8 Surface Processes
Mass Exchange
„
„
Volatilization
Reaeration
Momentum Transfer
„
Oil spills
Surface Heat Transfer
„
Lake temperature models
Air water exchange
Equilibrium: Henry’s Law
H=
c ge
cle
c ge
cle
Typical units for [H]:
atm-m3/mol (KH) or dimensionless (KH’)
For air KH’ ~ 42 KH
Two-film theory
J
zg
J l = k l (cl − cle )
Ex:
liquid
side
cle=cge/H
-zl
cl
cg cge
(cl − cle )
J l = Dl
zl
k l (cl − cle ) = k g (c ge − c e ) = k (cl − c g / H )
H=
c ge
cle
3 eqns, 3 unknows (cle, cge, k)
c
Two-film theory
1 1
1
= +
k k l Hk g
1
2
c ge − c g =
cl − cle =
k l ( Hcl − c g )
Hk g + k l
k g ( Hcl − c g )
Hk g + k l
Resistances in series:
kl << Hkg => 1 dominates (liquid side controlled)
kl >> Hkg => 2 dominates (gas side controlled)
Medium with lower equilibrium concentration controls
Typical values for air and water
“Typical” values for air (as gas)
Dg ~ 0.1 cm2/s, 0.1 < zg < 1 cm
=> kg = Dg/zg = 0.1 to 1 cm/s
“Typical” values for water (as liquid)
Dl ~ 2x10-5 cm2/s, 0.002 < zl < 0.02 cm
=> kl = Dl/zl = 10-3 to 10-2 cm/s
Kg ~ 100 kl so if H >> 0.01 then water side controlled
(think DO); if H << 0.01 then air side controlled (think
evaporation)
Example of liquid side control
H >> 0.01; assume H ~ 1 => cge = cle = ce
zg
ce
-zl
cg
cl
c
ce − c g =
ce-cg
cl-ce
cl − c e =
k l ( Hcl − c g )
Hk g + k l
≅ (c l − c g )
k g ( Hcl − c g )
Hk g + k l
kl
kg
≅ (c l − c g )
Liquid side control, cont’d
If we double kl (halve zl),
(ce-cg) doubles, both
gradients ~ double => twice
the mass flux; red line
zg
ce
-zl
cg
ce-cg
cl-ce
cl
c
If we double kg (halve zg),
(ce-cg) is halved, both
gradients ~ const => similar
mass flux; green line
Therefore mass flux
controlled by liquid side
Surface Renewal Theory
Described previously for streamreaeration formulae (Chapter 7)
zl (or zw or δ) not stagnant, but timedependent ~ [Dt]1/2, where t is
reciprocal of a renewal rate, depending
on bottom generated turbulence.
Thus kl (hence k) = Dl/zl ~ D1/2
Measurement of gas exchange
Gas-evasion experiment: introduce chemically
conservative gas (e.g., CO2, propane, radon)
at c > saturation, and watch c decline with
distance due to volatilization
In open water bodies (or rivers where you
don’t know flow rate) introduce a second,
non-volatile tracer such as salt.
Sometimes use tracer of opportunity
Application to rivers
x
Qr
m& v
(propane)
m& nv (salt)
cv m& nv
c nv m& v
1
e-1
0
0
hu/kl
m& nv
c nv =
Qr
m& v − kl x / hu
cv =
e
Qr
cv
m& v − kl x / hu
=
e
c nv m& nv
x
(stream reaeration
coefficient Ka = kl/h)
Gasses other than oxygen
Ka ~ D (stagnant film), D1/2 (surface
renewal), D2/3 (split the difference)
From Chapter 1, Sc = ν/D ~ MWb (b ~
0.35 to 0.4)
Ka/K ~ (DO2/D)2/3 ~ (32/MW) -1/4
Example: Propane C3H8, MW = 44
„
„
Ka/K = (32/44) -1/4 = 1.08
Calibrations actually shows Ka/K ~1.39
How far downstream must one go?
