Multi-Physics Model Development for Polonium Transport Behavior in a Lead-Cooled Fast Reactor

²¹⁰Po, a highly toxic element with strong volatility, is one of the main source terms of a Gen-IV lead-cooled fast reactor (LFR). Therefore, the radioactive safety caused by ²¹⁰Po has become an important topic in LFR-related research. In order to simulate the behavior of ²¹⁰Po in an LFR, this work developed a multi-physics model of an LFR from the perspective of radioactive transport. Considering the effects of nuclide decay, cover gas leakage, containment ventilation, and Po aerosol deposition, a comprehensive simulation was carried out to evaluate the sensitivity of those effects on the ²¹⁰Po distribution in detail. Preliminary results indicate that during normal operation, most of the ²¹⁰Po in the LBE exist in the form of PbPo, and around 10^–9 of ²¹⁰Po could evaporate from the LBE into the cover gas, and then further leak into the containment. In addition, even if the leakage rate of ²¹⁰Po in the cover gas into the containment is maintained at 5‰ per day, due to the deposition of Po aerosol, the ²¹⁰Po contamination on the inner surface of the containment is still below the radioactivity concentration limits.

Introduction

As one of the generation-IV (Gen-IV) nuclear power system, the lead-cooled fast reactor (LFR) is expected to be the first concept to achieve industrial demonstration (Lorusso et al., 2018; Nuclear Power, 2019). Lead–bismuth eutectic (LBE) is selected as the primary coolant for the LFR due to its neutronic economic and good thermal hydraulic properties (Dierckx et al., 2014). However, during normal operation of the LFR, ²¹⁰Po, a highly toxic element with strong volatility, will be produced in the LBE coolant, which will induce a new radioactive safety problem.

In the purpose of the occupational safety and public health, ICRP and nuclear power countries have specified healthful working conditions and given the exposure limits of ²¹⁰Po. In NRB-99 of Russia, the occupational annual limit on intake (ALI) is settled as 6.70 × 10³ Bq, and the safety permission of air concentration is below 2.7 Bq/m³ (Pankratov et al., 2004). For the U.S. NRC regulation of 10 CFR Part.20, the derived air concentration (DAC) of ²¹⁰Po is settled as 1.1 × 10⁵ Bq/m³, while one DAC is equal to allowable maximum air concentration at the breathing rate of 1.3 m³/h for 2,000 working hours per year. And the ALI for ingestion and inhalation are 1.11 × 10⁵ Bq and 2.22 × 10⁴ Bq, respectively (Nuclear Regulatory Commission 10CFR, 2017). In China, the National Standard GB18871-2002 rules mention the ALI of ²¹⁰Po as 8.3 × 10⁴ Bq, and the exemption concentration as 10⁴ Bq (Mao, 2014), the National regulation on ²¹⁰Po of RA, US, and CN are given in Table 1.

TABLE 1

TABLE 1. National regulation on ²¹⁰Po of RA, US, and CN.

Multi-Physics Model for Po Transport in the LFR

Source Term of Po in the LFR

²¹⁰Po has a short half-life of 138.4 days (Ram et al., 2019) and is the β decay product of ²¹⁰Bi (Loewen, 2005). During the normal operation of the LFR, ²¹⁰Po evaporates from the LBE coolant as a simple substance and a chemical form of PbPo, and then accumulates in the cover gas. Accompanied with the leakage of cover gas, ²¹⁰Po is released into the containment (Feuerstein et al., 1992), which results in significant radioactive contamination and potential occupational exposure during maintenance.

${}^{209}Bi+n{\to }^{210}Bi\underset{t_{1/2}=5\text{days}}{\overset{{\beta }^{-}}{\to }}Po\underset{t_{1/2}=138.4\text{days}}{\overset{\alpha }{\to }}Pb.(1)$

Po Behavior in the LFR

Physical and Chemical Properties of Po in the LFR

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.

FIGURE 1

FIGURE 1. Migration characteristics of polonium in the LBE reactor.

