Radon indoor concentration time-variation model

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Introduction
Radon Is well recognized as a major source for the cancer of lungs [1][2][3][4][5][6]. Being a gas, the radon atoms could escape from any surface out to the air where they decay to their progeny that may attach to dust particles to create radioactivedust , which we inhale. Early researches on radon were mainly concerned about mines of uranium, but it was realized later that radon levels have to be monitored at all work places and residences, because radon is emitted from everything surrounding us; from ground , walls , water, etc. With these investigations carried on over the years, many articles have been written that report radon concentrations at various places of work, homes and schools, at different cities from different countries. It is, however, worthy of mention that the risk of radon promoting a lung cancer is not only related to its concentration inside the place of concern, but it is also related to how many hours the person stays in that place, and whether they are smokers. Some surveys indicate that smokers could be ten times more susceptible to lung cancer than non − smokers [7][8][9][10][11][12][13]. Therefore, in addition to having reports about the concentrations of the indoor radon , it is necessary to have deliberate studies on its behavior. This comes only by carrying both theoretical and experimental researches on all the relevant elements, e.g. the concentration of radon in the constructing walls of the buildings and in the soil underneath. Other important influencing factors are also the levels of radon in the surrounding outside air , the radon dif fusive and advective characteristics, different contributors to the radon indoors , and temporal changes. These studies can help in finding and developing methods to minimize the exposure of the individuals to radon.
In this paper, one of the important factors that will be studied is the time variation of the radon indoor concentration . In previous researches, measurements on indoor radon usually focused on long − term temporal variations [14][15][16][17], while calculations usually assumed a steady state condition [18][19][20]. In this research, however, a theoretical investigation is performed on the radon indoor concentration short − term temporal variations ; at top, for one or two days, on an hourly basis. This is important for places that have very high radon concentrations , in particular when the residents remain for long time periods indoors , especially in winter with poor ventilation , or for workers at unventilated mines for example. Another remarkable side of the model is that, unlike most of the previous theoretical studies, where only some sources are considered, this model considers all known sources of indoor radon [18][19][20][21]. It additionally provides a prediction of possible existence of unknown sources . Furthermore, as different from previous schemes, this model gives the radon indoor concentration in an analytical form, not by numerical calculations [18][19][20][21]. Moreover, the model , with its semi empirical feature, is expected to give better description to the experimental measurements.

and of
Soil Is the most important radon source . Annual estimation of the radon flow from the ground worldwide is nearly 90 × 10 18 Bq [22]. Almost 50% of the radon indoors is from the soil (supposing the room is in the ground f loor ) [23].
Beside soil , there are other different sources of the indoor radon . These are outlined in Fig. 1. Considerable radon levels can be detected in water streams [24]. The higher layers of these waters constantly release radon to the surrounding atmosphere by volatilization [25]. This indicates that the water lower layers have more concentrations of radon than the higher ones. Similarly, because of radon release to the outside air by advection and dif fusion , bottom parts of the soil have higher concentrations of radon than the top parts [25]. From this argument, it is clear that radon concentration can differ widely from one place to another. For instance, in outside air it covers a range of 1Bq.m −3 − 100Bq.m −3 [26]. In poorly ventilated residences, it covers a range of 20Bq.m −3 − 2000Bq.m −3 , with a close range for mines with good ventilation [26]. Mines without good ventilation can have a much larger range [26]. Furthermore, radon concentration can broadly vary with dif ferentseasons and atmospheric conditions [16,17].

