Governing equations

A CFD software based on the finite volume method was employed to simulate the concentration distribution of the attached ²¹²Pb and ²¹²Bi within the ²²⁰Rn chamber. Mass- and momentum-conservation equations were used to model the flow-field mode. Then, various modules/equations describing the relevant physical processes governing the attached ²¹²Pb and ²¹²Bi are incorporated using a user-defined function to simulate the concentration distribution of the attached ²¹²Pb and ²¹²Bi in the ²²⁰Rn chamber. The predictive accuracy of this software was previously validated in various contexts [27].

In the simulation, the flow was assumed continuous and incompressible. The temperature inside the ²²⁰Rn chamber was considered constant and uniform and was set at 300 K. The governing equations for the mass and momentum conservation are as follows [28, 29, 30, 37]: $\frac{\partial \rho }{\partial t}+\frac{\partial \left(\rho u_i\right)}{\partial x_i}=\mathrm{0,}$ (1) $\frac{\partial \left(\rho u_i\right)}{\partial t}+\frac{\partial \left(\rho u_i{u}_j\right)}{\partial x_i}=-\frac{\partial P}{\partial x_i}+\frac{\partial \mu \left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right)}{\partial x_j}\mathrm{,}$ (2) where ρ is the fluid density (kg m^-3), t is the fluid time (s), xi is the spatial coordinates (m) with the numbering index xi = 1, 2, 3 for x, y, and z, respectively, ui is the velocity vector(m s^-1) with the numbering index i=1, 2, 3 for u, v, and w, respectively, P is the pressure(N m^-2), and μ is the fluid viscosity.

The release of β- and γ-rays does not generate sufficient recoil kinetic energy to free the ²²⁰Rn progeny from aerosols during the decay of ²¹²Pb to ²¹²Bi. Therefore, the impact of the recoil variables was not considered in this study. The activity concentration distribution of the ²²⁰Rn progeny in the ²²⁰Rn chamber is as follows [30, 37]: $\frac{\partial C_i}{\partial t}+\nabla \cdot \left(\left(U+{\nu }_{\text{s}}\right)C_i\right)=\nabla \cdot \left(D_{\text{e}}\nabla C_i\right)+{\lambda }_{i-1}{C}_{i-1}-{\lambda }_i{C}_i\mathrm{,}$ (3) where the numbering indices are i = 1, 2 for the attached ²¹²Pb and ²¹²Bi, respectively. U is the mean velocity(m s^-1), D_e is the effective diffusion coefficient(m² s^-1), Ci is the activity concentration (Bq m^-3) of the ith decay product, λi is the decay constant (s^-1) of the ith decay product. In Eq. (3), the parameters are estimated using the appropriate mathematical models, as discussed below. The decay constants (λi) are λ₂₁₂Pb=1.8×10^-5s^-1 and λ₂₁₂Bi=1.9×10^-4s^-1. ${\nu }_{\text{s}}$ is the settling velocity (m s^-1) from which the Stokes law can be obtained. ${\nu }_{\text{s}}=\frac{2r^{2}g}{9\mu }\left({\rho }_0-\rho \right)\mathrm{,}$ (4) where r is the radius of the particle (m); the radius of the aerosol particles is set to 100 nm. ρ₀ denotes the particle density (kg m^-3).

The effective diffusion coefficient D_e is the sum of the Brownian diffusion coefficient (D_b) and turbulent diffusion coefficient (D_t). The diffusion coefficient (D_b) is calculated using the Stokes-Einstein relationship: $D_{\text{b}}=\frac{K_{\text{b}}T{C}_{\text{u}}}{6\pi \mu r}\mathrm{,}$ (5) where C_u denotes the Cunningham slip correction factor that is calculated as follows [37, 38]: $C_{\text{u}}=1+\frac{l_{\text{m}}}{2r}\left(2.34+1.05e^{-0.39\frac{d}{l_{\text{m}}}}\right)\mathrm{,}$ (6) where k_b is the Boltzmann constant (m² kg s^-2 K^-1), l_m is the mean free path of the air molecules (m), μ is the viscosity of air (Pa s), and T is the temperature (K).