O’Connor-Dobbins at 20oC:
Ka = 3.9u 0.5/h1.5
u = 0.3 m/s, h = 1 m, Ka = 2.1 d-1
x~ u/Ka = (0.3 m/s)(86400s/d)/(2.1d-1)
~ 12 km
Application to open waters
h = water depth or
thermocline depth
x
u
m& v (propane)
c nv =
m& nv
cv m& nv
1
c nv m& v
e-1
0
2
2π σuh
m& v
− y 2 / 2σ 2 − kl x / hu
cv =
e
e
2π σuh
cv
m& v − kl x / hu
=
e
c nv m& nv
m& nv (salt)
0
e
− y 2 / 2σ
hu/kl
x
Mass transfer in lakes and
oceans
Most contaminants of concern are water side
controlled (e.g., DO, VOC)
In rivers, source of turbulence is bottom
roughness
In deep water bodies (lakes, oceans) it is
wind stress => uw* (water-side friction
velocity) which affects zl
Contaminants that are air side controlled also
affected by wind (through zg)
kl vs uw*
KL [cm/s]
4
4
2 3
10-3
2
2
3
1 33
3 3
333
[m/d]
10-2
5
5
5
(I) 10-3
1
10-1
100
101
5
w (Sc = 600)
41 2
4 22
2
41
10-2
w (Sc = 600)
4
4
44
44
4
5
10-1
100
101
Fraction vel. u*w [cm/s]
u*w [cm/s]
lab
field
Figure by MIT OCW.
uw* because transfer is water side controlled and uw*
is indicator of turbulence; yet uw* not easily measured
Wind Stress
z
τ = C10 ρ a u10 2 = ρ w u w* 2 u10 = 10 m wind
ua(z)
u w* =
C10 ρ a
ρw
speed;
u10
C10 = drag coef.
C10 = (0.8+0.065u10) x 10-3
uw(z)
[u10 > 1 m/s; Wu, 1980]
u10 -> C10 -> uw* -> kl
kl (or zl) vs u10
10-3
1000
n=1
10-4
n=1
Downing and
Truesdale
Juliano
n = 3/4
1
Mattingly
5
10
50
100
50
Central, Atlantic
(Broecker and Peng 1971)
10
5
Yu et al
10-6
Hoover and Berkshire (1969)
Thurber and Broecker (1970)
Liss (1973)
Zl
n = 3/2
10-5
Konwisher (1963)
Smooth surface
Konwisher (1963)
3 cm waves 60/mm
ELA Lakes
500
δ (microns)
KL, Overall Liquid Film Coefficient at 20o C, m/sec.
n=2
100
North Pacific
(Peng et al. 1974)
0
2
4
6
Yu and Hamrick (1984)
10
U10, Wind Velocity,(ms-1)
U10, Average Wind Speed, m/sec. at 10 m
Figures by MIT OCW.
8
Emerson (1075)
12
14
Example film coefficients
k l = 0.0004 + 0.00004u10
2
k g = 0.3 + 0.2u10
kl and kg in cm/s; u10 in m/s [Schwarzenbach et al, 1993]
Note that both depend on u10
Examples
Above eqns:
u10 = 5 m/s => kl = 1.4x10-3 cm/s (green dot);
kg = 1.3 cm/s
Figure 8.8:
zl = δ = 120 µm = 1.2x10-2 cm.
For DO, D = 2x10-5 cm2/s
kl = D/zl = 2x10-5/1.2x10-3 = 1.7x10-3 cm/s (red dot)
kl (or zl) vs u10
10-3
1000
n=1
Hoover and Berkshire (1969)
Thurber and Broecker (1970)
Liss (1973)
Zl
10-4
n = 3/2
100
50
Central, Atlantic
(Broecker and Peng 1971)
n=1
10-5
10
Downing and
Truesdale
Juliano
n = 3/4
5
Mattingly
1
5
10
North Pacific
(Peng et al. 1974)
0
2
4
6
U10, Wind
Yu et al
10-6
Konwisher (1963)
Smooth surface
Konwisher (1963)
3 cm waves 60/mm
ELA Lakes
500
δ (microns)
KL, Overall Liquid Film Coefficient at 20o C, m/sec.
n=2
50
8
100
U10, Average Wind Speed, m/sec. at 10 m
Figure by MIT OCW.
