Transport

You are on a train to NYC.

You are stirring the milk into your coffee.

The train and everything in it are moving toward NYC via directed or advective transport

As you stir, the milk is moving via turbulent diffusion. This is a random process.

Flux

ABBABA CCvF //

Flux: mass per unit area per unit time (ng/m2-day)

transfer or exchange velocity (m/day)

aka mass transfer coefficient

Concentration gradient (ng/m3)

Gradient flux law:

Fick’s First Law

One example of a gradient flux law is Fick’s First Law:

x

CDFx d

d

Relates the diffusive flux (Fx) of a chemical to its concentration gradient (dC/dx) and its molecular diffusion coefficient (D)

Fick’s Second Law

2

2

x

CD

t

C

The local concentration change with time (dC/dt) due to a diffusive transport process is proportional to the second spatial derivative of the concentration (concentration gradient)

Turbulent diffusionIn contrast to molecular diffusion, which arises due to thermal molecular motions, turbulent diffusion is based on the irregular patterns of currents in water and air.

Turbulent vs. laminar flow is defined by the Reynold’s number:

d = spatial dimension of the flow system or objects around which the flow occurs (m)v = typical flow velocityf = dynamic viscosity of the fluid (kg/m-s)f = density of the fluid (kg/m3)

For laminar flow Re < 0.1

ff /Re

dv

Turbulent diffusion

the effect of the turbulent velocity component on the transport of a dissolved substance can be described by an expression which has the same form as Fick’s first law:

x

CEF xx d

d

the molecular diffusivity (D) is now replaced by the turbulent or eddy diffusion coefficient, E

E >>> D

anisotropy

in natural systems, turbulent diffusion is usually anisotropic, meaning that the magnitude of E depends on the direction.

horizontal diffusion is usually much greater than vertical diffusion because:

1. natural systems extend horizontally

2. often the system (ocean, atmosphere) is density stratified

Transport through boundaries(Chapter 19)

What is a boundary = surface at which properties of a system change extensively or, discontinuously (interface)

air-water interface

sediment-water interface

epilimnion - hypolimnion (thermocline)

stratosphere – troposphere (tropopause)

What to boundaries do?

1. control the transport of energy and matter

2. control chemical process triggered by the contact of two systems with different chemical composition

What is the boundary condition?

may be defined by a value (i.e. concentration) or by a flux (i.e. mass flux across the boundary per unit time)

What types of boundaries are there?

1. bottleneck

2. wall

3. diffusive

classified according to the shape of the diffusivity (D) profile

bottleneck boundaries

bottleneck = mass crossing must squeeze itself through a zone in which transport occurs by molecular diffusion (usually interface)

example:

air-water interface

Like a toll booth on the turnpike

Dif

fusi

vity

D(x

)

bottleneck

molecular diffusivity

turbulent diffusion

distance

wall boundaryD

iffu

sivi

ty D

(x)

wall

turbulent diffusion

distance

molecular diffusion

at a wall boundary, a zone characterized by turbulent diffusion encounters a zone in which transport is dominated by a much slower process, such as molecular diffusion

example:

sediment-water interface

Like an icy stretch of road

diffusive boundaryD

iffu

sivi

ty D

(x)

diffusive boundary

distance

C (x)

D (x)

at a diffusive boundary, diffusivity is of similar magnitude on either side

diffusivity may be molecular or turbulent

example:

troposphere – stratosphere boundary (tropopause)

Air – Water Exchange (Chapter 20)

Inputs and outputs of PCBs (kg y-1)

Totten 2005

NY/NJ Harbor Estuary

Advection from Hudson River

260-470

STP effluents 32

Volatilization 317-846

Advection to Atlantic 130-190

Atm dep18-48

Stormwater 36-140

CSOs 67-146

Storage in sediments147-307

Dredging 150-290

Air – Water Exchange

the air-water interface can be thought of as a bottleneck boundary (if one phase is stagnant we can think of it as a wall boundary)

We already know, from our discussions of mass transfer, that the equation for the air-water exchange flux (Fa/w) should look like this: eq

wwwawa CCvF //

where va/w is a mass transfer coefficient or air-water exchange velocity (m/s)

the second term describes the fugacity gradient and the direction of air-water exchange:

aw

aeqw K

CC

Net air-water exchange flux

aw

awwawa K

CCvF //

sometimes we divide this into the absorption flux(“gross gas absorption”):

and the volatilization flux:

aw

awaabs K

CvF /

wwavol CvF /

total exchange velocitythe total exchange velocity can be interpreted as resulting from a two-component (air/water) interface with phase change. if water is the reference state, then:

awawwa Kvvv

111

/

(two resistances in series)

va typically is about 1 cm/s

vw typically is about 10-3 cm/s

Critical Kaw

thus if Kaw << 10-3

(dimensionless) or 0.025 L bar/mol then the air-side resistance (va) dominates

if Kaw >> 10-3

(dimensionless) or 0.025 L bar/mol then the water-side resistance (vw) dominates

both phases important

water-phase controlled

air-phase controlled

va derived from evaporation of water

Kaw (water) = 2.3 10-5, so air side resistance dominates

wind speed is important

va increases linearly with wind speed up to ~8 m/s

va (water) = 0.2u10 + 0.3

where u10 is the wind speed (m/s) at 10 meters

vw derived from tracers with high KH

O2, CO2, He, Rn, SF6

Influence of wind speed, but also wave field

Liss and Merlivat 1986

See Table 20.2 for equations

Air-water exchange models for lakes, oceans

Whitman Two-Film Model (1923)

considers two bottleneck boundaries, stagnant films on the air and water side of the interface where transport occurs by molecular diffusion

