Transport of a volatile contaminant in a free-surface wetland flow

Abstract

Presented in this paper is an analytical study of a pulsed volatile contaminant emission into a free-surface wetland flow. A simplified model is given for contaminant transport under the combined action of advection, mass dispersion, apparent reaction, and volatilization at the free water surface. The effect of periodic apparent reaction on contaminant transport is separated from the hydraulic effect via an extended transformation, with a limiting case covering the known transformation for constant apparent reaction rate. The analytical solutions of zeroth and first order concentration moments are rigorously derived and illustrated. It was found that the amount of contaminant decreases from the bottom bed to the free-surface under volatilization, and the total amount of contaminant decays with time. It was also found that the moving speed of the mass center of the whole contaminant cloud increases, as the ratio of volatilization coefficient to vertical effective mass dispersivity increases.

Keywords

contaminant transport / volatilization / reaction / wetland hydraulics

Introduction

The mechanism of contaminant transport in wetland flows has received increasing attention, due to its significant value for environmental risk assessment associated with wetlands (Mitsch and Gosselink, 1993; Costanza et al., 1997; Chen et al., 2009; Shao et al., 2012).

Contaminant transport in wetland flows is dependent on various factors, such as velocity distribution, apparent reaction, volatilization at the interface of water and air, infiltration at the bottom bed, etc. With respect to contaminant transport due to pure velocity distribution, some efforts have been made (Lightbody and Nepf, 2006a, b; Murphy et al., 2007; Nepf et al., 2007; Chen et al., 2010, 2012; Zeng, 2010; Wu et al., 2011a, b, 2012; Zeng et al., 2011, 2012a, b) based on Taylor’s classic analysis of dispersion (Taylor, 1953), the method of concentration moments (Aris, 1956), and the multi-scale analysis method (Mei et al., 1996). In realistic wetland flows, there are various complex reactions, such as absorption, desorption, hydrolysis, etc., which are important to contaminant transport. Zeng and Chen (2011) analyzed the effect of apparent reaction rate on contaminant dispersion based on an ecological transformation, in which apparent reaction rate is assumed to be constant. However, the apparent reaction rate may change due to changes of temperature, salinity, etc. The effect of variations in apparent reaction rate over time on contaminant transport in wetland flows has not been well understood.

Municipal sewage and industrial wastewater may contain various volatile contaminants, such as dimethylsulfide, dimethyldisulfide, etc. When the contaminants are discharged into a wetland flow, they may escape from the water into the air by volatilization. Up to now, the effects of volatilization on the transport of contaminants in wetland flows has not been well understood.

This paper presents a theoretical analysis of the transport of volatile contaminants for a typical case involving a pulsed volatile contaminant emission into a free-surface wetland flow. The specific objectives are: (ⅰ) to obtain a transformation to separate the effect of periodic apparent reaction from the hydraulic effect, (ⅱ) to present analytical solutions for the zeroth and first order concentration moments for volatile contaminants, and (iii) to illustrate the effect of characteristic parameters, associated with apparent reaction and volatilization, on contaminant transport.

Formulation

The governing equation for the mass transport of a volatile contaminant in typical wetland flows is shown below (Liu and Masliyah, 2005; Zeng and Chen, 2011).

(1)\varphi {\displaystyle \frac{\partial C}{\partial t}}+\nabla \cdot (uC)=\nabla \cdot (\kappa \lambda \varphi \nabla C)+\kappa \nabla \cdot (K\cdot \nabla C)-\varphi k_{a}C,

where Φ is porosity, t is time, C is contaminant concentration, u is velocity, \kappais tortuosity, λ is mass diffusivity, K is the mass dispersivity tensor, and k_a is apparent reaction rate. In realistic wetland flow situations, the apparent reaction rate may change with time. In this paper, we express a periodic apparent reaction rate with

(2)k_a=k_s+k_{o}f(t),

where k_s is the steady component of the apparent reaction rate, k_o is the amplitude of the oscillatory component of the apparent reaction rate, and f(t) is a periodic function with periodic average equal to zero.

Consider the transport of a volatile contaminant in a fully developed free-surface wetland flow in a Cartesian coordinate system, as shown in Fig. 1. The longitudinal x-axis is parallel to the flow direction, the vertical z-axis is upward, and the origin is located at the bottom bed. The solution for velocity distribution is (Zeng and Chen, 2011).