Yu and Hamrick (1984)
10
Velocity,(ms-1)
Emerson (1075)
12
14
Volatile Halogenated Organic
Compound (VHOC) Experiment
CH3Cl3 and other one carbon VOCs
(THMs) and two carbon VOCs (solvents)
discharged with waste water.
Used to
„
„
compute volatilization (assuming known
residence time) or
compute residence time (with known
volatilization)
TCE data in Boston Harbor
TCE loading from Deer & Nut
Island TPs: 24 m3/s at 11 µg/L
Ave harbor TCE
concentration
241 ng/L (all pts)
214 ng/L (excl. 840)
Harbor volume =
6x108 m2
Kossik, Gschwend & Adams, 1987
TCE Experiment, cont’d
Nominal residence time (w/o volatilization;
excluding presumed outlier)
cV
(214 x10 −9 kg / m 3 )(6 x10 8 m 3 )
τ* =
=
= 5.6d
3
−6
3
Qo co (24m / s )(86400 s / d )(11x10 kg / m )
TCE Experiment, cont’d
With volatilization
dc
V
= −kAc − Q f
dt
⎛k A Q⎞
dc
= −⎜⎜
+ ⎟⎟c
dt
V⎠
⎝ V
κ∗=1/τ*
k = piston velocity ~ kl (water
side control)
κ* = bulk removal rate (t-1)
V/Q = τ = hydraulic res time
kA/V = k/h
For CH3Cl3 H = 1.13 (dimensionless) >> 1 => ws control
D = 1.0x10-5 cm2/s
TCE Experiment, cont’d
From Figure 8.8 and u10 = 5 m/s, δ = 1.2x10-2 cm
k = D/δ =(1.0x10-5)/(1.2x10-2)= 0.00083 cm/s = 3
cm/hr = 0.72 m/d
1/t* = 1/t + k/h
1/t = 1/t* - k/h = (1/5.6d) – (0.72 m/d)/6m
=0.18 – 0.12 = 0.06d-1 => τ = 17 d
Estimated τ is too high; reason is likely extraneous
or under-accounted sources of CH3Cl3
Momentum Exchange
Chapters 2, 3 discussed surface shear
stress for eddy diffusivity and
hydrodynamic modeling
Previous section discussed stress as
source of turbulence governing mass
exchange
Also of interest in transporting floating
material, specifically spilled hydrocarbons
Oil Spills
Composition
Fate
Transport (spreading,
advection)
Clean-up
Marine Sources (103 MT/yr)
N. America
Global
160
600
Petroleum Extraction
3
38
Petroleum Transport
9
150
Petroleum Consumption
84
480
Total
260
1300
Natural Seeps
About half is anthropogenic
(Oil in the Sea III, NRC, 2003)
Composition
Crude and Refined
Oils
Always multiple
constituents
Characterized by
Boiling Point (or
distillation cut)
Fate
Volatilization (lighter
fractions)
Emulsification
(depending on oil)
Natural dispersion (if
enough energy)
Biodegradation
Dissolution
Photo-oxidation
Sediment particle
interaction
Output from NOAA’s ADIOIS
model; independent of transport
Transport Models
Spreading and Advection
Pre-planning (evaluate risk)
Real-time (assist clean-up; needs to be
quick and dirty)
Hind-cast (who is responsible, damage
assessment)
Simple advection model
z
ua(z)
us
uw(z)
fa
fw
fw
2
2
τa =
ρ aua =
ρ w (u s − u w ) =
ρ w (∆u s ) 2
2
2
2
ρa
∆u s
fw ≅ fa →
=
≅ 0.03
ua
ρw
Surface current speed ~ 3% of
wind speed. (Also explained by
Stokes Drift due to surface
waves)
In which direction?
