A periodic pumping technique of soil gas for 222Rn stabilization in large calibration chambers: part 2—theoretical formulation and experimental validation

In an adjoining publication, we demonstrated the novel technique to harvest soil gas of natural origin as a highly efficient source of 222Rn for calibration applications in a large volume 222Rn calibration chamber. Its advantages over the use of conventional high strength 226Ra sources, such as the capability to serve as a non-depleting reservoir of 222Rn and achieve the desired concentration inside the calibration chamber within a very short time, devoid of radiation safety issues in source handling and licensing requirements from the regulatory authority, were discussed in detail. It was also demonstrated that stability in the 222Rn concentration in large calibration chambers could be achieved within ± 20% deviation from the desired value through a semi-dynamic mode of injection in which 222Rn laden air was periodically pumped to compensate for its loss due to leak and decay. The necessity of developing a theory for determining the appropriate periodicity of pumping was realized to get good temporal stability with a universally acceptable deviation of ≤ ± 10% in the 222Rn concentration. In this paper, we present a mathematical formulation to determine the injection periods (injection pump ON and OFF durations) for the semi-dynamic operation to achieve long term temporal stability in the 222Rn concentration in the chamber. These computed pumping parameters were then used to efficiently direct the injection of soil gas into the chamber. We present the mathematical formulation, and its experimental validations in a large volume calibration chamber (22 m3). With this, the temporal stability of 222Rn concentration in the chamber was achieved with a deviation of ~ 3% from the desired value.

Inhalation of 222 Rn, 220 Rn and its decay products accounts for more than half of the annual effective dose from radiation sources of natural origin. Latest study carried out in Europe showed that residential 222 Rn accounts for about 9% of the deaths from lung cancer and 2% of all cancer deaths 1 . Measurements and dose assessments due to 222 Rn and progeny are performed using a wide variety of active and passive detectors and dosimeters. To set up standard protocols and maintain mutual conformity between the various detectors and instruments used by different laboratories, calibration facilities are established in different countries [1][2][3] .
Testing and calibration of 222 Rn detectors in large volume walk-in type calibration chambers are carried out, mainly by three operational modes: (1) dynamic mode-in which 222 Rn rich air from the source is continuously pumped to the chamber through the inlet and simultaneously maintaining the outlet in the open condition to attain the concentration levels [3][4][5] , (2) static mode-in which 222 Rn laden air is filled once to achieve desired initial concentration value and the experiment is performed during the decay of 222 Rn 5-7 , and (3) semi-dynamic mode-in which, initially 222 Rn is injected into the chamber to obtain the desired value and then switched over to a pulsed mode of injection so that concentration is maintained within a certain band of the desired value. The periodicity of 222 Rn injection is adjusted in such a way that the loss of 222 Rn due to leak and decay is just compensated 8,9 . Calibration exercise at a low level (~ 1,000-5,000 Bq m −3 ) 10-12 demand a steady 222 Rn concentration for considerably long-time durations, from a few weeks to a few months.
In our previous publications, we demonstrated the harvesting of soil gas as a highly efficient secondary source of 222 Rn for calibration experiments [13][14][15][16] as well as a substitute for 226 Ra source in research studies 17,18 . Advantages of soil gas 222 Rn over the conventional sources are instantaneous natural availability, zero regeneration time required, zero cost, and excellent source stability. These features of soil gas 222 Rn are well described in the previous publications 10,14,18 . Through the semi-dynamic mode of operation, it was shown that the 222 Rn concentration could be maintained with a deviation of ± 20% from the desired value using soil gas. This deviation from the desired value is large since universally recommended long term calibration exposures demand a steady 222 Rn concentration with an acceptable deviation of ≤ ± 10% for considerably long-time durations, from a few weeks to a few months for a large calibration chamber 11,12,19 . Higher deviation than the acceptable limit was because the periodicity of pumping was selected (1) by monitoring the concentration in the chamber using active monitors which may have some delayed response, (2) based on the knowledge of soil gas 222 Rn concentration and the chamber volume, and (3) total 222 Rn outflow from the chamber (sum of leak and decay) 8 and not based on a theoretical model involving the functional behaviour of various parameters affecting the 222 Rn concentration profile in the calibration chamber.
In this paper, we present a mathematical formulation to determine the exact injection periods (Pump ON and OFF durations) for 222 Rn laden air for the semi-dynamic operation to achieve good long-term temporal stability in the 222 Rn concentration in the chamber. These computed pumping parameters are then used to efficiently direct the injection of soil gas into the chamber. The experimental validations of the theoretical model and achievement of very good stability in 222 Rn concentration in the chamber are discussed here.

