Abstract
A mathematical modeling study is conducted to describe the mobility and retention of elemental copper nanoparticeles (nCu⁰) in one-dimensional saturated porous media in presence of humic acid (HA). Two numerical solution schemes were implemented in this study: (i) the finite differences (FD: Eulerian) and (ii) the random-walks particle-tracking (RWPT: Lagrangian). FD approach is the conventional method used in colloid filtration modeling for both parameter estimation and predictive modeling. RWPT removes an intrinsic limitation of FD in addressing non-uniform/stochastic particle characteristics (e.g., polydispersity) but its application is limited to predictive modeling. Inverse analysis of the experimental breakthrough concentrations (BTCs) of nCu0 was done using FD method by incorporating four different filtration modeling approaches: (i) the clean-bed filtration theory (CFT), (ii) maximum attainable retention capacity (MRC) model, and two subsets of two-kinematic attachment site model (2S). The 2S model considers a mix of CFT and MRC attachment sites under (iii) reaction-limited (unfavorable) and (iv) diffusion limited (favorable) attachment conditions for MRC sites. This was done for three nCu⁰ transport experiments at different background HA concentrations ranging from 1 to 10 mg/l. 2S model provided the best fit to the observed BTCs with a normalized sum of squared residual (NSSR) values lower than other two models by an order of magnitude. The reported NSSRs for the 2S model plunged by approx. 90−97% from the corresponding NSSRs for CFT model. Surprisingly, the NSSRs for 2S3P model also dropped sharply (approx. 75−97%) compared to corresponding NSSRs for MRC model across the experiments, though both of the models incorporated 3 fitting parameters. The special subclass of 2S model with three fitting parameters (2S3P model with favorable attachment to MRC site) showed almost similar normalized SSR values to those of unconstrained 2S model (2S4P model) for all tested HA concentrations. Predictive modeling of particle retention profiles was done with both FD and RWPT approaches and the four specified filtration models using model-specific nCu⁰ mobility parameters obtained from the inverse analysis of nCu⁰ BTCs. None of the Eulerian models, which incorporated an ensemble-averaged uniform particle size, could predict the observed non-exponential nCu0 RPs. The Lagrangian approach qualitatively reconciled the gap between the observed and predicted RPs. Hyper-exponential RPs were predicted when the reported particles size heterogeneity was incorporated in RWPT simulations. Physical straining was predicted to influence nCu⁰ RPs only in the first 1−1.5 cm of column length. Coupling the theoretical "shadow effect" on retention capacity and particles size heterogeneity with or without straining effects could not qualitatively capture the discrepancy between the predicted and observed RPs.