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Sensitivity analysis of an improved method for measuring the radon diffusion coefficient of porous materials
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Sensitivity analysis of an improved method for measuring the radon diffusion coefficient of porous materials W.H. van der Spoel a , M. van der Pal b a Faculty of Civil Engineering and Geosciences, Section Building Engineering, Delft University of Technology,
Stevinweg 1, 2628 CN Delft, The Netherlands b Faculty of Architecture, Building and Planning, Group Building Physics, Eindhoven University of Technology,
PO Box 513, 5600 MB Eindhoven, The Netherlands
Several techniques are available for determining the radon diffusion coefficient of porous materials. The common approach is to position a porous sample between two compartments. In one of the chambers a known high radon activity concentration is introduced while the radon concentration in the other chamber is measured. A steady state or transient analysis may be utilised. In this paper, a transient analysis is outlined for two experimental conditions in a cylindrical geometry, complemented with a rigorous treatment of the counting uncertainty in a continuous radon measurement. The sensitivity and accuracy of a least-squares regression to calculate the diffusion coefficient from artificially generated data is discussed.
1. Introduction Diffusion of radon in building materials is a major driving force for radon exhalation into buildings with relatively low radon concentrations. The common approach to measure this material parameter in the laboratory is to position a porous sample between two compartments. In one of the chambers a known high radon activity concentration is introduced while the radon concentration in, or flux into, the other (ventilated) chamber is measured. In case radon decay in the sample can be ignored and for steady-state conditions, this technique concerns a direct measurement of the diffusion coefficient. A possible drawback is that one has to wait for equilibrium. A quicker time-dependent method was introduced by Nielson et al. [1] and Zapalac [2]. In the experiments of Nielson et al., a known high radon activity concentration is introduced in one of the chambers while the radon concentration in the ‘detection’ chamber is monitored continuously using a scintillation cell. By fitting an analytical expression, corrected for RADIOACTIVITY IN THE ENVIRONMENT VOLUME 7 ISSN 1569-4860/DOI 10.1016/S1569-4860(04)07092-5
© 2005 Elsevier Ltd. All rights reserved.
Sensitivity analysis of an improved method for measuring the radon diffusion coefficient
741
the volume of the detection chamber, for the time-dependent alpha activity in the detection chamber, the diffusion coefficient is inferred. In the technique introduced by Zapalac, the integrated radon flux into the (ventilated) second chamber is measured between contiguous time intervals and, as decay of radon is ignored, may only be applied for thin samples. Several others have used time-dependent methods. Søgaard-Hansen and Damkjær [3] made the assumption of well-mixed chambers, allowing an exact analytical solution including decay in the sample. Tsai and Hsu [4,5] applied a weighted fit to the data obtained with the analytic method of Zapalac. The technique has also been used for radon diffusion through membranes. Wójcik et al. [6] made an analysis that included radon decay in the membrane. In this paper, we present and analyse two different time-dependent methods in a cylindrical geometry, using continuous measurements of the alpha activity in both chambers. The advantage of using a cylindrical sample is that leakage along the sample can be more easily avoided than for most other geometries. We outline a method to estimate the counting uncertainty of such a continuous measurement. These uncertainties are required to perform a weighted leastsquares regression. Finally, the strengths and weaknesses of the two methods are compared based on artificially generated data.
2. Model description Consider a hollow cylindrical sample with inner radius r1 and outer radius r2 . If we ignore radon generation and advective transport in the sample and assume that the pore–air radon concentration C (Bq m−3 ) only varies in the radial direction, the macroscopic radon transport equation may be written in polar coordinates: ∂C ∂ 2 C De ∂C = De 2 + − λC, ∂t r ∂r ∂r
(1)
where t is time (s), r is the polar coordinate (m), λ is the decay constant (s−1 ) and De is the effective diffusion coefficient (m2 s−1 ). We may write D = βDe ,
(2)
where D is the bulk diffusion coefficient1 and β a coefficient [7,8], also called the “partitioncorrected porosity”, that accounts for partitioning of radon in the air, water and adsorbed phase: β = ε(1 − m + Lm) + ρka ,
(3)
where ε is the material porosity, m the fraction of moisture saturation of the pores, L the Ostwald coefficient for partitioning between air and water, ρ the material density (kg m−3 ) and ka the radon surface adsorption coefficient (m3 kg−1 ). For dry materials and no adsorption, β equals the porosity ε. Two types of experimental conditions are considered. In the first type (1), a radon source is placed in the inner compartment with activity S1 (Bq). Assuming a well-mixed leak-tight 1 A reverse definition is also used.
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compartment, the boundary condition at r = r1 is ∂C ∂C (4) = 2πr1 hD − λV1 C + λS1 , r = r1 , t > 0, ∂t ∂r where V1 is the volume of the inner compartment and h the height of the sample. The terms at the right-hand side account for a diffusive flux through the inner surface, radon decay and production, respectively. Since a scintillation cell is attached to the compartment, the volume V1 is larger than the inner volume of the sample (πr12 h). Similarly, the boundary condition at r = r2 is written as ∂C ∂C = −2πr2 hD − λV2 C, r = r2 , t > 0. V2 (5) ∂t ∂r As initial condition we have V1
C = 0,
r1 r r2 , t = 0.
(6)
In the second type (2), the boundary condition describes an (often used) experiment in which the radon concentration in the inner compartment is constant: C = C0 ,
r = r1 , t > 0.
