Theory of radon diffusion and exhalation in porous media

One-dimensional steady-state diffusive migration of radon in a porous medium can be described using the following equation: $D\frac{{\partial }^{2}C}{\partial x^2}-{\lambda }_{\text{Rn}}C+\frac{\alpha }{\eta }=0,$ (1) where C is the radon concentration in the air, Bq m^-3; x is the vertical depth of the porous medium, m; D is the diffusion coefficient of radon in the porous medium, m² s^-1; η is the porosity, %; α is the free radon production rate, Bq m^-3 s^-1; and λ_Rn is the decay constant of radon, 2.1×10^-6 s [42].

The boundary conditions in Eq. (1) are C(0)=C_a and ${\partial C/\partial x|}_{x=h}=0$ at the top and bottom of the porous medium, respectively (Fig. 1). The radon concentration in the boundary air was significantly lower than that in the porous medium, making it possible to assume C_a=0 Bq m^-3. Upon substituting of $\sqrt{D/\lambda }$ for L, the analytical solution of Eq. (1) is: $C\left(x\right)=\frac{\alpha }{\lambda \eta }\frac{\left(e^{\frac{2h}{L}}-e^{\frac{2h-x}{L}}-e^{\frac{x}{L}}+1\right)}{\left(e^{\frac{2h}{L}}+1\right)}.$ (2)

Fig. 1Full-screen View

One-dimensional model of radon diffusion in porous medium

The radon exhalation rate for diffusion is then theoretically deduced: $E_{\text{a}-\text{d}}={\eta D\frac{\partial C}{\partial x}|}_{x=0}=\alpha L\frac{\left(e^{\frac{2h-x}{L}}-e^{\frac{x}{L}}\right)}{\left(e^{\frac{2h}{L}}+1\right)},$ (3) where E_a-d is diffusive radon exhalation rate, Bq m^-2 s^-1.

The closed-loop method and device for radon exhalation rate measurement

The closed-loop method is commonly used in the measurement of radon exhalation rate, which involves the accumulation of radon in the chamber (Fig. 2). Radon exhaled from the surface of the porous medium flowed into a radon detector (e.g., RAD7 from Durridge Company Inc., USA) driven by an air pump and returned to the chamber through the flow-in vent tube.

Fig. 2Full-screen View

Connection diagram of the system

Inside the accumulation chamber, the accumulation of radon is described as: $\frac{\text{d}C}{\text{d}t}=\frac{E{S}_{\text{a}}}{V}-{\lambda }_{\text{Rn}}C-{\lambda }_{\text{b}}C-{\lambda }_{\text{l}}C,$ (4) where t is the accumulation time, s; E is the radon exhalation rate, Bq m^-2 s^-1; S_a is the exhaled area of the porous medium, m²; λ_b is the back-diffusion rate of radon in the chamber, s^-1; λ_l is the leakage rate of radon in the chamber, s^-1; V is the volume of the chamber, m³. λ_e is the effective decay constant, a sum of radon decay, back-diffusion, and leakage rate (λ_e=λ_Rn+λ_b+λ_l), s^-1.

The boundary conditions in Eq. (4), C(0)=C₀. Upon substitution of C₀ for 0 Bq m^-3 in approximation, the curve fitting solution for Eq. (4) is $C\left(t\right)=\frac{E{S}_{\text{a}}}{{\lambda }_{\text{e}}V}\left(1-e^{-{\lambda }_{\text{e}}t}\right).$ (5)

The linear fitting solution for Eq. (4) is^[6] $C\left(t\right)=\frac{E{S}_{\text{a}}}{V}t.$ (6)

In subsequent calculations, the fitted radon exhalation rates of E_f-exp and E_f-lin were obtained by curve fitting (Eq. (5)) and linear fitting (Eq. (6)), respectively, with the criterion R² > 0.9: The calculation of R² involves the values of the residual sum of squares (RSS) and the total sum of squares (TSS), the equations of which are as follows: $R=1-\frac{\text{RSS}}{\text{TSS}}=1-\frac{{\displaystyle \sum _{i=1}^n{\left(y_i-f_i\right)}^2}}{{\displaystyle \sum _{i=1}^n{\left(y_i-\overline{y}\right)}^2}},$ (7) where n is the total sum of the data points, y_i is the actual data point, f_i is the value obtained from the fitted curve, and $\overline{y}$ is the mean value of all data points.

