Health Physics Pergamon Press 1976. Vol. 30 (March), pp. 263-270. Printed in Northern Ireland
EFFECTIVE DIFFUSION COEFFICIENT OF RADON IN CONCRETE, THEORY AND METHOD FOR FIELD MEASUREMENTS* MICHEL V. J. CULOT
Visiting Scientist, I.N.E.N., Centro Nuclear de Mexico, Apartado Postal 27-190, Mexico 18, D. F. HILDING G . OLSON
Associate Professor, Mechanical Engineering Dept, Colorado State University, Fort Collins, CO 8052 1 and KEITH J. SCHIAGER
Professor of Health Physics, Dept. of Radiation Health, Graduate School of Public Health, University of Pittsburgh, Pittsburgh, PA 15261
(Received 15 January 1975; accepted 21 August 1975) Abstract-A linear diffusion model serves as the basis for determination of an effective radon diffusioncoefficient in concrete. The coefficientwas needed to later allow quantitative prediction of radon accumulationwithin and behind concrete walls after application ofanimpervious radon barrier. A resolution of certain descrepancies noted in the literature in the use of an effective diffusion coefficient to model diffusionof a radioactive gas through a porous medium is suggested. An outline of factors expected to affectthe concrete physical structure and the effective diffusion coefficient of radon through it is also presented. Finally, a field method for evaluating effective radon diffusion coefficientsin concrete i s proposed and results of measurements performed on a concrete foundation wall are compared with similar published values of gas diffusion coefficients in concrete. INTRODUCTION
tailings contain small quantities of radium, the DIFFUSION of radon across concrete boundaries precursor of radon, and were used in commercial of structures is a phenomenon of interest to and residential structures as leveling courses those health physicists concerned with the radon and as backfill under and around concrete progeny levels inside ordinary buildings (YEATES foundations (CULOTet al., 1973). Suppression et al., 1972). The same diffusion process is a of radon influx by application of a radon barrier concern for those responsible for monitoring on the indoor faces of these structures’ foundaleaks in facilities handling alpha emitters, since tions was expected to result in increased radon spurious alarms can be triggered by detection concentrations in and around the foundations. of radon progeny when gross alpha monitors The necessity for a quantitative prediction of are used (FREIBURG,1973). Interest of the the associated, potentially prohibitive, enhanceauthors about radon diffusion arose during ment (build-up) in gamma radiation intensity development of a radon barrier as a remedial suggested the need to determine the diffusion action against high radon progeny concen- coefficient of radon in concrete. I t was also trations existing in many structures of western considered desirable to compare the experiColorado following the use of uranium mill mentally determined coefficient with previously tailings as a construction material. These reported values.
* Research supported by the Atomic Energy Com-
(1) EFFECTIVE DIFFUSION COEFFICIENT OF A RADIOACTIVE GAS IN A POROUS MEDIUM
mission, Contract No. AT(ll-1)-2273 and by the While the derivation of the diffusion characEnvironmental Protection Agency, Grant No. RO 1teristics of a gas through a substance requires ECOOl53. 263
264
EFFECTIVE DIFFUSION COEFFICIENT OF RADON IN CONCRETE
imposed on the diffusion process by the circuitous nature of pores is obviously a lengthened one. Considering alternatively only one of these two factors, or both, results in the definition of two different effective diffusion coefficients and lead to different diffusion equations. If the flux introduced in equation (1) is expressed per unit area of open pores, only = -k, grad C (1) the lengthening of the pathway is taken into account and a larger value of the effective where 4 = gas flux per unit area of porous diffusion coefficient, k,*, will result. If the medium (atoms/cm2-sec), same flux is measured per unit area of porous k, = effective diffusion coefficient of the medium, as in equation (l), a lower value, k,, gas in porous medium (cm2/sec), is obtained since both the restrictions in cross C = gas concentration per unit volume sectional area and the lengthening of the pathof interstitial space (atoms/cms). way are taken into account. Concurrently, as will be shown, the diffusion equation resulting While all authors agree on the form of equation from the second definition, k,, will necessarily (l), a lack of uniformity has been observed in contain an explicit reference to porosity of the the literature with respect to the definition of medium while in the balance equation resultthe effective diffusion coefficient (TANNER, 1964, ing from the first definition, ke*, explicit refKRANER et al., 1964; JUNGE, 1963). Use of an erence to porosity is only necessary for those effective diffusion coefficient accounts for the media that contain sources. restricted nature of passages left open for the Consider a reference volume of porous meddiffusion process across porous medium as op- ium (Fig. 1) with unit cross-sectional area and posed to the value of an ordinary diffusion thickness, ctr. This material emits P atoms/ coefficient which describes the rate at which sec-cms of porous medium from the solid matrix diffusing atoms would progress in a plenum portion into the interstitial space. The differfilled only with substance that occupies pores. ential equation resulting from the balance equaIn comparison to free diffusion, two factors tion for one dimensional diffusion of a decaying reduce the flux of a substance across a reference substance in such a medium is: volume of space filled by the solid matrix of a porous medium (Fig. 1): first, the cross-sectional area through which diffusion can take place is reduced by a fraction equal to the ratio of open pore area to the total cross-section of where p = porosity of the medium (interstitial volume/total volume), reference volume, and secondly the pathway A = decay constant of diffusing gas (sec-l)
study of the sizes, shapes and surface properties of diffusion pathways (GEANKOPOLIS, 1972), the model of a flux proportional to the concentration gradient can always be applied to materials as widely differing in densities as air and metals (HEWITTand SHARRATT, 1963). According to such a model:
4
I; u
u)
+ ._ c 3
X
K
x+dx
FIG.1. Reference volume of porous medium.
.