Mathematical formulation of the model and solutions
In the previous study 10 , it was demonstrated that the average value of natural 222 Rn concentration in soil gas, measured over one full year, in the premises of the Centre for Advanced Research in Environmental Radioactivity (CARER) was 78.0 ± 20.0 kBq m −3 . The average value of 226 Ra activity concentration in soil was 42.0 ± 4.2 Bq kg −1 .
Since soil gas 222 Rn concentration is significantly larger than the concentration to be maintained, for a semidynamic mode of operation it is required to determine the optimum pumping rate that yields highest 222 Rn stability over time at a predetermined concentration level. To achieve this, we set up 222 Rn concentration evolution equation considering periodic injection combined with various removal processes. Let us consider the process in which 222 Rn is injected for a time T 1 and the injection was turned OFF for a period T 2 . Let T (= T 1 + T 2 ) be the duration of ON/OFF cycle. Since the injection is carried out through a flow, it is assumed that the flow in the chamber exists for the period T 1 (it is zero during T 2 ). During the period T 2, only leak out of the chamber and radioactive decay processes are removal mechanisms. As a result, we can analyse the process in terms of the injection cycles denoted by "n" and time t (t, 0 < t < T) within each cycle. The running time corresponding to a time t of the nth cycle may be represented as follows: Since pumping exists for time T 1 and is zero for the time T 2 = T − T 1 within each period, the corresponding time-dependent flow rate f(t) has the following form: A schematic of the rise and fall of total 222 Rn activity in the chamber as a function of pumping cycles is shown in Fig. 1. The hatched rectangular pulses are 222 Rn injection rate sequences during the pumping process. The lines indicate an increasing and decreasing sequence of total air activity of 222 Rn in the chamber due to injection and decay respectively. For general flow rates and periods of injection, the activity in the chamber will either go on increasing or decreasing over long periods. However, there exists an optimum ON-OFF ratio at which it will maintain a constant average value modulated by small ripples in every cycle.
Equations (2a) and (2b) are a system of coupled first-order equations within each of the cycles. This system can be solved by specifying initial conditions at the beginning of the nth cycle. Let A n (0), as yet unknown, be the activity in the chamber at the beginning of nth cycle. We can now express the solution to Eqs. (2a, 2b) as follows: where we have introduced the concepts of "total removal rate" ( ON ) during pump flow ON condition and "total removal rate" ( OFF ) during pump OFF condition, defined as: The unknown quantity A n (0) is determined by invoking the continuity conditions at the periodic boundaries. That is, the activity attained at the end of the nth cycle should be equal to the initial activity for the (n + 1) th cycle. i.e.
Upon inserting Eq. (6a) in Eqs. (3a) and (3b), we obtain: The initial condition for the iteration sequence is provided by the activity A 0 injected for the first time (n = 0) into the chamber. i.e. Equation (6b) provides the iteration for obtaining A n (0) for all cycles. With this, one can write a recurrence relation in the following compact form: where condition for stable concentration. To proceed further, we make an important assumption that the injected activity spreads rapidly in the chamber due to a mixing element such as fan, leading to spatially uniform 222 Rn concentration. Detailed studies conducted with a 22.7 m 3 calibration chamber at CARER 8,13 with a mixing fan of capacity 3,620 m 3 h −1 have shown that time required for attaining uniform concentration is < 10 min, which is much smaller than the pump OFF period (T 2 ). With this experimental data we can convert the activities A n 's to concentrations C n ' as follows: With this Eq. (8) can be rewritten as In this periodic injection scenario, one cannot achieve perfectly uniform 222 Rn concentrations because of the finite time required for homogenization within the chamber. However, the presence of a strong mixing element like a fan is expected to rapidly spread the injected gas throughout the chamber. As the chamber has a certain volume of outflow per hour (due to leak and decay) it is possible only to maintain a stable mean concentration. From Eq. (10b) we see that if the mean concentration has to remain constant right from the first cycle, it should not have a dependency on 'n' Eq. (8) suggests that it is possible only when C n (0) remains independent of n. That means all C n (0)'s should be the same as the initial concentration C 0 i.e.
