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Sorption/desorption is important phenomena in various scientific and industrial fields, such as contaminants removal via activated carbon in water treatment, duration of soil remediation, gas separation in chemical engineering, pollutant transport by airborne particles (or suspended particles/sediments in rivers or batch or column tests for biodegradation or contaminant transport), etc. Sorption/desorption kinetics has to be considered for sorbents with large grain sizes (d), distribution coefficients (K_d) or small intraparticle porosities (ε). A coupled model was developed taking both external film diffusion and intraparticle diffusion into account. At early time, external mass transfer resistance dominates mass transfer, while the internal mass transfer resistance takes over at late time. Since no simple analytical solution in real time domain exists for the coupled model even for homogenous sorbent and numerical solutions need intricate discretization in time and space, I derived a semi-Laplace solution for coupled model where the (average) concentrations in each phase (e.g., solid and water) in real time domain can be obtained via a very fast and efficient inverse Laplace transformation. For practical use, a first order approximation solution for coupled model was derived, which can be easily applied for fractional solute uptake in the solid phase ranging from 50 to 91% of equilibrium achieved, representing scenarios typically encountered in batch experiments. The derived solutions showed that mass transfer shifts from external film diffusion to intraparticle diffusion depending on K_d, ε or Sherwood number (Sh).
Secondly, I extended the semi-Laplace solution to redistribution in heterogeneous systems allowing easy adaptation of different kinetics models for different sorbents in terms of their properties (particle size, geometry, intraparticle porosity, organic carbon content, distribution coefficient etc.) by using different transfer functions in Laplace domain; solute concentrations in each phase can be obtained by inverse Laplace transformation. The redistribution model was validated by two batch experiments (Phenanthrene/Anthracene d10 redistribution in mixtures of spherical microplastics of different sizes and a sediment suspension with polyethylene passive samplers). Sherwood numbers fitted to experimental data agree well with Sherwood number relationships for suspended particles based on boundary layer theory and Kolmogorov’s turbulence theory. Heterogenous materials (e.g., soils or sediments) with only few percent of strong sorbents might lead to pseudo-equilibration at early times, which rapidly lowers concentration gradients and thus slows down sorption in the remaining part of the mixture tremendously (even slower than in an equivalent homogeneous strong sorbent). The kinetics of this few percent of strong sorbents are super slow and their behaviors cannot be captured in short batch tests, which might explain discrepancies in solute mass recoveries in batch tests (mass release from strong sorbents not captured) and exhaustive solvent extraction methods.
The coupled model application also allows to nicely explain particle/gas distribution kinetics of polycyclic aromatic hydrocarbons (PAHs) in the atmosphere. Mass transfer of low molecular weight PAHs with low distribution coefficients (K_pg s) is dominated by intraparticle pore diffusion, while external film diffusion is limiting for high molecular weight compounds. The latter results in maximum observed distribution coefficients i.e. apparent bulk distribution coefficients (K_(pg,b,a)); appear independent on octanol-air partitioning coefficient (K_oa) or subcooled liquid vapor pressure (P_L^o) (slope = 0); for intraparticle pore diffusion they in-/decrease with the square root of K_oa or P_L^o. Moderate molecular weight compounds lie in between (slope shifts from 1/2 to 0) and both external and internal mass transfer resistances have to be considered.
Furthermore, I developed a numerical one-dimensional (1D) reactive solute transport model considering different sorption/desorption kinetic mechanisms to investigate the impacts of initial conditions (e.g., pre-equilibrium or after the first flooding of the column) and heterogenous materials on column leaching tests. Through numerical experiments, steep concentration gradients close to the outlet of column have to be expected for compounds with small K_d (< 1 L kg-1). Only for large K_d (> 10 L kg-1), initial conditions can be neglected. Longitudinal dispersion and non-linear sorption smooth the leaching curves and lead to smaller initial concentrations than expected under equilibrium conditions especially for compounds with small K_d values. Sample heterogeneity, including grain size and fractions of sorptive particles, strongly influences leaching curves. A small fraction (< 1%) of strongly sorbing particles with high K_d may cause very slow desorption rates compared to homogeneous sorbent having the same average K_d, especially if mass release is limited by IPPD, leading to non-equilibrium. Mixtures of strongly coarse particles with a small fraction (< 10%) of less sorbing fine material might lead to a stepwise concentration decline in the column effluent.
Finally, I introduced two mesh free methods (semi-Laplace solution and physical in¬formed neural networks (PINNs)) to solve 1D advective-dispersive transport with coupled film-intraparticle pore diffusion (ADE-FIPPD), which can replace the tedious numerical solutions if pre-equilibrium condition (uniform concentration) is initially achieved (e.g., K_d > 10 L kg-1). The semi-Laplace solution shows remarkable performance with deviations of normalized concentrations within the spatial-temporal domain of less than 1% when compared to the numerical solutions. PINNs exhibit slightly higher deviations (< 10%) than the semi-Laplace solution which are still acceptable across most of the spatial-temporal domain excluding the regions with stepwise concentration gradients. Since external mass transfer resistance and dispersion predominate for a very short time period initially, the mass transfer shift in ADE-FIPPD is more likely to be observed for coarse particles with large K_ds (e.g., d >2 cm and K_d > 1000 L kg-1) and intraparticle porosities (ε > 0.2). ADE-FIPPD of fine particles (e.g., d < 400 μm) show almost the same behavior of intraparticle pore diffusion (ADE-IPPD).
This thesis provides various solutions for sorption/desorption kinetics dominated by film diffusion or intraparticle diffusion including coupled models in both homogenous and heterogeneous systems. Those models help us to better understand sorption/desorption behavior in mixtures and the interactions of different sorbents (mass redistribution) within different compartments, which can be easily applied in many fields, such as groundwater remediation, chemical engineering, analytical chemistry, air pollution control, etc. |
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