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% (${\chi }_{Po,LBE}$) of Po in the LBE is left in the form of Po, and the remaining 99.8% (${\chi }_{PbPo,LBE}$) 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:

$2PbP{o}_{\left(sol\right)}+heat\leftrightarrow PbP{o}_{\left(g\right)}+P{o}_{\left(g\right)}+P{b}_{\left(s\right)},(2)$

$PbPo+H_{2}O\leftrightarrow H_{2}P{o}_{\left(g\right)}+Pb{O}_{\left(sol\right)}.(3)$

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

TABLE 2

TABLE 2. Po, PbPo, and H₂Po foundation properties (Feuerstein et al., 1992).

Mass Flow of Po in LFR

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.

FIGURE 2

FIGURE 2. The molecule flowchart of ²¹⁰Po in an LFR.

In LBE coolant,

• Accompanied with neutron irradiation, the isotope ²¹⁰Bi is produced, and then we have the concentration of Bi as c_Bi210 (mol/m³), with a decay rate ${\lambda }_{Bi210}$ of 1.6 × 10^–6 (1/s);

• ²¹⁰Bi decays into ²¹⁰Po, then we have the concentration of Po as $c_i^{LBE}$(mol/m³), where the species i means Po or PbPo;

• Define the decay of Po, $J_{decay}^{LBE}$= λ_Po·$c_i^{LBE}$ (mol/m³s);

• Define the purification rate of Po in LBE, $J_{puri}^{LBE}$= η_LBE·$c_i^{LBE}$ (mol/m³s);

• Define the Po and PbPo evaporate flux into cover gas at the LBE–gas interface as ${J_{i,vapor}|}_{interface}$ (mol/m²s);

Then the molecule balance equation for each species in LBE can be written as Eq. 4:

$\frac{d{c}_i^{LBE}}{dt}={\chi }_i{\lambda }_{Bi210}{c}_{Bi210}-\left(J_{puri}^{LBE}+J_{decay}^{LBE}+J_{leak}^{LBE}+{J_{i,vapor}|}_{interface}\right).(4)$

In cover gas.

• Define the decay rate of Po, $J_{decay}^{gas}$= λ_Po·$c_i^{gas}$ (mol/m³s);

• Define the purification rate of Po, $J_{puri}^{gas}$= η_gas·$c_i^{gas}$(mol/m³s);

• Define the leakage rate of Po into the containment, $J_{leak}^{gas}$ (mol/m³s);

Then the molecule balance equation for each species in cover gas can be written as Eq. 5:

$\frac{d{c}_i^{gas}}{dt}={J_{i,vapor}|}_{interface}-\left(J_{puri}^{gas}+J_{decay}^{gas}+J_{leak}^{gas}\right)(5)$

In containment.

• All the PbPo are assumed to be reacted with H₂O in air, to generate H₂Po in the containment, $c_{H_{2}Po}^{air}$ (mol/m³);

• Define the decay rate of H₂Po, $J_{decay}^{air}$ (mol/m³s);

• Define the deposition rate of H₂Po, $J_{dep}^{air}={\lambda }_{dep}\cdot c_{H_{2}Po}^{air}$ (mol/m³s);

• Define the purification rate $J_{puri}^{air}$ and leakage rate $J_{leak}^{gas}$ of H₂Po in the containment.

Then the molecule balance equation for each species in the containment can be written as Eq. 6:

$\frac{d{c}_{H_{2}Po}^{air}}{dt}=J_{leak}^{gas}-\left(J_{puri}^{air}+J_{dep}^{air}+J_{decay}^{air}+J_{leak}^{air}\right)(6)$

And due to the H₂Po aerosol deposition, there will be surface contamination on the ground of the containment, which is calculated as follows:

$\frac{d{c}_{H_{2}Po}^{ground}}{dt}=J_{dep}^{air}-J_{decay}^{air}(7)$

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.

$P_{i,sat}=x_{i,LBE}{P}_i.(8)$

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.