(3)u(z)=U{\displaystyle \frac{\cosh \alpha -\cosh [\alpha (\mathrm{1}-z/H)]}{(\alpha \cosh \alpha -\sinh \alpha )/\alpha }},

where U is the depth-averaged velocity, and α is a dimensionless characteristic parameter.

Consider a uniform instantaneous discharge of a volatile contaminant at the position of x=0 at time t=0. The corresponding initial condition can be written as (Zeng and Chen, 2011)

(4){C(x,z,t)|}_{t=\mathrm{0}}={\displaystyle \frac{Q}{\varphi H}}\delta (x),

where Q is the mass per unit width, and δ(x) is the Dirac delta function.

For the two-dimensional process, Eq. (1) reduces to

(5){\displaystyle \frac{\partial C}{\partial t}}+{\displaystyle \frac{u}{\varphi }}{\displaystyle \frac{\partial C}{\partial x}}=\kappa \left[\lambda +{\displaystyle \frac{K_L}{\varphi }}\right]{\displaystyle \frac{{\partial }^{\mathrm{2}}C}{\partial x^{\mathrm{2}}}}+\kappa \left[\lambda +{\displaystyle \frac{K_V}{\varphi }}\right]{\displaystyle \frac{{\partial }^{\mathrm{2}}C}{\partial z^{\mathrm{2}}}}-k_s[\mathrm{1}+\sigma f(t)]C,

where K_L and K_V are the longitudinal and vertical mass dispersivities, respectively; and σ=k_o/k_s is the ratio of the steady component to the amplitude of the oscillatory component.

Since the discharge amount of the volatile contaminant is definite, the concentration boundary conditions at x=±∞ are

(6)C(\pm \infty ,z,t)=\mathrm{0},

assuming that the ambient concentration of the volatile contaminant in the air is equal to zero. For the impermeable bottom bed at z=0, and the free surface at z=H, the concentration boundary conditions are

(7){{\displaystyle \frac{\partial C}{\partial z}}|}_{z=\mathrm{0}}=\mathrm{0},{\left[-\kappa \left(\lambda +{\displaystyle \frac{K_V}{\varphi }}\right){\displaystyle \frac{\partial C}{\partial z}}\right]|}_{z=H}={(h_{s}C)|}_{z=H},

where h_s is the volatility coefficient, which is similar to a heat transfer coefficient.

The governing equation, as well as the boundary and initial conditions, can be rewritten in dimensionless form as

(8){\displaystyle \frac{\partial \mathrm{\Omega }}{\partial \tau }}+{\displaystyle \frac{P{e}_x{R}_x^K}{\varphi }}\psi {\displaystyle \frac{\partial \mathrm{\Omega }}{\partial \xi }}=R_x^K{\displaystyle \frac{{\partial }^{\mathrm{2}}\mathrm{\Omega }}{\partial {\xi }^{\mathrm{2}}}}+{\displaystyle \frac{{\partial }^{\mathrm{2}}\mathrm{\Omega }}{\partial {\zeta }^{\mathrm{2}}}}-N[\mathrm{1}+\sigma \theta (\tau )]\mathrm{\Omega },

(9)\mathrm{\Omega }(\pm \infty ,\zeta ,\tau )=\mathrm{0},

(10){{\displaystyle \frac{\partial \mathrm{\Omega }}{\partial \zeta }}|}_{\zeta =\mathrm{0}}=\mathrm{0},{\left[R_s\mathrm{\Omega }+{\displaystyle \frac{\partial \mathrm{\Omega }}{\partial \zeta }}\right]|}_{\zeta =\mathrm{1}}=0,

(11){\mathrm{\Omega }(\xi ,\zeta ,\tau )|}_{\tau =\mathrm{0}}=H\delta (H\xi ),

where:

(12)$$\begin{gathered} \xi ={\displaystyle \frac{x}{H}},\zeta ={\displaystyle \frac{z}{H}},\tau ={\displaystyle \frac{\kappa (\lambda +K_V/\varphi )t}{H^{\mathrm{2}}}},\mathrm{\Omega }(\xi ,\zeta ,\tau )={\displaystyle \frac{\varphi C{H}^{\mathrm{2}}}{Q}}, \\ P{e}_x={\displaystyle \frac{UH}{\kappa (\lambda +K_L/\varphi )}},R_x^K={\displaystyle \frac{\lambda +K_L/\varphi }{\lambda +K_V/\varphi }},\psi ={\displaystyle \frac{\alpha \{\cosh \alpha -\alpha \cosh [\alpha (\zeta -\mathrm{1})]\}}{\alpha \cosh \alpha -\sinh \alpha }}, \\ \theta (\tau )=f\left({\displaystyle \frac{H^{\mathrm{2}}\tau }{\kappa (\lambda +K_V/\varphi )}}\right),R_s={\displaystyle \frac{H{h}_s}{\kappa (\lambda +K_V/\varphi )}}. \end{gathered}$$

The dimensionless parameter R_s reflects the relative strength of volatilization, and the effective mass dispersion in the vertical direction.

Separation of the effect of periodic apparent reaction from the hydraulic effect

The dimensionless concentration determined by Eqs. (8)–(11) is dependent on advection, mass dispersion, apparent reaction, and volatilization. For the case of constant apparent reaction rate, the effect of apparent reaction can be separated from the hydraulic effect by a simple exponential transformation (Zeng and Chen, 2011). For the case of periodic apparent reaction rate, an extended exponential transformation equation,

(13)\mathrm{\Omega }(\xi ,\zeta ,\tau )={\mathrm{\Omega }}^{*}(\xi ,\zeta ,\tau )\exp \left({\displaystyle {\int }_{\mathrm{0}}^{\tau }-N[\mathrm{1}+\sigma \theta ({\tau }_{\mathrm{1}})]d{\tau }_{\mathrm{1}}}\right),

was used to separate the effect of apparent reaction from the hydraulic effect, where Ω^*(ξ, ζ, τ) is only dependent on advection, mass dispersion, and volatilization. The transformation given by Zeng and Chen (2011) can be included as a special case of σ=0. With Eq. (13), we have

(14){\displaystyle \frac{\partial {\mathrm{\Omega }}^{*}}{\partial \tau }}+{\displaystyle \frac{P{e}_x{R}_x^K}{\varphi }}\psi {\displaystyle \frac{\partial {\mathrm{\Omega }}^{*}}{\partial \xi }}=R_x^K{\displaystyle \frac{{\partial }^{\mathrm{2}}{\mathrm{\Omega }}^{*}}{\partial {\xi }^{\mathrm{2}}}}+{\displaystyle \frac{{\partial }^{\mathrm{2}}{\mathrm{\Omega }}^{*}}{\partial {\zeta }^{\mathrm{2}}}},

(15){\mathrm{\Omega }}^{*}(\pm \infty ,\zeta ,\tau )=\mathrm{0},

(16){{\displaystyle \frac{\partial {\mathrm{\Omega }}^{*}}{\partial \zeta }}|}_{\zeta =\mathrm{0}}=\mathrm{0},{\left(R_s{\mathrm{\Omega }}^{*}+{\displaystyle \frac{\partial {\mathrm{\Omega }}^{*}}{\partial \zeta }}\right)|}_{\zeta =\mathrm{1}}=0,

(17){{\mathrm{\Omega }}^{*}(\xi ,\zeta ,\tau )|}_{\tau =\mathrm{0}}=H\delta (H\xi ).

To illustrate the effect of time-dependent apparent reaction rate on contaminant transport, three kinds of apparent reaction rate were considered. The first case is where\theta (\tau )=\left|\cos (\tau )\right|, which means that ambient factors strengthen the apparent reaction all the time. The second case is where\theta (\tau )=-\left|\cos (\tau )\right|, which means that the ambient factors weaken the apparent reaction all the time. The third case is where θ(τ)=cos(τ), which means that the ambient factors strengthen and weaken the apparent reaction periodically.

Figure 2 shows the variation of Λ₁ with τ for N=1, and σ=0.1, 0.2, and 0.3, where Λ₁ is expressed as

(18){\mathrm{\Lambda }}_{\mathrm{1}}={\displaystyle \frac{{\mathrm{\Omega }}_{\mathrm{1}}}{{\mathrm{\Omega }}_{\mathrm{0}}}}={\displaystyle \frac{\exp \left({\displaystyle {\int }_{\mathrm{0}}^{\tau }-N[\mathrm{1}+\sigma \left|\cos ({\tau }_{\mathrm{1}})\right|]d{\tau }_{\mathrm{1}}}\right)}{\exp \left({\displaystyle {\int }_{\mathrm{0}}^{\tau }-Nd{\tau }_{\mathrm{1}}}\right)}}.