Turbulence model

During the operation of the ²²⁰Rn chamber, when the fan operates at its lowest power setting, the minimum Reynolds number is approximately 2500. As the wind speed increases, the entire ²²⁰Rn chamber becomes turbulent. Consequently, this study chose the standard $\kappa -\epsilon$ model to conduct the simulation calculations. This equation is expressed as follows: $\frac{\partial \left(\rho \kappa \right)}{\partial t}+\frac{\partial \left(\rho \kappa u_i\right)}{\partial x_i}=\frac{\partial \left(\left(\mu +\frac{{\mu }_t}{{\sigma }_{\kappa }}\right)\frac{\kappa }{x_j}\right)}{\partial x_j}+G_{\kappa }-\rho \epsilon$ (7) $\frac{\partial \left(\rho \epsilon \right)}{\partial t}+\frac{\partial \left(\rho \epsilon u_i\right)}{\partial x_i}=\frac{\partial \left(\left(\mu +\frac{{\mu }_t}{{\sigma }_{\epsilon }}\right)\frac{\epsilon }{x_j}\right)}{\partial x_j}+\rho C_{1}E\epsilon -\rho C_2\frac{{\epsilon }^2}{\kappa +\sqrt{\nu \epsilon }}\mathrm{,}$ (8) where μ_t is the turbulent-flow velocity, Gκ is the generation term of the turbulent kinetic energy κ owing to the average velocity gradient, ${\sigma }_{\kappa }$ and ${\sigma }_{\epsilon }$ are the Prandtl numbers corresponding to the turbulent kinetic energy κ and dissipation rate σ, respectively, and C₁ and C₂ are constants: C₁=1.44, C₂=1.92.

Grid setting at boundary

To reduce the backflow at the inlet and outlet, they were extended by 10 cm. In this study, the grid is formed using hexahedral and tetrahedral methods, in which the fan, inlet, and outlet areas are tetrahedral grids, and the other areas are hexahedral grids. The flow field in the fan area is complex; thus, in this study local encryption is performed for the fan area. The total number of grids in the ²²⁰Rn chamber is approximately 586 000. The grid quality was evaluated using the orthogonal quality, and the orthogonal quality average was 0.81279. The mesh is illustrated in Fig. 2(a). Based on the boundary conditions described in Sect. 2.3, the deposition area was set near the wall. In this study, a grid layer near the wall was recorded as the deposition area. Figure 2(b) shows the deposition area setting of the x=0 plane, where the outermost light-yellow area is the deposition area.

Fig. 2Full-screen View

(Color online) Schematic of (a) meshing and (b) depositional area setting of ²²⁰Rn chamber

Grid independence test

In this study, a grid-independence test experiment was conducted with grid numbers of 295 000 and 788 000. In this experiment, the wind speed and concentration of ²²⁰Rn progeny at C22 in the ²²⁰Rn chamber were compared. M1, M2, and M3 represent grid numbers of 295 000, 586 000, and 788 000, respectively. The following results are shown in Fig. 3: (1) in the experiment comparing the wind speed of three kinds of grid flow fields, the relative deviation of wind speed at C22 in the ²²⁰Rn room is 3.18%; (2) an experiment on the changes of ²¹²Pb and ²¹²Bi concentrations with time was conducted under the condition of a wind speed of 0.05 m/s and progeny-complement rate of 7 L/min. When the progeny concentration of ²²⁰Rn in the ²²⁰Rn chamber reached a steady state, the relative deviation in the progeny concentration of ²²⁰Rn was 1.25%. This grid independence test showed that with an increase in the number of grids, the consistency of the concentration-change law of the ²²⁰Rn progeny is good, and the flow field inside the ²²⁰Rn chamber exhibits little change, which proves that the number of grids set in this study is appropriate.

Fig. 3Full-screen View

Comparison of a wind speed and b progeny concentration with the change of grid number

Decay products of ²²⁰Rn concentration measurements

In this study, the decay products of thoron inside the ²²⁰Rn chamber were measured using an alpha spectrometer method based on alpha spectroscopy. The measurement procedure involved sampling the decay products of thoron (²²⁰Rn) using a microporous membrane and self-made sampling rod. The structure of the sampling rod, illustrated in Fig. 4, includes a head, microporous membrane, metal mesh, washer, and thread. The tail of the sampling rod is a long metal rod. The figure shows that microporous membranes, metal meshes, and washers are stacked in sequence in the structure head, and the head and thread are connected to form the structure of the sampling head. The sampling-rod samples from the ²²⁰Rn chamber using an air pump, the air containing the ²²⁰Rn progeny enters the sampler through the head and then passes through microporous membrane, and the ²²⁰Rn progeny is adsorbed on the microporous membrane. Then, it passes through the metal mesh, which mainly prevents the microporous membrane from being broken owing to excessive sampling airflow. A washer is used to ensure that microporous membrane and metal mesh are tightly attached to the structure head, ensuring that the airflow passed through the microporous membrane. The structure duct is approximately 70 cm long and can be sampled at different positions along the axis of the ²²⁰Rn chamber-sampling hole. The rear end of the structural duct can be connected to a pipe to reintroduce gas into the ²²⁰Rn chamber.

