MKEP4 SS11 12 Gas Transfer

8/18/2019 MKEP4 SS11 12 Gas Transfer

1/21

1

Master Course „Environmental Physics“ (MKEP4)

http://www.iup.uni-heidelberg.de/institut/studium/lehre/MKEP4/

12. Gas and Heat Transfer

between Air and Water

Summer Term 2011

Werner Aeschbach-Hertig

Institut für Umweltphysik

Lecture Program of MKEP4

Part 1: Introduction and Fundamentals (4 sessions)1. Introduction to Environmental Physics and the Earth System

2. Global energy balance and structure of the atmosphere

3. Stratification and convection in air and water

4. Trans ort rocesses

Part 2: Geophysical Fluid Dynamics (7 sessions)5. Introduction to Geophysical Fluid Dynamics

6. Navier-Stokes equation and geostrophic approximation

7. Geostrophic Flow and Vorticity

8. Turbulence

9. Turbulent transport and flow near boundaries10. Global circulation of the atmosphere

11. Global circulation of the ocean

Part 3: Other Compartments and Fields (4 sessions)12. Gas and heat transfer between air and water

13. Freshwater systems

14. Soil and Groundwater 15. The cryosphere

2


2/21

2

Contents of Today's Lecture

Gas and heat transfer between air and water

• Heat exchange

Change of ocean heat content and temperature

• Gas solubility (Henry's law)

• Gas exchange: Stagnant laminar film model

Transfer velocity: Dependence on diffusion, Schmidt-no., wind

• Evaporation

3

Heat and Gas Exchange (Transfer)

Wind-wave facility at IUP

4


3/21

3

Processes at the Air/Water Interface

http://www.whoi.edu/ooi_cgsn/page.do?pid=532785

Processes of Heat Transfer

The total heat flux Q tot [W/m2] from water (ocean, lakes, etc.) into

the atmosphere (upward) can be divided into 5 contributions:

net SW LWA LWW S LQ Q Q Q Q Q

thermal radiation

of the atmosphereconvection

sensible heat

6

(long wave, IR)

thermal radiation

of the water

(long wave, IR)

evaporation

(latent heat)

solar radiation

(short wave, VIS)


4/21

4

0SW SW SWQ 1 Q 1 0.65B

Radiative Heat Fluxes

Solar radiation (short wave):

reflectivit of clear sk cloud cover

Parameterisations to estimate the radiative terms:

4LWA LW A AQ 1 T

17

water in SW

radiation

fraction

Long wave atmospheric radiation:

reflectivity of water

(in LW (IR), ≈0.03)

emissivity of

atmosphere

Parameterisation

= 5.67∙10‐8 W m‐2 K‐4

Stefan‐Boltzmann constant

2 A

A A1.24 1 0.17BT

4WLWW WQ T Long wave water radiation:

of emissivity A

:

emissivity of water (≈ 0.97)

water vapour

pressure in air

7

Non-Radiative Heat Fluxes

S air p S 10 W AQ c c u T T Convection:

latent heat of evap

cL, cS ≈ 0.001:

bulk transfer coefficients

for vapor and heat,

stabilit ‐de endent

specific heat of airwind

velocity

transfer

coefficients

L air e L 10 s W AQ L c u q T q Evaporation:

specific humidity of air

saturation specific humidity at TW

q = ρv / ρair= e / (RvTρair)