When using the other definition, k,*, of the effective diffusion coefficient, aplausible assumption needs to be made on the equality between the fraction of open pore area in a unit crosssection and porosity, p, to avoid introducing a supplementary parameter relating these two values. The resulting differential equation is : d2C - - LC -P = 0. (3) dx2 P Comparison between (2) and (3) reveals the relationship k, =pk,* which could also be k,*
+
M. V. J. CULOT, H. G. OLSON and K. J. SCHIAGER
directly established from the respective definitions of k, and k,*, provided that equality between porosity and the fraction of open pore area in a unit cross-sectional area is maintained. Solutions of equations (2) and (3) for diffusion of a gas from a plane source of concentration C, into a n infinite medium without sources are: C(X)= C, e-2/Ap/ka. x = C, e-2//l/ke.* x = C,,e-xlR
(4)
where R = relaxation distance. This relaxation distance is closely related to the effective diffusion coefficient and is a measure of the range of gas in a sourceless medium, since it is the distance necessary for either concentration or flux to drop by a factor e. TANNER (1964) discussing radon migration in the ground, represents the solution to diffusion from a plane source (C,) in a n infinite porous medium without sources by the equation
-
c(X) = C, e-‘\/YD*z
(5)
where D = diffusioncoefficient ofradon through soil. The absence of the porosity parameter from the relaxation distance identifies D with k,*. By contrast, in the same reference, but in another article studying effusion of radon from the ground, KRANERet al. (1964) show the equation €or diffusion of radon through ground as :
265
the diffusion equation for radon through soil: (7)
where d = diffusion constant (cma/sec), f = rate of production within the soil (atoms/cms-sec), C, = concentration of diffusing gas in soil air (atomslcm’) . Again, no mention is made of soil porosity and its omission from the equation is explained by the assumption that “if the soil is sufficiently porous, diffusion proceeds as if the soil were absent.” The parameter d corresponds to k,* and the production term, f, should represent a production rate per unit volume of interstitial space. With porosities on the order of lo%, a lack of distinction between k, and k,* will lead to discrepancies of a n order of magnitude between reported values of “effective diffusion coeficients.” (2) VARIABILITY IN CONCRETE PHYSICAL STRUCTURE
A second major factor which accounts for the wide range of values reported for diffusion coefficients in concrete is the widespread variance which occurs in porosity and physical structure of concrete as a function of type of aggregates that are used, waterlcement ratios used in the cement paste, and curing conditions. Cement paste in concrete (Design and Control 1956) of Concrete Mixtures, 1968; TROXELL, occupies from 23% (lean mix) to 36% (rich where D = bulk diffusion coefficient for radon mix) of the total volume, sand 25-30?;, and through volume of the porous med- aggregates the remainder. Overall porosity of concrete will, consequently, depend as much ium (cmlsec), on porosity of the cement paste as on the agS = porosity of the medium, porosity. It will always be less than B = a constant related to emanation gregates’ the volume-weighted average of these two compower of the medium into the inponents, due to the presence of sand in the mix. terstitial volume of soil. Porosity of the cement paste consists of two Despite use of a n identical symbol, definitions types. Gel porosity results from the gel-like et al. (1964) identify D structure assumed by hydration products of adopted by KRANER with k, instead of k,*. Assuming that B repre- various mineral oxides used in manufacturing sents a source term evaluated per unit volume dry cement. Pores of this gel structure conof interstitial space such as P/p, equations (2) stitute 28 % of hydrated paste and are filled with and (6) are in agreement. Finally in his mono- entrapped water molecules. Pores range in graph on airchemistry and radiochemistry, size from 1.5 to 8.0 nm (TROXELL, 1956; PowJUNGE (1963) presents the following version of ERS and VERBECK, 1966). Their ability to
266
EFFECTIVE DIFFUSION COEFFICIENT OF RADON IN CONCRETE
allow radon diffusion across concrete is very (3) VALUES REPORTED FOR THE DIFFUSION OF GASES IN CONCRETE low as evidenced by their sieving effect on gas Diffusion coefficient values most closely remolecules such as nitrogen, hydrogen and lated to the case of radon diffusion in conhelium (MILLS,1968). Capillary porosity recrete and which were available in the literasults from net reduction in volume of water-dry cement mixtures during the hydration process. ture appeared in a publication by SCHWEITE Capillary pores are much larger in size than et al. (1968). Studying porosity of mortars gel pores and although their diameter distri- and concretes by means of 0,-N, diffusion bution has a mean of approx 2 x lo-' cm measurements, they arrived at a value of the (VERBECK and HELMUTH, 1968), they can be as specific permeability, y : large as 1.3 x lo3 cm (POWERS and VERBECK, 1966). Immediately after mixing cement paste, capillary porosity is considered to be made of, and filled by, all the water which surrounds where Deft = bulk diffusion coefficient of N,-0, dry cement grains. As hydration of cement system in concrete (cm'/sec), grains proceeds, the capillary system is proDo = diffusion coefficient of 0,-N,. gressively replaced by gel pores. The water Values obtained were y = 4.56 x lo"', Do = cement ratio by weight (w/c) used in the mix 2 x 10-1 cm2/secand result in Deft = 9 x 10" dictates the amount of capillary porosity left crn'lsec. The same authors report data from in paste after all of the cement has been hyd- ZAGAR(1967) which states that, for lightweight rated. When a paste involving an optimum concretes (0.4. . . 2 g/cm3), y can vary from w/c = 0.42 is cured under dry conditions (i.e. 3 x lo4 to 5 x and that €or ordinary without additional supply of water) there will concretes ( 2 270 kg cement/m3) y is found be a minimum capillary porosity of about 8 %. over a range extending from as low as iO-5, If the w/c ratio is increased, which is the usual possibly lo-', up to lo-,. case in the field, capillary porosity increases also. As curing of cement paste proceeds, the (4) MEASUREMENT OF k,-EXPERIMENTAL ARRANGEMENT capillary pore system becomes segmented by The effective diffusion coefficient of radon in gel structures, unless the wlc ratio exceeds 0.7, in which case segmentation will never occur. concrete was measured using the basement According to cement mix calculational methods wall of an experimental building especially (CZERNIN, 1962), a w/c ratio of 0.7 results in a built to perform studies on radon progeny control. The structure (detailed in Fig. 2 and with capillary porosity of 33.3 %. Porosity of common aggregates ranges from inner dimensions of 264 x 600cm) was built a fraction of a per cent to 20 % for dense stones with a basement. Uranium mill tailings were and can reach 50 % for lightweight aggregates -7 /-zocm (POWERS and VERBECK,1966). Frequently, pores in aggregate are at least the size of paste capillaries. Thus the role of aggregates in allowing radon to diffuse through concrete is important. It is concluded that the type of aggregates used in concrete, the w/c ratio used in the mix, and curing conditions are the three dominant parameters that control permeability of concrete to radon. Also, overall concrete porosity which should be considered as relevant to the problem of radon diffusion can consequently I U,O"l"rn U r m v r n Milt range from a few per cent (optimum mix and Tailsr? Toilingi dense aggregate) to about 25 % (high w/c ratio FIG, 2. Basement of experimental building. and porous aggregates). I
M. V. J. CULOT, H. G. OLSON and K. J. SCHIAGER !jroniun Mill Concrete Wall
1
Silicone
Caulk
.
