Chapter 7 - Gas Flowlawr.ucdavis.edu/classes/ssc107/SSC107Syllabus/chapter7-00.pdf · Chapter 7 - Gas Flow ... w is amount of gas dissolved in water and S (g gas / cm3 soil) s is

SSC 107, Fall 2000 – Chapter 7 Page 7-1

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Chapter 7 - Gas Flow

• Mechanisms of flow• Fick's Law• Methods for measuring D• Diffusion equations• Diffusion to plant roots• Consumption or production of gases• Aeration• Other diffusion processes• Water vapor movement

______________________________________________________________________

Important Soil Gases

O2

CO2

O2 CO2 H2SSO2

NH3 N2 N2O CxHy


Mechanisms for transfer of gases

1. Mass flow in vapor phase

a. Barometric pressure changesb. Windc. Irrigation or raind. Production from other chemicals

2. Mass flow in dissolved (liquid) phase

a. Movement with water

3. Diffusion in liquid phase

a. 103 - 104 smaller than diffusion in vapor phaseb. Important in transport of O2 to roots and anoxic denitrification sites

4. Diffusion in vapor phase

a. Probably main mechanism

Fick's Law (for diffusion of gases)

dzdC D - = J =

Atm ga

ggg

Fick's law assumes that gases are dilute and that there is equimolar counter diffusion of each gas.

Dag is the diffusion coefficient in air without soil and Cg is concentration of gas.

What happens to Dag for soil?

Le is the effective length

L

Le


In soil, not air, the equation becomes

where Ae is the effective pore area. The soil-air content, a, is

L

L

A

voltotalgas vol =a

eeA= or

Thus

LC

LLDa - =

Atm

e

g

e

agg ∆

Multiplying top & bottom by L gives LC

LDLa - = J g2e

ag

2

g∆

Let D LL a = D a

g

2

e

sg

Thus dzdC D - = Jor

LC D- = J gs

gggs

gg∆

Dsg is the apparent diffusion coefficient for soil or soil-gas diffusivity

Self diffusion coefficient is for a gas into itself

Mutual (or binary) diffusion coefficient is for one gas diffusing into a different gas

LL

e

2

is called the tortuosity factor

For some cases Da 0.66 = D ag

sg (not too good when a is small)

Thus

e

gag

e

g

LCD

tAm ∆−=

ee

LaALA =


aircmsoilcm 0.66 =

LL

2

2

e

2

Penman

Other Relationships

D a = D ag

3/2sg Marshall

D a = D ag

4/3sg Millington

D a = D ag

sg

µγ Currie

D a = D a

g2

10/3sg

φ Millington and Quirk

D a a aa

Dgs

b

ga= +

+

2 0 041003

100100

2 3

./

Moldrup et al. (1999)

where a100 is the soil-air content at a soil-water pressure head of –100 cm of water and b is the slopeof the Campbell water retention function. If b is not known, it can be estimated from the clay fractiongiven by the following equation:

b = 13.5 CF + 3.5 where CF is the clay fraction.

Effect of temperature on Dg

Where T is in degrees Kelvin. The value of n has been found to be about 1.72-1.75.

n

1

2TT

TT

DD 12

=


An apparatus for measuring the diffusion coefficient for soil

Diagram giving initial and boundary conditions

X=0

X=-L

X=-(L+a)

SoilCore

C=C0 t=0

C=C0 t=0

C=Ci t=0

C=f(t) t›0

DiffusionChamber

SoilCore

Diffusion Chamber

Sample Port

Slide - Horizontal

Position A


Method of measuring Dsg in lab

down)(flow dz

dC D- = Atm - = J -

gsg

gg

mg - mass of gasCg - concentration in chamber

dzdC AD- =

dtdm - gs

gg V - volume of chamber

V C = m gg

dzdC

VAD + =

dtdC g

sgg∴

Co - Concentration of tracer gas

L-C-C

dzdC ogg ≅ at the top of the core

L - Length of soil core

)C-C( VL

AD - = dt

dCog

sgg

Rearranging

Diffusion Chamber

Sample Port

SoilCore

Position B


gintegratin anddt VL

AD - = C-C

dC sg

og

g

dt VL

AD - = C-C

dC t

o

sg

og

gc

c

g

i

∫∫

Where Ci is initial concentration in chamber at t=0

t

0

sgg

i0 t VL

AD - = )CC( n CCg

−l

tVL

AD- = )C-C( n - )C-C( nsg

oiog

ll

tVL

AD - = C-CC-C n

sg

oi

og

l

If Ci = 0

tVL

AD - = C

C-C nsg

o

go

l

Equation not valid for small times because L-C-C

dzdC ogg ≠

Also, equation does not take into account change in storage of gas in soil core.


0

-0.02

-0.04ln [(Cg - Co)/(Ci - Co)]

-0.06

-0.08

-0.10 0 0.5 1.0 Time (hours)

A plot of ln [(Cg - Co)/(Ci - Co)] vs. time using hypothetical data from a soil core with values of a=0.1 m3 m-3, L = 76 mm, A = 4540 mm2, and V = 0.5 L.

Two relationships of soil gas diffusion coefficient to soil-air content.

