Thermo-diffusional radon waves in soils

Science of the Total Environment 565 (2016) 1–7

Contents lists available at ScienceDirect

Science of the Total Environment journal homepage: www.elsevier.com/locate/scitotenv

Thermo-diffusional radon waves in soils Leonid Minkin a,⁎, Alexander S. Shapovalov b a b

Portland Community College, 12000 SW 49th Ave, Portland, OR 97219, USA Saratov State University, 83 Astrakhanskay Street, Saratov 410012, Russian Federation

H I G H L I G H T S

G R A P H I C A L

A B S T R A C T

• Temperature oscillations in atmosphere generate radon waves in soil. • Radon flux in atmosphere is a harmonic function of time. • Radon concentration waves in soil have the same frequency as the temperature waves.

a r t i c l e

i n f o

Article history: Received 6 January 2016 Received in revised form 17 April 2016 Accepted 18 April 2016 Available online xxxx Editor: D. Barcelo Keywords: Radon Thermo-transpiration Temperature waves Radon oscillations Atmosphere

a b s t r a c t A new theoretical framework for diurnal and seasonal oscillations of the concentration of radon in soil and open air is proposed. The theory is based on the existing temperature waves in soils and thermo-diffusional gas flux in porous media. As soil is a non-isothermal porous medium, usually possessing a large fraction of microscopic pores belonging to Knudsen's free molecular field, a thermo-diffusional gas flow in soil has to arise. The radon mass transfer equation in soil for sinusoidal temperature oscillations at the soil–atmosphere boundary is solved, which reveals that radon concentration behaves as a damped harmonic wave. The amplitude of radon concentration oscillations and phase shift between radon concentration oscillations and soil temperature depend on the radon diffusion coefficient in soil, rate of radon production, soil thermal conductivity, average soil temperature, decay constant, and heat of radon transfer. Primarily numerical calculations are presented and comparisons with experimental data are shown. Published by Elsevier B.V.

1. Introduction

⁎ Corresponding author. E-mail address: [email protected] (L. Minkin).

http://dx.doi.org/10.1016/j.scitotenv.2016.04.131 0048-9697/Published by Elsevier B.V.

The correlation between atmospheric temperature fluctuations and radon flux has been the subject of numerous investigations. Many researchers have observed annual and diurnal oscillations of 222Ra (radon) and 220Rn (thoron) fluxes. To our knowledge, the first paper

2

L. Minkin, A.S. Shapovalov / Science of the Total Environment 565 (2016) 1–7

which revealed correlation between outdoor radon concentration and air temperature is dated to 1956 (Okabe, 1956). Since then, monitoring of radon concentration in air is continuing all around the world. Besides the theoretical interest of this correlation, the knowledge of seasonal and diurnal radon fluctuations in soil and air has a practical application. It is of importance for the predictions of the natural disasters (radon concentration change can be a precursor indicative of earthquakes, tornados, hurricanes (Cigolini et al., 2001; Planinic et al., 2004; Tsvetkova et al., 2005; Crockett et al., 2006)), radiation monitoring over caves with tourist business (Barbosa et al., 2010), for risk assessment of radiation levels at sites of radioactive waste burial, and safety of mine workers (Kant et al., 2010). Atmospheric diurnal radon and thoron concentrations typically reach their maxima in early morning (Schery and Grumm, 1992; Podstawczyńska et al., 2010) and seasonal radon concentrations usually have maxima in winter. However, some authors found deviations from this general tendency. Schery et al. (1984) observed the enhancement of flux density in the afternoon and Gesell (1983) gives thirteen references of measurements of radon diurnal and annual variations made in different countries where reported maxima of seasonal values occurred in different seasons (winter, fall, and summer) depending upon the location. Reiter (1978); Grumm et al. (1990); Porstendörfer et al. (1991), and Sesana et al. (2003) accepted an eddy diffusion model to explain the periodic diurnal oscillations of radon concentration. But Schery et al. (1989) find it difficult to believe that such effects of turbulent mixing are significant. It is also not easy to explain the radon diurnal oscillations at the 0.57 m depth, recorded by Schery et al. (1984), based on the turbulent mechanism. Gesell (1983) also concluded that seasonal radon fluctuations could not be explained well by turbulent transfer model. As radon concentration oscillations correlate with temperature oscillations, a suggestion was made that thermal advection of air in the soil might be responsible. However calculations made by Schery and Petscek (1983) indicate that for real values of permeability and other soil parameters, advection is not a reasonable factor in this phenomenon. Since there is no consensus and there is no convincing analytical model, further research is needed in this topic. Merrill and AkbarKhanzaden (1998) and Jilek et al. (2014) investigated the effect of climatic conditions (such as barometric pressure, thermal air gradient, relative air humidity, wind speed and direction and solar radiation intensity) on atmospheric radon concentration diurnal and seasonal oscillations. They collected a significant amount of experimental data, but could not discover the mechanism responsible for periodic radon concentration oscillations in the air. There are also difficulties in explaining radon concentration fluctuations in soil and caves. Winkler et al. (2001) found that on average, at all sampling positions in the test field and nearly at all-times radon concentration at 0.5 m depth was significantly higher than at 1 m depth in contrast to theory and some field experiments (Al-Shereideh et al., 2006). Radon concentration oscillations in underground locations such as caves, tunnels, and basements of the buildings are often opposite to that above the ground level i.e., in winter, they were lower than in summer (Li et al., 2006; Perrier et al., 2007; Dueñas et al., 2011). Although extensive research efforts have been made to develop an adequate model of radon transport and numerous papers were devoted to diurnal and seasonal radon concentration oscillations in the atmosphere, the models' predictions are still very often in conflict with the experimental data. Clarification of radon movement in soils is needed to predict radon fluctuations, and new ideas have to be brought forth to explain the mechanism of seasonal and diurnal radon oscillations. Goldman et al. (1987, 1992) first proposed the idea that thermodiffusion (thermo-transpiration) is a dominant mechanism of air exchange in soil. Minkin (2001, 2002, 2003) developed this idea and theoretically and experimentally showed that thermogradient is a driving force of radon indoor entry. The review (Minkin and Shapovalov, 2008) of published papers related to radon transport revealed that

