flow field on the aerosol particles. Figure 4 shows the velocity vector distribution of the flow field at the top of the smoke cloud at t = 5 s. In Fig. 4, the top of the smoke cloud appears to have a significant buoyant vortex ring structure. The center of the vortex ring is upward, and the radius of the vortex ring increases gradually with time. A kidney-shaped vortex ring appears at the top of the smoke cloud in the velocity vector section. This occurs because the airflow generates a vortex that forms two counter-rotating vortices as it rolls up on one side. Additionally, there are many small vortex rings within the larger vortex ring. The generation of vorticity is likely linearly proportional to the horizontal density gradient. Given that the initial shape of the smoke cloud is either elliptical or circular, it is influenced by the horizontal density gradient in the opposite direction, resulting in a pair of counter-rotating streamline vortices. This process initiates a large-scale roll-up phenomenon and significantly alters the shape of the smoke cloud, which deviates from the shape predicted by the traditional Gaussian concentration distribution. The velocity field produced by the vortex structure aids the overall rise of the smoke cloud. Following the large-scale roll-up, the outer boundary of the smoke cloud exhibits notable stretching and distortion, which in turn generates secondary vortices with the same or opposite vorticity signs, causing the smoke cloud's movement to display turbulent characteristics.

FIG. 4.
Spatial distribution of velocity vectors at the top of the plutonium aerosol-smoke cloud at t = 5 s.
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Spatial distribution of velocity vectors at the top of the plutonium aerosol-smoke cloud at t = 5 s.

FIG. 4.
Spatial distribution of velocity vectors at the top of the plutonium aerosol-smoke cloud at t = 5 s.
View largeDownload slide

Spatial distribution of velocity vectors at the top of the plutonium aerosol-smoke cloud at t = 5 s.

Close modal

Figure 5 shows that the aerosol rate inside the plutonium aerosol-smoke cloud is the largest and decreases gradually toward the edge. The rate at the top of the smoke cloud is higher than that at the bottom, and the higher velocity region inside the smoke cloud is like a water droplet. Due to the continuous suction of external air into the smoke cloud, the upward velocity inside the smoke cloud decreases slowly with time, and its maximum rate decreases from 10.5 m/s at 1 s to 3.86 m/s at 5 s. At the same time, the maximum rates at 20 and 60 s are only 1.74 and 0.92 m/s, respectively.

FIG. 5.
Spatial distribution of plutonium-aerosol rates at various times. (a) t = 1 s, (b) t = 5 s, (c) t = 30 s, and (d) t = 60 s.
View largeDownload slide

Spatial distribution of plutonium-aerosol rates at various times. (a) t = 1 s, (b) t = 5 s, (c) t = 30 s, and (d) t = 60 s.

FIG. 5.
Spatial distribution of plutonium-aerosol rates at various times. (a) t = 1 s, (b) t = 5 s, (c) t = 30 s, and (d) t = 60 s.
View largeDownload slide

Spatial distribution of plutonium-aerosol rates at various times. (a) t = 1 s, (b) t = 5 s, (c) t = 30 s, and (d) t = 60 s.

Close modal

The study addresses the aerosol rise after a chemical explosion of an explosive device wrapped with plutonium material. This study first simulated the spatial distribution of the explosive products after a chemical explosion of 10 kg TNT. The explosion parameters were used as the input for the diffusion source of plutonium aerosol, and then its rise was simulated using DPM. The plutonium aerosol-cloud heights were compared with those of the GF2 experiment and fitted to the time-dependent curves based on the Chapman-Richards growth model and Levenberg–Marquardt algorithm. The spatial distribution and particle velocity of plutonium aerosol and the vortex ring structure of the plutonium aerosol-smoke cloud were analyzed. The main conclusions are summarized below:

  1. Based on the fitting results of the Chapman–Richards growth model, the height of the plutonium aerosol-smoke cloud after the chemical explosion had a logarithmic distribution as a function of time. The difference between the simulated value and the experimental value mainly appears in the early stage of the rising period of the smoke cloud.

  2. The plutonium aerosol-diffusion distribution showed a remarkable mushroom cloud shape, in which large particles were concentrated in the stem of the smoke cloud or rapidly settled to the ground, and the small aerosol particles were concentrated at the top of the smoke cloud.

  3. The simulation results show that there are obvious vortex ring structures on both sides of the top of the chemical explosion smoke cloud, and the internal rate of the chemical explosion smoke cloud is the largest and gradually decreases to the edge.

The authors have no conflicts to disclose.

Hongyi Yao: Data curation (equal); Investigation (equal); Software (equal); Writing – original draft (equal). Yonggang Huo: Formal analysis (equal); Project administration (equal); Writing – review & editing (equal). Xingfu Cai: Investigation (equal); Methodology (equal); Supervision (equal). Sufen Li: Software (equal); Supervision (equal). Haowei Wang: Data curation (equal); Validation (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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