Ω₀ is the contaminant concentration corresponding to θ(τ)=0, and Ω₁ is the contaminant concentration corresponding to \theta (\tau )=\left|\cos (\tau )\right|. It is shown that Λ₁ decreases with time. For a given N, Λ₁ decreases with the increase of σ.

Figure 3 shows the variation of Λ₂ with τ for N=1, and σ=0.1, 0.2, and 0.3, where Λ₂ is expressed as

(19){\mathrm{\Lambda }}_{\mathrm{2}}={\displaystyle \frac{{\mathrm{\Omega }}_{\mathrm{2}}}{{\mathrm{\Omega }}_{\mathrm{0}}}}={\displaystyle \frac{\exp \left({\displaystyle {\int }_{\mathrm{0}}^{\tau }-N[\mathrm{1}-\sigma \left|\cos ({\tau }_{\mathrm{1}})\right|]d{\tau }_{\mathrm{1}}}\right)}{\exp \left({\displaystyle {\int }_{\mathrm{0}}^{\tau }-Nd{\tau }_{\mathrm{1}}}\right)}},

and Ω₂ is the contaminant concentration corresponding to \theta (\tau )=-\left|\cos (\tau )\right|. It is shown that Λ₂ increases with time. For a given N, Λ₂ increases with increases of σ.

Figure 4 shows the variation of Λ₃ with τ for σ=0.1, 0.2, 0.3, and N=1, where Λ₃ is expressed as

(20){\mathrm{\Lambda }}_{\mathrm{3}}={\displaystyle \frac{{\mathrm{\Omega }}_{\mathrm{3}}}{{\mathrm{\Omega }}_{\mathrm{0}}}}={\displaystyle \frac{\exp \left({\displaystyle {\int }_{\mathrm{0}}^{\tau }-N[\mathrm{1}+\sigma \cos ({\tau }_{\mathrm{1}})]d{\tau }_{\mathrm{1}}}\right)}{\exp \left({\displaystyle {\int }_{\mathrm{0}}^{\tau }-Nd{\tau }_{\mathrm{1}}}\right)}},

and Ω₃ is the contaminant concentration corresponding to θ(τ)=cos(τ). It is shown that Λ₃ changes periodically with time. For a given N, the amplitude of Λ₃ increases with increases of σ.

Figure 5 shows the variation of Λ₃ with τ for σ=0.2, and N=0.5, 1, and 2. It is shown that as N increases, the amplitude of Λ₃ increases.

Concentration moment analysis

The method of concentration moments (Aris, 1956) was adopted to analyze the transport of a volatile contaminant in a free-surface wetland flow. The zeroth order concentration moment m₀ and the first order concentration moment m₁ are defined respectively as

(21)m_{\mathrm{0}}\equiv {\displaystyle {\int }_{-\infty }^{\infty }{\mathrm{\Omega }}^{*}(\xi ,\zeta ,\tau )d\xi },

and

(22)m_{\mathrm{1}}\equiv {\displaystyle {\int }_{-\infty }^{\infty }{\mathrm{\Omega }}^{*}(\xi ,\zeta ,\tau )\xi d\xi }.

The zeroth and first order concentration moments reflect the amount of volatile contaminant, and the movement of the mass center of the contaminant cloud, respectively.

The concentration moments satisfy (Fried, 1975; Barton, 1983)

(23){\xi {\mathrm{\Omega }}^{*}|}_{\xi =\pm \infty }={{\displaystyle \frac{\partial {\mathrm{\Omega }}^{*}}{\partial \xi }}|}_{\xi =\pm \infty }=\xi {{\displaystyle \frac{\partial {\mathrm{\Omega }}^{*}}{\partial \xi }}|}_{\xi =\pm \infty }=\mathrm{0.}

Applying the operator {\displaystyle {\int }_{-\infty }^{\infty }(\cdot )}d\xito Eqs. (8), (10), and (11), with the aid of Eqs. (9) and (23), we have