there are numerous misconceptions about the mechanisms of radon infiltrations into homes and thermo-diffusion should be considered as a competitive (Zafrir et al., 2013) and in some cases dominant mechanism of radon transport in soil (Dong et al., 2004; Minkin and Shapovalov, 2007). This fact makes it reasonable to apply the theory of thermodiffusion radon transport to explain diurnal and seasonal outdoor radon oscillations. This research article presents a novel theory of these fluctuations based on thermo-diffusion. All calculations below are made in the international system of units. 2. Transport equations Natural soils are generally not homogeneous. Mineral composition, structure, and water content vary with depth and location. However, the assumption of a homogeneous soil may still be reasonable and is widely used in soil physics and in solving problems of radon transport (Van Wijk and de Vries, 1963; Clements and Walkening, 1974; Marshall and Holmes, 1979; Sposito, 1976; Van Der Spoel et al., 1997). To facilitate the mathematical treatment some approximations are made: 1. The soil is homogeneous. 2. All variables change only in vertical direction (x is depth positive downward; at the soil surface x = 0). 3. No heat is generated in soil or converted into other form of energy. 4. Soil is a porous medium. 5. Diurnal and seasonal temperature oscillations on the soil– atmosphere border are periodic. 6. Soil's intrinsic properties are not functions of time (often used assumption: Clements and Walkening, 1974; Schery et al., 1984, 1989; Ota and Yamazawa, 2010). Using these assumptions, the temperature distribution in soil can be described as a damping harmonic wave function (Van Wijk, 1963; Van Wijk and de Vries, 1963; Marshall and Holmes, 1979; Hillel, 1982) T ðx; t Þ ¼ T 0 þ T 1 ðx; t Þ ¼ T 0 þ T 10 expð−αxÞ cosðωt−αxÞ where T0 is the average soil temperature, T10 is the amplitude of temperqffiffiffiffiffiffiffiffiffi ω is the damping ature oscillations at the earth surface, α ¼ C soil 2κ constant (angular wave number), κ is thermal conductivity, Csoil is volumetric heat capacity of soil, t is time, ω is angular seasonal or diurnal frequency (ωd = 2π day−1 = 2π / 86,400 s−1 = 7.27 × 10−5 s−1 for diurnal variations and ωs = 2π year−1 = 1.99 × 10−7 s−1 for seasonal variations). The variable part of the last equation can be written in a complex form T 1 ðx; t Þ ¼ T 10 exp½−αxð1 þ iÞ expðiωt Þ

ð1Þ

pffiffiffiffiffiffiffiffi where i ¼ −1 and T10(x, t) exp[−αx(1 +i)] is a complex amplitude of temperature oscillations at the depth x. This equation shows that at the depth x the amplitude of temperature oscillations is smaller than T10 by a factor exp(−αx) and that there is a phase shift, −αx, between temperature oscillations on the soil surface and at depth x. Thermal conductivity, κ, and volumetric heat capacity, Csoil, depend on the type of soil, porosity, humidity and have a large range. For typical soil κ = 1.2 J m−1 s−1 and Csoil = 2.0 × 106 J m−3 °C−1 (Van Wijk, 1963) and therefore one can calculate typical diurnal and seasonal damping lengths l = 1/α (at the depth x = l the amplitude of oscillations is 1/e ≈ 0.37 times of the amplitude at the surface); they are ld = 0.13 m and ls = 2.5 m. For real soils, parameters κ and Csoil have very large ranges and therefore damping length for the diurnal variation has approximately the range from 0.03 m to 0.16 m; the damping length pffiffiffiffiffiffiffiffiffi for seasonal variations is 365≅ 19 times greater than the diurnal one. Real soils have pore radius distributions that depend on many factors. In general, a large fraction of these pores have radius smaller

