Steady-State Measurement of Krypton-85-Air Diffusion Coefficients in Porous Media Parker C. Reist Department of Industrial Hygiene, Harvard School of Public Health, Boston, Mass. 02115
To evaluate the possible use of porous underground formations for the retention and disposal of waste krypton-85, the diffusion coefficient of Kr-air mixtures within these formations is often needed. This paper reports on the design and use of a “diffusion cell” to measure these coefficients under various conditions of porosity and moisture content. It is concluded that the diffusion cell measurements represent to a good approximation the actual diffusion coefficient of air-85Kr mixtures in porous media.
I
n investigating the possible use of porous, underground media for storage of waste krypton-85, leakage from the media by permeation and diffusion must be considered. To estimate diffusion losses, a knowledge of the diffusional behavior of krypton in the porous media is required. This behavior is generally expressed in terms of a diffusion coefficient. Conventional methods for measuring the diffusion coefficient of a gas in a porous medium require long sampling timzs, as in the case of the method used by Penman (1940), or complex data treatment, with the methods used by Papendick and Runkles (1965) or Dye and Dallavalle (1958). Presented here is a simple technique for measuring this coefficient for kryptonair mixtures, based on a method used by Evans, Truitt, and Watson (1961) to measure helium and argon diffusion through large-pore graphite.
dimensional system in a form similar to that given b) E\ a m , Watson, and Mason (1961) as
where CKris the concentration of krypton-85 (pCi ~ 1 1 7 . ~ 9 a t any distance x (cm.) within the medium, D is the diffusion coeficient for the krypton-air mixture (cm.2 set.-'), and inKris the mole fraction of krypton in air. Since in our range of interest (concentrations up to several millicuries per cubic centimeter) the mole fraction of krypton in air is very much smaller than 1, the second term of Equation 1 can be neglected and Equation 1 can be written: -
JKr
D
dCsr dx ~
Under this condition the concentration gradient dCh, dx a t the steady-state approximates a straight line, with a slope of (C, - C,)/L where C , and C, are the krypton-85 concentrations at the up- and downstream faces, respectively, and L is the thickness of the medium-Le., the distance between the two faces. The current through the porous medium at the steady state is then: J
= ( C , - CO)(D/L)
(3)
If the krypton-85 leaving the downstream face of the cell is swept away by a n air stream having a flow of Q,, the concentration of krypton-85 in the stream is (J)(A)/(Q,),where A is the total normal area through which the gas is diffusing.
Experimental Method Consider a porous medium of length L arranged as shown in Figure 1 with the two opposite faces exposed to separate gas streams of differing composition. When the pressure on the left-hand face (face 1, Figure 1, B ) is much greater than the pressure on the right-hand face (face 2), there is a net flow of gas toward face 2. If conditions are reversed, the flow is in the opposite direction. When the pressures at the two faces are exactly equal, the gas movement, or current, from one face to the other will be mainly by diffusion, if moleculewall collisions are assumed to be negligible compared to molecule-molecule collisions. The *6Kr current, JK,(pCi cm.-2sec.-1), can be considered to be the number of atoms of‘ krypton crossing a unit area normal to the flow per unit time (Jost, 1952). The usual diffusion equation for this current can be written for a one566 Environmental Science and Technology
,..- .... ......
at’
........-...,....
2
I
I
(AI
18)
Figure 1. Design of diffusion cell
effective diffusion coefficient: All of the surface area, A , is not available for diffusion and, on the average, a gas molecule must travel a distance somewhat greater than the thickness of the medium in order to pass through it. Using the well-known transient diffusion equations, the D e i [described can be used to estimate the diffusion of krypton85 within a porous medium. In addition, if for a given soil type the porosity and Deiican be measured and a value of D calculated using kinetic theory, the tortuosity of the particular soil can be estimated. Such a factor is extremely important in describing the permeation characteristics of soil.
Substituting this value into Equation 3 and solving for D gi\ es (4) In a porous medium the D measured is actually a so-called “effective D,” Deff,which Carman (1956) considers to be related to the ‘‘true’’ diffusion coefficient of the gas pair (that measured outside the porous medium) by the expression D,fr/D
=
(5)
€,IT
where E is the porosity of the medium and 7 is its tortuosityi.e.. the ratio of the path length through the porous medium to the direct or “crow-flight” distance through the medium. Thus two factors are taken into account in computing the
Procedure
An air stream containing krypton-85 from a large 11,000liter reservoir was passed across sample face 1 (Figure l),
Table I. Krypton-Air Diffusion Coefficients in Various Unconsolidated Porous Media (Corrected to STP)
No. Of
Sample Type
Samples
Porosit),
z
&”.
Std. Dev.,
Sq. Cm./Sec.
l 7
Tort uosltl
1 0
Air alone
6
100
0.134
0.021
6-mm. glass beads
9
...
0.035
0.007
Coarse sand, 0.0503
