Environmental Engineering Lecture Note Week 10 Transport Processes

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Environmental Engineering Lecture Note Week 10 (Transport Processes) Joonhong Park Yonsei CEE Department 2016.

Environmental Engineering Lecture Note Week 10 (Transport Processes) Joonhong Park Yonsei CEE Department 2016. 5. 11 CEE 3330 Y 2013 WEEK 3

Transport Processes (I) 4. A Basic Concepts and Mechanisms – – Contaminant Flux Advection

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 3330 -01 May 8, 2007 Joonhong Park Copy Right

Mechanisms of mass transport and transfer Y-direction Dispersion (by Momentum Gradient) Pollutant Mass in

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 CEE 3330 -01 May 8, 2007 Joonhong Park Copy Right

Definition: Flux of material i (Ni) = The number of moles of material i

Definition: 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 d. A * dt CEE 3330 -01 May 8, 2007 Joonhong Park Copy Right

Reynolds transport theorem: Mass continuity Magnitude of the molar flux normal to a differential

Reynolds transport theorem: Mass continuity Magnitude of the molar flux normal to a differential element of surface area, d. A ~. = |Ni| cos θ = Ni nv ~ Flux is a Vector: Ni = Nix ix + Niy iy + Niz iz CEE 3330 -01 May 8, 2007 Joonhong Park Copy Right

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

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

1 -D Advective Flux of Contaminant ΔL CIN (contaminant concentration) A t V (water

1 -D Advective Flux of Contaminant ΔL CIN (contaminant concentration) A t V (water velocity) X CIN V ΔL A Flux N @ X= C*∆L * A / AMay (∆L 8, /V) CEE 3330 -01 2007= C*V Joonhong Park Copy Right Assumption: Steady State Uniform Water Flow Field t + ΔL/V

Molecular Diffusion The random motion of fluid molecules causes a net movement of species

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 3330 -01 May 8, 2007 Joonhong Park Copy Right

1 -D Diffusive Flux of Contaminant t =0 t=t 1 t=t 2 t=t 3

1 -D Diffusive Flux of Contaminant t =0 t=t 1 t=t 2 t=t 3 CEE 3330 -01 May 8, 2007 Joonhong Park Copy Right

Fick’s 1 st Laws Di : Diffusion coefficient or diffusivity a property of the

Fick’s 1 st Laws Di : 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 10 -5 cm 2/s CEE 3330 -01 May 8, 2007 Joonhong Park Copy Right

Example Passive Dosimetry Adsorbent Ambient Concentration Co=? Mt: accumulated mass at t. (Assumption: Co

Example Passive Dosimetry Adsorbent Ambient Concentration Co=? Mt: accumulated mass at t. (Assumption: Co =constant) Co 0 L Diffusion Distance CEE 3330 -01 May 8, 2007 Joonhong Park Copy Right

Fick’s 1 st Laws Di : Diffusion coefficient or diffusivity a property of the

Fick’s 1 st Laws Di : 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 10 -5 cm 2/s CEE 3330 -01 May 8, 2007 Joonhong Park Copy Right

Albert Einstein’s Solution X: traveling distance t: traveling time D: Diffusion coefficient CEE 3330

Albert Einstein’s Solution X: traveling distance t: traveling time D: Diffusion coefficient CEE 3330 -01 May 8, 2007 Joonhong Park Copy Right

Example Passive Dosimetry Adsorbent Ambient Concentration Co=? Co 0 L Diffusion Distance CEE 3330

Example Passive Dosimetry Adsorbent Ambient Concentration Co=? Co 0 L Diffusion Distance CEE 3330 -01 May 8, 2007 Joonhong Park Copy Right

Dispersion The spreading of contaminants by nonuniform flow is called dispersion. This is not

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 3330 -01 May 8, 2007 Joonhong Park Copy Right

Types of Dispersion Processes Slowdispersion Rapid dispersion Side view Top view CEE 3330 -01

Types of Dispersion Processes Slowdispersion Rapid dispersion Side view Top view CEE 3330 -01 May 8, 2007 Joonhong Park Copy Right

Types of Dispersion Processes Taylor (Shear Flow) Dispersion: occurs in laminar flow (pipes and

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 3330 -01 May 8, 2007 Joonhong Park Copy Right

Shear Flow Dispersion (in a laminar flow) Fluid velocity profile Concentration gradient Injected pulse