:oni ,oiner
Filter
267
a fraction of the wall surface were negligible and that a one dimensional model was adequate to describe the diffusion of radon from the tailings into the container. (5) COMPUTATION
OF le,-THEORETICAL FOUNDATION
The diffusion coefficient of radon through concrete, k,, was calculated by equating the measured value of the radon flux, F, to its analytical expression which involves k, and the average radon concentration on both sides of the wall [C, (tailings side) and C, (container), with C, C,]. This led to a transcendental equation in k,, which was solved by a numerical iteration method attributed to Newton (CONTE, 1965). Formulation of the transcendental equation resulted from the following assumptions : Diffusion of radon through the concrete was described by a one dimensional equation without a source term:
>
' ~ o : e r Manometer
FIG.3. Experimental arrangement for the measurement of radon flux.
used as a fill under the basement floor slab and as a backfill around the foundations. The conk, d2C(x) (9) crete mix formulation was of the following comP d x a - K ( x ) = 0. position per m3, 1008 kg of 1.9 cm rocks, 808 kg of sand, 279 kg of cement and approx 99 1. of Using the constant concentration recorded on water. The apparatus used to record radon the source side, C,, and an average value, C,, of fluxes is shown in Fig. 3. In order to monitor concentrations recorded in the container during possible pressure gradients occurring during the flux measurements, the solution of (9) is: the flux measurements, a 1.9 cm hole was drilled 1 [(C, - Cte-'T)e*2 through the wall 102 cm above the basement C(X) = 2 sinh (rT) floor and a 91 cm long metal pipe was inserted (c,eTT- C,)e-C2], (10) in this opening (Figs. 2 and 3). The metal pipe was positioned 15 cm into the surrounding where r = l / @ / k e (cm-I), tailings and was then sealed at the wall with T = thickness of the wall (cm). silicone caulking. A water manometer connected between the pipe and the basement's F l u into the container, F, at x = T (Fig. 3) atmosphere indicated any pressure differential. was evaluated by The pipe was also used for measuring radon concentrations in the tailings(C,). To measure radon flux out of the wall, a square metal container ("flux can") covering an area symmetrically located around the pipe penetration was Using (10) to evaluate the derivative in (1 1) we find sealed with silicone caulk to the wall. Radon k,r[C, - C, cash (rT)] flux was determined by measuring radon conF= sinh ( r I") * (12) centration buildup in the container. The selected size of the container (122 x 122 x 2.5 The large difference between C, and C, renders cm) was the result of a theoretical analysis expression (12) quasi-independent of the averperformed to insure that modifications intro- aging procedure used in establishing C,. By duced in the normal radon flow pattern by defining : imposing a higher radon concentration over Y = (ks)l'z, (13)
+
268
EFFECTIVE DIFFUSION COEFFICIENT OF RADON IN CONCRETE
equation (12) transforms into the following transcendental equation in y :
- C, cosh (jZ1/2fi1/2T ')I, sinh (jZ1I2p1l2 Th)
F=
(14)
which is equivalent to
A stopping criterion of the iterative procedure depicted by (16) was set in terms of the ratio of the difference between two successive approximations to the magnitude of the last approximation. After the root of (17) has been found, k, is determined from (13).
F sinh (c') = a - b cosh (c')
( 6 ) EXPERIMENTAL RESULTS
The solution to a transcendental equation
f (y) = 0 by Newton's method necessitates a preliminary approximation, yo, of the solution while the second and successive approximations, until obtaining a root, are given by
An analysis was performed to determine the maximum time span, T,, over which the radon concentration buildup in the container could be approximated by a linear function of time. T, was found to be dependent on the container size. Flux into the container over a time span T, < T, was calculated by:
F = [C, - CilV in pCi/dm2-hr A Ts where f'(yj) is the first derivative off(y) evaluated a t y =yi. I n the case of equation (15) we have
f ( h ) = a * y - b - y cosh
where C,, Ci= final and initial concentration in container (pCi/l.),
V = volume of container (I.),
(ch)
A = surface area covered by container (dm2), T,= time lag between C, and Ci(hr).