Measuring device and geometric model

The three-dimensional geometry (Fig. 3) for the model of the accumulation chamber (AC) had internal dimensions of 30 cm×30 cm×10 cm with a thickness of 4 mm, inserted into uranium mill tailings (a height of 50 cm filled in a container of 100 cm×100 cm×60 cm). Two vent tubes (Ø6 mm) were designed on the diagonal of the chamber, with 3 cm from each hole to the inner wall of the chamber. Besides the AC model, a reference (R) model, a container dimension of 100 cm×100 cm×60 cm, was established to determine the reference radon exhalation rate by the closed-loop method for uranium mill tailings (a height of 50 cm) filled in the container being sealed but with two holes (Ø6 mm, 50 cm apart) on the top cover.

Fig. 3Full-screen View

(Color online) Three-dimensional geometric model of the accumulation chamber from (a) top (b) lateral, and (c) overall views. (Color figure online)

Considering the disturbance of the porous medium caused by the jet from the inlet vent tube, two vent tube schemes (open-ended and half-open) were designed. The first type is a vent tube with an open end. The other was sealed at the bottom end of the vent tube with four holes (Ø2mm, 2 cm apart, and a normal direction of 90° of two adjacent holes) open on the tube wall to avoid the direct impact of the jet from the vent tube on the porous medium.

The chamber had a thickness of 4 mm, and its volume was subtracted from that of the porous medium upon insertion.

The geometry was discretized into polyhedral and hexahedral elements using Fluent Meshing to reduce memory and computing time. The mesh was refined in the region (e.g., the gas–solid interface and local opening of the vent tube) where rapid changes in the radon concentration and air flow were expected, as shown in Fig. 4.

Fig. 4Full-screen View

(a) Discrete mesh from overall view. (b) The sectional plane of discrete mesh where vent tubes are included

Governing equations

The gas flow, which is regarded as incompressible, in the chamber and porous medium followed the laws of conservation of mass and momentum. Following the law of conservation of mass, the mass conservation equation is described by [43] $\nabla \cdot \overrightarrow{v}=0,$ (8) where $\overrightarrow{v}$ is the physical velocity vector, m s^-1.

Following the law of momentum conservation, the momentum conservation equation is described by [44] $\frac{\partial {\rho }_a\overrightarrow{v}}{\partial t}+\nabla \cdot \left({\rho }_a\overrightarrow{v}\overrightarrow{v}\right)=-\nabla p+{\rho }_a\overrightarrow{g}+\nabla \cdot \left(\overline{\overline{\tau }}\right)-\overrightarrow{F},$ (9) where ρ_a is air density, kg m^-3; p is the static pressure, Pa; $\overrightarrow{g}$ is the gravitational vector, m s^-2; and $\overrightarrow{F}$, the source term of external body forces, N m^-3, describes the viscosity of the porous medium, and is only present in the porous zone. The laminar flow is described as follows: $\overrightarrow{F}=\eta \overrightarrow{v}\frac{\mu }{K},$ (10) where K is permeability, m²; μ is the dynamic viscosity, Pa s; η is the porosity of medium.

The viscous stress tensor ($\overline{\overline{\tau }}$) is described by^[45] $\overline{\overline{\tau }}=\left(\mu +{\mu }_t\right)\left(\nabla \overrightarrow{v}+\nabla {\overrightarrow{v}}^T\right),$ (11) where μ_t is the turbulent viscosity, Pa s, defined by ${\mu }_{\text{t}}=\rho C_{\mu }\frac{k^2}{\epsilon }$ (C_μ is a constant; k is the turbulent kinetic energy, m² s^-2; and ε is the turbulent dissipation rate, m² s^-3).

The governing equation for radon migration is expressed as $\frac{\partial C}{\partial t}=D\frac{{\partial }^{2}C}{\partial x_{{}_i}^2}-v_i\frac{\partial C}{\partial x_i}+\frac{S_{Rn}}{\eta },$ (12) $v_i=\frac{K}{\eta \mu }\nabla P_i,$ (13) where D is the radon diffusion coefficient (m² s^-1); v_i corresponds to the physical velocity of convection in the x, y, and z directions (m s^-1); S_Rn is the source term (Bq m^-3 s^-1); P_i corresponds to the pressure in the x, y, and z directions (Pa).