(24){\displaystyle \frac{\partial m_{\mathrm{0}}}{\partial \tau }}={\displaystyle \frac{{\partial }^{\mathrm{2}}{m}_{\mathrm{0}}}{\partial {\zeta }^{\mathrm{2}}}},

(25){{\displaystyle \frac{\partial m_{\mathrm{0}}}{\partial \zeta }}|}_{\zeta =\mathrm{0}}=\mathrm{0},{\left(R_s{m}_{\mathrm{0}}+{\displaystyle \frac{\partial m_{\mathrm{0}}}{\partial \zeta }}\right)|}_{\zeta =\mathrm{1}}=\mathrm{0},

(26){m_{\mathrm{0}}(\zeta ,\tau )|}_{\tau =\mathrm{0}}=\mathrm{1.}

Solving Eqs. (24)-(26), we have

(27)m_{\mathrm{0}}(\zeta ,\tau )={\displaystyle \sum _{i=\mathrm{1}}^{\infty }{\displaystyle \frac{\mathrm{2}({\beta }_i^{\mathrm{2}}+R_s^{\mathrm{2}})}{{\beta }_i^{\mathrm{2}}+R_s^{\mathrm{2}}+R_s}}}{\displaystyle \frac{\sin {\beta }_i}{{\beta }_i}}\cos ({\beta }_i\zeta )\exp (-{\beta }_i^{\mathrm{2}}\tau ),

where β_i is determined by

(28){\beta }_i\tan {\beta }_i=R_s.

Table 1 presents the eigenvalues β_i (i= 1, 2, 3, 4, and 5) for R_s = 0.25, 0.5, 1.0, 2.0, and 4.0.

Figure 6 shows the variation of m₀ with ξ for τ=0.1, and R_s = 0.25, 0.5, 1.0, 2.0, and 4.0. Figure 7 shows the variation of m₀ with ξ for τ=1, and R_s = 0.25, 0.5, 1.0, 2.0, and 4.0. It is shown that m₀ decreases as ξ increases, which means that the amount of contaminant near the free-water surface is less than that near the bottom bed. The distribution of m₀ for the volatile contaminant is different from that for the non-volatile contaminant in the vertical direction, which stays constant (Zeng and Chen, 2011). At the initial stage, the volatilization has less effect on m₀ near the bottom bed. The effect of volatilization on m₀ gradually increases with time, and finally has an important impact on the whole cross-section.

The depth average of m₀ is

(29){\overline{m}}_{\mathrm{0}}={\displaystyle \sum _{i=\mathrm{1}}^{\infty }{\displaystyle \frac{\mathrm{2}({\beta }_i^{\mathrm{2}}+R_s^{\mathrm{2}})}{{\beta }_i^{\mathrm{2}}+R_s^{\mathrm{2}}+R_s}}}{\left({\displaystyle \frac{\sin {\beta }_i}{{\beta }_i}}\right)}^{\mathrm{2}}\exp (-{\beta }_i^{\mathrm{2}}\tau ).

Figure 8 shows the variation of {\overline{m}}_{\mathrm{0}}with τ for R_s = 0.25, 0.5, 0.75, 1.0, and 2.0. It is shown that {\overline{m}}_{\mathrm{0}}gradually decreases to zero. The dimensionless time scale for m₀ to decay fully is given by

(30){\tau }_c>>{\displaystyle \frac{\mathrm{1}}{{\beta }_{\mathrm{1}}^{\mathrm{2}}}},

and the corresponding dimensioned time scale is given by

(31)t_c>>{\displaystyle \frac{H^{\mathrm{2}}}{{\beta }_{\mathrm{1}}^{\mathrm{2}}\kappa (\lambda +K_V/\varphi )}}.

As R_s increases, {\overline{m}}_{\mathrm{0}} decreases. This result means that the total amount of volatile contaminant in the water body decreases, as volatilization increases.

