Where do these two models leave us?. F = K ol * ( C w – C a / H) Whitman two film model un-measurable parameters: z w & z a Surface renewal model un-measurable parameters: r w , & r a. Calculation of Flux. Net Flux = k ol * (C w – C a /H*)
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F = Kol * ( Cw – Ca / H)
Whitman two film model un-measurable parameters: zw & za
Surface renewal model un-measurable parameters: rw, & ra
Net Flux = kol * (Cw – Ca/H*)
1 / kol = 1 / kw + 1 / ka H*
Gross fluxes:
Volatilization: Water to air transfer
Fvolatilization = kol * Cw
Absorption: Air to water transfer
Fabsorption = kol * Ca/ H*
Figure from Schwarzenbach, Gschwend and Imboden, 1993
“Little” Mixing: Stagnant, 2-film model - “little” wind
“More” Mixing: Surface Renewal model – “more” wind
Wave Breaking: intense gas transfer ( breaking bubbles) – “even more” wind
Figure from Schwarzenbach, Gschwend and Imboden, 1993
Aerodynamic Transport Region
Transfer to broken surface region
~103m
Viscous Sublayer
~10-3m
Broken Surface
Unbroken Surface
Figure from Baker, 1996
Effect of Wind Speed seen only through changes in Mass Transfer Co-efficient
Net Flux = kol * (Cw – Ca/H*)
----
where
1 / kol = 1 / kw + 1 / ka H*
Figure from Schwarzenbach, Gschwend and Imboden, 1993
1 / kol = 1 / kw + 1 / ka H*
Wanninkhof et al. (1991) summarized SF6 experiments in five North American Lakes and found the gas transfer coefficient to be a power function of Wind Speed. The relationship is normalized to the appropriate Schmidt number to relate to the gas transfer of CO2 across the water side boundary layer.
kw,CO2 = 0.45 u 1.64
where kw,CO2 is the mass transfer coefficient (cm / hr) across the water layer for CO2 (SC = 600 at 20oC), and u is the wind speed at a 10 m reference height (m/s).
A two parameter Weibull Distribution can be used to describe the cumulative frequency distribution of wind speeds (Livingstone and Imboden, 1993. Tellus 45B 275-295).
F (u) = e ^ –(u / h)x
Where F (u) is the probability of a measured wind speed exceeding a given value, u.
the form of the distribution curve is determined by the shape parameter, x. The scale of the u-axis is determined by the scale parameter, h .
the negative derivative of F(u) yields the corresponding probability density function:
f(u) = ( x / u) ( u / h ) x e ^-( u / h ) x
Thus, the mean transfer coefficient over the whole range of wind speeds is obtained by integrating the product of the transfer velocity and wind speed probability density f(u).
kmean,CO2 = 0 to 8 f(u) *kw(u) * du
from Xhang et al. 1999
Instead of a single mean wind speed value during a sampling period, the wind speed probability density f(u) can be used to account for the effects of a nonlinear wind speed on mass transfer across the water layer over a range of wind speeds during a certain time period.
This Weibull distribution can be transformed into a linear form y = ax + b as follows:
x = ln u
y = ln [ -ln F(u) ]
and parameters a & b are determined by linear regression. They are related to the Weibull parameters as follows:
x = a
h= exp (-b / a)
Gas transfer of tracer CO2 can be up to 50% higher when accounting for the non-linear influence of wind speed. Calculations based on mean wind speeds underestimate the total PCB flux to Lake Michigan by as much as 20 to 40% depending upon wind speed variation.
from Xhang et al. 1999
1 / kol = 1 / kw + 1 / ka H*
Wanninkhof et al. (1991) summarized SF6 experiments in five North American Lakes and found the gas transfer coefficient to be a power function of Wind Speed. The relationship is normalized to the appropriate Schmidt number to relate to the gas transfer of CO2 across the water side boundary layer.
kw,CO2 = 0.45 u 1.64
where kw,CO2 is the mass transfer coefficient (cm / hr) across the water layer for CO2 (SC = 600 at 20oC), and u si the wind speed at a 10 m reference height (m/s).
kw,HOC =kw,mean,CO2 (ScHOC / ScCO2 ) -0.5
= cm s-1
Where Sc is the Schmidt number, the ratio of kinematic viscosity of water and the molecular diffusivity in water. Diffusivity of Hydrophobic organic contaminants (HOCs) through water are calculated using the emperical method of Wilke and Chang (1955). The Schmidt number for CO2 ranges from ~1900 to ~500 for temperatures from 0oC to 25oC according to the equation:
ln ScCO2 = -0.052 T +21.71
where ScCO2 is the Schmidt number for CO2 and T is the temperature in K.