- F - s i n h (ch) (17) and
f( J ) f'(y)
-
a - y - b * y cosh (c/y) - F sinh
a
b-c
(cb)
F-C
+sinh (ch)+ [ y- b ] Y cash
(ch).
(1%) A plot of the function f ( J ) = 0 was obtained to discard the possibility of the presence of several roots near the area of interest.
The water manometer consistently showed no pressure difference across the concrete wall. Average results of experiments performed on two different days (typical of many such runs) are reported in Table 1. Since the porosity of the building concrete was not known, two extreme values, 5 and 25%, were assumed. The corresponding values of k, for the average
Table I. Aueragr eafivimental results of radon flux measurements uifh uranium mill tailings behind a concrete wall (resvlfs of rnrnsurcrnmls on 2 different da.w)
pcilljter, radon concentration on tailings side of concrete wall
pCi/liter, i n i t i a l radon concentration on container s i d e of wall
(19)
pcilliter, f i n a l radon concentration on container s i d e of w a l l
hr., time span between i n i t i a l and
pCi/cm2-s, radon f l u x entering final the concentrations container
39,152
73
231
4.29
2.6~10-~
40,481
55
204
4.54
2.14~10-~
I
M. V. J. CULOT, H. G. OLSON and K. J. SCHIAGER
269
Tablc 2 . Diffusion coeficienf and relaxation distance of radon in ordinary concrete uring a source of uranium mill tailings
assumed porosity of the concrete barrier
cm2s, e f f e c t i v e diffusion c o e f f i c i e n t of the radon
cm, relaxation distance
0.05
1.69
10-5
12.7
0.25
3.08 x 10-5
7.6
values, F = 2.38 x pCi/cma-sec, C , = 40,000 pCi/l. and C, = 100 pCi/l. are presented in Table 2, along with the corresponding relaxation distances computed by (4). The experimental value of k, obtained with the concrete used for the foundation walls of our facility ( ~ 2 . 5X 10-‘cm2/sec) and the value of Deft measured by SCHWEITE et al. (1968) (9 x cm2/sec) are both located towards the lower end of the range of possible values reported by ZACAR (1967). An explanation lies in the composition of the concrete mix used in our building. The formulation given amounts to a w/c ratio of 0.46 which, according to cement mix calculational methods (ZAGAR,1967), corresponds, at most, to a 14.4 % capillary porosity of cement paste. Aggregates used by the contractor were visually observed to be of the dense type. If aggregates are assumed to have a 10 % porosity, overall porosity of the concrete amounted to 5.6 %. (7) CONCLUSIONS
Application of a linear diffusion theory to diffusion of radon across a concrete wall yielded experimental results indicating that the relaxation distance of radon in the concrete foundation wall of the experimental facility is of the order of 10 cm, with an associated effective diffusion coefficient of the order of 2 x lo-’ cm2/sec. This value obtained by an independent “flux can” method easily adapted for field work, compares well with the value of 9 x lo-’ cm2/sec reported by SCHWEITE et al. (1968). Application of these results to the modification of radon profiles in concrete walls sealed by a radon barrier is expected to yield valuable predictions on the associated build-up
of the gamma field. Remarks presented about the derivation of an effective diffusion coefficient will hopefully be of interest to those considering future similar experimental work. REFERENCES
CONTES. D., 1965, Elementary Nummical Analysis (New York: McGraw-Hill). CULOTM. V. J., OLSON H. G. and SCHIAGER K. J., 1973, Radon Progeny Control in Buildings. Final report on research supported by the Environmental Protection Agency (Grant No. R 0 1 EC00153 and the Atomic Energy Commission (Contract No. AT(11-1)-2273). CZERNIN W., 1962, Cement Chemistry and Physics for Ciuil Engineers (Chemical Publishing Co.). Design and Control of Concrete Mixtures, No. 3, 1968 (Portland Cement Association). K., 1973, personal communication. FREIBURG GEANKOPOLIS G. J., 1972, Mass Transport Phenomena (San Francisco: Holt, Rinehart & Winston). HEWITT G. F. and SHARRATT E. W., 1963, Nature, Lond. 198. JUNGE C. E., 1963, Air Chemistry and Radiochemistry, International Geophysics Series, Vol. 4 (New York: Academic Press). KRANER H. W., SCHROEDER G. L. and EVANS R. D., 1964, Measurements of the Effects of Atmospheric Variables on Radon-222 Flux and Soil Gas Cornentration, The Natural Radiation Ennironment (Univ. of Chicago Press). MILLSR. H., 1968, Molecular Sieve Effect in Concrete, Proc. 5th Int. Symp. on the Chemistry of Cement, Part 111,Japan T. C. and VERBECK G. T., 1966, American POWERS Society for the Testing of Materials. Significance of Tests and Properties of Concrete and Concrete Mixing Materials, STP 169-A. H. E., BOHOME H. J. and LUDWIG U., SCHWIETE 1968, Measuring Gas Diffusion for the Valuation of Open Porosity on Mortars and Concrete, Proc.
2 70
EFFECTIVE DIFFUSION COEFFICIENT OF RADON IN CONCRETE
5th Int. Symp. on the Chemistry of Cement, Part 111, Japan. TANNER A. B., 1964, Radon Migration in The Ground, A Review, The Natural Radiation Environment (Univ. of Chicago Press). TROXELL G. E., 1956, Combosition and Properties of Concrete (New York : McGraw-Hill),
VERBECK G . J. and HELMUTH R. H., 1968, Structures and Physical Properties of Cement Paste, Proc. 5th Int. Symp. Chemistry of Cement, Part 111, Japan. YEATES D. B., GOLDXN A. S. and MOELLER D. W., 1972, Nucl. Saf. 13, 4. ZAGAR L., 1967, S’rechsaal 100.