Shear Flow Dispersion (in a laminar flow) Fluid velocity profile Concentration gradient Injected pulse @ t=0 Dispersed pulse @ t=t Factors to cause the dispersion -Average effect -Concentration gradient due to velocity gradient (momentum gradient) CEE 3330 -01 May 8, 2007 Joonhong Park Copy Right

Turbulent Dispersion y y x U C Gaussian Normal Distribution CEE 3330 -01 May

Turbulent Dispersion y y x U C Gaussian Normal Distribution CEE 3330 -01 May 8, 2007 Joonhong Park Copy Right

Energy Balance and Bernoulli Eq. A 1 A 2

Energy Balance and Bernoulli Eq. A 1 A 2

Momentum Balance Momentum Flux Newtonian fluid y x z

Momentum Balance Momentum Flux Newtonian fluid y x z

Dispersion Equation Dispersivity (Tensor) Free-Liquid Molecular Diffusion Coefficient (Scalar) Identity Matrix CEE 3330 -01

Dispersion Equation Dispersivity (Tensor) Free-Liquid Molecular Diffusion Coefficient (Scalar) Identity Matrix CEE 3330 -01 May 8, 2007 Joonhong Park Copy Right

Dispersion Distance X: traveling distance t: traveling time Ddd: Dispersion coefficient CEE 3330 -01

Dispersion Distance X: traveling distance t: traveling time Ddd: Dispersion coefficient CEE 3330 -01 May 8, 2007 Joonhong Park Copy Right

Water Flow in Porous Media - History and equation. - Determination of K and

Water Flow in Porous Media - History and equation. - Determination of K and k. CEE 3330 -01 May 8, 2007 Joonhong Park Copy Right

Darcy’s Experiment (1856) Flow of water in homogeneous sand filter under steady conditions A:

Darcy’s Experiment (1856) Flow of water in homogeneous sand filter under steady conditions A: cross area Sand Porous Medium h 2 L h 1 Datum Q = - K * A * (h 2 -h 1)/L CEE 3330 -01 May 8, 2007 K= hydraulic conductivity Joonhong Park Copy Right

Darcy’s Experiment (1856) Flow of water in homogeneous sand filter under steady conditions A:

Darcy’s Experiment (1856) Flow of water in homogeneous sand filter under steady conditions A: cross area Sand Porous Medium h 2 L h 1 Datum Q = - K * A * (h 2 -h 1)/L CEE 3330 -01 May 8, 2007 K= hydraulic conductivity Joonhong Park Copy Right

Darcy’s Law Q = - K * A * (Φ 2 - Φ 1)/L

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/d. L] Hydraulic Conductivity, K (L/T) KΞk*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 3330 -01 May 8, 2007 Joonhong Park Copy Right

Typical values of K and k -log K (cm/s) -2 -1 Permeability 1 2

Typical values of K and k -log K (cm/s) -2 -1 Permeability 1 2 Permeable 3 4 Clean gravel Rocks 4 7 8 9 10 11 Impermeable None Very fine sand, silt, Clean sand or Sand gravel Loess, loam, solonetz Oil rocks 3 6 Poor Peats -log k (cm 2) 5 Semi-permeable Good Aquifer Soils 0 5 Stratified clay Unweathered clay Sandstone Good Breccia, Limestone granite dolomite 6 CEE 3330 -01 7 8 May 9 8, 2007 10 11 12 13 14 15 16 Joonhong Park Copy Right

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

Hydrodynamic Dispersion Mechanical Dispersion (Tensor) In one-D system, α, dispersivity V, pore velocity (=q/n) Molecular Diffusion (Scalar) Identity Matrix CEE 3330 -01 May 8, 2007 Joonhong Park Copy Right

Transport and dispersion of a fixed quantity of a nonreactive groundwater contaminant y t

Transport and dispersion of a fixed quantity of a nonreactive groundwater contaminant y t 3 t 1 t 2 X CEE 3330 -01 May 8, 2007 Joonhong Park Copy Right

Reading Assignment Read p. 159 -172 Practice EXHIBIT 4. A. 1 at p. 165

Reading Assignment Read p. 159 -172 Practice EXHIBIT 4. A. 1 at p. 165 EXAMPLE 4. D. 1 at p. 196 -197 CEE 3330 -01 May 8, 2007 Joonhong Park Copy Right

HW Problem 4. 1 Problem 4. 4 Problem 4. 6 Problem 4. 12 CEE

HW Problem 4. 1 Problem 4. 4 Problem 4. 6 Problem 4. 12 CEE 3330 -01 May 8, 2007 Joonhong Park Copy Right