As shown in Figure 1, ²¹⁰Po exists in the form of Po, PbPo, and H₂Po in a typical LFR. And it is immediately apparent that the migration behavior is affected by the key processes of evaporation, decay, diffusion, convection in flow, deposition in air, and chemical reactions with other substances.

In LBE coolant, once the Po is generated from the ²¹⁰Bi decay, it will have a reaction with Pb immediately and form the compound of PbPo with stable chemical properties. For example, when the coolant temperature is around 673K, only 0.2% ( χ P o , L B E ) of Po in the LBE is left in the form of Po, and the remaining 99.8% ( χ P b P o , L B E ) is in the form of PbPo. Both Po and PbPo can evaporate into the cover gas at a high temperature (Li et al., 1998; Buongiorno, 2001; Buongiorno et al., 2017), which is slow but non-neglectful from the perspective of radioactive safety.

For an accident condition, such as steam generator tube rupture (SGTR), PbPo may directly come in contact with steam, and highly volatile radioactive compounds such as H₂Po will be produced, which will make the Po behavior even more complicated. The chemical equations are as follows: 2 P b P o ( s o l ) + h e a t ↔ P b P o ( g ) + P o ( g ) + P b ( s ) , (2) P b P o + H 2 O ↔ H 2 P o ( g ) + P b O ( s o l ) . (3)

The fundamental properties of Po, PbPo, and H₂Po are summarized in Table 2.

To gain insight into the characteristic feature of ²¹⁰Po in LFR on a normal operation, the mass flow map of ²¹⁰Po had been clearly drawn in Figure 2.

In LBE coolant,

• Accompanied with neutron irradiation, the isotope ²¹⁰Bi is produced, and then we have the concentration of Bi as c _Bi210 (mol/m ³ ), with a decay rate λ B i 210 of 1.6 × 10^–6 (1/s);

Then the molecule balance equation for each species in LBE can be written as Eq. 4: d c i L B E d t = χ i λ B i 210 c B i 210 − ( J p u r i L B E + J d e c a y L B E + J l e a k L B E + J i , v a p o r | i n t e r f a c e ) . (4)

In cover gas.

• Define the decay rate of Po, J d e c a y g a s = λ _Po · c i g a s (mol/m ³ s);

Then the molecule balance equation for each species in cover gas can be written as Eq. 5: d c i g a s d t = J i , v a p o r | i n t e r f a c e − ( J p u r i g a s + J d e c a y g a s + J l e a k g a s ) (5)

In containment.

• All the PbPo are assumed to be reacted with H₂O in air, to generate H₂Po in the containment, c H 2 P o a i r (mol/m ³ );

Then the molecule balance equation for each species in the containment can be written as Eq. 6: d c H 2 P o a i r d t = J l e a k g a s − ( J p u r i a i r + J d e p a i r + J d e c a y a i r + J l e a k a i r ) (6) And due to the H₂Po aerosol deposition, there will be surface contamination on the ground of the containment, which is calculated as follows: d c H 2 P o g r o u n d d t = J d e p a i r − J d e c a y a i r (7)

From above Eqs. 4–7, a numerical relationship between the ²¹⁰Po transport and multi-physics environment of LFR had been finally established.

Then a particular emphasis should be put on the evaporation model of Po and PbPo, which dominates how much of the ²¹⁰Po source will escape from the restraint of LBE. According to Raoult's law for ideal gas, the vapor pressure of dilute solution is equal to the vapor pressure of a pure solvent multiplied by the mole fraction of the solvent in solution at the special temperature (Feuerstein et al., 1992; Loewen, 2005). And the saturated vapor pressure, which is frequently cited, is listed in Table 2. P i , s a t = x i , L B E P i . (8)

There is a general consensus that the evaporation equilibrium of Pb/PbPo can always be achieved in microscale on the surface of LBE coolant. Then the evaporation flux can be constructed as an equation of saturated vapor pressure andreal-time vapor pressure. J i , v a p o r | i n t e r f a c e = J i , 0 × c i , L B E × P i , s a t − P i , g a s P i , s a t , (9) where P i , g a s is the partial pressure of i in cover gas, P i is the saturated vapor pressure of pure i, x i is the mole ratio of solute i in solvent LBE, P i , s a t is the saturated vapor pressure of solute i in solvent LBE, and J i , 0 is the evaporation rate of Po in vacuum, and because of the cover gas in the LFR, the evaporation rate of Po and PbPo is four orders of magnitude lower than that in vacuum.