${J_{i,vapor}|}_{interface}=J_{i,0}\times c_{i,LBE}\times \frac{P_{i,sat}-P_{i,gas}}{P_{i,sat}},(9)$

where $P_{i,gas}$ 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, $P_{i,sat}$ 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.

Multi-Physics Frame and Numerical Models

Multi-Physics Frame

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 ($\overrightarrow{u}$ 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.

FIGURE 3

FIGURE 3. The multi-physics model of Po migration in an LFR.

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.

Fluid Flow Model

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:

$\rho \frac{\partial \mathrm{u}}{\partial t}+\rho \left(\mathrm{u}\cdot \nabla \right)\mathrm{u}=\nabla \cdot \left[-P\mathrm{l}+\mathrm{\kappa }\right]+\mathrm{F}+\rho \mathrm{g},(10)$

$\frac{\partial \rho }{\partial t}+\nabla \cdot \left(\rho \mathrm{u}\right)=0(11)$

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 $\rho$ is the density (kg/m³), $\mathrm{u}$ is the velocity vector (m/s), P is the pressure (Pa), $\mathrm{\kappa }$ is the Brinkman’s permeability (m²), $\mathrm{F}$ is the volume force vector (N/m³), and $\mathrm{g}$ 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:

${\kappa }_i=\frac{Q}{\rho }\mu L_i/\Delta P_i{A}_i,(12)$

where Q is the mass flow (kg/s), $\mu$ is the dynamic viscosity (Pa∙s), $L_i$ is the flow length of the core or HX (m), $\Delta P_i$ 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$\times$10^–9 m²∼1.04$\times$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 ($\overrightarrow{u}$) of LFR was obtained.

Heat Transfer Model

The heat transfer model is also based on a porous medium model, and the general governing equation of heat transfer is as follows:

$\left(\rho C_p\right)\frac{\partial T}{\partial t}+\rho C_p\mathrm{u}\cdot \nabla \cdot \mathrm{q}=Q_{heat},(13)$

$\mathrm{q}=-k_{eff}\nabla T,(14)$

 where $C_p$ is the specific heat capacity at a constant pressure (J/kg·K), $\mathrm{q}$ is the heat flux by conduction (W/m²), $Q_{heat}$ is the heat sources as the reactor core and HX (W/m³), and $k_{eff}$ 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:

$Q_{HX}=\frac{-K_{HX}\left(\frac{T_{in}+T_{out}}{2}-T_2\right)}{L_{HX}},(15)$

where $Q_{HX}$ is the total heat transfer power of the HX (W/m³), $K_{HX}$ is the total heat transfer coefficient (W/m²∙K), $T_{in}$ is the temperature that flows into the HX (K), $T_{out}$ is the temperature out of the HX (K), $T_2$ is the temperature of the secondary loop (K), and $L_{HX}$ is the flow length of HX (m).

Molecule Transfer Model

Since $\overrightarrow{u}$ 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:

$\frac{\partial c_i}{\partial t}+\nabla \cdot \left({\mathrm{J}}_{i,diff}+\mathrm{u}{c}_i\right)=R_i-{J_{i,vapor}|}_{interface},(16)$

where $c_i$ is the concentration of i (mol/m³), ${\mathrm{J}}_{i,diff}$ is the diffusive flux vector (mol/m²·s), and $R_i$ is a production or consumption rate expression (mol/m³·s).

Numerical Modeling for ²¹⁰Po in a Typical LFR

Modeling Condition

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.

TABLE 3

TABLE 3. Operation parameters of CLEAR, SVBR, MYRRHA, and CASE in this work (Zrodnikov et al., 2008; Zrodnikov et al., 2011; Didier et al., 2015).