Fig. 4Full-screen View

(a) Structure of self-made sampling rod and (b) measurement procedure of progeny concentration

During the sampling process, the ²²⁰Rn progeny concentration was first sampled for 10 min at a sampling flow rate of 5 L/min, and the sampled filter film was then placed in an alpha spectrometer for counting. Counts N₁ and N₂ were obtained by measuring the 8.78 MeV energy-peak counts of ²¹²Po using the α-spectrum approach [9, 36]. Finally, the activity concentrations of attached ²¹²Pb and ²¹²Bi were determined using Eqs. (12) and (13). The measurement procedure for the progeny concentration is shown in Fig. 4b. $C_{{}^{212}\text{Pb}}=\frac{1}{1.42Q{\eta }_1{\eta }_2{K}_a}\left(0.1073N_2-0.105N_1\right)\mathrm{,}$ (13) $C_{{}^{212}\text{Bi}}=\frac{1}{1.42Q{\eta }_1{\eta }_2{K}_a}\left(0.1593N_1-0.0377N_2\right)\mathrm{,}$ (14) where η₁ is the counting efficiency (η₁=0.127), η₂ is the filtration efficiency (η₂=0.99), and Ka is the self-absorption factor of the filter membrane (Ka=0.97) [9, 10].

Indicators for experimental- and simulation-error comparisons

The attached ²¹²Pb and ²¹²Bi abbreviations in the following text are written as ²¹²Pb and ²¹²Bi, respectively. Henceforth, progeny refers to the attached ²¹²Pb and ²¹²Bi. In this study, the alpha-spectrometer method mentioned in Sect. 2.5 was used to measure progeny concentrations with varying wind speeds and progeny-replenishment rates. The obtained experimental results ($c_i^{\text{e}}$) were then compared with the numerical simulation results ($c_i^{\text{s}}$) by calculating the relative deviation(RD). Verifying the uniformity of the progeny concentration through experimental research is difficult because it requires the simultaneous measurement of the progeny concentration at 27 spatial positions, as shown in Fig. 5. As a result, this study confirmed the consistency of progeny concentration between the simulation and experimentation at equilibrium. However, the uniformity of the progeny concentration between the simulation and experimentation at equilibrium has not yet been confirmed. $RD=\left|\frac{c_i^e-c_i^s}{c_i^e}\right|\mathrm{.}$ (15)

Fig. 5Full-screen View

(Color online) Distribution of monitoring sites in the ²²⁰Rn chamber

Performance-evaluation index of the ²²⁰Rn chamber

The distribution of monitoring points in the ²²⁰Rn chamber is illustrated in Fig. 5. Within the calibration chamber, the top, central, and bottom monitoring points were identified as T, C, and B, respectively. The coordinates of the central point are denoted as C22(0,0,0), which is the sampling area S₁. The distance between each site and the adjacent sites is 0.2 m. The sampling area S₂ is located at C12 in Fig. 5.

Two important indices, the progeny-concentration level and uniformity of progeny concentrations, are crucial in determining the regulation ability. The mean progeny concentration($\overline{C}$) at each monitoring point was calculated using Eq. (16). The uniformity of progeny concentrations within the ²²⁰Rn chamber can be assessed using the relative standard deviation (RSD) at these monitoring points, as expressed in Eq. (17). $\overline{C}=\frac{1}{9}{\displaystyle \sum _1^9{c}_i}\mathrm{,}$ (16) $RSD=\frac{\sqrt{\frac{1}{9}{\displaystyle \sum _1^9{\left(c_i-\overline{c}\right)}^2}}}{\overline{C}}\times 100\%\mathrm{,}$ (17) where ci denotes the concentration of ²¹²Pb or ²¹²Bi at the ith point.