(7)
The other conditions (5) and (6) still apply for this experiment. Note that since C = 0 at t = 0, the radon concentration in the inner compartment is described by a step function. The differential equation (1) with appropriate boundary conditions was solved using Laplace transformation. Basically, the radon concentration as a function of time in each of the compartments can be written as an infinite series: C(t) = b0 +
∞
bi exp(ai t).
i=1
Note that for the first type experiment, C(r, t = 0) = 0, thus type, this only holds for r > r1 , since C(r1 , t > 0) = b0 .
(8) ∞
i=1 bi
= −b0 . For the second
3. Experimental premises 3.1. Set-up In an experiment of the first type, a radon source is placed in the inner compartment. Before the start of an experiment, the set-up should be flushed for some time (> 3 h) with radonpoor ambient air to remove radon and its progeny. The airflow is preferably forced through the porous sample. After stopping the flow, the alpha-activity in both compartments is measured with attached scintillation cells that register the number of pulses due to alpha-decay in contiguous time intervals of typically an hour. In an experiment of the second type, a constant radon concentration in the inner compartment may for example be obtained by placing a radon source in an external closed canister with a large volume ( Vset-up ) for more than about a month to establish equilibrium alphaactivity. The set-up is similarly flushed as in the previous experiment, where after the large canister is connected to the inner compartment and air is continuously circulated between these volumes.
Sensitivity analysis of an improved method for measuring the radon diffusion coefficient
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3.2. Detection efficiencies The count rate measured with the two scintillation flasks depends on the alpha activity in the measuring volume, determined by the concentrations in the scintillation flask of 222 Rn and its alpha-emitting progeny. It therefore depends on the efficiencies η1 , η2 and η4 for detecting an alpha particle emitted in the decay of 222 Rn, 218 Po and 214 Po, respectively. The relative detection efficiencies η2 /η1 and η4 /η1 are usually > 1. In this paper, we use η2 /η1 = 1.15 and η4 /η1 = 1.33. The efficiency η1 for detecting an alpha from radon decay is about 0.5 for a common scintillation cell.
4. Error analysis The analytical expression for the number of counted pulses in the time intervals is fitted to the data using a weighted least-squares method. As a result, the statistical error in the number of counted pulses in each interval is required. Based on the work of Aldenkamp and Stoop [9] and Inkret et al. [10], we have derived the variance of the net number of counted pulses in the case that the radon concentration is described by equation (8). A statistical analysis taking account of correlations in the decay process has also been utilised by Sonoc and Sima [11]. 4.1. Expectation and variance of counted pulses Consider the experiment in which N0 radon atoms (and no progeny) are present at some time t in a scintillation cell and in which the detected pulses due to alpha decay of radon and its progeny are counted from time t1 to t2 (t2 > t1 > t). Let pj (t1 − t, t2 − t) denote the probability of detecting j (j = 0, 1, 2, 3) pulses during the counting interval, originating from a radon atom present at t. Then the expectation value of the number of detected pulses E(X) is, omitting the dependency on t, t1 and t2 for brevity: E(X) = N0 (p1 + 2p2 + 3p3 ) ≡ N0 pE , where we have defined pE ≡ p1 + 2p2 + 3p3 . The variance is given by [10] Var(X) = N0 pE − pE2 + 2p2 + 6p3 ≡ N0 pVar .
(9)
(10)
The alpha-activity of radon and its daughter atoms in the scintillation cell as a function of time are described by the Bateman equations. Integration of the activity of each nuclide over the counting interval from t1 to t2 gives the expected number of transitions in that interval: T1 transitions from 222 Rn take place, T2 from 218 Po to 214 Pb and T4 from 214 Po to 210 Pb. The expected number of detected pulses in the counter interval can be written as E(X) = η1 T1 + η2 T2 + η4 T4 = N0 pE ,
(11)
from which pE may be calculated. To calculate pVar we need the probabilities p2 and p3 . The probability of detecting 2 pulses during the counting interval, originating from a radon atom present at t, consists of 5 terms:
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W.H. van der Spoel, M. van der Pal
(1) radon atoms present at t1 that decay twice and both decays are recorded; radon atoms present at t1 that decay thrice and (2) the first and second decay is recorded and the third is not, (3) the first and the third decay is recorded and the second is not, (4) the second and third decay is recorded and the first is not; (5) 218 Po atoms present at t1 that decay twice: p2 =
n2 1 T2 − T4 η1 η2 + T4 η1 η2 (1 − η4 ) + T4 η1 η4 (1 − η2 ) + T4 η2 η4 (1 − η1 ) = N0 N0 (12) + T4 η2 η4 ,
where n2 is the number of radon atoms present at t from which two alpha decays are recorded in the counting interval from t1 to t2 , T2 is the number of transitions from 218 Po to 214 Pb originating from radon atoms present at t1 , T4 is the number of transitions from 214 Po to 210 Pb originating from radon atoms present at t , and T is the number of transitions from 1 4 214 Po to 210 Pb originating from 218 Po atoms present at t . The probability of detecting 3 pulses 1 consists of one term: radon atoms present at t1 that decay thrice and all decays are recorded: η1 η2 η4 T4 n3 (13) = . N0 N0 To apply the above model to a continuously changing radon concentration C(t) as described by equation (8), one should consider the rate of change of the number of new radon atoms Nn (t) in a scintillation flask, for which we may write p3 =
dNn (t) dN (t) 1 dAr (t) (14) = + λN(t) = + Ar (t), dt dt λ dt where N(t) is the total number a radon atoms, and Ar (t) is the radon activity in the scintillation cell which equals Vm C(t) with V m the volume of the cell. In the counting interval from t1 to t2 , decays may be recorded from radon atoms (and their progeny) formed in the period between the start of an experiment at t = 0 and t1 , and radon atoms formed in the counting period from t1 to t2 . The expectation of the number of counted pulses is obtained by integration and consists of a contribution from both periods: t1 t2 dNn (t) dNn (t) E X(t1 , t2 ) = (15) pE (t1 − t, t2 − t) dt + pE (0, t2 − t) dt. dt dt 0 t1 For the variance Var(X) we similarly have t1 t2 dNn (t) dNn (t) pVar (t1 − t, t2 − t) dt + pVar (0, t2 − t) dt. (16) Var X(t1 , t2 ) = dt dt 0 t1 Both equations can be written in terms of the radon activity Ar (t), and thus C(t), using equation (14). In the second type diffusion experiment, we have a step-wise concentration change in the inner compartment at t = 0. The expectation of the number of counted pulses and the variance are then given by (15) and (16) added with, respectively: Vm C0 pE (t1 , t2 ) λ
and
Vm C0 pVar (t1 , t2 ) . λ
(17)
Sensitivity analysis of an improved method for measuring the radon diffusion coefficient
745
Fig. 1. Ratio of variance and expectation of number of counted pulses in 1 h intervals as a function of time for different radon detection efficiencies η1 .