²¹⁰Po Distribution in LFR

Under a normal operation condition, the temperature field of LBE ranges from 610 to 670K, and on the surface for Po/PbPo evaporation, it is around 660K. The high-temperature LBE flows up out of the core and down into the HX and is then driven by the main pump flows back into the core. This is the primary cooling loop of the LBE in the reactor vessel. Then we coupled the LBE thermal fluid with cover gas to figure out the entire temperature field and velocity field in the LFR vessel, as shown in Figures 4, 5.

FIGURE 4

FIGURE 4. The temperature of LBE and cover gas in an LFR vessel at a steady state.

FIGURE 5

FIGURE 5. The flow field of LBE, Ar, and air in an LFR vessel and containment at a steady state.

Then the ²¹⁰Po concentration field and transport map were computed with environmental inputs of $\overrightarrow{u}$ and T. And the results at a steady state are shown in Figure 6. ²¹⁰Po is evenly distributed in the reactor vessel and containment, except for a small amount of high concentration areas.

FIGURE 6

FIGURE 6. The ²¹⁰Po transport in LFR vessel and containment at a steady state.

As shown in Figure 7, the average concentration of ²¹⁰Po in LBE vs. time is calculated during 100 days from start-up. Under normal conditions, the ²¹⁰Po concentration will reach equilibrium within 30 days. And the concentration of Po in LBE is around 1.08 × 10¹⁰ Bq/kg, which is far beyond the exemption concentration according to Chinese regulation (GB18871-2002). In cover gas, the Po concentration is 6.58 × 10³ Bq/L, which is also 2–3 orders of magnitude higher than one DAC value. The concentration of Po in the containment is ∼1 × 10^–3 Bq/L, which is below the limitation in law.

FIGURE 7

FIGURE 7. The average concentration of Po in LBE during 100 days.

Impact Factors of the Po Transport

Operation Temperature

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.

FIGURE 8

FIGURE 8. The average concentration of Po in cover gas and containment under different temperature of evaporating interface.

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.

Deposition Rate

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.

TABLE 4

TABLE 4. Influence factors on ²¹⁰Po concentration in the containment.

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.

FIGURE 9

FIGURE 9. The surface contamination due to ²¹⁰Po deposition in the containment (100 days).

Leakage Rate

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.

FIGURE 10

FIGURE 10. Average activity of ²¹⁰Po in the containment under different leakage rates of cover gas.

Purification System

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.

${\lambda }_{purify}=\frac{fm\eta }{M},(17)$

where m is the LBE mass flow rate (kg/s),$\eta$ 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.

FIGURE 11

FIGURE 11. The average activity of ²¹⁰Po in the containment under different purification fractions.

Discussion and Conclusion

As a summary of the aforementioned simulation results, for a typical LFR, most of the ²¹⁰Po accumulate in the LBE coolant can be restrained as a form of PbPo; only ∼10^–9 of them could evaporate into cover gas, which is strongly dependent on the temperature on the LBE–gas interface. The aerosol deposition seems to be a positive feature in that it takes the radioactive ²¹⁰Po down onto ground and restrains the DAC value in air. However, the surface contamination and aerosol resuspension are newly emerging issues. Ventilation combined with the atmosphere purification system is an artificial and reliable approach for Po decontamination.

In addition, some major safety challenges of polonium in LFR still exist. The unknown phenomena and their effects on polonium transport behavior need to be further and deeply explored, such as the oxygen control in LBE, material corrosion products, and impurity filtration. In the next step, more mechanisms of Po transport in the real operation condition and design base accident (DBA) condition will be taken into consideration.

Data Availability Statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

Author Contributions

DL: paper writing, data preparation, and systematic analysis. WY: program coding and sensitivity analysis. ZZ and HB: validation and verification. HJ and WJ: module coding. NM: theoretical model development.

Funding

This study was funded by the National Defense Pre-Research Foundation of China (grant no. 201-XXXX0404-fzsdx-Tm06) and the Fundamental Science on Nuclear Wastes and Environmental Safety Laboratory (grant no. 18kfhk04).

Conflict of Interest

The authors DL, ZZ, and HB were employed by the company Nuclear Power Institute of China.

The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s Note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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