L. Minkin, A.S. Shapovalov / Science of the Total Environment 565 (2016) 1–7

than the mean free path of air molecules (Marshall and Holmes, 1979; Wu et al., 1990) which is approximately 0.7 μm. For example, silty clays exhibit a bi-modal distribution with a pore diameter of approximately 0.2 μm and 4 μm (Garcia-Bengochen et al., 1979); Walla-Walla (coarse, silty mixed, mesic Typic Haploxerolls) and Bashaw (very fine, montmorillonitic, mesic Typic Pelloxereeta) soils have mean diameters 0.7 μm and 0.06 μm (Goldman and Minkin, 1994). As a large portion of soil pores have a diameter less than mean free path of gas molecules, the gas flow in these pores has to be described as Knudsen's free molecular gas flow (Patterson, 1956). It is well known that a temperature gradient causes gas transfer in these pores even if there is no pressure differential (Heer, 1972; Clifford and Hillel, 1986; Lifshitz and Pitaevski, 1986; Graur and Sharipov, 2009). Irreversible thermodynamics also predicts mass transfer in a multi-component non-isothermal system (Prigogine, 1955; De Groot, 1963; Waldram, 1987). Experimental data confirm this thermo-diffusion effect (Grew and Ibs, 1952; Oriani, 1969; Rutherford, 1973; Stark, 1976; Golubentsev et al., 1983, 1984; Sposito, 1976) and specifically thermo-diffusion gas transport in soils (Goldman et al., 1987, 1992). A phenomenological approach is used in this research to describe mass transfer in a two-component system — solid soil matrix and soil gas. Gas mass flux, J, in soil depends on a concentration gradient and temperature gradient (thermal diffusion) (Bokstein et al., 1974; Goldman et al., 1987, 1992) J ¼ −D

∂C DCQ  ∂T þ ∂x kT 2 ∂x

ð2Þ

where D is an effective diffusion coefficient, C is gas concentration in soil, k is Boltzman's constant, Q* is heat of radon transfer in soil. The effective diffusion coefficient and heat of transfer of a specific gas in porous media filled by air depend on pore size distribution and their connectivity, Knudsen's number, pore porosity, tortuosity, and internal microstructure of the pores (Pollard and Present, 1948; Mason and Malinauskas, 1983; Zalc et al., 2004; Mu et al., 2008). In the frame of non-equilibrium thermodynamics, these parameters cannot be found but must be defined from the kinetic theory or experimental data. For radon transport in soil the mass balance equation is (Culot et al., 1976; Nielson et al., 1997) ∂C ∂J ¼ − þ S−λC ∂t ∂x

ð3Þ

where S is the rate of radon production in soil and λ is the radon decay constant. Eqs. (2) and (3) give the differential equation for radon concentration distribution in soil   2 ∂C ∂ C ∂ CQ  ∂T þ SðxÞ−λC: ¼ þD 2 −D 2 ∂t ∂x ∂x kT ∂x

ð4Þ

Eq. (4) is linear and therefore its solution can be presented as the sum of time-independent and time-dependent parts. As J = J0(x) + J1(x,t), C = C0(x) + C1(x,t), T = T0 + T1(x,t), S = S0(x) + S1(x,t), and under the assumption that T0 ≫ T1, S1(x,t) = 0, 0 1 C0 ≫ C1 and ∂C ≫ ∂C , Eq. (4) can be separated into two equations: ∂x ∂x time-independent and time-dependent parts

2

D

d C 0 ðxÞ þ SðxÞ−λC 0 ¼ 0 dx2

  ∂C 1 ðx; t Þ ∂ C 1 ðx; t Þ ∂ ∂T 1 ðx; t Þ −λC 1 ðx; t Þ −DA ¼D C 0 ðxÞ 2 ∂t ∂x ∂x ∂x

ð5Þ

where A ¼

thermo-diffusion radon mass transfer. Eq. (5) is solved for different boundary conditions and the function C0(x) will be considered as the known function. Eq. (6) can be written in the form 2

2

∂C 1 ∂ C1 ∂T 1 ∂ T1 ¼ D 2 −a −b 2 −λC 1 ∂t ∂x ∂x ∂x

ð6Þ

Eq. (5) is the time stationary radon diffusion equation in

a temperature homogeneous field (Gadd and Borak, 1995; Van der Spoel et al., 1997) and C1(x,t) is radon concentration related to

ð7Þ

0 where ¼ DAB; B ¼ ∂C ; b ¼ DAC 0 . One has to keep in mind that the ∂x coefficients a and b in Eq. (7) depend on C0, i.e. solution of Eq. (5). 2

1 ¼ −αð1 þ iÞT 1 and ∂∂xT21 ¼ 2α 2 iT 1 ; the sum As ∂T ∂x

2

−a

∂T 1 ∂ T1 −b 2 ¼ αT 1 ½að1 þ iÞ−2bαi ¼ f ðx; t Þ ∂x ∂x

is a wave function with amplitude and phase modulation f ðx; t Þ ¼ α ½að1 þ iÞ−2baiT 10 exp½−αxð1 þ iÞ expðiωt Þ

ð8Þ

and therefore Eq. (7) is an nonhomogeneous linear differential equation with time-periodic term f(x,t) 2