4.2. Example Consider an experiment with a constant radon source in a closed compartment without radon and progeny initially. The radon concentration is described by C(t) = a(1 − exp(λt)) where a is a constant. For a diffusion experiment the time-dependent radon concentration is slightly different, but the main results are similar. The ratio of the variance and expectation of the number of counted pulses in 1 h contiguous intervals is shown in Fig. 1, where ratios η2 /η1 = 1.15 and η4 /η1 = 1.33 have been used. The time-dependent behaviour of the variance is mainly of concern at the start, for t < 5 h. Thereafter, the equilibrium value is nearly reached which is dependent on the radon detection efficiency η1 . In case η1 , η2 and η4 1, or for a counting interval of less than about 30 s, the number of detected pulses approaches a Poisson distribution, i.e. Var(X) ≈ E(X). 5. Sensitivity analysis Artificial measurement data have been generated for the two experimental conditions based on the models described in Section 2 for an experiment of 7 days and a counting interval of 3 h, giving a total of 112 data points (data from both scintillation cells are fitted simultaneously). A random error conforming to a Gaussian distribution with a calculated variance as outlined in Section 4 was added to each data point. The calculations were performed for a base case where h = 10.4 cm, r1 = 2.55 cm, r2 = 9 cm, V1 = 0.7 L and V2 = 2.8 L (of these, only V2 is varied in the sensitivity analysis). The volume of the scintillation cells is Vm = 0.3 L and the radon detection efficiency of both cells is set at η1 = 0.5. Further we assumed no background counts and the parameter β was set at 0.8. The regressions were carried out using the Levenberg–Marquardt method for four values of the effective diffusion coefficient (10−6 , 10−7 , 10−8 and 10−9 m2 s−1 ) and a source strength S1 = 1000 Bq for type-1 conditions. For type-2 conditions, the concentration C0 was set at either 104 Bq m−3 or 105 Bq m−3 . In all cases data points with less than 10 counts were omitted. 5.1. Type-1 experiment First we consider a regression with De and S1 as free parameters and no systematic error in the other quantities. For this regression, the diffusion coefficient is found with a relative
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W.H. van der Spoel, M. van der Pal
uncertainty between 0.04% (De = 10−8 m2 s−1 ) and 0.6% (De = 10−6 m2 s−1 ). These small uncertainties indicate that taking S1 as a free parameter does not pose any problem. However, if the radon detection efficiency of e.g. scintillation cell nr. 2 has a small systematic error of 2%, the fitted diffusion coefficient will contain a systematic error. For De = 10−6 m2 s−1 the systematic error is 20% and the relative uncertainty is 18%. For De = 10−7 m2 s−1 these values are 4% and 2%, respectively. For smaller values of De the errors are less than 1.5%. It is thus rather difficult to accurately determine the diffusion coefficient in this way when De > 10−7 m2 s−1 . The cause of this problem lies in the fact that D e is largely determined by the concentration difference between the inner and outer compartment, which is very sensitive to the relative efficiency of the scintillations cells. This unwanted sensitivity would be lower for a larger volume V2 . When V2 = 28 L instead of 2.8 L, the systematic error in De reduces to 5% for De = 10−6 m2 s−1 and even further for larger values of V2 . An optimum is found when V2 is several cubic metres. As an alternative to using a large outer compartment, one may also add the relative efficiency as a free parameter in the regression. In doing so, it was found that the effective diffusion coefficient in the range 10−6 –10−9 m2 s−1 is obtained without a systematic error and with a relative uncertainty < 1% when V2 = 2.8 L. The accuracy is better than 0.5% when V2 = 28 L. The accuracy of the method slightly degrades for V2 = 280 L. The above sensitivity analysis ignores the uncertainty in the partition-corrected porosity β, see equation (3). Since β depends on the porosity, water content, density and adsorption coefficient, these quantities should be accurately known to determine the diffusion coefficient. Alternatively, one could take β as a free parameter in the regression. If ηrel is accurately known, a regression with β, De and S1 as free parameter gives both β and De within 1.5% for De 10−7 m2 s−1 , β = 0.8 and V2 = 28 L (slightly worse for V2 = 2.8 L). For De = 10−6 m2 s−1 and V2 = 28 L, the accuracy is about 10%. In case ηrel is also a free parameter (giving a total of 4 free parameters), similar accuracies are obtained. 5.2. Type-2 experiment Similar tests were performed in case the concentration in the inner compartment is constant. For the regression with De and C0 as free parameters, it was found that the calculated diffusion coefficient is less sensitive to systematic errors of the relative efficiency. It was also observed that increasing the outer volume has a positive effect on the systematic error in De . From the experimental point of view it is, however, difficult to create a constant concentration in the inner compartment when the outer volume is large and the diffusion length is much larger than the sample thickness. In this respect, the outer volume should be small. The calculations have therefore been restricted to two volumes: V2 = 2.8 L and 28 L. For these values, the relative systematic error in De (10−6 m2 s−1 ) is about 4% when the systematic error in ηrel is 2%. The errors are smaller for lower values of De . Also for this experiment it is worthwhile to perform a regression with the relative efficiency as additional free parameter. It was found that in all cases (4 values of De ) with V2 = 2.8 L the calculated diffusion coefficient has no significant systematic error and the relative uncertainties are < 0.3%. With V2 = 28 L the results were slightly less accurate, but this is due to a 10 times lower value of C0 . With identical C0 the results are similar.
