The Effect of Pore Structural Factors on Diffusion in Compacted Sodium Bentonite
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ility that the composition of silica sand in bentonite insignificantly affect pore structural parameters in diffusion. Also for the effect of grain size on diffusion, Da values of 134 Cs (Cs+) and HTO in Na-montmorillonite have been reported for coarse and fine grain sizes, 75-150 µm (100-200 mesh) and 0, x = 0
Q(t ) De α 2α = 2 t− − 2 6 π A ⋅ L ⋅ Co L
C(t, x) = 0, t > 0, x = L
(−1) n De ⋅ n 2 ⋅ π 2 ⋅ t − exp 2 ∑ L2 ⋅ α n =1 n ∞
(3)
Where Q(t) is the accumulative quantity of the tracer permeated through bentonite (cpm), A is the cross-section area of the bentonite sample (m2), L is the thickness of the bentonite sample (m), Co is the concentration of the tracer in the tracer cell(cpm•m–3), and De is the effective diffusion coefficient (m2•s–1). At long time such as steady state, the exponentials fall away to zero. Therefore, equation (3) is approximately written by the following equation for steady state. De α Q(t ) = A ⋅ L ⋅ Co ⋅ 2 t − 6 L
(4)
The De is calculated from the slope of Q(t) with time in steady state based on equation (4). The concentration gradient of tracer in the filter, which was used to prevent the swelling of bentonite, is corrected to calculate true De in bentonite. This correction is made using the following equation for the steady state [16]. L + 2L f 2L f De = L − De De t f
(5)
where Det is the effective diffusion coefficient before correction (m2•s–1), Def is the effective diffusion coefficient in the filter (m2•s–1), and Lf is the thickness of the filter (m) (1 mm). Although De described above is the diffusion coefficient for steady state, Da is one for nonsteady state and the concentration profile gradually changes with time. Basic equation for a onedimensional non-steady state diffusion is as given by equation (2). For HTO, since solubility does not exist, an analytical solution in case of an instantaneous planar source can be applied for the calculation of Da. For one-dimensional diffusion of a planar source consisting of a limited amount of substance in a cylinder of infinite length, when it is assumed that the diffusion is independent of the position, equation (2) becomes as follows [14]. Initial condition: Boundary condition: C=
C(t, x) = 0, t = 0, x ≠ 0 C(t, x) = 0, t > 0, x = ± ∞
M x2 exp − 2 πDa ⋅ t 4 Da ⋅ t
(6)
Where M is the total amount of the tracer pipetted per unit bentonite sample area (dpm•m–2) and Da is the apparent diffusion coefficient (m2•s–1). The slope given by a plot of LnC versus x2 gives Da from the relation with diffusing time based on equation (6). The Da values of HTO were determined based on this method. RESULTS AND DISCUSSION Diffusion Direction Dependency on De and SEM Observations
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De [m2·s-1]
De [m 2 ･s-1 ]
The concentrations of HTO in the measurement cell as a function of time showed non-linear curves in transient state and increased in a straight line with time in steady state. The concentration profiles of HTO in bentonite approximately linearly decreased from the tracer cell
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