EFFECTIVE DIFFUSION COEFFICIENT OF RADON IN CONCRETE, THEORY AND METHOD FOR FIELD MEASUREMENTS* MICHEL V. J. CULOT
Visiting Scientist, I.N.E.N., Centro Nuclear de Mexico, Apartado Postal 27-190, Mexico 18, D. F. HILDING G . OLSON
Associate Professor, Mechanical Engineering Dept, Colorado State University, Fort Collins, CO 8052 1 and KEITH J. SCHIAGER
Professor of Health Physics, Dept. of Radiation Health, Graduate School of Public Health, University of Pittsburgh, Pittsburgh, PA 15261
(Received 15 January 1975; accepted 21 August 1975) Abstract-A linear diffusion model serves as the basis for determination of an effective radon diffusioncoefficient in concrete. The coefficientwas needed to later allow quantitative prediction of radon accumulationwithin and behind concrete walls after application ofanimpervious radon barrier. A resolution of certain descrepancies noted in the literature in the use of an effective diffusion coefficient to model diffusionof a radioactive gas through a porous medium is suggested. An outline of factors expected to affectthe concrete physical structure and the effective diffusion coefficient of radon through it is also presented. Finally, a field method for evaluating effective radon diffusion coefficientsin concrete i s proposed and results of measurements performed on a concrete foundation wall are compared with similar published values of gas diffusion coefficients in concrete. INTRODUCTION
tailings contain small quantities of radium, the DIFFUSION of radon across concrete boundaries precursor of radon, and were used in commercial of structures is a phenomenon of interest to and residential structures as leveling courses those health physicists concerned with the radon and as backfill under and around concrete progeny levels inside ordinary buildings (YEATES foundations (CULOTet al., 1973). Suppression et al., 1972). The same diffusion process is a of radon influx by application of a radon barrier concern for those responsible for monitoring on the indoor faces of these structures’ foundaleaks in facilities handling alpha emitters, since tions was expected to result in increased radon spurious alarms can be triggered by detection concentrations in and around the foundations. of radon progeny when gross alpha monitors The necessity for a quantitative prediction of are used (FREIBURG,1973). Interest of the the associated, potentially prohibitive, enhanceauthors about radon diffusion arose during ment (build-up) in gamma radiation intensity development of a radon barrier as a remedial suggested the need to determine the diffusion action against high radon progeny concen- coefficient of radon in concrete. I t was also trations existing in many structures of western considered desirable to compare the experiColorado following the use of uranium mill mentally determined coefficient with previously tailings as a construction material. These reported values.
* Research supported by the Atomic Energy Com-
(1) EFFECTIVE DIFFUSION COEFFICIENT OF A RADIOACTIVE GAS IN A POROUS MEDIUM
mission, Contract No. AT(ll-1)-2273 and by the While the derivation of the diffusion characEnvironmental Protection Agency, Grant No. RO 1teristics of a gas through a substance requires ECOOl53. 263
264
EFFECTIVE DIFFUSION COEFFICIENT OF RADON IN CONCRETE
imposed on the diffusion process by the circuitous nature of pores is obviously a lengthened one. Considering alternatively only one of these two factors, or both, results in the definition of two different effective diffusion coefficients and lead to different diffusion equations. If the flux introduced in equation (1) is expressed per unit area of open pores, only = -k, grad C (1) the lengthening of the pathway is taken into account and a larger value of the effective where 4 = gas flux per unit area of porous diffusion coefficient, k,*, will result. If the medium (atoms/cm2-sec), same flux is measured per unit area of porous k, = effective diffusion coefficient of the medium, as in equation (l), a lower value, k,, gas in porous medium (cm2/sec), is obtained since both the restrictions in cross C = gas concentration per unit volume sectional area and the lengthening of the pathof interstitial space (atoms/cms). way are taken into account. Concurrently, as will be shown, the diffusion equation resulting While all authors agree on the form of equation from the second definition, k,, will necessarily (l), a lack of uniformity has been observed in contain an explicit reference to porosity of the the literature with respect to the definition of medium while in the balance equation resultthe effective diffusion coefficient (TANNER, 1964, ing from the first definition, ke*, explicit refKRANER et al., 1964; JUNGE, 1963). Use of an erence to porosity is only necessary for those effective diffusion coefficient accounts for the media that contain sources. restricted nature of passages left open for the Consider a reference volume of porous meddiffusion process across porous medium as op- ium (Fig. 1) with unit cross-sectional area and posed to the value of an ordinary diffusion thickness, ctr. This material emits P atoms/ coefficient which describes the rate at which sec-cms of porous medium from the solid matrix diffusing atoms would progress in a plenum portion into the interstitial space. The differfilled only with substance that occupies pores. ential equation resulting from the balance equaIn comparison to free diffusion, two factors tion for one dimensional diffusion of a decaying reduce the flux of a substance across a reference substance in such a medium is: volume of space filled by the solid matrix of a porous medium (Fig. 1): first, the cross-sectional area through which diffusion can take place is reduced by a fraction equal to the ratio of open pore area to the total cross-section of where p = porosity of the medium (interstitial volume/total volume), reference volume, and secondly the pathway A = decay constant of diffusing gas (sec-l)
study of the sizes, shapes and surface properties of diffusion pathways (GEANKOPOLIS, 1972), the model of a flux proportional to the concentration gradient can always be applied to materials as widely differing in densities as air and metals (HEWITTand SHARRATT, 1963). According to such a model:
4
I; u
u)
+ ._ c 3
X
K
x+dx
FIG.1. Reference volume of porous medium.
.