Surface Renewal Model

interface is described as a diffusive boundary. parcels of air or water undergo a/w exchange to eqbm, then are replaced (air is replaced more often than water b/c less viscous)

Boundary Layer Model (Deacon, 1977)

considers changes in turbulence and molecular diffusivity (due to changes in T) separately

Whitman two-film model

Whitman Two-Film Modeleach stagnant boundary layer has a characteristic thickness :

w

iwiw

D

v

a

iaia

D

v

cm 3.0v ,

, awater

awatera

D cm 02.0

v ,

,

2

2 wCO

wCOw

D

If we assume that the layer thickness is the same for all chemicals then we can easily convert the transfer velocity for water or CO2 to a velocity for our chemical:

wCO

wi

wCO

i,w

D

D

v

v

,

,

, 22

aOH

ai

aOH

i,a

D

D

v

v

,

,

, 22

Diffusivity

In air:

In water:

cm/s)(in )]/1()/1[(

10 23/13/1

2/175.13

,

iair

iairai

VVp

MMTD

i of memolar voluV

/mol20.1cmair of memolar voluV

(atm) pressure catmospheri p

i ofMW M

g/mol) (28.97air ofweight molecular average M

Kin temp T

3

i

air

i

air

B.3 Tableappendix

centipoisein iscosity solution v

589.01.14,

0.0001326 cm/s)(in

i

wiV

D

Air-water exchange in flowing waters

Physics of boundary now influenced by both wind and water movement

Turbulence in rivers is primarily introduced by shear at the bottom

Water side: vw is affected by flow

Air side: va is not affected by flow

Two models:

Small Eddy Model (Lamont and Scott, 1970)

Large Eddy Model (O’Connor and Dobbins, 1958)

small

vs.

large

eddy

Small Eddy ModelThe turbulent eddies produced by water flowing over the rough bottom are small compared to the depth of the river(bottom is smooth and/or river is deep)

4/13*

2/1161.0v

h

uvSc w

iwiw

Sciw = Schmidt number = vi/Diw = viscosity/diffusivity

vw = kinematic viscosity of water

u* = shear velocity

h = water depth

Large Eddy Model

The turbulent eddies produced by water flowing over the rough bottom are large compared to the depth of the river (river is shallow and/or bottom is rough)

2/1

constant v

h

uDiwiw

Constant ~ 1

u = mean flow velocity of river

h = mean river depth

SummaryWe have moved from the smooth flow (small eddy) regime to the rough flow (large eddy) regime.

At even rougher flow, bubbles (foam) are formed which further enhance air-water exchange.

Note that when we apply either the large or small eddy model, we necessarily assume that air-water exchange is enhanced (greater than the stagnant flow models).

Thus the vw we calculated from either the large or small eddy model must be greater than the vw we get from the stagnant two-film model!

Study SiteNeed:

• Large fetch upwind of site

• Easy access

• Power

Description of the Micrometeorological Technique

• Uses two systems to determine turbulent fluxes in the near surface atmosphere:

– Aerodynamic Gradient (AG) Method

• determine profile of wind speed, temperature and water vapor, which along with concurrent measurements of PCB air concentration at two heights are use to determine vertical fluxes of PCB emanating from the water column.

– Eddy Correlation system

• directly measure fluxes of momentum, sensible heat and latent heat, which can be used for correction of PCB concentration profile for non-adiabatic conditions.

Calculation of Fluxes and va/w• Vertical PCB fluxes (FPCB) were calculated using the

Thornthwaite-Holtzman equation :

• Every term can be measured except C , which is the atmospheric stability factor. C can be determined from M, H, and W which are the atmospheric stability factors for momentum, heat and water vapor

• Calculate va/w from:

Czz

CCkuF

1

2

21

ln

* k = von Karman’s constantu* = friction velocityC1 = upper concentrationC2 = lower concentrationz1 = upper heightz2 = lower height

wwa CvF /w

wa C

Fv /

Need measurable conc gradient!

Micrometeorological Measurement

Results: PCB fluxesLow MW congeners have higher fluxes due to higher Cw and faster va/w

Heavier congeners volatilize more slowly b/c they are sorbed to solids and have slow va/w

Results: va/w

How to use va/w

aw

awwawa K

CCvF //

aw

awaabs K

CvF /

wwavol CvF /

It’s hard to use net flux, because it is dependant on both Ca and Cw, and is not, therefore, pseudo first order with respect to either of them.

By dividing the flux in to the absorption and volatilization fluxes, you can model each as a pseudo first order process.

Pseudo first order rate constants

ACvAFdt

dMwwaabs /

wwavol CvF /

To obtain a pseudo first order rate constant, you need to get va/w into units of 1/time:

Mass lost = volatilization flux times surface area

wawwwa

wwavol CkC

d

vC

V

Av

V

AF

dt

dC

/

/

To convert to concentration change, divide by volume:

Define a pseudo first order rate constant kaw = va/w/d (d = depth)