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4. Mathematical Model

  D

where C is particle concentration, D_Lon, D_Lat, and D_V are the longitudinal, lateral, and vertical dispersion coefficients, respectively, V is the mean surface-water velocity, and is a mass-transfer coefficient for particle immobilization. Interception by aquatic vegetation and adsorption of particles that diffused to the sediment were among the plausible mechanisms of particle immobilization in our experiment; however, sedimentation did not contribute significantly to TiO₂ removal because the settling velocity of these particles was very small (<10^-2 cm/h).

[8] We employed a finite-element method to solve equation (1) for a three-dimensional domain measuring 9.3 m long, 3 m wide (the channel width), and 0.6 m deep (Figure 1a). The model domain was discretized into 16,000 quadratic-Lagrange elements and the numerical solution to equation (1) was obtained for zero initial TiO₂ concentrations, a zero gradient in TiO₂ concentrations across the lateral boundaries, and zero total flux across both the free surface and ground surface. A specified TiO₂ flux across a planar internal boundary (0.05 m x 1.8 m) was used to simulate the injection source (Figures 1a and 1b).

[9] Observations from a separate experiment on the transport of bromide (a conservative tracer) revealed that the channel walls were permeable and that a cross-channel component of surface-water flow existed. While the bromide data could not be used to make quantitative determinations about TiO₂ transport (because the magnitude of the flow velocities varied between experiments), the bromide results did emphasize the need to account for cross-channel flow within our modeling framework. We accomplished this by computing the magnitude and direction of the mean surface-water velocity from the component velocities and then we rotated the coordinate system for the model domain such that the x axis was parallel with the direction of V. The magnitude and direction of the mean surface-water velocity are expressed by

  D

and

  D

where v₁ and v₂ are the components of the surface-water velocity parallel to the channel wall and perpendicular to the channel wall, respectively (Figure 1a).

[10] We applied the model in inverse mode in order to estimate v₁ and v₂, as well as the parameters that govern dispersion (D_Lon, D_Lat, and D_V) and particle-immobilization kinetics (). Best-fit parameter values were identified by using a Levenberg-Marquardt algorithm (as programmed in MATLAB) to minimize the sum-of-the-squared residuals between measured and modeled TiO₂ concentrations.