Transport of Radon in Still Water under Steady-State and Transient Conditions
Syahrir, S. Usman, H. Spitz, J. Weisman
University of Cincinnati, Department of Mechanical, Industrial and Nuclear Engineering, 598 Rhodes Hall,
Cincinnati, Ohio 45221-0072 USA. e-mail: [email protected]
Abstract. The transport of 222Rn through sealed, horizontal 2.54 cm diameter tubes filled with still water was
observed to be significantly greater than that predicted solely from molecular diffusion.
The rate of 222Rn
222
transport was also found to be related to the concentration of Rn in the water. A constant source of radon
producing an air concentration ranging from 1.1 to 176 Bq cm-3 was connected to one end of a horizontal,
insulated tube filled with water ranging from 30 to 52 cm in length. Continuous measurements of radon in air
under transient and steady-state conditions were obtained using ZnS(Ag) alpha scintillators that were connected
at each end of the water filled diffusion channel. Conventional values for pure molecular diffusion of radon in
water range from 1.14 x 10-5 to 1.56 x 10-5 cm2 sec-1. Values for radon transport observed in this study are from
30 to 50 times greater than the rates cited for molecular diffusivity. Microturbulence produced by alpha particles
from the decay of radon and its progeny may be responsible for eddy diffusivity observed in our experiments
ranging from 5.14 x 10-4 to 8.14 x 10-4 cm2 sec-1. Theoretical analysis demonstrates that the energy associated
with alpha decay of radon and its short-lived decay products in the water is sufficient to induce micro-turbulence
in the diffusion channels to explain the ten-fold enhancement of radon transport observed in this study. This
phenomenon of “radio-turbulence” may have significance for predicting the fate of radioactive wastes in soil and
ground water, the geological disposal of spent fuel, nuclear fuel integrity analysis, and for studying the fate of
radio-labeled pharmaceuticals in the body where the biokinetics could be affected by the energy associated with
radioactive decay.
Introduction
Disposal of approximately 4 x 107 kg of insoluble, radium-bearing waste material stored in two concrete silos at
the U. S. Department of Energy Fernald site is complicated by the emanation of 222Rn into the atmosphere [1].
The concentration of 222Rn measured in the air filling the enclosed dome of the storage silos in 1991 was 130
kBq/L [2]. The waste, which has a total estimated 226Ra activity of approximately 6 x 1013 Bq, was generated at
Fernald in the 1950s as a result of refining pitchblende ores obtained from the Belgian Congo [3]. Waste in the
old concrete silos will be mixed with water and pumped into 4 new stainless steel tanks, each having a volume of
approximately 3 x 106 liters. An air treatment system is connected to the new tanks for collecting radon on
charcoal to avoid any release of the gas to the environment. How the concentration of radon in air above the
waste in the new tanks will be effected by residual water covering the insoluble, radium-bearing waste was the
initial motivation for the research described in this paper.
A series of preliminary laboratory measurements were conducted to determine the relationship between the
thickness of the water barrier covering a quantity of insoluble, radium-bearing waste and the radon concentration
in a fixed volume of air above the water [4]. The percentage of radon emanation directly from the radiumbearing waste material was found to be 7 % or 13 % depending upon whether the waste was dry or wet,
respectively. The initial measurements of radon transport in water used a set of 5 vertical 15.24 cm diameter
PVC plastic tubes containing approximately 40 g of the insoluble radium bearing waste enclosed in a fiberglass
filter envelope generating approximately 7.4 x 105 Bq of 222Rn. The waste was covered with water ranging in
depth from 10 to 61 cm. The length of each PVC tube was adjusted so that the volume of air in the enclosed
space above the water column was approximately 18 L. Results of the initial experiments indicated that for
every 25 cm of water covering the waste, the radon concentration in a fixed volume of air above the water will
be reduced by approximately 50%. However, the observed transport of radon through the water barrier was
significantly more rapid than expected, so additional experiments were conducted to investigate the
phenomenon.
The molecular diffusion of radon in water at room temperature cited in the literature ranges from 1.14 × 10-5 to
1.56 × 10-5 cm2 sec-1 [5,6]. The traditional method for determining the molecular diffusivity of a gas dissolved in
water is to measure the time necessary for the gas to cross a barrier, such as a glass frit, capillary tube, or
molecular-sized membrane of known thickness. No diffusion barriers were used in our preliminary experiments
because the original research objective was to simulate conditions that would be present in the large waste
storage tanks. However, the experimental apparatus was insulated from changes in room temperature and
vibration to avoid any external forces that could confound the results.
1
A possible explanation for the greater-than-expected transport of radon in water that was observed in the vertical
channel experiments was the potential for turbulence produced by the energy associated with alpha decay of
radon and its short-lived progeny. Such turbulent energy would normally be absorbed by the frit or membrane in
conventional diffusion experiments and would not be available to interfere with molecular diffusion.