∂C 1 ∂ C1 ¼ D 2 −λC 1 þ f ðx; t Þ: ∂t ∂x

ð9Þ

Eqs. (7) and (9) are found in assuming the existence of temperature waves in soil, presented by Eq. (1). The temperature wave in this form was obtained from a model for which a transient part of solution of the thermo-conductivity equation in soil is not the subject of interest, and only the periodic part of solution is found. It means that at the soil–atmosphere interface, harmonic temperature oscillations exist infinitely long (boundary condition) and Eq. (1) describes the temperature distribution in soil after a long time regardless of initial conditions. After a long period of time, all oscillations in soil will die off except for the driven temperature oscillations caused by temperature oscillations at the soil–atmosphere interface. Definitely, the same temperature wave equation will hold for the initial condition defined by Eq. (1) for t = 0. The temperature oscillations in soil have the same frequency as the frequency of temperature oscillations at the soil–atmosphere interface (Farlow, 1982). Eq. (9) is a nonhomogeneous partial differential equation of parabolic type. It is reasonable to assume that the “almost periodic force”, f(x,t), causes driven concentration oscillations (Petrovskii, 1967; Farlow, 1982). A transient part of the solution of Eq. (7) is not of interest in this research (this part exponentially decays in the course of time) and therefore, the initial condition is not important. Only the stationary part of the solution of Eq. (7) will be analyzed. 3. Stationary radon concentration oscillations in soil (forced radon oscillations) The central purpose of this research is to find the radon concentration distribution in soil and radon flux in atmosphere caused by temperature gradient oscillations. Eq. (9) does not have an analytical solution and should be solved numerically. Now in attempt to solve this equation some substantial simplifications are assumed. By neglecting the transient term, it is reasonable to find the solution of Eq. (9) in a quasi-wave form C 1 ðx; t Þ ¼ C 10 exp½−αxð1 þ iÞ expðiβÞ expðiωt Þ ¼

2

Q . kT 20

3

T1 C 10 expðiβÞ ð10Þ T 10

where C10 is the amplitude of radon concentration oscillations at x = 0, and β is a complex and slow changing (x-derivatives of this function are neglected) function of x. This last assumption is chosen to simplify the problem and make preliminary conclusions about the hypothesis proposed. We are looking for the solution of Eq. (9) in the wave form

4

L. Minkin, A.S. Shapovalov / Science of the Total Environment 565 (2016) 1–7

Eq. (10) with space modulated amplitude and phase. Eq (10) is a wave function that hasing the same frequency as the temperature wave but a different phase. Substituting Eqs. (8) and (10) in Eq. (9) gives the relation between C1(x,t) and T1(x,t)   h i 2 C 1 iω−2Diα 2 þ λ ¼ T 1 aα ð1 þ iÞ−2bα i which can be written in the form μ ðxÞ ¼

C1 aα ð1 þ iÞ−2bα 2 i ¼ =μ= expðiβÞ ¼ : T1 iω−2Dα 2 i þ λ

ð11Þ

Here μ is a complex function of x, which presents information about the magnitude of radon concentration and phase shift β between temperature and radon oscillations (β ¼ tan−1 ½ImðμÞ ReðμÞ ; ReðμÞ and ImðμÞ are real and imaginary parts of μ for every specific depth). In general, parameters a and b are functions of x, but not t and therefore, the parameter β depends only on x. Thus for every specific depth, radon concentration oscillations are sinusoidal in time. At the soil–atmosphere interface, these oscillations are sinusoidal as well and, therefore, according to Eq. (2), radon flux and concentration in open air, caused by thermo-diffusion, are harmonic functions of time. So, it is shown that the wave function, defined by Eq. (10), is the solution of the differential equation of radon mass transfer Eq. (7). This wave has the same frequency as the temperature wave in soil and Eq. (11) defines its amplitude and phase. The measurements of outdoor radon concentration (Figs. 1, 2) confirm that diurnal and seasonal radon oscillations can be very close to a sinusoidal function predicted by the theory proposed. Strictly speaking, the parameters a and b are functions of x. However, qffiffiffi for x much greater than the radon diffusion length in soil L ¼ Dλ , C0 is 0 constant, ∂C ≈0, parameter a ≈ 0, and b is constant. For example, for ∂x semi-infinite soil and constant S and D, the solution of Eq. (5) with the boundary condition C0(0) = 0 (Schery et al., 1984) is an exponential function

C 0 ðxÞ ¼ C 0 ð∞Þ½1− expð−x=LÞ

ð12Þ

0 ¼ 0, parameter a = 0 ,C0(x) = C0(∞) where C0(∞) = S/λ. If x ≫ L, B ¼ ∂C ∂x and parameter b is constant. The diffusion length, L, depends on a radon diffusion coefficient, D, which has a very large range. Its upper limit is the radon diffusion coefficient in open air, which is about

Fig. 2. Sinusoidal fit of monthly radon outdoor concentrations (solid line — Miles et al., 2012, values refer to the middle of each month; dashed line — Pinel et al., 1995, values refer to the first day of each month).

1.1 × 10−6 m2 s−1. At the lower extreme, in fully saturated soil, the radon diffusion coefficient may be as low as 1 × 10−10 m2 s−1. Decay constants for 222Rn and 220Rn (thoron) are λ222 = 2.1 × 10− 6 s− 1 and λ220 = 1.3 × 10− 3 s− 1. Taking a typical D = 2.6 × 10− 6 m2 s− 1 (Schery et al., 1984), one can calculate diffusion length for 222Rn and 220Rn: L222Rn = 1.1 m, L220Rn = 0.045 m. 4. Sample calculation To support the model proposed, a comparison with experimental data is demonstrated. As many parameters are involved to make calculations of radon fluctuations in soil, it is reasonable not to consider the absolute values of temperature and concentration, but instead their normalized oscillations T1(x,t)/T10 and C1(x,t)/C10 are used, for which the amplitudes are equal to one and dimensionless. Fig. 3 gives the graph of normalized diurnal theoretical and experimental temperature and radon concentration oscillations in soil, which are close to harmonic functions and well correlated. Fig. 4 illustrates the experimental and theoretical seasonal normalized radon concentrations in soil. One can see that the radon concentration variations are close to the harmonic functions both for diurnal and seasonal oscillations as predicted by the model proposed in this research. Figs. 3 and 4 illustrate the radon and temperature variations for the given coordinate (depth) x, while Fig. 5 shows the radon concentration variation with depth for the given time. Parameters of the theoretical

Fig. 1. Diurnal outdoor radon concentration in January and July 2000 (Sesana et al., 2003).