Sensitivity analysis of an improved method for measuring the radon diffusion coefficient
747
As mentioned, one could also take β as an additional free parameter in the regression. If ηrel is accurately known, a regression with β, De and C0 as free parameters gives both β and De within 1.5% for De 10−7 m2 s−1 , β = 0.8 and V2 = 2.8 L (somewhat worse for V2 = 28 L). In case ηrel is also a free parameter (giving a total of 4 free parameters), similar accuracies are obtained. 5.3. Discussion The above sensitivity analysis ignores any uncertainty in the sample dimensions (h, r1 and r2 ) and other experimental errors such as varying detection efficiencies during the experiment due to, e.g, temperature fluctuations, small leaks in the set-up, sample inhomogeneities, etc. In addition, the model assumes well-mixed compartments, which may not reflect reality. These effects could be the subject of a further sensitivity analysis. However, most of these errors can be minimised in a carefully prepared and executed experiment.
6. Conclusions Using the regression analysis for the first type experiment, a diffusion coefficient in the range 10−6 –10−9 m2 s−1 can accurately be determined using a 6.5 cm thick sample in an experiment of 7 days. For smaller values of De a thinner sample should preferably be used. Further, it was found that the outer compartment should not be too small, i.e., generally V2 > 10V1 , but not so large that the radon concentration becomes too low to be easily detected. Under these circumstances, a regression with S1 , De and the relative detection efficiency of the two scintillation cells as free parameters yields accurate results. This means that no information is required on the radon source strength and the relative detection efficiency of the scintillation cells to determine the radon diffusion coefficient. A regression with β, De , S1 and ηrel as free parameters gives both β and De within 1.5% for 10−9 De 10−7 m2 s−1 . For larger De , a thicker sample should be used to determine β and De simultaneously. The second type experiment has a slightly better performance and is less sensitive to the volume of the outer compartment. From the experimental point of view, and if β is also a free parameter, a small volume is however preferred. For very small values of De (< 10−9 m2 s−1 ) thinner samples should be used. Again, no information is required on the radon concentration in the inner compartment and the relative detection efficiency of the scintillation cells to determine the radon diffusion coefficient. It is concluded that both methods give similarly accurate estimates of De and β. The advantage of the first method lies in its experimental simplicity since there is no need to maintain a constant radon concentration in the inner compartment. Also, pressure differentials due to air circulation that may influence radon transport through the sample are excluded. The regression technique is especially shown to be a useful tool since it does not require data for the radon source strength, or concentration C0 , and the relative detection efficiency of the scintillation flasks.
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References [1] K.K. Nielson, D.C. Rich, V.C. Rogers, D.R. Kalkwarf, Report NUREG/CR-2875, US Nuclear Regulatory Commission, 1982. [2] G.H. Zapalac, Health Phys. 45 (1983) 377. [3] J. Søgaard-Hansen, A. Damkjær, Health Phys. 53 (1987) 455. [4] S.-C. Tsai, C.-N. Hsu, Geophys. Res. Lett. 20 (24) (1993) 2917. [5] C.-N. Hsu, S.-C. Tsai, S.-M. Liang, Appl. Radiat. Isot. 45 (8) (1994) 845. [6] M. Wójcik, Wlazło, G. Zuzel, G. Heusser, Nucl. Instrum. Methods Phys. Res. A 449 (2000) 158. [7] W.H. van der Spoel, Radon transport in sand, PhD thesis, TU Eindhoven, The Netherlands, 1998. [8] C.E. Andersen, Sci. Total Environ. 272 (2001) 33. [9] F.J. Aldenkamp, P. Stoop, Sources and transport of indoor 226 Radon – measurements and mechanisms, PhD thesis, Rijksuniversiteit Groningen, The Netherlands, 1994. [10] W.C. Inkret, T.B. Borak, D.C. Boes, Radiat. Prot. Dosim. 32 (1990) 44. [11] S. Sonoc, O. Sima, Radiat. Prot. Dosim. 45 (1992) 51.