In a porous medium, S_Rn is determined by the source and decay terms ($S_{Rn}=-\lambda \eta C_{Rn}+\alpha$), and D=D_m. In the accumulation chamber, the porosity is set to 1; S_Rn is determined by the decay term ($S_{Rn}=-\lambda \eta C_{Rn}$) and D=D_m+D_t (D_t is the turbulent diffusion coefficient, m² s⁻¹, defined by $\frac{{\mu }_t}{\rho S{c}_t}$ where turbulent Schmidt number (Sc_t) is 0.7 ^[46]; D_m is molecular diffusion coefficient of radon in air, 1.05×10^-5 m² s^-1).

Parameters for simulation

The simulated scenarios involved reference (R) and accumulation chamber (AC) models. In the AC model, the effects of the vent tube structure and permeability of the porous medium on radon exhalation were analyzed. Furthermore, the effects of insertion depth (H1/H3/H5 corresponding to insertion depths of 1/3/5 cm), flow rate (0.5/1/2 L min^-1), and effective diffusion coefficient (D1/D2/D3/D4 corresponding to 2.77×10^-7/7.77×10^-7/2.77×10^-6/7.77×10^-6 m² s^-1) were also tested. The effective diffusion coefficient of radon in a porous medium is defined as [42] $D_e=\eta D=\eta \tau D_m$ (14) where τ is the tortuosity factor, 0.66 (dimensionless) [42, 47].

In the R model, the radon exhalation rate obtained from the reference container is used to evaluate the rate measured using the AC model. The design of the R model helps improve the reliability of the simulation, considering that the reference radon exhalation rate is commonly used in laboratory tests to evaluate the radon exhalation rate obtained from the AC model.

The free radon production rate (α) is defined by^[42] $\alpha =\lambda {\rho }_s{A}_{Ra}{E}_{Rn},$ (15) where ρ_s is the density of the porous medium, kg m^-3; A_Ra is the activity concentration of radium, Bq kg^-1; and E_Rn is the radon emanation coefficient, dimensionless.

Radium activity concentration of uranium mill tailings commonly ranges from less than 5 kBq kg^-1 to as high as 10 kBq kg^-1 [48], with the emanation coefficient ranging from 0.1 to 0.35[42], dry-bulk density of 1800 kg m^-3, and averaged porosity of 0.4 measured using the drainage method for samples in the laboratory.

Laminar flow in a porous medium follows Darcy’s law, and permeability, which is an intrinsic property of a porous medium, is defined by the Kozeny-Carman empirical equation [49] $K=\frac{D_p^2}{180}\frac{{\eta }^3}{{\left(1-\eta \right)}^2},$ (16) where D_p is the average grain size (mm) and K is the permeability, m².

After sieving the tailings sand in the laboratory, D_p was determined to be 0.5 mm and K was calculated to be 2.47×10^-10 m². The physical parameters for the simulations were determined, as listed in Table 1.

The finite volume method (FVM) was adopted for numerical calculations. The k-ε turbulence model was set for the chamber zone due to confined jet from the four openings on the surface of vent tube, whereas the flow in the porous media was laminar model. The empirical constants in the k-ε turbulence model equations [45] are as follows: C_μ = 0.09, C_1ε = 1.44, C_2ε = 1.92, σ_k = 1.00, and σ_ε = 1.30.

To simplify the model, the following assumptions were made: the leakage of the container was not considered, and the porous medium was homogeneous, with air as an incompressible gas, ignoring the impact of gravity and temperature.

Initial value and boundary settings

The radon concentration distribution of the porous medium in the steady state was calculated before simulating the radon accumulation in the measurement, in which the radon concentration in the air was set to 0 Bq m^-3. Transient mode was adopted upon the measurement, with initial radon concentration at 0 s in the chamber set at 0 Bq m^-3. The detailed boundary settings are listed in Table 2.

Mesh independence

The three-dimensional geometric model was discretized in coarse, medium, and refine (0.5 M, 1.2 M, and 2.3 M cells, respectively) mesh and the radon exhalation rate for each mesh was further calculated as 1.4676593, 1.4674347, 1.4674092 Bq m^-2 s^-1, respectively, shown in Fig. 5. The decline in the radon exhalation rate slowed with an increase in the sum of cells. The criterion for discretizing the mesh is to minimize the error between the numerical and analytical results while maintaining the calculation at a lower cost. Thus, a medium mesh was used in subsequent simulations.

Fig. 5Full-screen View

Mesh sensibility analysis

The calculation is considered converged with the residual less than 10^-10 and the differential value of radon exhalation rate of two consecutive iterations less than 10^-7.