L. Minkin, A.S. Shapovalov / Science of the Total Environment 565 (2016) 1–7

5

Fig. 3. Normalized diurnal temperature and 222Rn concentration oscillations in soil at the depth 0.57 m in October 1982. Series 1 — experimental data of 222Rn concentration oscillations (Schery et al., 1984). Series 2 — experimental data of temperature oscillations (Schery et al., 1984). Series 3 — theoretical temperature oscillations.

curve for Fig. 5 are also chosen to adjust the experimental data and show that the diffusion and thermo-diffusion transport mechanism can explain the radon concentration distribution in soil. To make numerical calculations for radon concentration oscillations in soil, one has to use Eqs. (11) and (12). Many parameters must be defined in this case. However, some simple calculations can be done for limited cases. For example, for 220Rn ω ≪ λ, 2Dα2 ≪ λ and for x ≫ 0.03 m and x ≫ L, therefore parameter a ≈ 0. In this case, according to Eq. (11), parameter μ is equal

μ ðxÞ ¼

−2bα 2 i : λ

ð13Þ

Inspection of Eq. (13) shows that μ is a negative imaginary number. Therefore, in the considered case the phase shift between temperature and radon oscillations is π/2, i.e. quantities T1(x,t) and C1(x,t) are onequarter-cycle out of phase. Furthermore, we can see that the T1(x,t) oscillations leads C1(x,t).

Fig. 5. Typical 220Rn concentration distribution for summer time (profile of June 29, 1987) in central Pennsylvania soil. Solid line — experimental data (Rose et al., 1988). Dash line — theoretical distribution: C = C0 + C1, C0 = 29.000[1− exp(− 25×)], C1 = 77700exp(−x)sin(x + 2.7).

Unfortunately, there is a difficulty in making numerical calculations because there is no experimental data for the heat of transfer, Q*, of radon in soil. It involves a three component system, containing air, radon and solid soil matrix and specific experiments must be performed to find this parameter. Q* has a very large range. For example, for thermo-diffusion neon in glass and carbon in austenite Q* = 4 × 10− 19 J (Stark, 1976; Golubentsev et al., 1984) while for T0 = 298 K heat transfer of air in clay Q* = 0.8 × 10− 21 J (Minkin and Shapovalov, 2007). Let us calculate seasonal radon fluctuations and compare them to time independent concentration by choosing the following typical parameters of soil (Marshall and Holmes, 1979) D = 2.6 × 10−6 m2 s−1, κ = 1.2 J m−1 s−1 °C−1, Csoil = 2.0 × 106 J m−3 s−1 °C−1, T0 = 298 K and T10 = 10 K and Q* = 1.0 × 10− 20 J. Substituting these data in Eq. (12) defines time independable radon concentration C 0 ðxÞ ¼ C 0 ð∞Þ½1− expð−0:90xÞ

ð14Þ

and parameters a, b, A, α, and B can be found Q

∂C 0 Bq ¼ 0:90C 0 ð∞Þ expð−0:90xÞ 4 ; b m ∂x Bq ¼ DAC 0 ¼ 2:1  10−8 C 0 ðxÞ ; a ¼ DAB sKm Bq ; ∝diurnal ¼ 1:9  10−8 C 0 ð∞Þ expð−0:90xÞ s K m2 ¼ 7:8 m−1 ; ∝seasonal ¼ 0:41 m−1 :

A¼

Fig. 4. Normalized seasonal 222Rn concentration oscillations at the depth 1.07 m in 1987/ 1988 in a Central Pennsylvania soil. •Experimental data (Rose et al., 1988). ■Theoretical harmonic function.

kT 20

¼ 8:1  10−3 K −1 ; B ¼

As an example, for seasonal radon fluctuations and x = 0.50 m one can find C0(0.50) = 0.36C0(∞) , a = 1.15 × 10− 8C0(∞) , B = 0.57C0(∞) , b = 0.75 × 10− 8C0(∞), and according to Eq. (11), μ = (0.025 + 0.0089i) C0(∞), that allows us to calculate μ = 0.026C0(∞) and phase shift between temperature and radon concentration oscillation β = 0.34. Using Eqs. (1, 11, and 14) oscillations of temperature

6

L. Minkin, A.S. Shapovalov / Science of the Total Environment 565 (2016) 1–7

and radon concentration and time-independent radon concentration component at 0.50 m depth can be written as T 1 ðx; t Þ ¼ T 1 ð0:1; t Þ ¼ T 10 expð−αxÞ cosðωt−αxÞ   ¼ 8:1 cos 2:0  10−7 t−0:21   C 1 ðx; t Þ ¼ C 1 ð0:5; t Þ ¼ 0:26C 0 ð∞Þ cos 2:0  10−7 t þ 0:13 C 0 ð0:5Þ ¼ 0:36C 0 ð∞Þ: For this particular case the ratio of amplitude of radon oscillations C10(0.5) = 0.26C0(∞) to time independent radon concentration C0(0.5)=0.36C0(∞) is CC100 ¼ 0:72. The example indicates that forced radon oscillations, caused by the temperature gradient oscillations, can be substantial in comparison with the time independent radon concentration in soil. It also has to be mentioned that, although an amplitude of radon oscillations C10(x) can be much smaller than C0(x), the influence of radon concentration variations in soil on the radon flow into the atmosphere can be large even in this case because radon flux into atmosphere from the soil depends, according to Eq. (2), on four terms J ðx; t Þ ¼ −D