When using the other definition, k,*, of the effective diffusion coefficient, aplausible assumption needs to be made on the equality between the fraction of open pore area in a unit crosssection and porosity, p, to avoid introducing a supplementary parameter relating these two values. The resulting differential equation is : d2C - - LC -P = 0. (3) dx2 P Comparison between (2) and (3) reveals the relationship k, =pk,* which could also be k,*
+
M. V. J. CULOT, H. G. OLSON and K. J. SCHIAGER
directly established from the respective definitions of k, and k,*, provided that equality between porosity and the fraction of open pore area in a unit cross-sectional area is maintained. Solutions of equations (2) and (3) for diffusion of a gas from a plane source of concentration C, into a n infinite medium without sources are: C(X)= C, e-2/Ap/ka. x = C, e-2//l/ke.* x = C,,e-xlR
(4)
where R = relaxation distance. This relaxation distance is closely related to the effective diffusion coefficient and is a measure of the range of gas in a sourceless medium, since it is the distance necessary for either concentration or flux to drop by a factor e. TANNER (1964) discussing radon migration in the ground, represents the solution to diffusion from a plane source (C,) in a n infinite porous medium without sources by the equation
-
c(X) = C, e-‘\/YD*z
(5)
where D = diffusioncoefficient ofradon through soil. The absence of the porosity parameter from the relaxation distance identifies D with k,*. By contrast, in the same reference, but in another article studying effusion of radon from the ground, KRANERet al. (1964) show the equation €or diffusion of radon through ground as :
265
the diffusion equation for radon through soil: (7)
where d = diffusion constant (cma/sec), f = rate of production within the soil (atoms/cms-sec), C, = concentration of diffusing gas in soil air (atomslcm’) . Again, no mention is made of soil porosity and its omission from the equation is explained by the assumption that “if the soil is sufficiently porous, diffusion proceeds as if the soil were absent.” The parameter d corresponds to k,* and the production term, f, should represent a production rate per unit volume of interstitial space. With porosities on the order of lo%, a lack of distinction between k, and k,* will lead to discrepancies of a n order of magnitude between reported values of “effective diffusion coeficients.” (2) VARIABILITY IN CONCRETE PHYSICAL STRUCTURE
A second major factor which accounts for the wide range of values reported for diffusion coefficients in concrete is the widespread variance which occurs in porosity and physical structure of concrete as a function of type of aggregates that are used, waterlcement ratios used in the cement paste, and curing conditions. Cement paste in concrete (Design and Control 1956) of Concrete Mixtures, 1968; TROXELL, occupies from 23% (lean mix) to 36% (rich where D = bulk diffusion coefficient for radon mix) of the total volume, sand 25-30?;, and through volume of the porous med- aggregates the remainder. Overall porosity of concrete will, consequently, depend as much ium (cmlsec), on porosity of the cement paste as on the agS = porosity of the medium, porosity. It will always be less than B = a constant related to emanation gregates’ the volume-weighted average of these two compower of the medium into the inponents, due to the presence of sand in the mix. terstitial volume of soil. Porosity of the cement paste consists of two Despite use of a n identical symbol, definitions types. Gel porosity results from the gel-like et al. (1964) identify D structure assumed by hydration products of adopted by KRANER with k, instead of k,*. Assuming that B repre- various mineral oxides used in manufacturing sents a source term evaluated per unit volume dry cement. Pores of this gel structure conof interstitial space such as P/p, equations (2) stitute 28 % of hydrated paste and are filled with and (6) are in agreement. Finally in his mono- entrapped water molecules. Pores range in graph on airchemistry and radiochemistry, size from 1.5 to 8.0 nm (TROXELL, 1956; PowJUNGE (1963) presents the following version of ERS and VERBECK, 1966). Their ability to
266
EFFECTIVE DIFFUSION COEFFICIENT OF RADON IN CONCRETE
allow radon diffusion across concrete is very (3) VALUES REPORTED FOR THE DIFFUSION OF GASES IN CONCRETE low as evidenced by their sieving effect on gas Diffusion coefficient values most closely remolecules such as nitrogen, hydrogen and lated to the case of radon diffusion in conhelium (MILLS,1968). Capillary porosity recrete and which were available in the literasults from net reduction in volume of water-dry cement mixtures during the hydration process. ture appeared in a publication by SCHWEITE Capillary pores are much larger in size than et al. (1968). Studying porosity of mortars gel pores and although their diameter distri- and concretes by means of 0,-N, diffusion bution has a mean of approx 2 x lo-' cm measurements, they arrived at a value of the (VERBECK and HELMUTH, 1968), they can be as specific permeability, y : large as 1.3 x lo3 cm (POWERS and VERBECK, 1966). Immediately after mixing cement paste, capillary porosity is considered to be made of, and filled by, all the water which surrounds where Deft = bulk diffusion coefficient of N,-0, dry cement grains. As hydration of cement system in concrete (cm'/sec), grains proceeds, the capillary system is proDo = diffusion coefficient of 0,-N,. gressively replaced by gel pores. The water Values obtained were y = 4.56 x lo"', Do = cement ratio by weight (w/c) used in the mix 2 x 10-1 cm2/secand result in Deft = 9 x 10" dictates the amount of capillary porosity left crn'lsec. The same authors report data from in paste after all of the cement has been hyd- ZAGAR(1967) which states that, for lightweight rated. When a paste involving an optimum concretes (0.4. . . 2 g/cm3), y can vary from w/c = 0.42 is cured under dry conditions (i.e. 3 x lo4 to 5 x and that €or ordinary without additional supply of water) there will concretes ( 2 270 kg cement/m3) y is found be a minimum capillary porosity of about 8 %. over a range extending from as low as iO-5, If the w/c ratio is increased, which is the usual possibly lo-', up to lo-,. case in the field, capillary porosity increases also. As curing of cement paste proceeds, the (4) MEASUREMENT OF k,-EXPERIMENTAL ARRANGEMENT capillary pore system becomes segmented by The effective diffusion coefficient of radon in gel structures, unless the wlc ratio exceeds 0.7, in which case segmentation will never occur. concrete was measured using the basement According to cement mix calculational methods wall of an experimental building especially (CZERNIN, 1962), a w/c ratio of 0.7 results in a built to perform studies on radon progeny control. The structure (detailed in Fig. 2 and with capillary porosity of 33.3 %. Porosity of common aggregates ranges from inner dimensions of 264 x 600cm) was built a fraction of a per cent to 20 % for dense stones with a basement. Uranium mill tailings were and can reach 50 % for lightweight aggregates -7 /-zocm (POWERS and VERBECK,1966). Frequently, pores in aggregate are at least the size of paste capillaries. Thus the role of aggregates in allowing radon to diffuse through concrete is important. It is concluded that the type of aggregates used in concrete, the w/c ratio used in the mix, and curing conditions are the three dominant parameters that control permeability of concrete to radon. Also, overall concrete porosity which should be considered as relevant to the problem of radon diffusion can consequently I U,O"l"rn U r m v r n Milt range from a few per cent (optimum mix and Tailsr? Toilingi dense aggregate) to about 25 % (high w/c ratio FIG, 2. Basement of experimental building. and porous aggregates). I
M. V. J. CULOT, H. G. OLSON and K. J. SCHIAGER !jroniun Mill Concrete Wall
1
Silicone
Caulk
.