Experiments performed in this research to study radon transport in water did not include diffusion barriers
because of the desire to simulate conditions that exist in the waste tanks and duplicate, albeit on a bench scale,
our preliminary laboratory measurements using vertical PVC tubing.
Materials and Methods
Several experimental arrangements involving eight horizontal 2.54 cm diameter water channels ranging in length
from 30 cm to 50 cm and 1.8 cm diameter U-tubes having vertical arms ranging in length from 10 cm to 20 cm
were used in performing laboratory-scale measurements of radon transport in water. Horizontal channels (Figure
1) were preferred to eliminate vertical mixing that could arise because of gravitational forces or buoyancy.
Figure 1: Laboratory arrangement for measuring the transport of 222Rn through water
A constant source of radon ranging from 1.1 Bq cm-3 to 176 Bq cm-3 was generated using either a sample of the
radium-bearing waste material or a standard RaCl solution. This radon source was attached to one end of the
water channel. Experimental objectives included measuring the time necessary for radon to travel through the
water channel and enter a fixed volume of air at the far end of the channel, identified as the “sink”. Continuous,
passive measurements of radon in fixed volumes of air at the source and sink ends of the channel were
performed throughout the duration of the experiment using alpha scintillation cells coated with ZnS(Ag). Once
a steady state radon concentration was achieved in the air at the sink end of the channel, samples of air and water
were extracted from both the source and sink ends of the channel using a precision, 1 mL gas-tight glass syringe.
The air and liquid samples were injected into a liquid scintillation counting vial containing 16 mL of scintillation
fluid. Both the liquid scintillation counter (LSC) and the scintillation flasks were calibrated using 222Rn gas
generated by 226RaCl standard solutions obtained from the National Institute for Standards and Technology
[7][8].
Theory
A one-dimensional steady-state diffusion equation was first used to determine the radon diffusion coefficient in
water at steady-state because a minimum concentration gradient was expected across the water barrier. The
steady-state differential equations for radon diffusion in water (w) and air (a) are
DW
DA
∂ 2CW
− λ CW = 0
∂x 2
∂ 2C A
− λC A = 0
∂x 2
(1)
where λ is radon decay constant, CW and CA refer to radon concentrations at a location, x, in the water or air with
the corresponding diffusion coefficient DW and DA. Derivation of the two-region equation is based on the
conditions shown in Figure 1.
2
+a
DA
0, Ch
DW
-h, C0
Figure 2. Steady-state model.
Radon from a constant source, Co, will diffuse in the water barrier through a height, h. Likewise, once released
from water, radon will diffuse through a column of air to a height, a. The concentration of radon in water at the
air-water interface is Ch or CW(0). The boundary conditions and requirements for the continuity of radon flux are
CW (− h ) ≡ C0
DW
∂CW
∂x
x =0
= DA
∂C A
∂x
x =0
C A (0 ) = H CW (0 )
∂C A
∂x
where the concentration gradient
x=a
=0
∂C
is the same at the interface (x = 0) and is equal to 0 at x = . H is the
∂x
Henry’s law constant for radon in water.
Applying the boundary conditions for water and air to the general solutions for equation (1), the solution for the
two region steady state diffusion equations for radon in the water at the interface (x = 0) is
Ch =
where k A =
λ
DA
and kW =
C0 k A
k A cosh(kW h) + H kW sinh(kW h) Tanh(k A a)
λ
DW
. Newton's method is used to determine kW for a particular measured
value of Ch/Co in order to obtain the radon diffusion coefficient in water, DW . The constant H is obtained from
the ratio of the radon concentration at the interface of the air and water.
3
When a = 0, the solution for Ch involves only one region (i.e., water) and leads to
Ch =
C0
.
cosh(kW h)
The equation can be solved to obtain DW,
DW =
λ h2
C
Cosh  0 
 Ch 
−1
2
.
Results
Table 1 lists values of DW determined using the one- and two-region solutions to the diffusion equation
Table 1: Steady-State Diffusion Coefficients (One- and Two-Region Models)
Dw, (cm2 s-1)
Two Region
One Region
h (cm)
C0, (Bq/cc)
Ch/C0
30
4
0.42
8.9E-04
8.2E-04
35
15
0.33
8.6E-04
8.2E-04
36
1
0.30
7.9E-04
7.7E-04
36
3
0.18
4.8E-04
4.7E-04
40
2
0.22
7.0E-04
7.0E-04
52
44
0.14
8.0E-04
7.9E-04
52
177
0.06
4.6E-04
4.6E-04
53
2
0.05
4.5E-04
4.4E-04
6.8E-04
6.6E-04
28%
26%
Avg. D (% SD)
The change in the radon concentration with distance in water from a constant source of radon, determined using
the two-region diffusion model, is illustrated in Figure 2 and leads to an average value for Dw equal to 6.8 x 10-4
cm2 sec-1.