Estimate of latent flux:

w

E P Adt

eL e w

L dmQ L P

A dt

For P = 1 m/a: Q L = 71 W/m2

8


5/21

5

Ocean – Atmosphere Heat Fluxes

Q LW = Q LWA + Q LWW

Marshall and Plumb, 2008 9

Seasonal Heat Fluxes

in a Lake

• Long-wave terms

(thermal radiation of air

and water) are largest

Q LWA

Q SW

contributions, but nearly

cancel

• Net effect in IR is a heatloss of the water

• Sensible heat flux is

u wards heat loss of

Q S

Q L

Q net

water) most of the time,

as usually TW > T A

• Net flux: Upwards in

winter, downwards in

summer

Q LWW

10


6/21

6

Annual Mean Net Upward Heat Flux

Marshall and Plumb, 2008 11

Change of the Ocean Heat Content

Development of heat content of the upper ocean

1955 to 2003

Heat uptake of different compartments between

1955 and 1998 (in 1022 J)

from Levitus et al., 2005, Geophys. Res. Lett. 32, doi:10.1029/2004GL021592

These data are outdated due to recent corrections to ocean temperature data series ‐ see next slide.

12


7/21

7

Change of the Ocean Heat Content

upper 700 m

old

Levitus et al., 2009. Geophys. Res.

Lett. 36: L07608,

doi:07610.01029/02008GL037155.

new

upper 700 m

upper 100 m

Domingues et al., 2008. Nature 453: 1090‐1093.

SST

http://www.nodc.noaa.gov/

OC5/3M_HEAT_CONTENT/

Trend global heat gain: 4.0∙1021 J/a = 1.3∙1014 W

AOcean = 3.6∙1014 m2

Imbalance: 0.35 W/m213

Imbalance of Global Radiation Budget

Hansen et al., 2005.

Science 308: 1431-1435.

Net radiative forcing in

2003 rel. to 1880:

Partly compensated by

warming of ~ 0.7 °C.

Global energy imbalance

in the year 2003:

+ 0.85 ± 0.15 W/m2

+ 1.80 ± 0.85 W/m2

14

Compare global energy

consumption in 2008:

4.7·1020 J/a = 1.5·1013 W

= 15 TW = 0.03 W/m2

(AEarth = 5.1∙1014 m2)


8/21

8

Temperature Response of the Ocean

Simple box model (strictly applicable only to mixed surface layer):

Change of heat content Eth (resp. temperature T) of a well mixed

water body of volume V due to a heat flux Q through the surface A

Q A

th w wdE V c dT Ah c dT

thdE QAdt

w Ah c dT QAdt

V Th

w

dT 1 Q

dt h c

Eth

Example: Q net = 0.35 W/m2, h = 100 m: dT/dt = 0.026 K/a

(Note: cw = 4.18∙106 J m‐3 K‐1; 1 a = 3.15∙107 s) 15

Temperature Trends in the Ocean

IPCC, 4th Assessment Report, 2007 16


9/21

9

Gases: Air/Water Solubility Equilibrium

Equilibrium:

different

concentrations

equal fluxes!

17

Gas Solubility in Water: Henry's Law

At equilibrium, the concentration cw of a gas in solution is propor‐

tional to its concentration cg (partial pressure p) in the gas phase:

KH: Henry coefficient ("constant")g H wc K c

If both concentrations molar (mol/L): KH dimensionless = K'HIn practice, many concentration units occur (mol/kg, mg/L, cm3STP/g, ….)

Conversion of units for cw in mol/L and p in atm via the ideal gas law:

gcn atm L pRT RTc K RT RTK

H wp cor

w wV mol c c

KH is the inverse of a solubility (large KH little gas dissolved)→ (Ostwald) solubility: L = 1/K'H (sometimes also defined as Henry const.)

w

g H

c 1L

c K

w

H

c 1

p K or

18


10/21

10

Dependence of Solubi lity on T and S

KH (or L) is specific for each gas i and depends on temperature and salinity of the water: , ,H H iK K T S

Temperature epen ence o ows rom Van t Ho equat on:

H2

d lnK H

dT RT

H: Enthalpy change for dissolution(H = U + pV)

Integrated with const. H: H H 00

H 1 1K T K T exp

R T T

Salinity dependence ("salting out") described by Setschenow

relation:

k: Setschenow or salting coefficient

kSH HK T,S K T,0 e

19

Temperature Dependence of Solubil ities

0.9

1

0.6

0.7

0.8

He NeAr Kr

c i (

T ) / c i ( 0 ° C )

L i (

T ) / L

i ( 0 ° C )

0.4

.