Ekman Model
Linearized equations of motion; constant viscosity
∂u
∂ 2u
− Ωv = E 2
∂t
∂z
∂v
∂ 2v
+ Ωu = E 2
∂t
∂z
w=0
∂w τ sx + iτ sy
=
E
∂z
ρw
Ω = 2ω sin φ Coriolis parameter
w = u + iv
Complex velocity
At depth (z = - oo)
At surface (z = 0)
Ekman Model, cont’d
w=
τ sy
ρw
⎧ Ω
iπ ⎫
exp ⎨
(1 + i ) z + ⎬
4⎭
EΩ
⎩ 2E
y
Surface drift 45o to right;
τs
Depth average drift 90o to right
z=0
x
z
Field experiments show surface
drift ~ 10o to right. Explained by
variable vertical viscosity E ~ z
(Madsen, 1977)
Other effects of wind: Coastal
Upwelling/Downwelling
Other effects of wind: Langmuir
Circulation
Oil Streaks
Wind
Figure by MIT OCW.
Idealized Spreading (Fay, 1969)
D
h
δ
dD
~
dt
1
g'h ~
g 'V
D
dD g 'V 2 t
~
dt
D5 υw
fr
dD
t
~
dt
ρw D υw
Gravity-Inertia
Gravity-Viscous
Surface Tension-Viscous
Idealized spreading, cont’d
Regime
Gravity-Inertia D = 2k1[g’Vt2]1/4
K1=1.14
Gravity-Viscous D = 2k2[g’V2t3/2/νw 1/2]1/6 K2=0.98
to 1.45
D=2k3[frt3/ρw2νw]1/4
Surface
Tension-Viscous
K3=1.6
Comments
Theory applies down to slick thickness of
about 0.1 mm
Additional spreading due to
„
„
„
Time-varying spillage
Wind, waves and non-uniform currents
Dispersion of submerged (slower moving) oil
droplets
Field experiments show oil often very nonuniform (90% of volume in 10% of area)
Oil Transport Models
Slick advected with underlying surface current
plus 3% of wind speed (~10% deflection to
right)
(3-D) models simulate transport of subsurface dispersed oil.
Currents can be observed or predicted
(sophistication depends on application—
available time)
Fate processes often computed independently
from transport
Model Simulations
NOAA’s 3D GNOME; ANS Crude off Coast of Florida
Mechanical Clean-up
Chemical Dispersion
Surfactants that reduce
interfacial tension
Create dispersed
droplets
Subsurface/bottom
impacts vs
surface/shoreline
Air (large spills) or boat
application
Window of opportunity
Chemical Dispersion, cont’d
NRC, 1989
In situ Burning
Considered secondary option (like chemical
dispersants)
Most appropriate for offshore spills (reduced
AQ impacts)
Surface Heat Transfer
and Temperature Modeling
Surface heat fluxes
Linearized surface heat transfer
Cooling ponds
Natural lakes and reservoirs
Importance of Temperature
Important WQ parameter
„
„
Thermal pollution
Species preference (fish habitat)
Affects rate constants
„
K=K20θT-20
Produces density stratification
„
ρ = ρ(T)
Important tracer (e.g., Ez)
Surface Heat Transfer (W-m-2)
φs
φa
φsr
φbr
φe
φc
φar
Net solar, φsn
Net atmospheric, φan
Back radiation, φr
Evaporation, φe
“Conduction”, φc
60 to 300
200 to 450
250 to 500
0 to 350
-70 to 200
φ n = φ sn + φ an − φ br − φ e − φ c
Solar Radiation
Short wave length (< 3µm)
Direct plus diffuse (scattered, reflected)