Effect of vent tube structure on radon concentration

Prior to the experimental simulation, the vent tube structures of the chamber were designed and reconstructed. Two schemes were established, the vent tube with end opening for Case 1 and the vent tube with four side wall openings (Ø2mm) on the wall for Case 2 (both under 1 L min^-1, 1×10^-9 m²). The growth curves of the radon concentrations for these two cases are shown in Fig. 6a. In addition, the permeability in Case 2 was changed to 1×10^-11 m² (Case 3 with four openings) to test the effect of permeability.

Fig. 6Full-screen View

(a) Growth curves of C_s-avg and C_v-avg in the first 600 s (case 1 and case 2), (b) the full growth curve of radon concentration for the R model, and (c) exhibition of all sorts of radon exhalation rate

Prior to reconstruction (case 1), C_s-avg was larger than C_v-avg in the first 100 s, with the gap narrowing and remaining for the rest of the time. However, C_v-avg was reckoned to grow faster than C_s-avg in view of the initial radon concentration being equal to 0 Bq m^-3, implying defects in the vent tube structure in Case 1. Upon modification (Case 2), C_s-avg increased more slowly than C_v-avg and the discrepancy remained constant. C_v-avg, representing the actual trend of radon concentration in the chamber, was used for subsequent simulations to study the migration of radon in the chamber.

The radon concentration contours for Cases 1 and 2 at 600 s are shown in Fig. 7. The streamlines in Case 1 show that the jet from the vent tube enters the porous medium and then returns to the chamber, directly affecting the radon concentration distribution in the porous medium, thus leading to the inhomogeneity of the radon concentration in the chamber. The streamlines in Case 2 mainly staying in the chamber indicate few impacts on the radon concentration distribution in the porous medium, in which the maximum, minimum, and average radon concentration in the chamber for Case 2 is 8676.49, 6208.08, and 7089.7 Bq m^-3, respectively, while those for Case 1 are 9473.19, 4444.73, and 6575.9 Bq m^-3. The NUI for Cases 2 and 1 were 34% and 76%, respectively, revealing that the uniformity of the radon concentration distribution in case 2 significantly improved.

Fig. 7Full-screen View

(Color online) The streamlines and distribution of the radon concentration along the diagonal section of the accumulation chamber at 600 s for (a) Case 1, (b) Case 2, and (c) Case 3(Color figure online)

Permeability effect on radon exhalation rate

The E_n on the surface covered by the chamber at 600, 1200, and 1800 s were 1.03988, 0.84632, and 0.69968 Bq m^-2 s^-1 for Case 2, and 1.03943, 0.84595, and 0.69937 Bq m^-2 s^-1 for Case 3. The approximate similarity of the radon exhalation rate as well as the contours and streamlines for Cases 2 and 3 imply that the effect of permeability on the closed-loop measurement of the radon exhalation rate can be neglected. Therefore, the permeability is set to 1×10^-10 m² in the subsequent simulations.

Reference radon exhalation rate

For assessment of radon exhalation rate under AC model, reference exhalation rate (E_ref) under R model was calculated. The C_v-avg growth curve from 0 to 1.2×10⁷ s was obtained by radon accumulation until saturation in the sealed container (Fig. 6b). The initial (E_a), fitted (E_f-exp), and total transient radon exhalation rates (E_n) were calculated, as shown in Fig. 6c. As the transient radon exhalation rate for convection remains at 0 Bq m^-2 s^-1 in approximation, the effect of convection on the exhalation rate measurement can be neglected. Thus, the total transient radon exhalation rate was determined by the transient radon exhalation rate for diffusion (E_n-d), which is represented by the overlap of E_n and E_n-d in Fig. 6c.

The fitted radon exhalation rate characterized the average exhalation for a given duration. Upon saturation for radon accumulation, Ef-exp was approximately equal to E_n, suggesting the consistency of the fit, as well as the back-diffusion phenomenon, with E_f-exp (0.520835 Bq m^-2 s^-1) being only 35.59% equivalent to E_a (1.463235 Bq m^-2 s^-1).

Flow rate and insertion depth effect on radon exhalation rate

The increased insertion depth of the chamber has been experimentally proven to reduce leakage during measurements. The effects of insertion depth (H1/H3/H5) and flow rate (0.5/1/2 L min^-1) on the radon concentration and exhalation rate are analyzed in this section. The radon distribution in the accumulation chamber at 1800 s is shown in Fig. 8. At a low flow rate (0.5 L min^-1), the radon concentration distribution is subject to notable spatial variation, while it gradually becomes uniform as the flow rate increases. In addition, lateral diffusion is responsible for the average concentration in the chamber for H1 being lower than that for H5, implying that the effect of insertion depth on the measurement cannot be neglected.