:oni ,oiner
Filter
267
a fraction of the wall surface were negligible and that a one dimensional model was adequate to describe the diffusion of radon from the tailings into the container. (5) COMPUTATION
OF le,-THEORETICAL FOUNDATION
The diffusion coefficient of radon through concrete, k,, was calculated by equating the measured value of the radon flux, F, to its analytical expression which involves k, and the average radon concentration on both sides of the wall [C, (tailings side) and C, (container), with C, C,]. This led to a transcendental equation in k,, which was solved by a numerical iteration method attributed to Newton (CONTE, 1965). Formulation of the transcendental equation resulted from the following assumptions : Diffusion of radon through the concrete was described by a one dimensional equation without a source term:
>
' ~ o : e r Manometer
FIG.3. Experimental arrangement for the measurement of radon flux.
used as a fill under the basement floor slab and as a backfill around the foundations. The conk, d2C(x) (9) crete mix formulation was of the following comP d x a - K ( x ) = 0. position per m3, 1008 kg of 1.9 cm rocks, 808 kg of sand, 279 kg of cement and approx 99 1. of Using the constant concentration recorded on water. The apparatus used to record radon the source side, C,, and an average value, C,, of fluxes is shown in Fig. 3. In order to monitor concentrations recorded in the container during possible pressure gradients occurring during the flux measurements, the solution of (9) is: the flux measurements, a 1.9 cm hole was drilled 1 [(C, - Cte-'T)e*2 through the wall 102 cm above the basement C(X) = 2 sinh (rT) floor and a 91 cm long metal pipe was inserted (c,eTT- C,)e-C2], (10) in this opening (Figs. 2 and 3). The metal pipe was positioned 15 cm into the surrounding where r = l / @ / k e (cm-I), tailings and was then sealed at the wall with T = thickness of the wall (cm). silicone caulking. A water manometer connected between the pipe and the basement's F l u into the container, F, at x = T (Fig. 3) atmosphere indicated any pressure differential. was evaluated by The pipe was also used for measuring radon concentrations in the tailings(C,). To measure radon flux out of the wall, a square metal container ("flux can") covering an area symmetrically located around the pipe penetration was Using (10) to evaluate the derivative in (1 1) we find sealed with silicone caulk to the wall. Radon k,r[C, - C, cash (rT)] flux was determined by measuring radon conF= sinh ( r I") * (12) centration buildup in the container. The selected size of the container (122 x 122 x 2.5 The large difference between C, and C, renders cm) was the result of a theoretical analysis expression (12) quasi-independent of the averperformed to insure that modifications intro- aging procedure used in establishing C,. By duced in the normal radon flow pattern by defining : imposing a higher radon concentration over Y = (ks)l'z, (13)
+
268
EFFECTIVE DIFFUSION COEFFICIENT OF RADON IN CONCRETE
equation (12) transforms into the following transcendental equation in y :
- C, cosh (jZ1/2fi1/2T ')I, sinh (jZ1I2p1l2 Th)
F=
(14)
which is equivalent to
A stopping criterion of the iterative procedure depicted by (16) was set in terms of the ratio of the difference between two successive approximations to the magnitude of the last approximation. After the root of (17) has been found, k, is determined from (13).
F sinh (c') = a - b cosh (c')
( 6 ) EXPERIMENTAL RESULTS
The solution to a transcendental equation
f (y) = 0 by Newton's method necessitates a preliminary approximation, yo, of the solution while the second and successive approximations, until obtaining a root, are given by
An analysis was performed to determine the maximum time span, T,, over which the radon concentration buildup in the container could be approximated by a linear function of time. T, was found to be dependent on the container size. Flux into the container over a time span T, < T, was calculated by:
F = [C, - CilV in pCi/dm2-hr A Ts where f'(yj) is the first derivative off(y) evaluated a t y =yi. I n the case of equation (15) we have
f ( h ) = a * y - b - y cosh
where C,, Ci= final and initial concentration in container (pCi/l.),
V = volume of container (I.),
(ch)
A = surface area covered by container (dm2), T,= time lag between C, and Ci(hr).