Upon applying this condition to Eq. (8), we obtain. i.e., i.e., This is an exact equation which relates T 1 (the injection period) to T 2 (the pump OFF period) for a given set of the quantity This equation may be further simplified under the assumption that the period of pumping is much smaller than the mean residence times i.e. λ ON T 1 << 1. This is justified because typically for a large chamber (in the present case 22.7 m 3 ) λ ON is controlled by the flow rate. For a flow rate of 60 L min −1 , λ ON is about 0.16 h −1 which is lesser than the frequency of pumping 1 T 1 +T 2 . The injection period is determined by stipulating that the concentration deviation from the stipulated mean due to decay/leak during the OFF period should not exceed, say 5%. As explained in Sect. 3 (Fig. 5), typically, the frequency would be 1 min pumping after every 59 min. This will, of course, change depending upon the soil gas concentration and the concentration required in the chamber, as indicated in Eq. (12c). In such a case, the exponential terms can be approximated by linear terms and one obtains: This results in the following formula: For future purposes, we denote this ratio by "R", where Since λ ON is related to λ OFF through flow rate, Eq. (12b) may be written as: Equation (12c) is the desired condition for attaining stable concentration right from the beginning with periodic injection. It is expressed as the ratio of duration of the pump OFF (T 2 ) and pump "ON" (T 1 ) periods in terms of two groups of system parameters, namely leak rate ratio and concentration ratio It is important to note the following points: In most situations, C SG >> C 0 and f 0 V OFF >> 1, and hence T 2 T 1 >> 1. This means the fraction of OFF period is much larger than the fraction of ON period in a given injection cycle. In practice, Eq. (12c) is useful for fixing the OFF period (T OFF ) for a given soil gas pumping ON period. Practical consideration (such as, the response time of the pump, the time required for the soil gas to reach from the point from where it is drawn to the pump through tubing which is maintained sufficiently long to minimise the 220 Rn concentration) demands that the ON period cannot be too short say less than one minute. Hence one would prefer the ON period to be at least for about a minute. Once this is done, then the next injection should be done after time T 2 = R T 1 and the cycle period will naturally be T = (1 + R) T 1 for concentration stabilization. The formula also offers us constraints on the highest concentration that can be stabilized for a given soil gas concentration and flow rate. For example, for continuous injection T 2 = 0 and the concentration that gets stabilized is: Figure 2(a,b) shows the variations of T 2 T 1 for concentration ratio BB and leak rate ratio AA at different flow rates. T 2 range can vary over 100 times much than T 1 if soil gas concentration is higher. There will be fluctuations or wiggles due to the periodicity of the injection process. We can decrease the wiggles by increasing the intervals of injection. www.nature.com/scientificreports/ It predicts that the ratio of the pump OFF (T 2 )/Pump ON (T 1 ) time is related to the removal rates in the chamber with and without flows and the ratio of soil gas 222 Rn concentration C SG to the stipulated concentration C 0 to be maintained in the chamber. However, it should be emphasized that Eq. (12c) can only predict the ratio of the times T 2 T 1 but not the total period (T = T 1 + T 2 ) of the cycle. To arrive at an optimal value for the period one has to stipulate additional constraints. For this, we impose that the fluctuations around the stabilized mean value lie within pre-specified limits and the variations of stabilized mean 222 Rn concentration with values of R is shown below in Fig. 3. It illustrates how the stabilization is achieved when the system is operated at T 2 = RT 1 ; The concentration either overshoots or undershoots to higher or lower steady-state value when T 2 < RT 1 and T 2 > RT 1 , respectively. The steady-state is reached as a result of an average balance between the injection rate and the decay + leak rates. It is observed from Fig. 3 that the stability of the concentration of 222 Rn to be maintained in the chamber depends on the ratio of T 2 and T 1 which gives an optimized value of R = R optimum , which may be estimated from Eq. (12b). At this value of R, it achieves maximum stability followed by its deterioration for other orders of R optimum .   www.nature.com/scientificreports/

Materials and method
Set up for extraction and periodic pumping of soil gas. The details of soil gas extraction were described in a previous publication 10 . Since the 222 Rn concentration in soil gas increases exponentially with depth and saturates to an equilibrium concentration at a depth greater than about 0.8 to 1 m 20,21 , it was extracted from a depth of about a meter. The extraction was performed using a hand-driven soil gas probe (STITZ, Germany) inserted inside the ground as shown in Fig. 4a. The total flow is bifurcated into multiple channels from each of which a probe is inserted into the soil and this is done to eliminate the possibilities of soil particles choking the inlet of the probes at higher flow rates. The separation between the probes was generally kept at a distance of about 1 m. The bifurcation reduces the suction velocity at the probe inlet and the possibility of large area perturbations of 222 Rn concentration in the soil matrix. The outlets of multiple soil probes were combined and are connected in series to a progeny filter, a dehumidifier, and a buffer volume of 0.052 m 3 (for mitigation of 220 Rn present in the soil gas). The detailed schematic diagram of the calibration chamber and soil gas probe arrangement to draw soil gas is shown in Fig. 4b. for maintaining temporal stability in 222 Rn concentration in the calibration chamber was achieved with the help of a programmable timer switch, which controlled the switching ON/OFF of the pump used to draw soil gas from the probe insertion point. Initially, the soil gas was pumped into the chamber continuously till a desired 222 Rn concentration was attained in the chamber. Once the desired 222 Rn concentration level was attained, the pumping was switched over to "periodic pumping mode". The duration of pumping (T 1 ), the period of pumping cycle (T) and duration of pump OFF (T 2 = T − T 1 ) were determined based on the ratio T 2 T 1 = R using Eq. (12c). The process of switching ON and OFF of the pump was performed automatically by the programmable timer switch to maintain temporal stability of concentration of 222 Rn in the chamber. Volumetric average 222 Rn concentration in the chamber was measured both during the transient build-up of initial desired level and periodic injections at an inlet port of the chamber, with a Smart RnDuo online 222 Rn monitor. Besides, continuous monitoring of the concentration was also performed using two AlphaGuard systems placed inside the chamber at two different points at a height of 1 m from the floor of the chamber.