Applying the operator {\displaystyle {\int }_{-\infty }^{\infty }(\cdot )}\xi d\xito Eqs. (8), (10), and (11), with the aid of Eqs. (9) and (23), we have

(32){\displaystyle \frac{\partial m_{\mathrm{1}}}{\partial \tau }}={\displaystyle \frac{{\partial }^{\mathrm{2}}{m}_{\mathrm{1}}}{\partial {\zeta }^{\mathrm{2}}}}+{\displaystyle \frac{P{e}_x{R}_x^K\psi }{\varphi }}{m}_{\mathrm{0}},

(33){{\displaystyle \frac{\partial m_{\mathrm{1}}}{\partial \zeta }}|}_{\zeta =\mathrm{0}}=\mathrm{0},{\left(R_s{m}_{\mathrm{1}}+{\displaystyle \frac{\partial m_{\mathrm{1}}}{\partial \zeta }}\right)|}_{\zeta =\mathrm{1}}=\mathrm{0},

(34){m_{\mathrm{1}}(\zeta ,\tau )|}_{\tau =\mathrm{0}}=\mathrm{0.}

Solving Eqs. (32)-(34), we have

(35)$$\begin{gathered} m_{\mathrm{1}}(\zeta ,\tau )={\displaystyle \sum _{j=\mathrm{1}}^{\infty }{\displaystyle \sum _{i=\mathrm{1}}^{\infty }{\displaystyle \frac{P{e}_x{R}_x^K}{\varphi }}{\displaystyle \frac{\mathrm{2}({\beta }_j^{\mathrm{2}}+R_s^{\mathrm{2}})}{{\beta }_j^{\mathrm{2}}+R_s^{\mathrm{2}}+R_s}}}{\displaystyle \frac{\mathrm{2}({\beta }_i^{\mathrm{2}}+R_s^{\mathrm{2}})}{{\beta }_i^{\mathrm{2}}+R_s^{\mathrm{2}}+R_s}}{\displaystyle \frac{\sin {\beta }_i}{{\beta }_i}}\cdot (P_{\mathrm{1}}+P_{\mathrm{2}})P_{\mathrm{3}}\cos ({\beta }_j\zeta ),} \\ \end{gathered}$$

where P₁, P₂ and P₃ can be expressed respectively as

(36)P_1=\{\begin{array}{cc}{\displaystyle \frac{\left({\beta }_i+{\beta }_j\right)\sin \left({\beta }_i-{\beta }_j\right)+\left({\beta }_i-{\beta }_j\right)\sin \left({\beta }_i+{\beta }_j\right)}{2\left({\beta }_i^2-{\beta }_j^2\right)\left(\cosh \alpha -\sinh \alpha /\alpha \right)/\cosh \alpha }}& \left(i=j\right)\\ {\displaystyle \frac{\sin \left(2{\beta }_i\right)+2{\beta }_i}{4{\beta }_i\left(\cosh \alpha -\sinh \alpha /\alpha \right)/\cosh \alpha }}& \left(i\ne j\right)\end{array},

(37)$$\begin{gathered} P_{\mathrm{2}}=-{\displaystyle \frac{({\beta }_i+{\beta }_j)[{({\beta }_i-{\beta }_j)}^{\mathrm{2}}+{\alpha }^{\mathrm{2}}]\sin ({\beta }_i+{\beta }_j)}{[\mathrm{2}{\alpha }^{\mathrm{4}}+\mathrm{2}{({\beta }_i^{\mathrm{2}}-{\beta }_j^{\mathrm{2}})}^{\mathrm{2}}+\mathrm{4}{\alpha }^{\mathrm{2}}({\beta }_i^{\mathrm{2}}+{\beta }_j^{\mathrm{2}})](\cosh \alpha -\sinh \alpha /\alpha )}} \\ -{\displaystyle \frac{({\beta }_i-{\beta }_j)[{({\beta }_i+{\beta }_j)}^{\mathrm{2}}+{\alpha }^{\mathrm{2}}]\sin ({\beta }_i-{\beta }_j)}{[\mathrm{2}{\alpha }^{\mathrm{4}}+\mathrm{2}{({\beta }_i^{\mathrm{2}}-{\beta }_j^{\mathrm{2}})}^{\mathrm{2}}+\mathrm{4}{\alpha }^{\mathrm{2}}({\beta }_i^{\mathrm{2}}+{\beta }_j^{\mathrm{2}})](\cosh \alpha -\sinh \alpha /\alpha )}} \\ -{\displaystyle \frac{\mathrm{2}\alpha ({\beta }_i^{\mathrm{2}}+{\beta }_j^{\mathrm{2}})\sinh \alpha +\mathrm{2}{\alpha }^{\mathrm{3}}\sinh \alpha }{[\mathrm{2}{\alpha }^{\mathrm{4}}+\mathrm{2}{({\beta }_i^{\mathrm{2}}-{\beta }_j^{\mathrm{2}})}^{\mathrm{2}}+\mathrm{4}{\alpha }^{\mathrm{2}}({\beta }_i^{\mathrm{2}}+{\beta }_j^{\mathrm{2}})](\cosh \alpha -\sinh \alpha /\alpha )}}, \end{gathered}$$