∂C 0 ðxÞ ∂C 1 ðx; t Þ DC 0 Q  ∂T ðx; t Þ DC 1 Q  ∂T ðx; t Þ −D þ þ ∂x ∂x ∂x ∂x kT 2 kT 2

three of which are connected with the temperature gradient (thermodiffusion process). These three terms depend not only on the absolute values of the oscillations of radon concentration and temperature but also on their derivatives, which can be significantly large. As C1(x, t) and T(x,t) are periodic functions of time, J(x, t) is also a periodic function of time. Therefore periodic radon flux at the soil–atmosphere interface J(0, t), may be an alternative explanation of diurnal and seasonal radon oscillations in atmosphere. 5. Conclusion • Harmonic temperature oscillations at the soil–atmosphere interface generate seasonal and diurnal fluctuations in radon concentration that behave as damped sinusoidal waves in soil. In this phenomenological approach, the temperature gradient is a thermodynamic force, which causes radon flux in soil. Kinetic theory also establishes that the temperature gradient is the driving force of radon flux in soil. • Radon flux in atmosphere is a harmonic function of time and therefore the radon diurnal and seasonal concentration variations in the atmosphere are also harmonic. • Radon concentration waves in soil have the same frequency as the temperature waves. • Amplitude and phase of radon waves in soil depend on radon diffusion coefficient in soil, soil thermal conductivity, average soil temperature, decay constant, heat of radon transfer, and rate of radon production; therefore, the maximum radon concentration in open air depends on these factors as well, and can be different depending on the time of day and season. The lack of experimental data for the parameter of the heat transfer of radon in soils (Q⁎) is a point of some uncertainty in numerical calculations. Future research efforts of finding this parameter for different soils and moisture content should be conducted to prove proposed theory. • Comparison with experimental data confirms the existence of harmonic radon concentrations in soil and the plausibility of using the concept presented to explain diurnal and seasonal radon variations in soil and atmosphere. • The findings of this research are grounded in the solution of the partial differential equation of the radon mass transfer and show that radon concentration in soil is really a wave function defined by Eqs. (9) and (10). Some simplifications were made to solve the differential equation.

• This research is a novel attempt to explain the numerous experimental data of radon concentration oscillations in both soil and atmosphere. It presents mostly a conceptual point of view and does not have the intention to give the precise magnitude of the thermodiffusion gas flux in soil. One of the reasons for this is that the equation of mass transfer is taken from the theory of irreversible thermodynamics and the coefficients of heat and mass transfer cannot be derived in the frame of this theory but have to be taken from the kinetic theory or experiment. The assumption of time independence of soil parameters is also contrived but widely used in radon transport theory. • Theory proposed is applicable not only to radon movement in soil but also to volatile organic compounds and gaseous exchange between soil and atmosphere.