Results and discussion
Stability of 222 Rn concentration in the walk-in chamber was achieved by periodic injections of soil gas. The amount of 222 Rn injected was equal to the sum of volume outflow of 222 Rn per hour due to leak and decay. We demonstrate the experiments for the fan ON condition in particular for two reasons, (1) fan ON condition is the situation of www.nature.com/scientificreports/ maximum 222 Rn depletion in the chamber due to the outward pressure exerted by air inside the chamber, (2) the circulation fan in the calibration chamber facility will always be switched ON to maintain spatial uniformity of 222 Rn concentration, relative humidity (RH) and temperature during the calibration experiments.
Initially, experiments were conducted by randomly selecting the duration and intervals of pumping. The experiments were performed for three different combinations of the period of pumping cycle T, pumping duration T 1 , non-pumping duration T 2 and soil gas flow rates as given in Table 1. The 222 Rn concentration in the Relative humidity and temperature inside the chamber were maintained at 70% and 28 °C during the experiment. It was observed from the results obtained (Fig. 5) that the stability of 222 Rn concentration in the chamber was better for case 1 (Table 1) in which the deviation observed from the desired concentration level was ~ 1% for a particular combination of T, T 1 , T 2 and f 0 . But, for other combinations of pump ON and OFF (cases 2 and case 3) the average 222 Rn concentration achieved in the chamber were 6,926 ± 777 Bq m −3 and 6,499 ± 788 Bq m −3 respectively which correspond to deviations of 17% and 20% from the desired values. Also, as evident from Fig. 5 the concentration values show a decreasing trend with time since the 222 Rn pumped during the selected pumping period intervals was not able to compensate for decay and leak. This confirms the fact that one cannot choose a random combination of pumping parameters. Hence, for achieving good temporal stability of 222 Rn concentration, an accurate pumping algorithm should be adopted.
In the theoretical model-based approach the parameters of pumping such as T, T 1 and T 2 for different soil gas flow rates were calculated using Eq. (12c). The experiments were carried out for different values of R, with an average soil gas 222 Rn concentration, C SG , of 78.0 ± 20.0 kBq m −3 (2σ) and the flow rate, f 0, was fixed at 30 L min −1 . For validation of the theory, let us stipulate that a particular calibration experiment is performed by maintaining a desired 222 Rn concentration of ~ 8,000 Bq m −3 . The details of the parameters of pumping, determined based on the theory developed in this study (Eq. 12c), for maintaining the desired concentration are presented in Table 2.
The temporal stability of 222 Rn concentration achieved in the chamber, for the parameters given in Table 2, are presented in Fig. 6. As discussed earlier, the ratio R is an important parameter in deciding the temporal stability of concentration, and this is demonstrated in Fig. 6. Experimental results (Fig. 6) confirm the predictions of the theory (Fig. 3) for all values of R. As evident from the results, when T 2 = R T 1 (i.e. R = R optimum ) excellent temporal stability in the concentration was achieved throughout the experimental duration of 9 days with an average value of 7782 ± 376 Bq m −3 , which corresponds to a deviation of < 3% from the desired value. This deviation is well within the standard error in the measurements. For all the other values of R (T 2 = R/6 T 1 , R/4 T 1 , R/2 T 1 and 2R T 1 ) there was either a build-up or a depletion of concentration in the chamber. These experimental results validate the predictions of the theory and the algorithm developed here will find immense application in large calibration chamber facilities for maintaining good temporal stability of 222 Rn concentration.
To summarise, the theoretical model developed in the present study allows optimization of the pumping rate, pumping duration and pump ON/OFF time for achieving excellent long term stability in the desired concentration of 222 Rn, with deviation ≤ ± 3%, in the chamber for calibration experiments. The technique is highly advantageous due to its technical simplicity and economic considerations. When coupled with the optimized periodic pumping algorithm the technique of harvesting of soil gas as an un-depleting source of 222 Rn would eliminate the need for expensive radioactive sources.  Table 1. Pumping parameters for periodic injections of soil gas into the chamber (parameters selected randomly).