Effect of wind speed on the performance parameter of ²²⁰Rn chamber

(1)Profile plots of the concentration of 212Pb and 212Bi

Figure 7a illustrates the distribution of the progeny concentration at x = -1 m, -0.3 m, 0.3 m, and 1 m at different wind speeds, and Fig. 7b shows the progeny-concentration profile at z = 0 m for various wind speeds. The progeny-concentration profile at x = 1 m in Fig. 7a shows that the progeny concentration is significantly higher in the bottom section than in the middle and upper sections. Conversely, at x = -1 m in Fig. 7a, the progeny concentration in the middle was higher and indicated a more uniform distribution than that at x=1 m. Furthermore, Fig. 7b demonstrates that the tail end of the progeny-concentration distribution is more uniform than that at the front end. As the wind speed increases, the uniformity of the distribution improves.

Fig. 7Full-screen View

(Color online) Simulated concentration profile of ²¹²Pb/²¹²Bi for different wind speeds for a fixed progeny-complement rate of 7 L/min at x= -1 m, -0.3 m, 0.3 m, 1 m, and z = 0 cm. The distribution of a ²¹²Pb concentration and b ²¹²Bi concentration in relation to wind speed respectively

Figure 8a displays the streamlines color-coded according to the velocity magnitude. Initially, progeny with a high concentration enters the calibration chamber from the inlet. Subsequently, the progeny circulate within the calibration chamber, particularly along the bottom wall, when the wind speed is low. Eventually, most of the progeny flow into the pipeline along the bottom wall of the calibration chamber, whereas a small fraction continues to circulate within the chamber.

Fig. 8Full-screen View

(Color online) a Streamlines colored by the velocity magnitude (m/s) of flow for wind speed of 0.05 m/s. b Progeny velocity vector diagram around progeny-complement entrance area

Furthermore, the concentration profiles of ²¹²Pb and ²¹²Bi are higher below the inlet than at other places, particularly when the wind speed is relatively low. Figure 8b provides a detailed kinematic explanation of the motion characteristics of the progeny displayed in Fig. 8a, c. v₁ is the fluid velocity and v₂ represents the combined velocity between the initial velocity ($v_{\text{G}}^{*}$) and the velocity produced by gravity (v_G). Figure 8b shows that the progeny are initially injected ($v_2^{*}$) with a downward velocity (in the negative Y direction). When the injection velocity ($v_2^{*}$) is significantly higher than the fluid velocity (v₁), the progeny particles tend to accumulate in region B. This phenomenon is more likely to occur when the wind speed is low. This explains why the concentration of progeny at the bottom of the calibration chamber is higher than that in the other sections.

(2)Effect of wind speed on progeny concentration

The variation in the concentration of ²¹²Pb with wind speed at a fixed progeny-complement rate of 7 L/min is plotted in Fig. 9a. Figure 9a shows that the ²¹²Pb concentration decreases with wind speed from 0.05 m/s to 0.5 m/s. When the wind speed is less than 0.25 m/s, the progeny concentration is highly sensitive to changes in wind speed. When the wind speed exceeds 0.25 m/s, the sensitivity decreases. For a detailed analysis, at a wind speed of 0.05 m/s, the ²¹²Pb concentration is 2513 Bq/m³, and the ²¹²Pb concentration is 378 Bq/m³ at a wind speed of 0.5 m/s. In comparison, the ²¹²Pb concentration in the latter is reduced to 15% of that in the former. In addition, at a wind speed of 0.05 m/s, the ²¹²Bi concentration is 2285 Bq/m³, and the ²¹²Bi concentration is 165 Bq/m³ at a wind speed of 0.5 m/s. In comparison, the ²¹²Bi concentration of the latter is reduced to 7.2% of that of the former. Progeny concentration is greatly affected by wind speed. This is primarily because higher wind speeds result in a more pronounced deposition of progeny [9]. Therefore, wind speed significantly affects progeny concentration and is a key parameter in its regulation.

Fig. 9Full-screen View

(Color online) Variation of (a) ²¹²Pb concentration and (b) ²¹²Bi concentration with wind speed at a fixed progeny-complement rate of 7 L/min

