Mass Confusion About Air-Water Mass Transfer
Damon Turney
Bren School of Environmental Science and Management
Committee Members:Jeff Dozier, Sanjoy Banerjee, Sally MacIntyre, Jordan
Clark
Fate and Transport
- Carbon- Toxins (DDT, Hg, Dieldrin, PCBs, many more)- Aeration of Hypoxic Water- DeAeration of Water Below Dam Spillways- Industrial Cleaning of Water in Air Spargers- Bioreactors and Other Reactors in Industry- Steam Condensation in Power Plants
from Wanninkhof et al. 1992
from Rosso 2006
from Kelly 1997
Convection + Diffusion
Convection + Diffusion = Confusion
€
∂[c]
∂t+
r v ⋅
r ∇[c] = D
r ∇2[c]
€
rF =
r v [c] + D
r ∇[c]
advection diffusion
€
∂[c]
∂t+
r v ⋅
r ∇[c] = D
r ∇2[c]
€
rF =
r v [c] + D
r ∇[c]
advection diffusion
€
rF ∝ ΔCr F = kΔC
Air
Water
Advection into thewater is not possible
Air
Water
Advection into thewater is not possible
Diffusion
Air
Water
Diffusion is ~10-9 m2/s
Diffusion is ~10-5 m2/sAdvection into thewater is not possible
Diffusion
Air
Water
Surface Renewal Theory
€
F =D
tNo unambiguous way of
determining a value for average t€
F = Δ[c]D
π t
Surface Divergence Theory
Surface Divergence Theory
1 mm
€
F = D∂[c]
∂y
Surface Divergence Theory
100 μm
€
∂w
∂z+
∂u
∂x ⎛ ⎝
⎞ ⎠i
= β
Surface Divergence Theory
1 mm
€
v =∂w
∂z+
∂u
∂x ⎛ ⎝
⎞ ⎠y = βy
€
∂w
∂z+
∂u
∂x ⎛ ⎝
⎞ ⎠i
= β
Surface Divergence Theory
€
∂[c]
∂t+
r v ⋅
r ∇[c] = D
r ∇2[c]
€
rF =
r v [c] + D
r ∇[c]
€
β
€
t
€
F
€
t
Surface Divergence Theory
€
∂[c]
∂t+
r v ⋅
r ∇[c] = D
r ∇2[c]
€
rF =
r v [c] + D
r ∇[c]
€
F = kΔC = 0.5 β 'D
How do we visualize (or measure) the velocities in the top ~100 microns of the water column accurately?
Surface Divergence Theory
from Turney et al. 2005
from McKenna & McGillis 2004
Surface Divergence Theory
from Xu et al. 2006
from Turney et al. 2005
Experimental Conditions
No Wind, Channel Flow (Straight Channelized River) 4 water heights 3 water velocities
Reynolds numbers ranging from 5000 to 30,000
Wind Waves, Microscale Wave Breaking10 wind speeds
fetch constant ~9 meters
Measurements
No Wind, Channel Flow- Mean Velocity, Depth, Temperature,- PIV Turbulence in Middle of Water Column- Friction of Water on Channel Bottom - PIV of Surface Divergence Motions- Oxygen Transfer Measurements
Wind Waves- Mean Velocity, Depth, Temperature,- PIV Turbulence in Middle of Water Column- Friction of Water on Channel Bottom- Wind Speeds (~1mm apart) at Locations Above Interface- Friction of Wind on Air-Water Interface- Stereo PIV of Surface Divergence Motions- Oxygen Transfer Measurements- Standard Water Height Measurements and Shadowgraphs
u* (cm/s)
k 1
0-5
(cm
/s)
€
β 'D
k=3.3 x 10-5 m/s
F=3.3 x 10-5 ΔC
Conclusions:
• It’s possible that the Surface Divergence Models is too simple, i.e., it ignores organized flows such as waves. Perhaps it is time for a new theory.
•Time-history effects are necessary for an adequate understanding of how the chemical is transported away from the interface.
• Accurate knowledge of velocity near the interface is very valuable. These measurement techniques should be pushed forward.
The Bigger Picture and Future Work
Pg C: 3.2 (+/-0.2) = 6.3 (+/-0.4) + 2.2(+/-1.3) - 2.4(+/-1.2) – 2.9 atmosphere fossil fuel terrestrial oceanic error
from Sabine et al. 2004
from Woods Hole Website
1 cmAir
Water
from Herlina & Jirka 2004
Air
Water
1 cm
from Tokoro et al 2007
from Belanger 1991
Surface Divergence Theory
1 mm
€
v = v i + y∂v
∂yi
+ y 2 ∂ 2v
∂y 2
i
+ y 3 ∂ 3v
∂y 3
i
+ H.O.T.
Surface Divergence Theory
1 mm€
v − v i = y∂v
∂yi
+ y 2 ∂ 2v
∂y 2
i
+ y 3 ∂ 3v
∂y 3
i
+ H.O.T.€
v = v i + y∂v
∂yi
+ y 2 ∂ 2v
∂y 2
i
+ y 3 ∂ 3v
∂y 3
i
+ H.O.T.
Surface Divergence Theory
1 mm€
v − v i = y∂v
∂yi
+ y 2 ∂ 2v
∂y 2
i
+ y 3 ∂ 3v
∂y 3
i
+ H.O.T.€
v = v i + y∂v
∂yi
+ y 2 ∂ 2v
∂y 2
i
+ y 3 ∂ 3v
∂y 3
i
+ H.O.T.