Acknowledgment The authors thank Dr. Sikes for the careful review of the paper and for making valuable suggestions. References Al-Shereideh, S.A., Bataina, B.A., Ershaidat, N.M., 2006. Seasonal variations and depth dependence of soil radon concentration levels in different geological formations in Deir Abu-Said District Irbid—Jordan. Radiat. Meas. 41, 703–707. Barbosa, S.M., Zafrir, H., Malik, U., Piatibratova, O., 2010. Multiyear to daily radon variability from continuous monitoring at Amram tunnel, Southern Israel. Geophys. J. Int. 182, 829–842. Bokstein, B.S., Bokstein, S.Z., Zhukovitsky, A.A., 1974. Thermodynamics and Diffusion Kinetics in Solids (in Russian). Metallurgia, Moscow. Cigolini, C., Salierna, S., Gervino, G., Bergese, P., Marino, C., Russo, M., Prati, P., Ariola, V., Bonetti, R., Begnini, S., 2001. High-resolution radon monitoring and hydrodynamics at mount Vesuvius. Geophys. Res. Lett. 28 (#21), 4035–4038. Clements, W.E., Walkening, M., 1974. Atmospheric pressure effect on 222Rn transport across the earth–air interface. J. Geophys. Res. 79, 5025–5029. Clifford, S.M., Hillel, D., 1986. Knudsen diffusion: effect of small pore size and low gas pressure on gaseous transport in soil. Soil Sci. 141, 289–297. Crockett, R.G., Gillmore, G.K., Phillips, P.S., Denman, A.R., Groves-Kirkby, C.J., 2006. Radon anomalies preceding earthquakes which occurred in the UK, in summer and autumn 2002. Sci. Total Environ. 64, 1–3. Culot, M.V.J., Olson, H.G., Schiager, K.J., 1976. Effective diffusion coefficient in concrete, Theory and method for field Measurements. Health Phys. 30, 263–270. De Groot, S.R., 1963. Thermodynamics of Irreversible Processes. North-Holland Publishing Company. Dong, L.F., Qiu, L., Forest, L., 2004. The thermos-diffusion model explains the changes of radon concentration with temperature. Journal of Nanhua University. Science and Engineering Edition. Vol. 18, pp. 54–57 (Mar). Dueñas, C., Fernández, M.C., Cañete, S., Pérez, M., Gordo, E., 2011. Seasonal variations of radon and the radiation exposure levels in Nerja cave, Spain. Radiat. Meas. 46, 1181–1186. Farlow, S.J., 1982. Partial Differential Equations for Scientists & Engineers. John Wiley & Sons. Gadd, M.S., Borak, T.B., 1995. In situ determination of the diffusion coefficient of 222Rn in concrete. Health Phys. 68, 817–822. Garcia-Bengochen, I., Lovell, C.W., Altschaeffl, A.G., 1979. Pore distribution and permeability of silty clay. J. Geotech. Eng. Div. 7, 839–856. Gesell, T.F., 1983. Background atmospheric 222Rn concentration outdoors and indoors: a review. Health Phys. 45, 289–302. Goldman, S., Minkin, L., 1994. Non-isothermal sorption gaseous exchange in a synthetic soil. J. Environ. Qual. 23, 180–187. Goldman, S., Minkin, L.M., Makarov, V.P., 1992. Non-isothermal gaseous exchange between soil and atmosphere. Chemosphere 24, 1961–1988. Goldman, S., Minkin, L.M., Myasnikov, N.G., 1987. Non-isothermal rotational air exchange in soil. Soviet Soil Sci. #5, 61–71. Golubentsev, A.F., Goldman, S.Y., Minkin, L.M., Rabinovich, E.M., 1984. Spectroscopic determination of thermodiffusion constant of binary gas mixture. J. Eng. Phys. Thermophys. 47, 628. Golubentsev, A.F., Goldman, S.Y., Minkin, L.M., 1983. Diffusion of He and Ne through the glass walls of the gas ballon. J. Eng. Phys. Thermophys. 45, 332. Graur, I., Sharipov, S., 2009. Non-isothermal flow of rarefied gas through a long pipe with elliptic cross section. Microfluid. Nanofluid. 6, 267–275. Grew, K.E., Ibs, T.L., 1952. Thermal Diffusion in Gases. Cambridge University Press. Grumm, D., Schery, S., Whitteletone, S., 1990. Two-filter continuous monitor for low level 220 Rn and 220Rn. Proceeding of the 1990 International Symposium on Radon and Radon Reduction Technology. EPA/600/9-90/0056 US Environmental Protection Agency, Research Triangle Park, NC, Paper B-III-4. Heer, S.V., 1972. Statistical Mechanics, Kinetic Theory, and Stochastic Processes, NY. Hillel, D., 1982. Introduction to Soil Physics. Academic Press.

L. Minkin, A.S. Shapovalov / Science of the Total Environment 565 (2016) 1–7 Jilek, K., Slezákova, M.J., Tomas, J., 2014. Diurnal and seasonal variability of outdoor radon concentration in the area of the NRPI Prague. Radiat. Prot. Dosim. 160, 57–61. Kant, K., Kuriakose, S., Rashmi, Sharma, G.S., 2010. Radon activity and radiation dose level in the slate mines in aravali range in India. Indian J. Pure Appl. Phys. 48, 463–465. Li, X., Zheng, B., Wang, Y., Wang, X., 2006. A study of daily and seasonal variations of radon concentrations in underground buildings. J. Environ. Radioact. 87, 101–106. Lifshitz, E.M., Pitaevski, L., 1986. Statistical Physics. Pergamon Press. Marshall, T.J., Holmes, J.W., 1979. Soil Physics. Cambridge University Press. Mason, E.A., Malinauskas, A.P., 1983. Gas Transport in Porous Media. Elsevier, The Dusty Gas Model. Merrill, E.A., Akbar-Khanzaden, F., 1998. Diurnal and seasonal variations of radon level. Effects of climate conditions, and radon exposure assessment in a former uranium metal production facility. Health Phys. 74, 6568–6573. Miles, J.C.H., Howard, C.B., Hunter, N., 2012. Seasonal variation of radon concentrations in UK homes. J. Radiol. Prot. 32, 275–287. Minkin, L., Shapovalov, A.S., 2007. Heat of transport of air in clay. Radiat. Prot. Dosim. 123, 221–225. Minkin, L., 2001. Thermodiffusion in concrete slab as a driving force of indoor radon entry. Health Phys. 80, 151–156. Minkin, L., 2002. Is diffusion, thermodiffusion, or advection a primary mechanism of indoor radon entry? Radiat. Prot. Dosim. 15, 153–162. Minkin, L., 2003. Thermal diffusion of radon in porous media. Radiat. Prot. Dosim. 106, 267–272. Minkin, L., Shapovalov, A.S., 2008. Indoor radon entry: 30 years later. Iran. J. Radiat. Res. 6, 1–6. Mu, D., Liu, Z., Huang, C., 2008. Determination of the effective diffusion coefficient in porous media including Knudsen effects. Microfluid. Nanofluid. 4, 257–260. Nielson, K.K., Rogers, V.C., Rodger, B.H., Pugh, T.D., Grondzik, W.A., Meijer, R.J., 1997. Radon penetration of concrete slab cracks, joints, pipe Penetration, and sealants. Health Phys. 73, 668–678. Okabe, S., 1956. Time variation of the atmospheric radon-content near the ground surface with relation to some geophysical phenomena. Mem. Coll. Sci. Univ. Kyoto 28A, 99–115. Oriani, R.A., 1969. Thermodiffusion in solid metals. J. Phys. Chem. 30, 339–351. Ota, M., Yamazawa, H., 2010. Forest floor CO2 and radon concentrations. Atmos. Environ. 44, 4529–4535. Patterson, G.N., 1956. Molecular Flow of Gases. John Wiley & Son, Inc., New York. Perrier, F., Richon, P., Gautam, U., Tiwari, D.R., Shrestha, P., Sapkota, S.N., 2007. Seasonal variation of natural ventilation and radon-222 exhalation in a slightly rising deadend tunnel. J. Environ. Radioact. 97, 220–235. Petrovskii, I.G., 1967. Partial Differential Equations. W. B. Saunders Co., Philadelphia. Pinel, J., Fearn, T., Darby, S.C., Miles, J.C.H., 1995. Seasonal correction factors for indoor radon measurements in the United Kingdom. Radiat. Prot. Dosim. 58, 127–132. Planinic, J., Radolic, V., Vukovic, B., 2004. Radon as an earthquake precursor. Nucl. Instrum. Methods Phys. Res., Sect. A 530, 568–574. Podstawczyńska, A., Kozak, K., Pawlak, W., Mazur, J., 2010. Seasonal and diurnal variation of outdoor radon (222Rn) concentrations in urban and rural area with reference to meteorological conditions. Nucleonika 55, 543–547.