and

(38)P_3=\{\begin{array}{cc}{\displaystyle \frac{-exp\left(-{\beta }_j^2\tau \right)+exp\left(-{\beta }_j^2\tau \right)}{{\beta }_j^2-{\beta }_j^2}}& \left(i\ne j\right)\\ \tau exp\left(-{\beta }_j^2\tau \right)& \left(i=j\right)\end{array},

Then we have

(39){\overline{m}}_{\mathrm{1}}(\tau )={\displaystyle \sum _{j=\mathrm{1}}^{\infty }{\displaystyle \sum _{i=\mathrm{1}}^{\infty }{\displaystyle \frac{P{e}_x{R}_x^K}{\varphi }}{\displaystyle \frac{\mathrm{2}({\beta }_j^{\mathrm{2}}+R_s^{\mathrm{2}})}{{\beta }_j^{\mathrm{2}}+R_s^{\mathrm{2}}+R_s}}}{\displaystyle \frac{\mathrm{2}({\beta }_i^{\mathrm{2}}+R_s^{\mathrm{2}})}{{\beta }_i^{\mathrm{2}}+R_s^{\mathrm{2}}+R_s}}{\displaystyle \frac{\sin {\beta }_i}{{\beta }_i}}}{\displaystyle \frac{\sin {\beta }_j}{{\beta }_j}}\cdot (P_{\mathrm{1}}+P_{\mathrm{2}})P_{\mathrm{3}}.

Figure 9 shows the variation of m₁/m₀ with ξ forP{e}_x{R}_x^K/\varphi =\mathrm{1.0}, α=1.0, τ=1.0, and R_s = 0.25, 0.5, 1.0, 2.0, and 4.0. It is shown that m_{\mathrm{1}}/{\overline{m}}_{\mathrm{0}} increases with ξ, which means that the mass center of a contaminant moves quickly on the streamline through ξ=1.0.

Figure 10 shows the variation of {\overline{m}}_{\mathrm{1}}/{\overline{m}}_{\mathrm{0}}with τ forP{e}_x{R}_x^K/\varphi =\mathrm{1.0}, α=1.0, and R_s = 0.25, 0.5, 1.0, 2.0, and 4.0. The mass center of the whole contaminant cloud will move longitudinally. The slope of the curves in Fig. 10 reflects the moving speed of the mass center. As volatilization increases, the moving speed decreases.

Conclusions

For free-water surface wetland flows, a simplified model has been given to reflect the effects of advection, mass dispersion, apparent reaction, and volatilization on contaminant transport.

The effect of periodic apparent reaction on contaminant transport is separated from the hydraulic effect via an extended transformation. The transformation for constant apparent reaction rate can be included as a special case of σ=0. For three cases,\theta (\tau )=\left|\cos (\tau )\right|, -\left|\cos (\tau )\right|, and cos(τ), the effects of periodic reaction on contaminant transport are illustrated. It is shown that the oscillatory component of periodic apparent reaction rate may increase or decrease the contaminant concentration.

The method of concentration moments was used to obtain the zeroth and first order concentration moments. The distribution of m₀ for the volatile contaminant was different from that for the non-volatile contaminant in the vertical direction. The former kept constant, while the latter decreased from the bottom bed to the free water surface. The total amount of contaminant cloud decays with time. At the initial stage, the influenced range of volatilization for m₀ is limited to the bottom bed. For the long term evolution of a volatile contaminant, the influenced range spreads to the whole cross-section. It was found that the moving speed of the mass center of the contaminant on the streamline increases from the bottom bed to the free water surface. The moving speed of the mass center for the whole contaminant cloud decreases with increases of the ratio of volatilization coefficient to vertical effective mass dispersivity.

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