Environmental Engineering Lecture Note Week 10 (Transport Processes) Joonhong Park Yonsei CEE Department CEE3330 Y2013 WEEK3

CEE May 8, 2007 Joonhong Park Copy Right Transport Processes (I) 4.A Basic Concepts and Mechanisms – Contaminant Flux – Advection – Diffusion – Dispersion 4.D Transport in Porous Media – Fluid Flow through Porous Media – Contaminant Transport in Porous Media

CEE May 8, 2007 Joonhong Park Copy Right Mechanisms of mass transport and transfer Pollutant Mass in Bulk Fluid Control Element Porous Solid Intra-phase Diffusion Boundary Layer Inter-phase Mass Transfer Advection (by Water Flow) Bulk-phase diffusion (by Conc. Gradient) Dispersion (by Momentum Gradient) X-direction Y-direction by Diffusion

CEE May 8, 2007 Joonhong Park Copy Right Definition: Flux of material i Flux of material i (Ni) = The number of moles of material i transported per unit cross-sectional area per unit time = # mole of material i dA * dt

CEE May 8, 2007 Joonhong Park Copy Right Reynolds transport theorem: Mass continuity Magnitude of the molar flux normal to a differential element of surface area, dA = |N i | cos θ = N i. n v Flux is a Vector: N i = N ix i x + N iy i y + N iz i z ~ ~

Characterization of Flow Steady Flow Unsteady Flow Uniform Flow Nonuniform Flow Steady State For Turbulent Flow

CEE May 8, 2007 Joonhong Park Copy Right 1-D Advective Flux of Contaminant C IN (contaminant concentration) V (water velocity) Flux X= C*∆L * A / A (∆L /V) = C*V ΔLΔL A C IN V XΔLΔL A t t + ΔL/V Assumption: Steady State Uniform Water Flow Field

CEE May 8, 2007 Joonhong Park Copy Right Molecular Diffusion The random motion of fluid molecules causes a net movement of species from regions of high concentration to regions of low concentration. The rate of movement depends on the spatial gradient of concentration of a solute. Our discussion is restricted to conditions in which the diffusing species is present at a low mole fraction (the infinite dilution condition).

CEE May 8, 2007 Joonhong Park Copy Right 1-D Diffusive Flux of Contaminant t =0 t=t 1 t=t 2 t=t 3

CEE May 8, 2007 Joonhong Park Copy Right Fick’s 1 st Laws D i : Diffusion coefficient or diffusivity a property of the diffusing species For molecules in air, typically D values is 0.1 cm 2 /s For molecules in water, typically D values is cm 2 /s

CEE May 8, 2007 Joonhong Park Copy Right Example Passive Dosimetry Ambient Concentration Co=? (Assumption: Co =constant) Adsorbent Co 0 Diffusion Distance L M t : accumulated mass at t.

CEE May 8, 2007 Joonhong Park Copy Right Fick’s 1 st Laws D i : Diffusion coefficient or diffusivity a property of the diffusing species For molecules in air, typically D values is 0.1 cm 2 /s For molecules in water, typically D values is cm 2 /s

CEE May 8, 2007 Joonhong Park Copy Right Albert Einstein’s Solution X: traveling distance t: traveling time D: Diffusion coefficient

CEE May 8, 2007 Joonhong Park Copy Right Example Passive Dosimetry Ambient Concentration Co=? Adsorbent Co 0 Diffusion Distance L

CEE May 8, 2007 Joonhong Park Copy Right Dispersion The spreading of contaminants by nonuniform flow is called dispersion. This is not a fundamentally distinct transport process. Instead, dispersion is caused by nonuniform advection and influenced by diffusion. A phenomenon caused by the gradient of momentum, which is expressed by a tensor.

CEE May 8, 2007 Joonhong Park Copy Right Types of Dispersion Processes Slow dispersion Rapid dispersion Slow dispersion Side view Top view

CEE May 8, 2007 Joonhong Park Copy Right Types of Dispersion Processes Taylor (Shear Flow) 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)

CEE May 8, 2007 Joonhong Park Copy Right Shear Flow Dispersion (in a laminar flow) Injected t=0 Dispersed t=t Factors to cause the dispersion -Average effect -Concentration gradient due to velocity gradient (momentum gradient) Fluid velocity profile Concentration gradient

CEE May 8, 2007 Joonhong Park Copy Right Turbulent Dispersion y x U y C Gaussian Normal Distribution

Energy Balance and Bernoulli Eq. A1A1 A2A2

Momentum Balance Momentum Flux x y z Newtonian fluid

CEE May 8, 2007 Joonhong Park Copy Right Dispersion Equation Dispersivity (Tensor) Free-Liquid Molecular Diffusion Coefficient (Scalar) Identity Matrix

CEE May 8, 2007 Joonhong Park Copy Right Dispersion Distance X: traveling distance t: traveling time Ddd: Dispersion coefficient

CEE May 8, 2007 Joonhong Park Copy Right Water Flow in Porous Media - History and equation. - Determination of K and k.

CEE May 8, 2007 Joonhong Park Copy Right Darcy’s Experiment (1856) Flow of water in homogeneous sand filter under steady conditions Datum h1h1 h2h2 Sand Porous Medium L A: cross area Q = - K * A * (h 2 -h 1 )/L K= hydraulic conductivity

CEE May 8, 2007 Joonhong Park Copy Right Darcy’s Experiment (1856) Flow of water in homogeneous sand filter under steady conditions Datum h1h1 h2h2 Sand Porous Medium L A: cross area Q = - K * A * (h 2 -h 1 )/L K= hydraulic conductivity

CEE May 8, 2007 Joonhong Park Copy Right Darcy’s Law Q = - K * A * (Φ 2 - Φ 1 )/L Φ piezometric head In a 1-D differential form, Darcy’s law may be: q = Q/A = - K * [dΦ/dL] Hydraulic Conductivity, K (L/T) K Ξ k * ρ * g / μ Here, k = intrinsic permeability (L 2 ) ρ: fluid density (M L -3 ); g: gravity (LT -2 ) μ: fluid viscosity (M L -1 T -1 )

CEE May 8, 2007 Joonhong Park Copy Right Typical values of K and k PermeableSemi-permeableImpermeable Permeability Aquifer Soils Rocks Good PoorNone Clean gravel Clean sand or Sand and gravel Very fine sand, silt, Loess, loam, solonetz Unweathered clay Stratified clayPeats Oil rocks SandstoneGood Limestone dolomite Breccia, granite -log K (cm/s) -log k (cm 2 )

CEE May 8, 2007 Joonhong Park Copy Right Hydrodynamic Dispersion Mechanical Dispersion (Tensor) In one-D system, Molecular Diffusion (Scalar) Identity Matrix α, dispersivity V, pore velocity (=q/n)

CEE May 8, 2007 Joonhong Park Copy Right Transport and dispersion of a fixed quantity of a nonreactive groundwater contaminant X y t1 t2 t3

CEE May 8, 2007 Joonhong Park Copy Right Reading Assignment Read p Practice EXHIBIT 4.A.1 at p.165 EXAMPLE 4.D.1 at p

CEE May 8, 2007 Joonhong Park Copy Right HW Problem 4.1 Problem 4.4 Problem 4.6 Problem 4.12