7

Pollard, W.G., Present, R.D., 1948. On gaseous self-diffusion in long capillary tubes. Phys. Rev. 73, 762–774. Porstendörfer, J., Butterweck, G., Reineking, A., 1991. Diurnal variation of the concentrations of radon and its short-lived daughters in the atmosphere near the ground. Atmos. Environ., Part A 25 (3–4), 709–713. Prigogine, I., 1955. Introduction to Thermodynamics of Irreversible Processes. M. Thomas. Reiter, E.R., 1978. Atmospheric Transport Processes. US Department of Energy, Springfield, Va. Rose, A.W., Washington, J.W., Greeman, D.J., 1988. Variability of radon with depth and season in a Central Pennsylvania. Soil Developed on Limestone. Northeastern Environmental Science Vol. 7 pp. 35–39. Rutherford, W.M., 1973. Isotopic thermal diffusion factor of Ne, Ar, Kr, Xe from column measurements. J. Chem. Phys. 58, 1613–1618. Schery, S.D., Petscek, A.G., 1983. Exhalation of radon and thoron: the question of the effect of thermal gradient in soil. Earth Planet. Sci. Lett. 56–60. Schery, S.D., Grumm, D.M., 1992. Thoron and its progeny in the atmospheric environment. In: Nriagu, J.O. (Ed.), Gaseous Pollutant: Characterization and Cycling. John Wiley & Sons. Schery, S.D., Gaeddert, D.H., Wilkening, M.H., 1984. Factors affecting exhalation of radon from gravelly sandy loam. J. Geophys. Res. 89, 7299–7309. Schery, S.D., Whittlestone, S., Hart, K.P., Hill, S.S., 1989. The flux of radon and thoron from Australian soils. J. Geophys. Res. 94, 8567–8576. Sesana, L., Capriolli, E., Marcazzan, G.M., 2003. Long period study of outdoor radon concentration in Milan and correlation between its temporal variations and dispersion properties of atmosphere. J. Environ. Radioact. 65, 147–160. Sposito, G., 1976. Solid State Diffusion. Oxford University Press. Stark, J.P., 1976. Solid State Diffusion. John Wiley & Sons. Tsvetkova, T., Przylibski, T.A., Nevinsky, I., Nevinsky, V., 2005. Measurement of radon in the East Europe under the ground. Radiat. Meas. 40, 98–105. Van der Spoel, W.H., Van der Graaf, E.R., De Meijer, R.J., 1997. Diffusive transport of radon in a homogeneous column of dry sand. Health Phys. 72, 766–777. Van Wijk, W., 1963. Physics of Plant Environment. John Wiley & Son, Inc. Van Wijk, W.R., de Vries, D., 1963. Periodic temperature variations in a homogeneous soil. In: Van Wijk, W.R. (Ed.), Physics of Plant Environment. North-Holland Publ. Co., Amsterdam, pp. 102–143. Waldram, J.R., 1987. Theory of Thermodynamics. Cambridge University Press. Winkler, R., Ruckerbauer, F., Bunzl, K., 2001. Radon concentration in soil gas: a comparison of the variability resulting from different methods, spatial heterogeneity and seasonal fluctuations. Sci. Total Environ. 272, 273–282. Wu, L., Vomocil, J.A., Childs, S.W., 1990. Pore size, particle size, aggregate size, and water retention. Soil Sci. Soc. Am. J. 54, 952–956. Zafrir, H., Barbosa, S.M., Malik, U., 2013. Differentiation between the effect of temperature and pressure on radon within the subsurface geological media. Radiat. Meas. 49, 39–56. Zalc, J.M., Reyesb, S., Iglesia, E., 2004. The effects of diffusion mechanism and void structure on transport rates and tortuosity factors in complex porous structures. Chem. Eng. Sci. 59, 2947–2960.