rate of of the
To have an appropriate evaluation of the inhaled dose of the radon indoors , it is sometimes necessary to observe the exposures closely, which in turn means having convenient methods to estimate the changes of the concentrations of the radon indoors in short terms. Due to several factors, the radon concentration is a function of time and position, and therefore to estimate the indoor exposure, an integration must be performed over the dimensions of the room and over the time interval of interest. However, this study concerns about the temporal variation , and so, for simplicity, the radon concentration at a given time is assumed to be the same at all positions in the room , i.e. position independent, and hence the integration over the dimensions will just yield the room 's volume. Thus, the exposure to an individual in a time interval Δt = t 2 − t 1 is taken as is the radon indoor concentration at a given time t(h) . To have a theoretical evaluation of C i (t) , we begin by the radon mass balance form that describes the different sources of indoor radon and its sinks is from sources that might be unknown, and Radon release from walls adds up to the concentration of radon inside the room . In a very short time interval, the rate of that addition can be put in the for m is the dif fusive transfer coefficient through the walls , and C bm Bq m 3 is the concentration of radon in the walls including both the emanated radon atoms and the non-emanated ones. On the other hand, in a very short time interval, the addition rate to the concentration of radon inside the room , by radon release from the soil , can be put in the form Radon in a room = radon sources + radon sinks, . 1 Outline of the indoor sources to the inside of the room . This flux, taking place by advection a n d dif fusion , i s g i v e n b y is the soil radon concentration , ΔP si (Pa) is the difference in pressure between the inner of the room and the soil , A m h.Pa and D s m h are the advective and dif fusive transfer coefficients through the soil , respectively. Thus [18][19][20][21] and hence Eq.
is the concentration of radon in the surrounding outdoor, and (h −1 ) is the constant of radon decay. In Eq. (6), the exchange with the outside air is assumed to be due to ventilation only, without taking into account the negligible contribution from the leakage processes.

Solution of the equation
The mass balance equation in its form as given by Eq. (5) has a number of unspecified functions. This makes it hard to attain its solution. To make it less complicated, and put it in a more convenient form, we approximate the involved functions by making use of some of the data from relevant measurements [14][15][16][17][27][28][29]. We may develop the formulas where a bm is a parameter that can relate the concentration of radon inside the building material to that of the indoor air , a s is a parameter that can relate the concentration of radon in the soil to that of the indoor air , and a o is a parameter that can relate the concentration of radon in the outdoor air to that of the indoor air . All these parameters are dimensionless and they are to be evaluated from the experimental observations.
For not long durations, the term can be assumed to be almost constant Substituting Eqs. (8)(9)(10)(11) in Eq. (5) or where Putting Equation (13) becomes which has the solution If C i (t) has an initial value C i (0) at t = 0 , then and therefore