In order to provide the specific parameters in the multi-physical field, the multi-physics coupling frame was established and the causal link between each physics had been found out in Figure 3. The flow field and the heat transfer field are coupled strongly, while the flow heat transfer field and the concentration field are coupled weakly. By taking the output of thermal fluid ( u → and T) as the input of the ²¹⁰Po molecule transfer, the ²¹⁰Po concentration and flux in both the steady state and the dynamic state can finally be obtained.

A two-dimensional axisymmetric geometric structure is utilized in this model, according to the typical LFR concept. The main components that have an effect on the source term transport behavior are taken into consideration, including reactor core, heat exchanger, pump, purification system, cover gas, and containment. The material library covers the main candidate materials used in the LFR.

In order to improve the utilization of computing resources, a porous medium model is modified to the descript ²¹⁰Po behavior in core and heat exchanger (HX), and the algebraic y + model is utilized to compute the LBE turbulence, which can couple porous media and laminar flow automatically. In reality, a porous media model is frequently adopted in the nuclear industry to simulate the reactor core or HX, which validates a good agreement with the operation condition (Koloszar et al., 2014). The governing equations of the LBE flow are shown as follows: ρ ∂ u ∂ t + ρ ( u ⋅ ∇ ) u = ∇ ⋅ [ − P l + κ ] + F + ρ g , (10) ∂ ρ ∂ t + ∇ ⋅ ( ρ u ) = 0 (11)

The first equation is the momentum conservation equation, and the second equation is the mass conservation equation. The Brinkman equation is used to describe the fast flow in saturated porous media, where ρ is the density (kg/m³), u is the velocity vector (m/s), P is the pressure (Pa), κ is the Brinkman’s permeability (m²), F is the volume force vector (N/m³), and g is the acceleration of gravity (m/s²).

The real reactor core is assembled by of tens of thousands of fuel rods vertically. In the simulation, we reasonably simplified the structure of the reactor core and HX as anisotropic porous media, and assumed that LBE flows out mainly from the z-axial direction but rarely from the r-axial direction. Then properties of porous media are determined by two physical quantities, that is, porosity and permeability. The real porosity of the LBE reactor core is 0.2–0.4 (Koloszar et al., 2015). In this model, the porosity of the core is 0.3 and the porosity of the HX is 0.62 (Koloszar et al., 2014). And the permeability is calculated by Darcy’s formula, which is given as follows: κ i = Q ρ μ L i / Δ P i A i , (12) where Q is the mass flow (kg/s), μ is the dynamic viscosity (Pa∙s), L i is the flow length of the core or HX (m), Δ P i is the pressure drop of the core or HX (Pa), and A i is the cross-sectional area of the core or HX (m²). The z-axial leakage rate of the core is calculated as 9.4 × 10^–9 m²∼1.04 × 10^–8 m² and the z-axial permeability of the HX is 3.65 × 10^–8 m²∼4 × 10^–8 m²; since it is assumed that rare permeation occurs in the r-axial direction, the r-permeability is set as 1 × 10^–12 m².

Finally, the LBE coolant circulation in a reactor vessel was established. Heated by the reactor core and driven by buoyance, the LBE coolant flows up. Then driven by the pump, the LBE coolant will flow down, through HX, and back to the core. The entire flow field ( u → ) of LFR was obtained.

The heat transfer model is also based on a porous medium model, and the general governing equation of heat transfer is as follows: ( ρ C p ) ∂ T ∂ t + ρ C p u ⋅ ∇ ⋅ q = Q h e a t , (13) q = − k e f f ∇ T , (14)  where C p is the specific heat capacity at a constant pressure (J/kg·K), q is the heat flux by conduction (W/m²), Q h e a t is the heat sources as the reactor core and HX (W/m³), and k e f f is the effective thermal conductivity (W/m·K). The HX takes all the thermal power generated by the core to the secondary loop. The heat source of the HX can be calculated theoretically as follows: Q H X = − K H X ( T i n + T o u t 2 − T 2 ) L H X , (15) where Q H X is the total heat transfer power of the HX (W/m³), K H X is the total heat transfer coefficient (W/m²∙K), T i n is the temperature that flows into the HX (K), T o u t is the temperature out of the HX (K), T 2 is the temperature of the secondary loop (K), and L H X is the flow length of HX (m).

Since u → and T of the whole LFR are obtained from the computation of Eqs. 10, 13, the governing equation for the ²¹⁰Po transport model can be written as follows: ∂ c i ∂ t + ∇ ⋅ ( J i , d i f f + u c i ) = R i − J i , v a p o r | i n t e r f a c e , (16) where c i is the concentration of i (mol/m³), J i , d i f f is the diffusive flux vector (mol/m²·s), and R i is a production or consumption rate expression (mol/m³·s).

