Diffusion of organic contaminants in triple-layer composite liners: an analytical modeling approach

Abstract

This paper presents an analytical solution for modeling the one-dimensional diffusion of organic contaminants through a composite liner, comprising a geomembrane, a geosynthetic clay liner and a soil liner system. The Laplace transformation technique is used to obtain a dimensionless analytical solution to the diffusion of organic chemicals through the triple-layer composite liners. The solution presented is verified against two alternative numerical solutions. The analytical solution is then adopted to provide a series of graphical design charts which can assist with the assessment and design of composite liners. Design examples are included for a composite liner having a geomembrane, a geosynthetic clay liner and a soil liner system. The proposed analytical solution provides a practical and relatively simple tool to assist with the design of composite liners and the validation of numerical models.

Appendix

The general solution to the governing Eqs. (1) and (2) can be written as:

$$ C_{\text{gm}} \left( z \right) = A_{1} z + A_{2} $$

(22)

$$ C_{\text{gcl}} \left( z \right) = A_{3} z + A_{4} $$

(23)

where A ₁ to A ₄ are the parameters to be determined by substituting the above equations into Eqs. (6) to (9) as follows:

$$ A_{1} = C_{0} S_{\text{gf}} $$

(24)

$$ A_{2} = - \frac{{D_{\text{gcl}} n_{\text{gcl}} [C_{0} S_{\text{gf}} - C_{\text{sl}} (L_{\text{gm}} + L_{\text{gcl}} ,t)S_{\text{gf}}^{'} ]}}{{D_{\text{gcl}} n_{\text{gcl}} L_{\text{gm}} + D_{\text{gm}} S_{\text{gf}}^{'} L_{\text{gcl}} }} $$

(25)

$$ A_{3} = - \frac{{C_{0} D_{\text{gm}} (L_{\text{gm}} + L_{\text{gcl}} ){\text{S}}_{\text{gf}}}} {{D_{\text{gcl}} n_{\text{gcl}} L_{\text{gm}} + D_{\text{gm}} S_{\text{gf}}^{'} L_{\text{gcl}} }} - \frac {{C_{\text{sl}} (L_{\text{gm}} + L_{\text{gcl}} ,t)L_{\text{gm}} (D_{\text{gcl}} n_{\text{gcl}} - D_{\text{gm}} S_{\text{gf}}^{'} )}} {{D_{\text{gcl}} n_{\text{gcl}} L_{\text{gm}} + D_{\text{gm}} S_{\text{gf}}^{'} L_{\text{gcl}} }} $$

(26)

$$ A_{4} = - \frac{{D_{\text{gm}} [C_{0} S_{\text{gf}} - C_{\text{sl}} (L_{\text{gm}} + L_{\text{gcl}} ,t)S_{\text{gf}}^{'} ]}}{{D_{\text{gcl}} n_{\text{gcl}} L_{\text{gm}} + D_{\text{gm}} S_{\text{gf}}^{'} L_{\text{gcl}} }} $$

(27)

Substituting Eqs. (22) to (27) into Eq. (10), the relationship between f _sl and C _sl can be obtained as follows:

$$ C_{\text{sl}} \left( {L_{\text{gm}} + L_{\text{gcl}} ,t} \right) = \eta_{1} + \eta_{2} f_{\text{sl}} \left( {L_{\text{gm}} + L_{\text{gcl}} ,t} \right) $$

(28)

where

$$ \eta_{1} = \frac{{C_{0} S_{\text{gf}} }}{{S_{\text{gf}}^{'} }} $$

(29)

$$ \eta_{2} = n_{\text{sl}} D_{\text{sl}} \left( {\frac{{L_{\text{gcl}} }}{{n_{\text{gcl}} D_{\text{gcl}} }} + \frac{{L_{\text{gm}} }}{{S_{\text{gf}}^{'} D_{\text{gm}} }}} \right) $$

(30)

The Eq. (28) is the upper boundary condition for the governing Eq. (3).

A Laplace transformation technique is used to obtain the solution to Eq. (3). A linear form of the governing Eq. (3) can be obtained with respect to time by introducing a Laplace transform, given as:

$$ \overline{{C_{\text{sl}} \left( {z ,p} \right)}} = \int\limits_{0}^{\infty } {{\text{e}}^{ - pt} } C_{\text{sl}} \left( {z ,t} \right){\text{d}}t $$

(31)

where $\overline{{C_{\text{sl}} \left( {z ,p} \right)}}$ is the Laplace transform of $C_{\text{sl}} \left( {z ,t} \right)$ and p is the Laplace parameter.

The governing equation for contaminant concentration in the soil liner can then become:

$$ p\overline{{C_{\text{sl}} \left( {z,p} \right)}} - C_{i} = \frac{{D^{ * } }}{{R_{\text{d}} }}\frac{{\partial^{2} \overline{{C_{\text{sl}} \left( {z,p} \right)}} }}{{\partial z^{2} }} $$

(32)

The Laplace transforms of the boundary conditions considered can be written as follows:

$$ \overline{{C_{\text{sl}} (L_{\text{gm}} + L_{\text{gm}} ,p)}} = \frac{{\eta_{1} }}{p} + \eta_{2} \frac{{\partial \overline{{C_{\text{sl}} (L_{\text{gm}} + L_{\text{gm}} ,p)}} }}{\partial z} $$

(33)

$$ \frac{{\partial \overline{{C_{\text{sl}} (\infty ,p)}} }}{\partial z} = 0 $$

(34)

The general solution to Eq. (32) can then be expressed as:

$$ \overline{{C_{\text{sl}} (z,p)}} = Ge^{\lambda z} + \frac{{C_{i} }}{p} $$

(35)

where G is to be determined from the boundary conditions. λ is the characteristic root which is determined by the following equation:

$$ \lambda = - \sqrt {\frac{{pR_{\text{d}} }}{D}} $$

(36)

Substituting Eqs. (35) and (36) into Eq. (33), the following expression can then be obtained:

$$ G = \frac{{(C_{i} - \eta_{1} )}}{{pe^{{(L_{\text{gm}} + L_{\text{gcl}} )\lambda }} \left( {\eta_{2} \lambda - 1} \right)}} $$

(37)

Using Eq. (37), the solution to Eq. (32) can be expressed as:

$$ \overline{{C_{\text{sl}} \left( {z,p} \right)}} = \frac{{C_{i} }}{p} + \frac{{(\eta_{1} - C_{i} )\exp [ - \sqrt {pR_{\text{d}} /D_{\text{sl}} } (z - L_{\text{gm}} - L_{\text{gcl}} )]}}{{{\text{p}}\eta_{2} (1/\eta_{2} + \sqrt {pR_{\text{d}} /D_{\text{sl}} } )}} $$

(38)

Following Carslaw and Jaeger [3], the inverse Laplace transform of $\overline{{C_{\text{sl}} (z,p)}}$ can be obtained as given in Eqs. (13) to (17).