Environmental Engineering Course Note 8

(Transport Processes II)

Joonhong Park

Yonsei CEE Department

2015. 5. 20

CEE3330 Y2013 WEEK3

CEE3330-01 May 8, 2007 Joonhong Park Copy Right

Transport Processes (II)

4.B Particle Motion– Drag on Particles– Gravitational Settling– Brownian Diffusion

4.C Mass Transfer at Fluid Boundaries– Mass-Transfer Coefficient– Transport across the Air-Water Interface

Size of environmental particles

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10-3 10-2 10-1 100 101 102

Unit (micro meter)

Aerosol

Hydrosol

Soil

Microbial

Fine mode Coarse mode

Nucleation mode Accumulation mode

Colloidal solids Suspended solids

Settleable solids

Clay Silt Sand

Virus Bacteria

Algae

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Particle Motion

Drag on Particle

Gravitational Settling

Brownian Diffusion

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Particle Motion

Drag on Particle

dfpd CVdF

21

42

fp

p

VdRe

Drag Force

Reynolds no.of particle

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Drag Coefficient (Cd) vs. Particlel Reynolds Number(Rep)

0.1 0.3 1.0 10 100 1000 10000

300

100

10

1

0.3

Rep

Cd

Stoke’s lawCd=24/Re

Newton’s lawCd=0.445

Intermediate regime

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Fluid properties influencing drag force

VdF pd 3Stoke’s law(Rep < 0.3)

Newton’s law(1000 < Rep < 350,000)

22173.0

VdF pfd

Drag force is a function of viscosity

Drag force is a function of fluid density

Correction for Small Particles(dp< 1µm) in air

g

p

p

gc

c

pd

d

dC

C

VdF

55.0exp80.051.21

3Cc = Cunningham slip correction factor

λg = the mean free path (the average distance that a gas molecule travels before colliding with another molecule.) Ex. λg = 0.066 micrometer at 293 oK and 1 atm

fVFd f = friction coefficient

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Gravitational Settling(a particle is denser than fluid)

dbg FFF

6

3p

g

dgF

6

3p

fb

dgF

)( VfunctionFd Terminal settling velocity (Vt) =

satisfyingV

2/1

3

4

f

f

d

pt C

gdv

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dgb FFF

6

3p

g

dgF

6

3p

fb

dgF

)( VfunctionFd

Terminal settling velocity (Vt) =

satisfyingV

2/1

3

4

f

f

d

pt C

gdv

Gravitational Settling(a particle is lighter than fluid)

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Particle Motion

Brownian Diffusion (following Fick’s 1st Law)

Diffusivity (Stoke-Einstein relation)

fkT

D

XC

DN diffusionb

k: Boltzmann’s constantT: absolute temp.f:fricktion coefficient

fVFd

pd

D

0.001 1

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Comparing Particle Transport by Settling and Diffusion

pd0.3 0.7 micro-meter

TravelingDistance

(cm)

Diffusion-air

Diffusion-water

0.03

3 x 10-5

Settling-airSettling-water

When a particle is lager than 1 micro-meter, settling becomes dominant

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Inter-phase Mass Transfer

Phase 1

Boundary Layer Inter-phase Mass Transfer

X-direction

Y-direction

Phase 2

Co

C

)(, imyi CCkN

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Mass-transfer coefficient

)(, iyi CCLD

N

)(, imyi CCkN

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Film Theory

fm L

Dk

Ci C

Well-mixedfluid

StagnantFluid film

Boundary

Flux

Lf

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Penetration Theory

2/1

tD

km

Ci C

Well-mixedfluid

StagnantFluid film

Boundary

Flux

Lf

t3

t2

t1instantaneous

Water-Air Interface (Double-Layer Theory)

Pi P Partial Pressure

Turbulently mixed zone

Stagnant film layer

Boundary

Flux

La

Stagnant film layer

Turbulently mixed zone

C Ci

Lw

Air

Water

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Reading Assignment

Environmental Engineering & Sciences (EES)

4.B Particle Motion– Drag on Particles– Gravitational Settling– Brownian Diffusion

4.C Mass Transfer at Fluid Boundaries– Mass-Transfer Coefficient– Transport across the Air-Water Interface