Transcript Transport - Department of Environmental Sciences

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
Gradient flux law:
FA/ B  vA/ B CB  CA 
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)
Fick’s First Law
One example of a gradient flux law is Fick’s First Law:
dC
Fx   D
dx
Relates the diffusive flux (Fx) of a chemical
to its concentration gradient (dC/dx) and its
molecular diffusion coefficient (D)
Fick’s Second Law
C
C
D 2
t
x
2
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 diffusion
In 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:
dv
Re 
f / r f
d = spatial dimension of the flow system or objects around which
the flow occurs (m)
v = typical flow velocity
f = dynamic viscosity of the fluid (kg/m-s)
rf = density of the fluid (kg/m3)
For laminar flow Re < 0.1
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:
dC
Fx   E x
dx
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
Diffusivity D(x)
bottleneck
turbulent
diffusion
bottleneck = mass
crossing must squeeze
itself through a zone in
which transport occurs
by molecular diffusion
(usually interface)
molecular
diffusivity
distance
example:
air-water interface
Like a toll booth on the
turnpike
wall boundary
Diffusivity D(x)
wall
turbulent
diffusion
molecular
diffusion
distance
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 boundary
Diffusivity D(x)
diffusive
boundary
C (x)
D (x)
at a diffusive boundary,
diffusivity is of similar
magnitude on either side
diffusivity may be
molecular or turbulent
example:
distance
troposphere – stratosphere
boundary (tropopause)
Air – Water Exchange
(Chapter 20)
Inputs and outputs of SPCBs (kg y-1)
Atm dep
18-48
Volatilization
317-846
Advection to
Atlantic 130-190
CSOs 67-146
Stormwater 36-140
STP effluents 32
NY/NJ Harbor
Estuary
Advection from
Hudson River
260-470
Dredging
150-290
Storage in sediments
147-307
Totten 2005
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

Fa / w  va / w Cw  Cw

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:
C
eq
Cw 
a
K aw
Net air-water exchange flux
Fa / w

Ca 

 va / w  C w 
K aw 

sometimes we divide this into the absorption flux
(“gross gas absorption”):
Fabs
 Ca 

 va / w 
 K aw 
and the volatilization flux:
Fvol  va / w Cw 
total exchange velocity
the 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:
1
va / w
1
1
 
vw va K aw
(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 waterside resistance (vw)
dominates
both phases important
air-phase
controlled
water-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 Model
each stagnant boundary layer has a characteristic thickness :
via 
a 
Dwater,a
v water,a
Dia
a
 0.3 cm
viw 
Diw
w
w 
DCO2 , w
vCO2 , w
 0.02 cm
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:
vi,a
vH 2 O , a

Di ,a
vi,w
DH 2O,a
vCO2 ,w

Di ,w
DCO2 ,w
Diffusivity
In air:
3
Di ,a  10
T 1.75 [(1 / M air )  (1 / M i )]1/ 2

p V air
1/ 3
V i

1/ 3 2
(in cm/s)
T  tempin K
M air  averagemolecularweight of air (28.97g/mol)
M i  MW of i
p  atmospheric pressure (atm)
V air  molar volume of air  20.1cm3 /mol
V i  molar volume of i
In water:
Di , w (in cm/s) 
0.0001326
  solution viscosity in centipoise
appendix Table B.3
0.589
 1.14 V i
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 Model
The 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)
v iw  0.161Sciw 
1 / 2
 v wu

 h
*3
1/ 4



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)
1/ 2
 Diw u 

viw  constant 

h


Constant ~ 1
u = mean flow velocity of river
h = mean river depth
Summary
We 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 Site
Need:
• 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 :
ku * C1  C2 
F 
z
ln 2 C
 z1 
k = von Karman’s constant
u* = friction velocity Need
C1 = upper concentration
measurabl
C2 = lower concentration
e conc
z1 = upper height
gradient!
z2 = lower height
• Every term can be measured except C , which
F be
is the atmospheric stability factor.v C can
F  va / w  Cw
a/w
determined from M, H, and W whichCare
the
w
Micrometeorological
Measurement
PCB Concentration gradients
6
5
4
Concentration ng m-3 3
Upper Sample
Lower Sample
2
1
0
PCB samples
Results: PCB fluxes
20
Heavier congeners
volatilize more slowly b/c
they are sorbed to solids
and have slow va/w
15
10
5
0
1
3
7+9
5+8
15
18
24+27
25
28
22
53
52+43
47+48
42
64
63
70+76
55
91
101+89+90
110
154
147
146
132
138+163
159
157
178
185
177
172+192
Fluxes pg/m2 s
25
Low MW congeners
have higher fluxes due
to higher Cw and faster
va/w
PCB Congeners
Congeners
180
174+181
157
138+163
153
151
101+89+90
93+95
70+76
40
42
49
53
20+21+33
25
17
11
7+9
1
Results: va/w
PCB MTC
14
12
10
8
MTC m/d
6
4
2
0
How to use va/w
Fa / w
Fabs

Ca 

 va / w  C w 
K aw 

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.
 C a  By dividing the flux in to the

 va / w 
 K aw  absorption and volatilization fluxes,
Fvol  va / w Cw 
you can model each as a pseudo first
order process.
Pseudo first order rate constants
Fvol  va / w Cw 
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
dM
 Fabs  A  va / w  Cw  A
dt
To convert to concentration change, divide by volume:
v
dC Fvol  A
A

 va / w   Cw  a / w  Cw  k aw  Cw
dt
V
V
d
Define a pseudo first order rate constant kaw = va/w/d (d = depth)