Absorbed & re-radiated (> 3µm) by clouds
Measured by pyranometer
Incident clear sky radiation calculated from
latitude, date and time of day
Corrections for cloud cover and reflection
Net Solar Radiation (cont’d)
φsr/φs
(%)
Jan Feb Mar Apr
May Jun Jul
Aug Sep Oct
Nov Dec
9
6
6
9
7
7
6
6
6
6
7
φ sn = φ s − φ sr ≅ 0.94φ sc (1 − 0.65C
C = fractional cloud cover
10
2
)
Depth-variation of solar radiation
Measured with
Secchi disk or in-situ
pyranometer
φ z = (1 − β )φ sn e − ηz
1.7
η=
dD
β ~ 0.5
Atmospheric Radiation
Long wave length (> 3µm)
Re-radiated from atmosphere
Measured by pyrgeometer
Incident clear sky radiation calculated from
absolute air temperature, vapor pressure
Corrections for cloud cover and reflection
Incident Radiation Formulae
φ ac = εσ (Ta + 273)
4
σ = Stefan-Boltzman const (5.7x10-8 W/m2-oK4)
ε = emissivity (dimensionless)
ε = 0.92 x10 −5 (Ta + 273) 2
Swinbank (1963)
ε = {1.0 − 0.26 / exp[7.77 x10 (Ta ) ]}
−5
⎛
⎞
e
⎟⎟
ε = 1.24⎜⎜
⎝ (Ta + 273) ⎠
1
2
Itso-Jackson
(1969)
7
e = vapor pressure, mbar
Brutsaert (1975
Net Atmospheric radiation
φ an = 0.97εσ (Ta + 273) (1.0 + 0.17C
4
C = fractional cloud cover
~3% reflection
2
)
Back Radiation
Water surface is nearly a black body
„
(ε ~ 0.97)
φ br = 0.97σ (Ts + 273) 4 = 5.5 x10 −8 (Ts + 273) 4
Evaporative Heat Flux
Measured
„
„
eddy flux (short term)
evaporation pans (long term)
Computed from mass transfer formulae
E = ρ f(Wz) (es-ez)
Dalton’s Law
es = vapor pressure at surface
ez = vapor pressure at elevation z
f(Wz) = wind speed function = a + bWz
(kg)
Evaporative Heat Flux (cont’d)
Mass transfer => heat transfer using
latent heat of vaporization
Lv = ( 2493. − 2.26Ts ) ×10 3
J/Kg
φ e = Lv E = f (W z )(e s − ea )
φ e = 3.72W2 (es − e2 )
“Lake Hefner”, Marciano
and Harbeck (1954)
(W/m2; W2 in m/s; es, e2 in mb)
φ e = 5 .1 A
−0.05
(A in ha)
W 2 (e s − e 2 )
“Fetch-dependence”
Harbeck, (1962)
z
z
ez
Wz
es
ez and Wz vary vertically (height above water) and
horizontally (above water or on-shore)
Evaporation from non-natural
water bodies
es increases with
temperature
„
Heated water bodies have
increased evaporation
(water vapor also lighter
than air)
es decreases with salinity
„
Saline bodies have
decreased evaporation
es decreases with pressure
ez
es
φe = f(Wz) (es-ez)
Conductive Heat Flux
Computed from evaporative flux using
Bowen Ratio
φ c = Rbφ e
(Ts − Tz )
Rb = Cb
(e s − e z )
Cb =0.61 mb/oC;
Summary
φ n = φ sn + φ an − φ br − φ e − φ c
functions of Ts
functions of external factors (met
and astronomical conditions)
Strategies for computation: table look up
Self regulation: errors in calculations compensate
Linear Heat Transfer
−φe
Equilibrium Temp, Te
„
„
Ts for which φn = 0
Function of met
−φe