Sensitivity analysis of an improved method for measuring the radon diffusion coefficient of porous materials W.H. van der Spoel a , M. van der Pal b a Faculty of Civil Engineering and Geosciences, Section Building Engineering, Delft University of Technology,
Stevinweg 1, 2628 CN Delft, The Netherlands b Faculty of Architecture, Building and Planning, Group Building Physics, Eindhoven University of Technology,
PO Box 513, 5600 MB Eindhoven, The Netherlands
Several techniques are available for determining the radon diffusion coefficient of porous materials. The common approach is to position a porous sample between two compartments. In one of the chambers a known high radon activity concentration is introduced while the radon concentration in the other chamber is measured. A steady state or transient analysis may be utilised. In this paper, a transient analysis is outlined for two experimental conditions in a cylindrical geometry, complemented with a rigorous treatment of the counting uncertainty in a continuous radon measurement. The sensitivity and accuracy of a least-squares regression to calculate the diffusion coefficient from artificially generated data is discussed.
1. Introduction Diffusion of radon in building materials is a major driving force for radon exhalation into buildings with relatively low radon concentrations. The common approach to measure this material parameter in the laboratory is to position a porous sample between two compartments. In one of the chambers a known high radon activity concentration is introduced while the radon concentration in, or flux into, the other (ventilated) chamber is measured. In case radon decay in the sample can be ignored and for steady-state conditions, this technique concerns a direct measurement of the diffusion coefficient. A possible drawback is that one has to wait for equilibrium. A quicker time-dependent method was introduced by Nielson et al. [1] and Zapalac [2]. In the experiments of Nielson et al., a known high radon activity concentration is introduced in one of the chambers while the radon concentration in the ‘detection’ chamber is monitored continuously using a scintillation cell. By fitting an analytical expression, corrected for RADIOACTIVITY IN THE ENVIRONMENT VOLUME 7 ISSN 1569-4860/DOI 10.1016/S1569-4860(04)07092-5
© 2005 Elsevier Ltd. All rights reserved.
Sensitivity analysis of an improved method for measuring the radon diffusion coefficient
741
the volume of the detection chamber, for the time-dependent alpha activity in the detection chamber, the diffusion coefficient is inferred. In the technique introduced by Zapalac, the integrated radon flux into the (ventilated) second chamber is measured between contiguous time intervals and, as decay of radon is ignored, may only be applied for thin samples. Several others have used time-dependent methods. Søgaard-Hansen and Damkjær [3] made the assumption of well-mixed chambers, allowing an exact analytical solution including decay in the sample. Tsai and Hsu [4,5] applied a weighted fit to the data obtained with the analytic method of Zapalac. The technique has also been used for radon diffusion through membranes. Wójcik et al. [6] made an analysis that included radon decay in the membrane. In this paper, we present and analyse two different time-dependent methods in a cylindrical geometry, using continuous measurements of the alpha activity in both chambers. The advantage of using a cylindrical sample is that leakage along the sample can be more easily avoided than for most other geometries. We outline a method to estimate the counting uncertainty of such a continuous measurement. These uncertainties are required to perform a weighted leastsquares regression. Finally, the strengths and weaknesses of the two methods are compared based on artificially generated data.
2. Model description Consider a hollow cylindrical sample with inner radius r1 and outer radius r2 . If we ignore radon generation and advective transport in the sample and assume that the pore–air radon concentration C (Bq m−3 ) only varies in the radial direction, the macroscopic radon transport equation may be written in polar coordinates: ∂C ∂ 2 C De ∂C = De 2 + − λC, ∂t r ∂r ∂r
(1)
where t is time (s), r is the polar coordinate (m), λ is the decay constant (s−1 ) and De is the effective diffusion coefficient (m2 s−1 ). We may write D = βDe ,
(2)
where D is the bulk diffusion coefficient1 and β a coefficient [7,8], also called the “partitioncorrected porosity”, that accounts for partitioning of radon in the air, water and adsorbed phase: β = ε(1 − m + Lm) + ρka ,
(3)
where ε is the material porosity, m the fraction of moisture saturation of the pores, L the Ostwald coefficient for partitioning between air and water, ρ the material density (kg m−3 ) and ka the radon surface adsorption coefficient (m3 kg−1 ). For dry materials and no adsorption, β equals the porosity ε. Two types of experimental conditions are considered. In the first type (1), a radon source is placed in the inner compartment with activity S1 (Bq). Assuming a well-mixed leak-tight 1 A reverse definition is also used.
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compartment, the boundary condition at r = r1 is ∂C ∂C (4) = 2πr1 hD − λV1 C + λS1 , r = r1 , t > 0, ∂t ∂r where V1 is the volume of the inner compartment and h the height of the sample. The terms at the right-hand side account for a diffusive flux through the inner surface, radon decay and production, respectively. Since a scintillation cell is attached to the compartment, the volume V1 is larger than the inner volume of the sample (πr12 h). Similarly, the boundary condition at r = r2 is written as ∂C ∂C = −2πr2 hD − λV2 C, r = r2 , t > 0. V2 (5) ∂t ∂r As initial condition we have V1
C = 0,
r1 r r2 , t = 0.
(6)
In the second type (2), the boundary condition describes an (often used) experiment in which the radon concentration in the inner compartment is constant: C = C0 ,
r = r1 , t > 0.