ScHOC = Ab. Viscosity / (Water Density * Diffusivity of HOC in Water)
(g cm-1 sec-1) / ( g cm-3 ) * cm2 sec-1
Viscosity, m ( g cm -1 s-1) Density of Water (g cm-1)
Diffusivity of PAH in water:
Dw,PAHs= 0.0001326/ ((100 * m1.14) * Mol Vol. 0.589)
= cm2 sec-1
When air temperatures are lower than water temperatures, then air at the water surface is warmer than the ambient air, and thus a density difference exists between the two air layers. The resulting buoyancy can affect the mass transfer at the air/water interface. This influence is not important when average wind speeds are in excess of ~3 m/s at 10 m reference height.
1 / kol = 1 / kw + 1 / ka H*
Increasing wind speed increases the air-layer mass transfer coefficient approximately linearly:
ka,H2O = 0.2 u +0.3
= cm s-1
where ka,H2O is the mass transfer coefficient across the air layer for the tracer H2O (cm / hr) and u is the wind speed. the tracer is related to HOCs through the ratio of air diffusivities (H2O vs. HOC) and can be estimated by the Fuller method:
ka,HOC = ka,H2O (DHOC,air / DH2O,air)0.61
where ka,H2O is the mass transfer coefficient (cm / hr) across the air layer for H2O, and u is the wind speed at a 10 m reference height (m/s).
Da,PAH = (0.001* (Temp1.75) * ( 1/28.97 + 1/ mol wgt.)0.5) / (20.10.33 + Mol Vol0.33)2
= cm2 s-1
Da,H2O = (0.001* (Temp1.75) * ( 1/28.97 + 1/18)0.5) / (20.10.33 + 180.33)2
= cm2 s-1
Net Flux = kol * (Cw – Ca/H*)
---- ----
1 / kol = 1 / kw + 1 / kaH*
----
Gross fluxes:
Volatilization: Water to air transfer
Fvolatilization = kol * Cw
Absorption: Air to water transfer
Fabsorption = kol * Ca/ H*
Net Flux = kol * (Cw – Ca/H*)
----
1 / kol = 1 / kw + 1 / kaH*
--------
Temperature affects on Kol:
Viscosity and Density of water
Diffusivity of HOC in water
Sc of HOC
Sc of CO2
Diffusivity of H20 and HOC in air
plus Henry’s law associated effects on Kol (via air side)
Net Flux = kol * (Cw – Ca/H*)
---- ----
1 / kol = 1 / kw + 1 / kaH*
----
Gross fluxes:
Volatilization: Water to air transfer
Fvolatilization = kol * Cw
Absorption: Air to water transfer
Fabsorption = kol * Ca/ H*
Net Flux = kol * (Cw – Ca / H*)
----
Temperature dependence of ambient atmospheric concentrations (and G/P distributions)
Gross fluxes:
Volatilization: Water to air transfer
Fvolatilization = kol * Cw
Absorption: Air to water transfer
Fabsorption = kol * Ca/ H*
log Cair = ao + a1 / T + a2 sin(wd) + a3 cos(wd)
after Zhang et al. 1999
Note: possible (and likely) interdependence of T and wind direction
_____________________________________________________________________________
RegionFlux (ng/m2 day)PeriodReference
_____________________________________________________________________________
PCBs
Lake Superior+19, +141AugustBaker and Eisenreich, 1990
1986+63AnnualJeremiason et al., 1994
1992+13 AnnualJeremiason et al., 1994
+8.3AnnualHornbuckle et al., 1994
Lake Michigan+8.5-22AnnualSwackhamer and Armstrong,1986
+244AnnualStrachan and Eisenreich,1988
N. Lake Michigan+34 (24-220)AnnualHornbuckle et al., 1995
Lake Ontario+81AnnualMackay, 1989
Green Bay+13-1300June-OctAchman et al., 1993
+111 avgJuneAchman et al., 1993
Open ocean -40June-AugIwata et al., 1993
+5Nov-MarIwata et al., 1993
Nearshore Chicago-32 to +59whole year Zhange et al. 1999
northern Chesapeak BayNelson et al., 1998
PAHs
Baltimore Harbor (1996-1997)Bamford et al., 1999
fluorene+2200
phenanthrene -140(-140 to 7500)
pyrene+590
Southern Chesapeake Bay
fluorene+570FebruaryDickhut and Gustafson, 1995
phenanthrene -570
fluoranthene -140
pyrene -50
Table from Nelson, 1996
Bamford et al. (1999) predicted gas exchange in Baltimore Harbor using Temperature dependent Henry’s Law values, and measured air and water concentrations, along with ambient meteorological parameters (wind speed, water Temperature, etc).
Using estimated uncertainties of 40%, 20%, 20% and 10% for Kol, Cw, Ca, and H resulted in propagated errors of between 43% and 910% of the calculated fluxes. Errors were larger when the fluxes were smallest (i.e. near equilibrium or nearly 0 ng/m2 day net flux), with average error of 64%. Most of the error associated with the fluxes are related to the estimation of Koldue to the effect of wind speed on the kw.