Surface Heat Exchange
Coefficient, K
„
Slope K
Slope of φn vs Ts
φn = -K(Ts-Te)
K ~ 20-50 W/m2oC
Te
Ts
T
Example: Periodic Heat Loss
ρcV
dT
dt
= Asφ n
φn
dT
= −k (T − Te )
dt
Te = Te + ∆Te e
iwt
T = T + ∆T * e iωt
T = Te + ∆Te iθ e iωt
k = K/ρcph
ω = 2π/P
As
V
h=
V/As
Periodic Heat Loss (cont’d)
T = T + ∆T * e
∆T * = ∆Te iθ
iωt
T = Te + ∆Te iθ e iωt
t L = (θ 2π ) P
T = Te + ∆Te iω (t −t L )
k
∆T *
=
=
k + iω
∆Te
k
k2 +ω2
e
Amplitude
i tan −1 ( −ω / k )
Phase lag
θ = tan −1 (−ω / k )
P
tL =
tan −1 (ω / k )
2π
Te
T
Examples
K/ρc = 1m/d*; h = 10m, k=K/ρch = 0.1d-1
P
1 day
365
ω=2π/P
0.17d-1
6.28d-1
∆T/∆Te
=k/(k2+ω2)0.5
θ=tan-1(-ω/k)
0.016
0.986
-89o
-10o
tL =
P/2πtan-1(ω/k)
0.247 d
10 d
* K ~ 48 W/m2oC
Cooling Lakes and Ponds
Used to cool electric power plants
Shallow (vertically well-mixed)
„
„
Erected with dikes
T = T(x,y) + T(t)
Deep reservoirs
„
Damming of reservoirs
Cooling capacity
„
r=KAp/ρcQo
Cooling Ponds
Deep
Stratified
Shallow
Dispersive
Shallow
Recirculating
Plan View
Elevation
Example: shallow-longitudinal
dispersive
2
d 2T
dT
ρcQo
= ρcWHEL
− K (T − TE )W
2
dx
dx
T1 - TE
∆T0
Single pass
Ti − TE
=
To − TE
1
1 2E *
L
4ae
(1 + a )2 e
a 2E *
L
− (1 − a )2 e
−a 2 E *
L
(To = Ti + ∆To)
0
0
1 2E *
L
a 2E *
L
− (1 − a )2 e
−a 2E *
L
0 0.1 0.2
1
2
Completely
mixed
1
0.5
3
r
Figure by MIT OCW.
4ae
(1 + a )2 e
5
2
Plug
Flow
Continuous operation
Ti − TE
=
∆To
EL*
8
1 2E *
L
− 4ae
Jirka et al. (1978)
4
5
Stratification in Lakes & Reservoirs
Factors causing vertical stratification
„
„
Differential absorbtion
Reduced vertical mixing
Factors causing horizontal stratification
„
„
„
Strong through flow
Strong wind
Differential absorbtion
Reservoir classification based on
horizontal through flow
(Orlob, 1969)
Through flow velocity = L/(V/Q)
Int’l wave speed ~ (g∆ρ/ρh)0.5 ~ Nh
„
„
N = buoyancy freq = [(g/ρ)(dρ/dz)]0.5
L = length; Q = flow; h = depth; V = vol
Fr = LQ/VNh
„
„
Fr << 1 vertically stratified
Fr >> 1 vertically mixed
1-D Reservoir Modeling
z
qout
h
∆z
qin
Q
∂Q
= qin − q out
∂z
0
∂T 1 ∂
1 ∂ ⎡
∂T ⎤ qin Tin − q out T
+
(QT ) =
AE z
+
⎢
⎥
∂t A ∂z
∂z ⎦
A ∂z ⎣
A
T
Surface Layer
Well mixed layer
„
„
Convective mixing
Wind mixing
Wind mixing algorithm
for surface
„
Oceans (Kraus-Turner)
1-D model below
T
WML
1-D
Surface Layer (cont’d)
∆PE
h ∆ρgh
= (∆ρg∆h ) =
u e ∆t
2
2
A
∆KE
2
3
= ρu* u s ∆t ~ ρu* ∆t
A
∆PE
= const
∆KE
2
ue
u*
~
= Ri −1
u * ∆ρ
gh
ρ
Many variants
τ=ρu*2
T
h
∆h
∆ρ
Lake stability
Stability index (PE of water body with equivalent mass
and heat content but uniform density – PE of stratified
body)
h
SI = ∫ [ ρ − ρ ( z )][ z − z c ]gA( z )dz
0
h
ρ = ∫ ρ ( z ) A( z )dz
0
h
z c = ∫ ρ ( z ) A( z ) zdz
0
h
∫ A( z)dz
Average density
0
h
∫ ρ ( z ) A( z )dz
0
Center of mass