Applying the and discussing the
We start by estimating the parameters a bm , a s and a o of Eqs. (8)(9)(10). They are dimensionless, with the evaluations Eqs. (22) and (8) develop from using the formula [27] together with Fig. 2. C bm i is the radon indoor concentration occurring by radon release from the constructing material that has an observed areal release rate E s ( Bq m 2 h ). The plot of Fig. 2 is a relation between E s and concentration of radon in a material [28]. 34 samples were analyzed, where E s and concentration of radon were measured by applying the sealed can technique [28]. A linear correlation between E s and concentration of radon was developed, with correlation coefficient 0.95. The straight-line used to describe the data is In accordance with ref [23], constructing materials shares by ∼ 20% of the radon indoor concentr , i.e. C bm i ≈ 0.2C i . Therefore, using Eqs. (25) and (26) or which gives Eqs. (8) and (22), where Eqs. (23) and (9) develop from Fig. 3 that relates the concentration of radon indoors to that in soil , for ground-f loor rooms [29]. The radon concentration indoor measurements were made by the alpha scintillation cells ASC and the portable AlphaGuard radon monitor . The concentrations of radon in soil were made by the EDA − ASC and the portable RDA − 200 detector [29]. 53 samples were analyzed. From  Fig. 3, one can see that most of the data are gathered at small concentrations of the indoor radon . Therefore, in our model , the dots are described by the following straight-line relation or which means that a s has the value 100. Eqs. (24) and (10) develop from taking the average of concentrations of radon outdoors recorded at various seasons in relation to that of (26) E s = 0.433C bm . Fig. 2 Radon areal release rate from a material in relation to its radon concentration, according to the measurements in ref [28] Fig. 3 Relating the concentration of radon indoors to that in soil, for ground-floor rooms, according to the measurements in ref [29] indoors [15][16][17]. Better evaluation of a o could be obtained by taking into account only the season relevant to the radon indoors in study.
Beside a bm , a s and a o , there are some parameters that describe the studied room . Because the values of these parameters are not provided with the data of the measured indoor concentrations [30], they are given typical numbers. The length, width and height of the room are taken as 5m, 4mand2.8m , respectively. The volume of the room is therefore 56m 3 , with S bm V ≈ 1.6m −1 . The rate of ventilation and the soil-indoor difference in pressure are assumed to be v = 0.8h −1 and ΔP si = 4Pa.
All the needed parameters to calculate C i (t) are now set. Only left, to be evaluated from fitting Eq. (21) to the measured concentrations , are the dif fusive transfer coefficient through the walls D bm , the share from unknown-sources U , the advective and dif fusive transfer coefficients through the soil , A and D s , respectively. Figure 4 shows a set of experimental data [30] for the variation of the indoor radon concentration over a couple of days. The measurements were performed by the PQ2000PRO AlphaGuard . 33 samples were analyzed. To fit Eq. (21) to the measurements of ref [30], the D bm , A , D s , and U parameters are found to be These are significant parameters for describing the indoor radon-entry. The good fit demonstrated in Fig. 4 implies the model 's efficacy in giving a description of the temporal change of radon indoor concentration . However, the time span during which the model is applicable is not clearly known. Trying to fit Eq. (21) to less data, or more, than those in Fig. 4, causes a slight change in the resulting parameters . Therefore, in order to have a better argument, several sets of data should be involved to test the range of applicability of the adopted approximations that constrain the time validity of the model (Eqs. 10 and 11). For the time being, however, this cannot be carried on as the available radon indoor concentration data are usually from long term observations (not on an hourly basis).
It is important to remark two points; first, the values of the parameters in Eqs. (22)(23)(24) are not exactly unique, but can have some ranges. However, it has been tested that, within these ranges, the results of Eqs. (32-35) and Fig. 4 are not so sensitive, given the uncertainties in the experimental data. Second, the results of Fig. 4 and Eqs.
(32-35) are not just about a transition from an initial radon concentration to a steady state final one, but it is about how the transition takes place.
The semi empirical side of the presented model gives it the advantage of being able to be refined and updated when further data are provided. This allows the improvement of the underlying approximations and assumptions, and hence the model results . Moreover, it is important to know the exact conditions and descriptions of the room under study, in order to have outcomes that are more precise. Additionally, as suggested by ref [31], the model can be polished further by putting in the scheme the radon areal release rate E s as observed from the constructed walls in the room instead of using the one measured from a sample of the constructing material.

Conclusion
Time variation of the radon indoor concentration has been studied by the application of the mass balance equation on the radon indoors . Each source and sink of radon was considered. The resultant equation was solved after simplifying its form by involving some approximate relations that were developed from empirical observations. An analytical formula was reached that well described the data of radon indoor concentration . As a result, some significant parameters that describe the entrance of radon into buildings were predicted. It is one of the merits of this model that it can predict radon dif fusion coefficients without going through their dependence on the structural specifications of the soil and the building material , like porosity, water saturation fraction, etc. The scheme is designed for describing not Fig. 4 Results of the model for estimating the radon indoor concentration as compared to the data of ref [30] long time-span, which is mostly around two days. The model has the important advantage that it can easily be upgraded any time , once new empirical data are available.
In spite of the encouraging results , the model 's outcomes still require a comparison to be made to some measurements . In a prospective future work, the model applicability will be tried to be extended by applying it successively on consecutive short periods to describe a long period overall. Meanwhile, if suitable measurements are to be available, a comparison will be made against the results expected by the model.

Conflict of interest
The author declares that there is no conflict of interest.
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