- F - s i n h (ch) (17) and
f( J ) f'(y)
-
a - y - b * y cosh (c/y) - F sinh
a
b-c
(cb)
F-C
+sinh (ch)+ [ y- b ] Y cash
(ch).
(1%) A plot of the function f ( J ) = 0 was obtained to discard the possibility of the presence of several roots near the area of interest.
The water manometer consistently showed no pressure difference across the concrete wall. Average results of experiments performed on two different days (typical of many such runs) are reported in Table 1. Since the porosity of the building concrete was not known, two extreme values, 5 and 25%, were assumed. The corresponding values of k, for the average
Table I. Aueragr eafivimental results of radon flux measurements uifh uranium mill tailings behind a concrete wall (resvlfs of rnrnsurcrnmls on 2 different da.w)
pcilljter, radon concentration on tailings side of concrete wall
pCi/liter, i n i t i a l radon concentration on container s i d e of wall
(19)
pcilliter, f i n a l radon concentration on container s i d e of w a l l
hr., time span between i n i t i a l and
pCi/cm2-s, radon f l u x entering final the concentrations container
39,152
73
231
4.29
2.6~10-~
40,481
55
204
4.54
2.14~10-~
I
M. V. J. CULOT, H. G. OLSON and K. J. SCHIAGER
269
Tablc 2 . Diffusion coeficienf and relaxation distance of radon in ordinary concrete uring a source of uranium mill tailings
assumed porosity of the concrete barrier
cm2s, e f f e c t i v e diffusion c o e f f i c i e n t of the radon
cm, relaxation distance
0.05
1.69
10-5
12.7
0.25
3.08 x 10-5
7.6
values, F = 2.38 x pCi/cma-sec, C , = 40,000 pCi/l. and C, = 100 pCi/l. are presented in Table 2, along with the corresponding relaxation distances computed by (4). The experimental value of k, obtained with the concrete used for the foundation walls of our facility ( ~ 2 . 5X 10-‘cm2/sec) and the value of Deft measured by SCHWEITE et al. (1968) (9 x cm2/sec) are both located towards the lower end of the range of possible values reported by ZACAR (1967). An explanation lies in the composition of the concrete mix used in our building. The formulation given amounts to a w/c ratio of 0.46 which, according to cement mix calculational methods (ZAGAR,1967), corresponds, at most, to a 14.4 % capillary porosity of cement paste. Aggregates used by the contractor were visually observed to be of the dense type. If aggregates are assumed to have a 10 % porosity, overall porosity of the concrete amounted to 5.6 %. (7) CONCLUSIONS
Application of a linear diffusion theory to diffusion of radon across a concrete wall yielded experimental results indicating that the relaxation distance of radon in the concrete foundation wall of the experimental facility is of the order of 10 cm, with an associated effective diffusion coefficient of the order of 2 x lo-’ cm2/sec. This value obtained by an independent “flux can” method easily adapted for field work, compares well with the value of 9 x lo-’ cm2/sec reported by SCHWEITE et al. (1968). Application of these results to the modification of radon profiles in concrete walls sealed by a radon barrier is expected to yield valuable predictions on the associated build-up
of the gamma field. Remarks presented about the derivation of an effective diffusion coefficient will hopefully be of interest to those considering future similar experimental work. REFERENCES
CONTES. D., 1965, Elementary Nummical Analysis (New York: McGraw-Hill). CULOTM. V. J., OLSON H. G. and SCHIAGER K. J., 1973, Radon Progeny Control in Buildings. Final report on research supported by the Environmental Protection Agency (Grant No. R 0 1 EC00153 and the Atomic Energy Commission (Contract No. AT(11-1)-2273). CZERNIN W., 1962, Cement Chemistry and Physics for Ciuil Engineers (Chemical Publishing Co.). Design and Control of Concrete Mixtures, No. 3, 1968 (Portland Cement Association). K., 1973, personal communication. FREIBURG GEANKOPOLIS G. J., 1972, Mass Transport Phenomena (San Francisco: Holt, Rinehart & Winston). HEWITT G. F. and SHARRATT E. W., 1963, Nature, Lond. 198. JUNGE C. E., 1963, Air Chemistry and Radiochemistry, International Geophysics Series, Vol. 4 (New York: Academic Press). KRANER H. W., SCHROEDER G. L. and EVANS R. D., 1964, Measurements of the Effects of Atmospheric Variables on Radon-222 Flux and Soil Gas Cornentration, The Natural Radiation Ennironment (Univ. of Chicago Press). MILLSR. H., 1968, Molecular Sieve Effect in Concrete, Proc. 5th Int. Symp. on the Chemistry of Cement, Part 111,Japan T. C. and VERBECK G. T., 1966, American POWERS Society for the Testing of Materials. Significance of Tests and Properties of Concrete and Concrete Mixing Materials, STP 169-A. H. E., BOHOME H. J. and LUDWIG U., SCHWIETE 1968, Measuring Gas Diffusion for the Valuation of Open Porosity on Mortars and Concrete, Proc.
2 70
EFFECTIVE DIFFUSION COEFFICIENT OF RADON IN CONCRETE
5th Int. Symp. on the Chemistry of Cement, Part 111, Japan. TANNER A. B., 1964, Radon Migration in The Ground, A Review, The Natural Radiation Environment (Univ. of Chicago Press). TROXELL G. E., 1956, Combosition and Properties of Concrete (New York : McGraw-Hill),
VERBECK G . J. and HELMUTH R. H., 1968, Structures and Physical Properties of Cement Paste, Proc. 5th Int. Symp. Chemistry of Cement, Part 111, Japan. YEATES D. B., GOLDXN A. S. and MOELLER D. W., 1972, Nucl. Saf. 13, 4. ZAGAR L., 1967, S’rechsaal 100.