(7)
The other conditions (5) and (6) still apply for this experiment. Note that since C = 0 at t = 0, the radon concentration in the inner compartment is described by a step function. The differential equation (1) with appropriate boundary conditions was solved using Laplace transformation. Basically, the radon concentration as a function of time in each of the compartments can be written as an infinite series: C(t) = b0 +
∞
bi exp(ai t).
i=1
Note that for the first type experiment, C(r, t = 0) = 0, thus type, this only holds for r > r1 , since C(r1 , t > 0) = b0 .
(8) ∞
i=1 bi
= −b0 . For the second
3. Experimental premises 3.1. Set-up In an experiment of the first type, a radon source is placed in the inner compartment. Before the start of an experiment, the set-up should be flushed for some time (> 3 h) with radonpoor ambient air to remove radon and its progeny. The airflow is preferably forced through the porous sample. After stopping the flow, the alpha-activity in both compartments is measured with attached scintillation cells that register the number of pulses due to alpha-decay in contiguous time intervals of typically an hour. In an experiment of the second type, a constant radon concentration in the inner compartment may for example be obtained by placing a radon source in an external closed canister with a large volume ( Vset-up ) for more than about a month to establish equilibrium alphaactivity. The set-up is similarly flushed as in the previous experiment, where after the large canister is connected to the inner compartment and air is continuously circulated between these volumes.
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3.2. Detection efficiencies The count rate measured with the two scintillation flasks depends on the alpha activity in the measuring volume, determined by the concentrations in the scintillation flask of 222 Rn and its alpha-emitting progeny. It therefore depends on the efficiencies η1 , η2 and η4 for detecting an alpha particle emitted in the decay of 222 Rn, 218 Po and 214 Po, respectively. The relative detection efficiencies η2 /η1 and η4 /η1 are usually > 1. In this paper, we use η2 /η1 = 1.15 and η4 /η1 = 1.33. The efficiency η1 for detecting an alpha from radon decay is about 0.5 for a common scintillation cell.
4. Error analysis The analytical expression for the number of counted pulses in the time intervals is fitted to the data using a weighted least-squares method. As a result, the statistical error in the number of counted pulses in each interval is required. Based on the work of Aldenkamp and Stoop [9] and Inkret et al. [10], we have derived the variance of the net number of counted pulses in the case that the radon concentration is described by equation (8). A statistical analysis taking account of correlations in the decay process has also been utilised by Sonoc and Sima [11]. 4.1. Expectation and variance of counted pulses Consider the experiment in which N0 radon atoms (and no progeny) are present at some time t in a scintillation cell and in which the detected pulses due to alpha decay of radon and its progeny are counted from time t1 to t2 (t2 > t1 > t). Let pj (t1 − t, t2 − t) denote the probability of detecting j (j = 0, 1, 2, 3) pulses during the counting interval, originating from a radon atom present at t. Then the expectation value of the number of detected pulses E(X) is, omitting the dependency on t, t1 and t2 for brevity: E(X) = N0 (p1 + 2p2 + 3p3 ) ≡ N0 pE , where we have defined pE ≡ p1 + 2p2 + 3p3 . The variance is given by [10] Var(X) = N0 pE − pE2 + 2p2 + 6p3 ≡ N0 pVar .
(9)
(10)
The alpha-activity of radon and its daughter atoms in the scintillation cell as a function of time are described by the Bateman equations. Integration of the activity of each nuclide over the counting interval from t1 to t2 gives the expected number of transitions in that interval: T1 transitions from 222 Rn take place, T2 from 218 Po to 214 Pb and T4 from 214 Po to 210 Pb. The expected number of detected pulses in the counter interval can be written as E(X) = η1 T1 + η2 T2 + η4 T4 = N0 pE ,
(11)
from which pE may be calculated. To calculate pVar we need the probabilities p2 and p3 . The probability of detecting 2 pulses during the counting interval, originating from a radon atom present at t, consists of 5 terms:
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(1) radon atoms present at t1 that decay twice and both decays are recorded; radon atoms present at t1 that decay thrice and (2) the first and second decay is recorded and the third is not, (3) the first and the third decay is recorded and the second is not, (4) the second and third decay is recorded and the first is not; (5) 218 Po atoms present at t1 that decay twice: p2 =
n2 1 T2 − T4 η1 η2 + T4 η1 η2 (1 − η4 ) + T4 η1 η4 (1 − η2 ) + T4 η2 η4 (1 − η1 ) = N0 N0 (12) + T4 η2 η4 ,
where n2 is the number of radon atoms present at t from which two alpha decays are recorded in the counting interval from t1 to t2 , T2 is the number of transitions from 218 Po to 214 Pb originating from radon atoms present at t1 , T4 is the number of transitions from 214 Po to 210 Pb originating from radon atoms present at t , and T is the number of transitions from 1 4 214 Po to 210 Pb originating from 218 Po atoms present at t . The probability of detecting 3 pulses 1 consists of one term: radon atoms present at t1 that decay thrice and all decays are recorded: η1 η2 η4 T4 n3 (13) = . N0 N0 To apply the above model to a continuously changing radon concentration C(t) as described by equation (8), one should consider the rate of change of the number of new radon atoms Nn (t) in a scintillation flask, for which we may write p3 =
dNn (t) dN (t) 1 dAr (t) (14) = + λN(t) = + Ar (t), dt dt λ dt where N(t) is the total number a radon atoms, and Ar (t) is the radon activity in the scintillation cell which equals Vm C(t) with V m the volume of the cell. In the counting interval from t1 to t2 , decays may be recorded from radon atoms (and their progeny) formed in the period between the start of an experiment at t = 0 and t1 , and radon atoms formed in the counting period from t1 to t2 . The expectation of the number of counted pulses is obtained by integration and consists of a contribution from both periods: t1 t2 dNn (t) dNn (t) E X(t1 , t2 ) = (15) pE (t1 − t, t2 − t) dt + pE (0, t2 − t) dt. dt dt 0 t1 For the variance Var(X) we similarly have t1 t2 dNn (t) dNn (t) pVar (t1 − t, t2 − t) dt + pVar (0, t2 − t) dt. (16) Var X(t1 , t2 ) = dt dt 0 t1 Both equations can be written in terms of the radon activity Ar (t), and thus C(t), using equation (14). In the second type diffusion experiment, we have a step-wise concentration change in the inner compartment at t = 0. The expectation of the number of counted pulses and the variance are then given by (15) and (16) added with, respectively: Vm C0 pE (t1 , t2 ) λ
and
Vm C0 pVar (t1 , t2 ) . λ
(17)
Sensitivity analysis of an improved method for measuring the radon diffusion coefficient
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Fig. 1. Ratio of variance and expectation of number of counted pulses in 1 h intervals as a function of time for different radon detection efficiencies η1 .