(3) Effect of wind speed on uniformity

Figure 7a and b in Sect. 3.2.1 qualitatively show that as the wind speed increases, the uniformity of the progeny concentration in the ²²⁰Rn chamber gradually improves. In addition, Fig. 8 quantitatively shows the relationship between the uniformity of the progeny concentration and wind speed. Based on the results presented in Fig. 10, an increase in wind speed leads to a gradual decrease in the RSD of the progeny concentration. This decrease indicates an improvement in the uniformity of the progeny concentration within the ²²⁰Rn chamber, which aligns with the conclusions presented in Sect. 3.2.1. Specifically, when the wind speed ranges from 0.05 m/s to 0.50 m/s, the RSD of ²¹²Pb in sampling area S₁ decreases from 10.51% to 0.60%, whereas the RSD of ²¹²Bi decreases from 7.57% to 0.57%. Similarly, in sampling area S₂, the RSD of ²¹²Pb decreases from 9.08% to 0.54%, and the RSD of ²¹²Bi decreases from 7.97% to 0.51% within the same wind-speed range. These significant changes in RSD indicate the strong influence of wind speed on the uniformity of progeny concentration. Notably, when the wind speed exceeds 0.2 m/s, it has little effect on the uniformity of the progeny concentration in the two sampled regions, with an RSD of 1.6%.

Fig. 10Full-screen View

(Color online) RSD of progeny with wind speed at a fixed progeny-complement rate of 7 L/min

Effect of progeny-complement rate on performance parameter of ²²⁰Rn chamber

(1) Profile plots of the concentration of 212Pb and 212Bi

Section 3.2.1 shows that when the wind speed is 0.05 m/s, the progeny-concentration gradient is the most obvious. Therefore, to clearly analyze the distribution of progeny concentration in Sect. 3.2.2, the wind speed in this section is set to 0.05 m/s. Figure 11a illustrates the distribution of progeny concentration at x = -1  m, -0.3 m, 0.3 m, and 1 m for different progeny-complement rates, and Fig. 11b shows the progeny-concentration profile at z = 0 m for various progeny-complement rates. Compared with the middle and upper sections of the progeny-concentration distribution at x = 1  m, the progeny concentration at the bottom is significantly higher, and at x = -1  m in Fig. 11a, the progeny concentration in the middle area is higher. Compared with the effect of wind speed on the progeny-complement rate mentioned in Sect. 3.2.1, the impact of wind speed on progeny-concentration uniformity is significantly greater.

Fig. 11Full-screen View

(Color online) Simulated concentration profile of progeny for different progeny-complement rates for a fixed wind speed of 0.05 m/s at x = -1 m, -0.3 m, 0.3 m, 1 m, and z = 0 m. The distribution of a ²¹²Pb concentration and b ²¹²Bi concentration with progeny-complement rates respectively

(2) Effect of progeny-complement rate on progeny concentration

The variation in ²¹²Pb concentration with the progeny-complement rate at a fixed wind speed of 0.05 m/s is plotted in Fig. 12a. Figure 12a and Fig. 12b show that the ²¹²Pb and ²¹²Bi concentrations gradually increase as the progeny-complement rate increases from 1 L/min to 7 L/min. The concentration of ²¹²Pb increased from 1132 Bq/m³ to 2513 Bq/m³, an increase of 122%, while the concentration of ²¹²Bi increased from 981 Bq/m³ to 2285 Bq/m³, an increase of 133%. Notably, Eqs. (9) and (10) show that when the progeny-complement rate increases, the progeny concentration in the air charged into the ²²⁰Rn chamber from the inlet decreases. However, the progeny-complement amount per unit time increases; thus, the progeny-concentration level can be effectively improved. Therefore, in addition to wind speed, the progeny-complement rate is a key parameter for regulating progeny concentration.

Fig. 12Full-screen View

(Color online) Variation of a ²¹²Pb concentration and b ²¹²Bi concentration with progeny-complement rate at a fixed wind speed of 0.05 m/s

(3) Effect of progeny-complement rate on uniformity

Figure 11a in Sect. 3.2.2 qualitatively show that as the progeny complement increases, the uniformity of the progeny-concentration change in the ²²⁰Rn chamber is not obvious. In addition, Fig. 13 quantitatively shows the relationship between the uniformity of the progeny concentration and progeny-complement rate. Based on the data presented in Fig. 13, the RSD of the ²¹²Pb concentration ranges from 8.90% to 10.51% in sampling area S1, whereas the RSD of the ²¹²Bi concentration ranges from 7.55% to 7.87% in the same area. In sampling area S2, the RSD of the ²¹²Pb concentration varies between 8.40% and 8.92%, whereas the RSD of ²¹²Bi concentration ranges from 6.97% to 7.45%. The RSD value of the progeny concentration exhibits little change and was relatively stable. Thus, we can conclude that the impact of the progeny-complement rate on progeny-concentration uniformity is negligible compared with the influence of wind speed.

Fig. 13Full-screen View

　RSD of progeny with progeny-complement rate at a fixed wind speed of 0.05 m/s