€
u − u i = y∂u
∂yi
+ y 2 ∂ 2u
∂y 2
i
+ y 3 ∂ 3u
∂y 3
i
+ H.O.T.
Surface Divergence Theory
1 mm€
v − v i = y∂v
∂yi
+ y 2 ∂ 2v
∂y 2
i
+ y 3 ∂ 3v
∂y 3
i
+ H.O.T.€
v = v i + y∂v
∂yi
+ y 2 ∂ 2v
∂y 2
i
+ y 3 ∂ 3v
∂y 3
i
+ H.O.T.
€
∂w
∂z+
∂u
∂x= −
∂v
∂y
Surface Divergence Theory
1 mm€
v − v i = −y∂w
∂z+
∂u
∂x ⎛ ⎝
⎞ ⎠i
+ y 2 ∂ 2v
∂y 2
i
+ y 3 ∂ 3v
∂y 3
i
+ H.O.T.€
v = v i + y∂v
∂yi
+ y 2 ∂ 2v
∂y 2
i
+ y 3 ∂ 3v
∂y 3
i
+ H.O.T.
€
∂w
∂z+
∂u
∂x= −
∂v
∂y
Surface Divergence Theory
1 mm
€
v = v i + y∂v
∂yi
+ y 2 ∂ 2v
∂y 2
i
+ y 3 ∂ 3v
∂y 3
i
+ H.O.T.
€
v − v i ≈ −y∂w
∂z+
∂u
∂x ⎛ ⎝
⎞ ⎠i
€
∂w
∂z+
∂u
∂x= −
∂v
∂y
Surface Divergence Theory
1 mm
€
v = v i + y∂v
∂yi
+ y 2 ∂ 2v
∂y 2
i
+ y 3 ∂ 3v
∂y 3
i
+ H.O.T.
€
v − v i ≈ −y∂w
∂z+
∂u
∂x ⎛ ⎝
⎞ ⎠i
= −yβ
€
∂w
∂z+
∂u
∂x ⎛ ⎝
⎞ ⎠i
=∂v
∂yi
= β
Surface Divergence Theory
1 mm
€
v = v i + y∂v
∂yi
+ y 2 ∂ 2v
∂y 2
i
+ y 3 ∂ 3v
∂y 3
i
+ H.O.T.
€
v − v i ≈ −y∂w
∂z+
∂u
∂x ⎛ ⎝
⎞ ⎠i
= −yβ
€
∂w
∂z+
∂u
∂x ⎛ ⎝
⎞ ⎠i
=∂v
∂yi
= β€
u − u i ≈ 0
Depth (cm) Flow Speed (cm/s) Macro Re Turbulent Re
5 2.3 1150 5 3.0 1510 5 10.0 5000 100 5 24.0 12000 170 5 54.5 30000 310 7 7.1 5000 100 7 17.1 12000 170 7 42.9 30000 280 9 5.6 5000 80 9 13.3 12000 140 9 33.3 30000 250 10 5.8 5800 10 10.7 10700 13 3.8 5000 50 13 9.2 12000 100 13 23.1 30000 190
No Wind
Depth (cm) Wind Speed (cm/s) Wave Height (cm)
5 75 - 5 75 - 5 165 - 5 224 - 5 291 - 10 319 - 10 410 0.2 10 499 0.4 10 599 0.6 10 712 0.7 10 706 0.7 10 799 0.9 10 894 1.1
Windy
from Takahashi et al. 2008
from Richey et al. 2002
Chemical Budgets
Change in Amount per Unit Time = Flux In - Flux Out
Surface Renewal Theory
Surface Divergence Theory
1 mm
€
F = D∂[c]
∂y
€
Δy = DΔ[c]
F
Surface Divergence Theory
100 μm
€
∂w
∂z+
∂u
∂x ⎛ ⎝
⎞ ⎠i
= β
Surface Divergence Theory
1 mm
€
v =∂w
∂z+
∂u
∂x ⎛ ⎝
⎞ ⎠y = βy
€
∂w
∂z+
∂u
∂x ⎛ ⎝
⎞ ⎠i
= β
Surface Divergence Theory
1 mm
Surface Divergence Theory
€
∂[c]
∂t+
r v ⋅
r ∇[c] = D
r ∇2[c]
€
rF =
r v [c] + D
r ∇[c]
Surface Divergence Theory
€
∂[c]
∂t+
r v ⋅
r ∇[c] = D
r ∇2[c]
€
rF =
r v [c] + D
r ∇[c]
€
β
€
t
€
F
€
t
Pg C: 3.2 (+/-0.2) = 6.3 (+/-0.4) + 2.2(+/-1.3) - 2.4(+/-1.2) – 2.9 atmosphere fossil fuel terrestrial oceanic error
Chemical Budgets
Change in Amount per Unit Time = Flux In - Flux Out
from Blais et al. 2001
from Schimpf et al. 2004
from Herlina & Jirka 2004
Surface Renewal Theory
Surface Renewal Theory
“Renewed” Spot
Surface Renewal Theory
“Renewed” Spot
Surface Renewal Theory
€
F = Δ[c]D
π t
Surface Renewal Theory
€
F = Δ[c]D
π t
Surface Renewal Theory
€
F =D
t€
F = Δ[c]D
π t