4.2. Example Consider an experiment with a constant radon source in a closed compartment without radon and progeny initially. The radon concentration is described by C(t) = a(1 − exp(λt)) where a is a constant. For a diffusion experiment the time-dependent radon concentration is slightly different, but the main results are similar. The ratio of the variance and expectation of the number of counted pulses in 1 h contiguous intervals is shown in Fig. 1, where ratios η2 /η1 = 1.15 and η4 /η1 = 1.33 have been used. The time-dependent behaviour of the variance is mainly of concern at the start, for t < 5 h. Thereafter, the equilibrium value is nearly reached which is dependent on the radon detection efficiency η1 . In case η1 , η2 and η4 1, or for a counting interval of less than about 30 s, the number of detected pulses approaches a Poisson distribution, i.e. Var(X) ≈ E(X). 5. Sensitivity analysis Artificial measurement data have been generated for the two experimental conditions based on the models described in Section 2 for an experiment of 7 days and a counting interval of 3 h, giving a total of 112 data points (data from both scintillation cells are fitted simultaneously). A random error conforming to a Gaussian distribution with a calculated variance as outlined in Section 4 was added to each data point. The calculations were performed for a base case where h = 10.4 cm, r1 = 2.55 cm, r2 = 9 cm, V1 = 0.7 L and V2 = 2.8 L (of these, only V2 is varied in the sensitivity analysis). The volume of the scintillation cells is Vm = 0.3 L and the radon detection efficiency of both cells is set at η1 = 0.5. Further we assumed no background counts and the parameter β was set at 0.8. The regressions were carried out using the Levenberg–Marquardt method for four values of the effective diffusion coefficient (10−6 , 10−7 , 10−8 and 10−9 m2 s−1 ) and a source strength S1 = 1000 Bq for type-1 conditions. For type-2 conditions, the concentration C0 was set at either 104 Bq m−3 or 105 Bq m−3 . In all cases data points with less than 10 counts were omitted. 5.1. Type-1 experiment First we consider a regression with De and S1 as free parameters and no systematic error in the other quantities. For this regression, the diffusion coefficient is found with a relative
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uncertainty between 0.04% (De = 10−8 m2 s−1 ) and 0.6% (De = 10−6 m2 s−1 ). These small uncertainties indicate that taking S1 as a free parameter does not pose any problem. However, if the radon detection efficiency of e.g. scintillation cell nr. 2 has a small systematic error of 2%, the fitted diffusion coefficient will contain a systematic error. For De = 10−6 m2 s−1 the systematic error is 20% and the relative uncertainty is 18%. For De = 10−7 m2 s−1 these values are 4% and 2%, respectively. For smaller values of De the errors are less than 1.5%. It is thus rather difficult to accurately determine the diffusion coefficient in this way when De > 10−7 m2 s−1 . The cause of this problem lies in the fact that D e is largely determined by the concentration difference between the inner and outer compartment, which is very sensitive to the relative efficiency of the scintillations cells. This unwanted sensitivity would be lower for a larger volume V2 . When V2 = 28 L instead of 2.8 L, the systematic error in De reduces to 5% for De = 10−6 m2 s−1 and even further for larger values of V2 . An optimum is found when V2 is several cubic metres. As an alternative to using a large outer compartment, one may also add the relative efficiency as a free parameter in the regression. In doing so, it was found that the effective diffusion coefficient in the range 10−6 –10−9 m2 s−1 is obtained without a systematic error and with a relative uncertainty < 1% when V2 = 2.8 L. The accuracy is better than 0.5% when V2 = 28 L. The accuracy of the method slightly degrades for V2 = 280 L. The above sensitivity analysis ignores the uncertainty in the partition-corrected porosity β, see equation (3). Since β depends on the porosity, water content, density and adsorption coefficient, these quantities should be accurately known to determine the diffusion coefficient. Alternatively, one could take β as a free parameter in the regression. If ηrel is accurately known, a regression with β, De and S1 as free parameter gives both β and De within 1.5% for De 10−7 m2 s−1 , β = 0.8 and V2 = 28 L (slightly worse for V2 = 2.8 L). For De = 10−6 m2 s−1 and V2 = 28 L, the accuracy is about 10%. In case ηrel is also a free parameter (giving a total of 4 free parameters), similar accuracies are obtained. 5.2. Type-2 experiment Similar tests were performed in case the concentration in the inner compartment is constant. For the regression with De and C0 as free parameters, it was found that the calculated diffusion coefficient is less sensitive to systematic errors of the relative efficiency. It was also observed that increasing the outer volume has a positive effect on the systematic error in De . From the experimental point of view it is, however, difficult to create a constant concentration in the inner compartment when the outer volume is large and the diffusion length is much larger than the sample thickness. In this respect, the outer volume should be small. The calculations have therefore been restricted to two volumes: V2 = 2.8 L and 28 L. For these values, the relative systematic error in De (10−6 m2 s−1 ) is about 4% when the systematic error in ηrel is 2%. The errors are smaller for lower values of De . Also for this experiment it is worthwhile to perform a regression with the relative efficiency as additional free parameter. It was found that in all cases (4 values of De ) with V2 = 2.8 L the calculated diffusion coefficient has no significant systematic error and the relative uncertainties are < 0.3%. With V2 = 28 L the results were slightly less accurate, but this is due to a 10 times lower value of C0 . With identical C0 the results are similar.