Fig. 8Full-screen View

(Color online) Radon concentration distribution in the accumulation chamber at 1800s (flow rate with 0.5/1/2 L min^-1 and insertion depth with 1/3/5 cm) (Color figure online)

Excluding the points in the initial 60 s, the average value for every 60 s was calculated as the concentration at the midpoint, and further fitted to obtain the exhalation rate. For example, the average radon concentration from 61 s to 120 s was calculated as the value at 90 s. The duration for curve fitting included 600 s, 720 s, 900 s, 1020 s, 1200 s, 1500 s, and 1800 s, while it was 240 s, 360 s, 480 s, 600 s, 720 s, 840 s, 960 s, 1080 s, 1200 s, 1320 s, 1440 s, and 1560 s for linear fitting.

E_f-exp and E_f-lin were obtained by fitting the points to the curve and linearly fitting the equations using different time ranges, as shown in Fig. 9. A long duration (1800 s) is unfavorable for curve fitting, the discrepancy of which with E_a enlarged (0.9% to 3.3% increase in deviation for various insertion depths compared with 600 s). In contrast, the linear fitting value declined significantly with increasing duration (14.7% to 22.8% increase in deviation for various insertion depths from 240 s to 1800 s).

Fig. 9Full-screen View

The radon exhalation rate for the fitting value (E_f-exp in curve fitting and E_f-lin in linear fitting) under various duration, the transient value in numerical simulation (E_n), and the analytical solution (E_a) (flow rate with 0.5/1/2 L min^-1, insertion depth with 1/3/5 cm, and permeability with 1×10^-10 m²)

Both E_f-exp and E_f-lin increased with the flow rate for a fixed insertion depth closer to E_a. Throughout the accumulation duration, E_n maintained a decreasing trend and was only 48–59% (for various insertion depths) of E_a at 1800 s, indicating that E_f is an averaged value for a certain duration and is unable to represent E_n at a specific moment. Moreover, E_f was more susceptible to back-diffusion for longer fitting durations.

For an in-depth analysis of the insertion depth effect (1 L min^-1), E_f-exp and E_f-lin, as well as the effective decay constant (λ_e), were calculated, shown in Fig. 10. λ_e (2.23582×10^-6 s^-1) is comparable to λ_Rn during closed-loop measurement in R model, in view of the assumption of radon-impermeable wall of the sealed container. A significant decrease of 27.2%–34.2% is shown for λ_e with the increased insertion depth, implying that the leakage was effectively reduced in line with the previous experimental observations.^[21]

Fig. 10Full-screen View

(Color online) (a) Effective decay constant (λ_e) for H1, H3, H5, and R model using curve fitting (1 L min^-1, λ_Rn=2.1×10^-6 s^-1). E_f-to-E_a ratios for H1, H3, H5, and R model using (b) curve fitting (E_f-exp) and (c) linear fitting (E_f-lin) (1 L min^-1)

Furthermore, the ratios of E_ref and E_f to E_a were calculated (Fig. 10b and 10c). The ratio for the curve fitting (E_f-exp) is found to be positively correlated with λ_e. Increased insertion depth results in lower λ_e and the ratios, suggesting an obvious back-diffusion effect. Unlike curve fitting, the ratios for linear fitting showed slight growth. Specifically, the exhalation rates of curve fitting using 1800 s and linear fitting using 480 s for H1 were 0.89952 and 0.81364 Bq m^-2 s^-1 respectively, while they were 0.89241 and 0.83551 Bq m^-2 s^-1 for H5. The effect of increased insertion depth on lower leakage is demonstrated by linear fitting (i.e., the fitted exhalation rate is closer to E_a obtained by the increased insertion depth), whereas curve fitting for longer durations is more affected by back-diffusion.

In general, a uniform mixture of radon inside the chamber facilitates the radon exhalation rate measurement as the flow rate increased from 0.5 to 2 L min^-1. Although the increased insertion depth helps lower the leakage rate, the side effect is the decreased accuracy of the curve fitting owing to back diffusion.

Validation of the numerical model

The reliability and accuracy of the numerical model are validated. In previous experiments, Gutiérrez-Álvarez^[21] deployed two reference boxes (RB1 and RB2) to obtain reference radon exhalation rate. Cylindrical (CX, where X denotes the insertion depth in centimeters) and rectangular (referred to as VX, where X has the same definition) accumulation chambers were designed to evaluate the influence of insertion depth. V10, C0, C3, C6, C9 were tested on RB1, whereas V10, C0, C3, C6, C9, C12, and C14 were tested on RB2.

