Transient diffusion--breakthrough time estimation

Kov As

(B21)

Ke A5

The concentration at the cap-water interface, which would be indicative of the

level of exposure of bottom-surface dwelling organisms, is defined by the

balance of the flux through the cap with the flux through the benthic boundary

layer. The contaminant concentration at the cap-water interface is:

Kov C0

Ccw

(B22)

Kbl

Either of these concentrations or the estimated fluxes may be compared with

applicable criteria for the chemical in question to determine if a specified cap

thickness is adequate.

Transient diffusion--breakthrough time estimation

The simple steady-state analysis presented above is not capable of predicting

the time required for the contaminant(s) to migrate through the cap layer. Until

sorption and migration in the cap is complete, the flux to the water column will

be less than predicted by Equation B17. Addressing this problem requires

incorporation of time explicitly in the differential mass balance. The following

partial differential equation represents a differential mass balance on the con-

taminant in the pore water of the cap as it diffuses from the contaminated sedi-

ment below.

Cpw

4/3

(B23)

The conditions of a constant concentration at the sediment-cap interface are

applied as specified by Equation B15 and the concentration of the overlying

water at the height Leff in the cap. Carslaw and Jaeger (1959) present a solution

to the equivalent heat transfer problem that in terms of concentration and mass

diffusion can be written

Deff {n }2 t

(C0 Cw) Deff

( 1) exp

Fdiff

(B24)

Leff

Rf Leff

n 1

where Deff represents Dw 4/3. This solution is also given in this form by Thoma

et al. (1993). Note that as t , the exponential term approaches zero and the

flux approaches the value obtained by the approximation Kov Deff Leff as

indicated by Equation B17. From Equation B24, one can obtain relations for the

breakthrough time and the time required to approach the steady-state flux.

Breakthrough time, b, is defined as the time at which the flux of contaminant

from the contaminated sediment layer has reached 5 percent of its steady-state

B13

Appendix B Model for Chemical Containment by a Cap