Sensitivity analysis of an improved method for measuring the radon diffusion coefficient
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As mentioned, one could also take β as an additional free parameter in the regression. If ηrel is accurately known, a regression with β, De and C0 as free parameters gives both β and De within 1.5% for De 10−7 m2 s−1 , β = 0.8 and V2 = 2.8 L (somewhat worse for V2 = 28 L). In case ηrel is also a free parameter (giving a total of 4 free parameters), similar accuracies are obtained. 5.3. Discussion The above sensitivity analysis ignores any uncertainty in the sample dimensions (h, r1 and r2 ) and other experimental errors such as varying detection efficiencies during the experiment due to, e.g, temperature fluctuations, small leaks in the set-up, sample inhomogeneities, etc. In addition, the model assumes well-mixed compartments, which may not reflect reality. These effects could be the subject of a further sensitivity analysis. However, most of these errors can be minimised in a carefully prepared and executed experiment.
6. Conclusions Using the regression analysis for the first type experiment, a diffusion coefficient in the range 10−6 –10−9 m2 s−1 can accurately be determined using a 6.5 cm thick sample in an experiment of 7 days. For smaller values of De a thinner sample should preferably be used. Further, it was found that the outer compartment should not be too small, i.e., generally V2 > 10V1 , but not so large that the radon concentration becomes too low to be easily detected. Under these circumstances, a regression with S1 , De and the relative detection efficiency of the two scintillation cells as free parameters yields accurate results. This means that no information is required on the radon source strength and the relative detection efficiency of the scintillation cells to determine the radon diffusion coefficient. A regression with β, De , S1 and ηrel as free parameters gives both β and De within 1.5% for 10−9 De 10−7 m2 s−1 . For larger De , a thicker sample should be used to determine β and De simultaneously. The second type experiment has a slightly better performance and is less sensitive to the volume of the outer compartment. From the experimental point of view, and if β is also a free parameter, a small volume is however preferred. For very small values of De (< 10−9 m2 s−1 ) thinner samples should be used. Again, no information is required on the radon concentration in the inner compartment and the relative detection efficiency of the scintillation cells to determine the radon diffusion coefficient. It is concluded that both methods give similarly accurate estimates of De and β. The advantage of the first method lies in its experimental simplicity since there is no need to maintain a constant radon concentration in the inner compartment. Also, pressure differentials due to air circulation that may influence radon transport through the sample are excluded. The regression technique is especially shown to be a useful tool since it does not require data for the radon source strength, or concentration C0 , and the relative detection efficiency of the scintillation flasks.
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References [1] K.K. Nielson, D.C. Rich, V.C. Rogers, D.R. Kalkwarf, Report NUREG/CR-2875, US Nuclear Regulatory Commission, 1982. [2] G.H. Zapalac, Health Phys. 45 (1983) 377. [3] J. Søgaard-Hansen, A. Damkjær, Health Phys. 53 (1987) 455. [4] S.-C. Tsai, C.-N. Hsu, Geophys. Res. Lett. 20 (24) (1993) 2917. [5] C.-N. Hsu, S.-C. Tsai, S.-M. Liang, Appl. Radiat. Isot. 45 (8) (1994) 845. [6] M. Wójcik, Wlazło, G. Zuzel, G. Heusser, Nucl. Instrum. Methods Phys. Res. A 449 (2000) 158. [7] W.H. van der Spoel, Radon transport in sand, PhD thesis, TU Eindhoven, The Netherlands, 1998. [8] C.E. Andersen, Sci. Total Environ. 272 (2001) 33. [9] F.J. Aldenkamp, P. Stoop, Sources and transport of indoor 226 Radon – measurements and mechanisms, PhD thesis, Rijksuniversiteit Groningen, The Netherlands, 1994. [10] W.C. Inkret, T.B. Borak, D.C. Boes, Radiat. Prot. Dosim. 32 (1990) 44. [11] S. Sonoc, O. Sima, Radiat. Prot. Dosim. 45 (1992) 51.