The effective decay constant measured on RB1 (approximately 4.88×10^-6 s^-1) is 62% greater than that on RB2 (approximately 3×10^-6 s^-1) measured by RAD7, whereas it is 58% greater by Alphaguard, implying that the container of RB2 is better sealed during measurement. The effective decay constant for the accumulation chamber peaked at an insertion depth of 0 cm (C0), approximately 1.88×10^-4 s^-1 on RB1 and 2.5×10^-4 s^-1 on RB2. As the insertion depth increased, the effective decay constant declined by 38%–52% to almost 2.9×10^-5 s^-1, indicating a lower leakage rate. The variation in the effective decay constant for the reference boxes (RB1 and RB2) and the accumulation chamber fit well with the simulated results shown in Fig. 10a. The effective decay constant decreased by 27.2%–34.2% with increasing depth from 1 to 5 cm.

Furthermore, the ratios of the curve-fitted radon exhalation rate to the reference values for C0, C3, and C6 on RB2 were chosen for comparison with those for H1, H3, and H5 because the airtightness of RB2 was better than that of RB1. The ratios for C0, C3, and C6 were 1.65, 1.22, 1.1 measured by Alphaguard, and 1.18, 1.04, 1.02 by RAD7, respectively. An evident downward trend was observed for both Alphaguard and RAD7, which agrees well with the trends of the ratios for H1, H3, and H5 by exponential fitting in Fig. 10b.

Although the reference radon exhalation rate (equivalent to the value obtained from the R model in this study) was used as an intrinsic value for comparison, the potential discrepancy caused by back diffusion should be noted. Inaccurate reference values can potentially render comparisons unreliable.

In summary, the simulation not only validates the variation in the experimental data but also further reveals the positive correlation between the back-diffusion phenomenon and the increase in insertion depth.

Diffusion coefficient effect on radon exhalation rate

In view of the inevitable back-diffusion phenomenon mentioned in Section 5.4 (i.e., the curve-fitted radon exhalation rate declines with an increase in the insertion depth and with a lower effective decay constant), the effect of the radon diffusion coefficient for the porous medium on E_a, E_f, and E_n was additionally analyzed. According to Eq. (3), the radon exhalation rate is positively correlated with the effective diffusion coefficient, explaining why the E_a (1.2081 Bq m^-2 s^-1) for D1 scenario (the lowest effective diffusion coefficient among the four scenarios) is apparently lower than that for the other three scenarios (the former equivalent to 81.3%–87.7% of the latter in Fig. 11). Lower radon exhalation needs longer time for the radon concentration to reach saturated, evidenced by lower value of λ_e for D1 (Fig. 12a). Increasing the effective diffusion coefficient causes E_f or E_n to deviate from E_a, suggesting that the effective diffusion coefficient is an important factor in the back-diffusion phenomenon.

Fig. 11Full-screen View

The radon exhalation rate for the fitting value (E_f-exp in curve fitting and E_f-lin in linear fitting) under various duration, the transient value in numerical simulation (E_n), and the analytical solution (E_a) (D1/D2/D3/D4 corresponding to 2.77×10^-7/7.77×10^-7/2.77×10^-6/7.77×10^-6 m² s^-1, 1 L min^-1, 1×10^-10 m², and 1 cm insertion depth)

Fig. 12Full-screen View

(Color online) (a) Effective decay constant (λ_e) obtained by applying curve fitting for D1, D2, D3, and D4 scenario (1 L min^-1, λ_Rn=2.1×10^-6 s^-1). Ratios of E_f to E_a for D1, D2, D3, and D4 scenario using (b) curve fitting (E_f-exp) and (c) linear fitting (E_f-lin) (1 L min^-1)

The ratios of E_f to E_a using various fit durations for the four effective diffusion coefficient scenarios are shown in Fig. 12b and 12c. The ratios decrease with the increasing effective diffusion coefficient, going from 0.98 to 0.84 for curve fitting (E_f-exp, 1800 s) and from 0.96 to 0.67 for linear fitting (E_f-lin, 480 s). The deviations between the curve (1800 s) and linear (480 s) fitting were 2.08%, 3.88%, 9.55%, and 20.68% for D1, D2, D3, and D4, respectively, making the exhalation rate for a lower effective diffusion coefficient that can be calculated approximately using linear fitting in a time-saving manner to replace curve fitting.