4
1.0
Ch/C0
0.8
0.6
0.4
0.2
0.0
0
10
20
30
40
50
60
h, cm
Figure 3: Variation of Ch/C0 with water depth observed in the steady state experiment
Continuous, passive measurements of radon were performed on the source and sink ends of the water column
until the concentration of radon in the air at the sink reached a constant value, usually within approximately 30
days. The longest and shortest duration of measurements was 55 days and 8 days, respectively. These
experiments were designed to investigate the influence of alpha decay of radon and its short-lived progeny on the
diffusion phenomenon which is generally presumed to be strictly molecular in the absence of any convective
forces, such as turbulence, natural convection or mechanical vibration. However, results of these measurements
suggest a diffusion rate at least an order of magnitude greater than that expected solely from molecular motion.
Discussion
The decay of 222Rn and its short-lived, alpha-emitting progeny will produce, at equilibrium, a total of 19.18 MeV
of energy that will be deposited in a very small volume comparable to the range of the alpha particles. This
locally-deposited energy may be the reason for the enhanced transport of radon observed in these experiments by
transitioning from molecular diffusion to convective transport due to the alpha energy dissipation rate.
A theoretical calculation for the eddy diffusivity associated with the alpha energy dissipation can be developed
using a simple scaling law expression.
DE =
(V
E
.PL )
2
where D E is the eddy diffusivity, V E is the average eddy velocity, and PL is the Prandtl mixing length. The
average eddy velocity ( V E ) can be estimated using the relationship
VE = (ε .l )
1/ 3
where ε is the rate of energy dissipation per unit mass and l is the length scale of the dominant turbulence.
Assuming that the rate of energy dissipation is the same as the rate of alpha decay energy production, then the
alpha energy dissipation rate is
ε=
 decay 
1
 19.18
 sec .cc 
 MeV 

 1.6 10−6
decay


 gm 
1

 cc 
 erg 


 MeV 
 cm2 
≅ 3 x 10−5  3 
 sec 
5
and the length scale is the range of the alpha particles in water, viz.
l = 4.5 x 10 −3 cm.
Then, the average eddy velocity per unit radon activity concentration (Bq mL-1) can be estimated using the
following:
VE = (ε .l )
1/ 3
= (3.0 10 −5 4.5 10 −3 )1 / 3
VE = 5.13 x10 −3
cm
sec
Assuming that the mixing length is related to the diameter of the diffusion channel (i.e., 2.54 cm diameter; PL ~
0.127 cm), eddy diffusivity is equal to
DE ≅ 3.25 x 10 −4 cm 2 / sec
Note that this prediction is for a unit radon concentration (viz., 1 Bq cm-3). Values listed on Table 1 reflect much
larger radon concentrations achieved in the laboratory experiments. Therefore, this value for DE is in relatively
good agreement with the experimental results shown on Table 1 suggesting that the alpha energy associated with
the decay of radon and its short-lived, alpha-emitting progeny is sufficient to enhance the diffusion rate as
observed in the experiments. It is likely that the diffusion rate under the influence of convective forces
associated with alpha decay energy will no longer be constant throughout the fluid media. In fact, at the source
side, where the concentration of radon is high, the value of eddy diffusivity would also be higher than that found
at the sink side where the radon concentration and diffusion rate would also be low. In order to properly model
the diffusion phenomenon under these conditions a finite differencing scheme may be employed which uses a
concentration dependent diffusion coefficient.
The impact of “alpha convection” is important not only for radon transport but may also play an important role
in the methods used for testing nuclear fuel integrity by measuring concentration of fission product in primary
coolant. Likewise the presence of alpha emitting contaminants in high-level radioactive liquid wastes can also
be making a significant contribution to micro-turbulence in the solutions. There may even be applications of this
phenomenon to methods for monitoring xenon and krypton for nuclear non-proliferation compliance.
Conclusions
The literature cites that the diffusion coefficient of radon in water is 1.13 x 10-5 cm2 sec-1. However,
observations from experiments lead to a steady state and transient transport coefficient equal to 6.79 x 10-4 cm2
sec-1 and 3.97 x 10-4 cm2 sec-1, respectively. Observations that the transport of radon through water in a series of
laboratory experiments was greater than that predicted by molecular diffusion has led to a theoretical explanation
that involves energy deposition from the alpha decay of 222Rn and its short-lived decay products as a source of
eddy diffusivity. Theoretical analysis of the transport of radon through water that includes the eddy diffusivity
associated with the decay of radon and its alpha-emitting decay products support the measurement results
observed in these experiments and the introduction of a new phenomenon, radio-turbulence. The process of
radio-turbulence may also have application in other areas, such as in methods for testing nuclear fuel integrity by
the fission product inventory in primary coolant and the analysis of the stability of high level liquid radioactive
wastes.
References
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to the Fernald Environmental Restoration Management Corporation. Cincinnati, Ohio; 30 September 1994.
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Materials Co. of Ohio; Cincinnati, Ohio; 1987.
6
4. Spitz, H.: Evaluation of Water as a Fluid Diffusion Barrier to Reduce Radon Emanation from K-65 Waste
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Ohio; 1999.
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