0 5 10 15 20 25 30

e

N2

O2

T [°C]

20


11/21

11

Salinity Dependence of Solubi lities

0.95

1

He NeAr Kr

0.8

0.85

0.9

Xe

N2

O2

c i (

S ) / c i (

0 )

L i (

T , S

) / L i (

T , 0

)

0.75

0 5 10 15 20 25 30 35 40

S [g/kg]

21

Atmospheric Equilibrium Concentrations

The equilibrium (or air saturation) concentration cs,i is the dissolved

concentration of gas i in equilibrium with moist (saturated) air:

c e x i: solubility of gas i in water, ,s a m s

cs depends on T, S and p:

,i atm s ip p e x

, , , ,i eq i s ic T S p T S p e T x

Atmospheric pressure p depends on altitude z: sz z

0p z p e

pi ,air: par a pressure o gas n mo s a r

p: total atmospheric pressure

es: saturation water vapour pressure

xi: mixing ratio of gas i in air

, . /2 2 2s O O s Oc p e x 10 6mg l

Examples O2, N2:

altitude z = 400 m, p = 0.95 atm

S = 0, T = 10°C, es = 0.012 atm

O2 = 53.7 mg l‐1 atm‐1, xO2 = 0.209 N2 = 23.1 mg l‐1 atm‐1, xN2 = 0.781 , . /2 2 2s N N s Nc p e x 17 1mg l

22


12/21

12

Solubil ity and Equilibr ium of some Gases

Solubilities i in pure water (S = 0) for various temperatures in g l‐1

atm

‐1

:

, . /

2s COc 1 3 mg lxCO2 = 3.9∙10‐4

, . /

2s Nc 22 8 mg l

s,i

p = 1 atm, T = 0°C

xN2 = 0.781

, . /

2s Oc 14 4 mg lxO2 = 0.209

, . /s Ar c 0 89 mg lxAr = 9.3∙10‐3

23

Equilibrium, Saturation, Oversaturation

Equilibrium conc.: Equilibrium at local air pressure (water surface)

Saturation conc. (absolute): Maximum dissolved conc. at in situ

pressure (incl. phyd) before bubbles are formed

i toti p pOversaturation:

, ,i H i W ip K c

'tot 0p p gz z': water depth

Henry's law relates dissolved concentrations to partial pressures.

I t e sum o a part a pressures excee s t e externa pressure

(oversaturation), bubbles are formed and gas escapes.

Typical example: Methane bubbles rising from lake sediments.


13/21

13

Gas Contents of some Volcanic Lakes

Lake Monoun, Cameroon Lake Kivu, Kongo/Ruanda

80 % Sat.

Schmid et al., 2005, G3, doi:10.1029/2004GC000892Halbwachs et al., 2004, Eos 85 (30): 281

Lake Nyos, Cameroon: Artificial Degassing of a Lake

Principle


14/21

14

Lake Nyos, Cameroon: Artificial Degassing of a Lake

Halbwachs et al., 2004, Eos 85 (30): 281

Gas Transfer: Molecular Boundary Layer

Free atmosphere or open water: Efficient transport by turbulence

Near the boundary: Molecular‐viscous boundary layer

• laminar flow, no turbulence

• Transport only by molecular diffusion: "Resistance"

For heat and most gases, the water‐side boundary layer dominates

(most strongly restricts exchange): One‐layer model

Flux of gas i by molecular diffusion in water boundary layer:

ii i

dc j D

dz Di: molecular diffusion constant of gas i in water

ci: concentration of gas i in water

In steady state, j must be constant, thus dc/dz = constant

(linear concentration profile)28


15/21

15

“ Stagnant Film” Model

z turbulent

cg, p

Air

Water

s g Hc p c K sc

molecular

C

c

turbulent

29

Transfer Velocity in the Film-Model

For gases with low solubility, the main transport resistance is on the

water side: 1‐Film model, water‐side control

Flux in the laminar film is determined by molecular diffusion

Dk

S Sc cdc j D D k c cdz

Transfer coefficient:

