Mechanisms of mass transport and transfer Ydirection Dispersion

Mechanisms of mass transport and transfer Y-direction Dispersion (by Momentum Gradient) Pollutant Mass in Control Bulk Fluid Element Advection (by Water Flow) Bulk-phase diffusion (by Conc. Gradient) Boundary Layer Porous Solid Inter-phase Mass Transfer by Diffusion Intra-phase Diffusion X-direction

Material (Mole) Balance Molar rate of input of material i across the control volume boundaries Molar rate of output of material i across the control volume boundaries - = ± Net rate of transformation of material i within the control volume Net rate of change (either accumulation or depletion) of material i within the control volume

Differential Form of Material Balance in Vc Have you heard about Gauss Divergence theorem?

Types of Dispersion Processes Taylor Dispersion: occurs in laminar flow (pipes and narrow channels); transverse direction of solute movement driven by solute concentration gradient Turbulent (eddy) dispersion: velocity fluctuations created by fluid turbulence acting across large advection-dominated fields; large channels, rivers, streams, and lakes. Hydrodynamic and mechanical dispersion: flow in porous media (activated carbon filters; groundwater)

Molecular Diffusivity Fick’s 1 st Law Fick’s 2 st Law

Prandtl Hypothesis

(1) Molecular exchange of momentum in laminar flow (2) Macroscopic exchange in turbulent flow Turbulent shear stress

Eddy Momentum Dispersivity Eddy Viscosity (time-averaged) Kinematic Eddy viscosity (time-averaged) UNIT?

Eddy Mass Dispersivity

(Mechanical) Dispersion Coefficient Eddy Dispersivity (Tensor) Free-Liquid Molecular Diffusion Coefficient (Scalar) Identity Matrix

Dispersion Coeff. Determination Independent and direct measurement Dependent and indirect measurement

Hydrodynamic Dispersion Mechanical Dispersion (Tensor) In one-D system, α, dispersivity V, pore velocity (=q/n) Molecular Diffusion (Scalar) Identity Matrix

Dispersion Number

Differential Form of Material Balance in Vc Have you heard about Gauss Divergence theorem?

Advection-Dispersion-Reaction (ADR) Equation CX CX+ΔX CIN A V (water velocity) is varying! X ΔX

1 -D Solute Advection-Dispersion. Reaction (ADR) Eq. In a porous medium, q = n. V

Analytical Solution of 1 -D Solute ADR Eq. Analytical solution is available 1 - and 2 -D for homogeneous systems with uniform velocity Initial and Boundary Conditions C=Co at x=0 t>0 x =>∞ t>0 C=0 at t=0 0 ≤ x ≤ ∞

Analytical Solution of 1 -D Solute ADR Eq.

Sorption Effect Dry bulk density (mass-soil/volume) Q Sorbed phase concentration (mass-solute/mass-soil)

Sorption Effect-Retardation Factor

Simulations C A+D => ? A+D+S=>? A+D+S+R=>? X Continuous Source C A+D+S+R=>? X Slug Release