In order to provide input for a simulation case, several typical LFR concepts in the world are investigated, and the main operation parameters are listed in Table 3 (Zrodnikov et al., 2008; Zrodnikov et al., 2011; Didier et al., 2015). The typical ventilation rate is one volume turnover per hour according to the PWR operation experience. In this work, we assume an atmosphere purification system for the containment, which has a ventilation rate of 0.5/h and ²¹⁰Po remove efficiency of 95%. According to IPPE experience, lifetimes of ²¹⁰Po aerosol range from 60–80 s on equipment to 100–150 s in the containment room. And practically, the aerosol lifetimes are usually selected as 100 s for calculation (Yefimov et al., 1997). Besides, the influence of leakage is also the key factor that has to be taken into consideration. As a baseline, the leakage rate of cover gas is assumed as 5‰ per day, and the leakage rate of the containment is assumed as 4‰ per day.

Since the Po/PbPo evaporation rate is very sensitive to temperature, the influence of operation temperature is taken into analysis. Six different temperature ranges for LFR operation were simulated by adjusting the efficiency of the HX. And the corresponding average concentrations of ²¹⁰Po in cover gas and containment were obtained, which is shown in Figure 8.

Apparently, from Figure 8, a simple conclusion can be made that the evaporation rate of ²¹⁰Po from LBE to cover gas will increase rapidly with the higher temperature on the LBE–gas interface. While the ²¹⁰Po concentration in cover gas is increasing, the ²¹⁰Po concentration in the containment is increasing in the same proportion. However, at high operation temperature like 401–467°C, the concentration of ²¹⁰Po in the containment is 63.2 Bq/L, which is a bit higher than one DAC value.

A comprehensive analysis was made for all the influential factors (including deposition, ventilation, leakage, and decay) on the ²¹⁰Po concentration in the containment, which is shown in Table 4. It is obvious that the phenomenon of aerosol deposition is the key process affecting aerosol concentration in containment, based on the assumption in this work. However, the aerosol deposition rate is strongly dependent on the dynamic particle size and density, which will guide our future experiment.

We simulated the total surface contamination due to ²¹⁰Po deposition in the containment and its accumulation during 100 days normal operation. As shown in Figure 9, after 10 days operation, the ²¹⁰Po activity on the ground of the containment exceeds the limit value of 4 × 10³ Bq/m² specified in the China regulation GB18871-2002. Therefore, it is very important that special decontamination measures must be developed to control the ²¹⁰Po and avoid radioactivity harms to maintenance staff.

In the real operation, the problem of cover gas leakage is inevitable. In order to better study the influence of leakage from the safety perspective, the leakage fraction per day of cover gas was varied from 0.01% to 0.02, 0.04, 0.08, 0.16, and 0.32%; the ²¹⁰Po activity in the containment was calculated; and the parameter sensitivity was taken in the analysis.

In Figure 10, the average activity of ²¹⁰Po in the containment is very sensitive to the leakage rate of the cover gas. Once the leakage rate is higher than 0.01, the ²¹⁰Po activity in air will exceed the one DAC limit. In order to control the ²¹⁰Po release and ensure the LFR in a safety state, the leakage rate of the cover gas should be kept below 0.01 strictly, and as low as possible. In case of serious radioactive leakage, the purification system should be quickly started to control the ²¹⁰Po activity.

IPPE developed the techniques of the alkaline extraction of polonium from LBE (Yefimov et al., 1997). And MIT developed the polonium extraction techniques of rare-earth filtering and polonium hydride stripping (Larson, 2002). All these research studies show that the extraction and purification efficiency of ²¹⁰Po in a real LFR have the potential to reach 90–99%. Therefore, it is reasonable to assume a purification efficiency of 95% to analyze the sensitivity of purified fraction to the ²¹⁰Po activity in the containment. While the leakage rate is fixed, we modified the flow fraction into the purification system ( f ), from 0.01 to 0.2. λ p u r i f y = f m η M , (17) where m is the LBE mass flow rate (kg/s), η is the purification efficiency (%), and M is the total LBE mass (kg).

Apparently, in Figure 11 above, the purification fraction has a significant influence on the activity of ²¹⁰Po in the containment. A high purification fraction could reduce the ²¹⁰Po contamination in the containment effectively, but significantly increase the load of the LBE purification system and the operation cost. While the purification fraction is less than 0.01, the ²¹⁰Po concentration will exceed one DAC limit. And while the purification fraction is more than 0.05, the decrease rate of ²¹⁰Po in the containment slowed down gradually. To make a compromise between nuclear safety and economy, a purification fraction of ∼0.05 is recommended.