(D: Diffusion coefficient in water)

2L T L

kL T

Dependence of k on D: k D

Model parameter: Film thickness , which depends on strength of turbulence, e.g. expressed by friction velocity u* (related to u10)

k = transfer/exchange

velocity, piston velocity

30


16/21

16

Friction Velocities in Air and Water

Friction velocity:

z u

2 2

10 D *u c u

2

* x zu v v Wind velocity:

air

water

u(z)

2xz W *w0 u

2 2xz air D 10 air *0 c u u

x

Shear stress continuous at z = 0 (conservation of momentum), thus:

31

air *w *air

w

u u

Measure of turbulence in the water,

dependent on wind velocity

Wind Veloci ty and Film Thickness

Typical exchange velocities k for some gases and correspon-

ding film thicknesses w for different wind velocities u10.

gas exchange velocity k (m d-1)

Molecular diffusion coefficient D

at 25 °C (m2

s-1

)

Film thickness decreases with increasing wind speed.

Typical values are on the order of 100 m32


17/21

17

Dependence of k on D: Schmidt Number

The stagnant film model is the simplest gas exchange model. More

complex models show different dependences of k on D:

n

To compare exchange of heat, momentum, and gases, the transfer

velocity is often parameterised by the so called Schmidt number:

Schmidt number: ScD

nk Sc with 1 2 n 1 Thus we get:

Measurements of k are often normalised to Sc = 600 (CO2 at 20°C)

33

Transfer Velocity in Different Regimes

nk Sc

1 2 n 1

smooth surface:

n = 2/3 (or n = 1)

rough surface:

Liss and Merlivat, 1986

n = 1/2

34


18/21

18

Transfer Velocity and Wind Speed

Dependence of

k on

wind speed

wind speed, but how

exactly?

Frequently used

parametrisation:

2

10k u

Lab measurements

with fit curves

normalised to Sc = 600

35


Wanninkhof and Bliven 1991

Liss and Merlivat 1984

36


19/21

19


Various models

37

Ocean – Atmosphere CO2 Flux

Based on 940.000 pCO2 measurements, assuming k u2. Balance: ‐ 1.6 ± 1 GtC/a. IPCC AR4, 2007 38


20/21

20

Importance for CO2-Budget

39

Gas Exchange for Vapour: Evaporation

Water vapour saturation pressure as Henry‐equilibrium:

2H O s H wp e K c .

-1

w -1

1000 g L molc 55 55

M 18 g mol L

Vapour pressure p and "Henry coefficient" of water for different temperatures

o concen ra on gra en n wa er: vapora on s con ro e y a r‐s e m.

Sensible heat flux: Also air‐side controlled, flux given by th w j k c T

40


21/21

Evaporation and Wind Speed

41

Summary

• Air/water heat transfer

– Radiative terms dominate, SW downward, IR terms nearly cancel

– Latent and sensible heat flux: upward, analogous to gas transfer

– Eva oration: Gas exchan e for water va our air-side controlled

• Solubility equilibrium for gases in water

– Henry's Law: Concentrations in both phases proportional

– Equilibrium concentration in water: cs(T,S,p)

• Air/water gas exchange

– Laminar boundary layer (usually water-side) transport resistance

– - = ,

– Schmidt number parameterisation: k Sc-n, ½ < n < 1

• Transfer velocity and wind speed

– strong but not exactly known relationship (e.g. k u2)

– important for correct estimation of ocean CO2 uptake

42

Documents

MKEP4 SS11 12 Gas Transfer