1. Environmental Engineering Lecture Note Week 10 (Transport Processes) Joonhong Park Yonsei CEE Department 2016. 5. 11 CEE3330 Y2013 WEEK3

2. CEE3330-01 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

3. CEE3330-01 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

4. CEE3330-01 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

5. CEE3330-01 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 ~ ~

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

7. CEE3330-01 May 8, 2007 Joonhong Park Copy Right 1-D Advective Flux of Contaminant C IN (contaminant concentration) V (water velocity) Flux N @ 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

8. CEE3330-01 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).

9. CEE3330-01 May 8, 2007 Joonhong Park Copy Right 1-D Diffusive Flux of Contaminant t =0 t=t 1 t=t 2 t=t 3

10. CEE3330-01 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 10 -5 cm 2 /s

11. CEE3330-01 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.

12. CEE3330-01 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 10 -5 cm 2 /s

13. CEE3330-01 May 8, 2007 Joonhong Park Copy Right Albert Einstein’s Solution X: traveling distance t: traveling time D: Diffusion coefficient

14. CEE3330-01 May 8, 2007 Joonhong Park Copy Right Example Passive Dosimetry Ambient Concentration Co=? Adsorbent Co 0 Diffusion Distance L

15. CEE3330-01 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.

16. CEE3330-01 May 8, 2007 Joonhong Park Copy Right Types of Dispersion Processes Slow dispersion Rapid dispersion Slow dispersion Side view Top view

17. CEE3330-01 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)

18. CEE3330-01 May 8, 2007 Joonhong Park Copy Right Shear Flow Dispersion (in a laminar flow) Injected pulse @ t=0 Dispersed pulse @ t=t Factors to cause the dispersion -Average effect -Concentration gradient due to velocity gradient (momentum gradient) Fluid velocity profile Concentration gradient

19. CEE3330-01 May 8, 2007 Joonhong Park Copy Right Turbulent Dispersion y x U y C Gaussian Normal Distribution

20. Energy Balance and Bernoulli Eq. A1A1 A2A2

21. Momentum Balance Momentum Flux x y z Newtonian fluid

22. CEE3330-01 May 8, 2007 Joonhong Park Copy Right Dispersion Equation Dispersivity (Tensor) Free-Liquid Molecular Diffusion Coefficient (Scalar) Identity Matrix

23. CEE3330-01 May 8, 2007 Joonhong Park Copy Right Dispersion Distance X: traveling distance t: traveling time Ddd: Dispersion coefficient

24. CEE3330-01 May 8, 2007 Joonhong Park Copy Right Water Flow in Porous Media - History and equation. - Determination of K and k.

25. CEE3330-01 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

26. CEE3330-01 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

27. CEE3330-01 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 )

28. CEE3330-01 May 8, 2007 Joonhong Park Copy Right Typical values of K and k PermeableSemi-permeableImpermeable Permeability Aquifer Soils Rocks -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 3 4 5 6 7 8 9 10 11 12 13 14 15 16 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 )

29. CEE3330-01 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)

30. CEE3330-01 May 8, 2007 Joonhong Park Copy Right Transport and dispersion of a fixed quantity of a nonreactive groundwater contaminant X y t1 t2 t3

31. CEE3330-01 May 8, 2007 Joonhong Park Copy Right Reading Assignment Read p.159-172 Practice EXHIBIT 4.A.1 at p.165 EXAMPLE 4.D.1 at p.196-197

32. CEE3330-01 May 8, 2007 Joonhong Park Copy Right HW Problem 4.1 Problem 4.4 Problem 4.6 Problem 4.12