0

0.5

Dgs/Dga

0 0.5Soil-air content (a)

Penman

Millington &Quirk


Diffusion equationsWith concentration on fluid basis

Steady State

dzdC D - = J

gsgg

Cg is concentration on fluid basis and units are g gas/cm3 soil air

soil cmair cmgas/ g D - =

s soil cmgas g 3

sg2

s soil cmair cm = D

3sg∴

Transient State

∆ storage = ∆ flux

dzJ - =

tCa gg ∂∂

∂

zC D =

tCa

2

g2

sg

g

∂∂

∂∂

aD = D

zC D =

tC s

gm2

g2

mg

∂∂

∂∂

ssoil cm = D

2

m∴


With concentration on soil basis use Cm

Steady State

dzdC D - = J m

mg

Cm - g gas/cm3 soil

Transient State

zC D =

tC

2m

2

mm

∂∂

∂∂

soil cm soil cm g

ssoil cm =

s soil cmg

23

2

3


Dissolution of gas in soil water and adsorption on soil

where soil) cm / gas (g S 3w is amount of gas dissolved in water and soil) cm / gas (g S 3

s is amount ofgas adsorbed to soil solids.

The relationships between gas phase and water phase concentrations are

where KH is the "dimensionless" Henry's coefficient (cm3 water/cm3 air).

The relationship between the gas dissolved in the liquid phase and that in the sorbed phase is

where soil) water /gcm( K 3d is the liquid/soil partition coefficient. Substituting Sw from the equation

above gives

Taking the derivatives of S w and S s with respect to time and substituting into the original equationgives

Dividing both sides by a gives

tS -

tS -

zC D =

tC a sw

2

g2

sg

g

∂∂

∂∂

∂∂

∂∂

K / C = Sor / S K = C HvgwvwHg θθ

θρ vwdbs / S K = / S

K / C K = S Hgbds ρ

z C D =

tC

KK

Ka 2

g2

sg

g

H

bd

H

v

∂∂

∂∂

ρ+

θ+

z C D =

tC

aKK

+ aK

+ 1 2g

2

mg

H

bd

H

v

∂∂

∂∂

ρθ


or

Consumption or Production of Gases

- O2 consumed

- CO2 produced

- some gases adsorbed

- some gases react

t)(z, rg is sink or source terms to take consumption or production into account

z C

RD =

tC

2

g2

mg

∂∂

∂∂

by givent coefficien nretardatio theis Randa / D = D where sgm

H

bd

H aKK

+ aK

+ 1 = R v ρθ

t)(z, r + zC

D = tCa g2

g2

sg

g

∂∂

∂∂


A steady-state solution for gas diffusion and consumption

For O2

where S(z,t) = rg(z,t)/a and rg is the gas reaction rate.

Assume

S(z, t) = α, α is a constant O2 consumption rate

assume steady state, 0 = tC g

∂∂

Integrate once with respect to z

where c1 is a constant of integration.

D =

dzCd , =

dzCd D

m2

g2

2g

2

mα

α

c + z D

= dz

dC dz, D

= dz dz

Cd 1m

g

m2

g2 αα∫∫

dzdC

DJgs

gg −=

)t,z(SzCD

tC

2

2g

mg −

∂∂=

∂∂


Integrate again with respect to z

to give

where c2 is an additional constant of integration.

At z = 0, Cg = Co. Therefore, Co = C2, and

C + z c + z D2

= C o12

mg

α

Assume we have a finite column with a closed bottom or a water table at depth L, flux at depthL = 0 or dCg/dz = 0. Substituting dCg/dz = 0 in equation determined after first integrationgives

dz c + zdz D

= dz dzCd

1m

g ∫∫∫ α

c + z c + z D 2

= C 212

mg

α

C + z D

L - z

D2 = C

LD

- = c

c + LD

= 0

om

2

mg

m1

1m

αα

α

α


Oxygen profiles in soils as related to degree of biological activity and the soil gaseousdiffusion coefficient.

Curve Degree of Activity Dgs/Dg

a

(liters/m2 day) 1 10 0.06 2 5 0.06 3 10 0.25 4 5 0.25

0

1

SoilDepth(m)

15 21Oxygen (%)

1 2

3

4


Aeration - Effects on Plants

1. O2 needed for root respiration - critical values of flux

2. Good aeration is essential for maximum H2O absorption.

Sudden reduction of O2 will cause growing plant to wilt.

3. CO2 retards uptake of nutrients.

Reduction follows K>N>P>Ca>Mg

4. CO2 & H2O form carbonic acid which increases the solubilityof many soil minerals. Some ions may become toxic to plants

5. Growth of roots limited by either lack of O2 or buildup of CO2

6. O2 needs increase with temperature

7. O2 needs increase as soil-water pressure head increases

- physical process-meaning at lower air content, gradients must be increased

8. Rate of O2 flux (supply) and CO2 removal is most important

Diffusion to plant roots

- O2 must diffuse through water films to reach root

- Mechanism simulated by measuring O2 diffusion to a microelectrode

AF*nI = J t

g ′ This is the oxygen diffusion rate (ODR)

It is current (amps) in time t n* = 4 for O2 molecules F' is the Faraday constant = 96,500 coulombs A is the surface area of the electrode Ds

g cannot be determined.


Several figures related to aeration follow:

Oxygen diffusion rates at a given soil depth as a function of depth of water table. (Williamson and vanSchilfgaarde, 1965).



Above figure from Glinski andStepniewski (1983)



Other Diffusion Processes

-Flooded soil or sediments

• O2 diffusion

• NH3 volatilization

• Solute diffusion

A diagram showing diffusion processes in flooded soil (Reddy et al.)


- Denitrification

Aggregate or anoxic pocket or "hot spot"

Also

- Diffusion of radon gas from soil into dwellings

- Volatilization of pesticides or volatile organics from soil

__________________________________________________

Stagnant Air Layer

________________________________________________Soil Surface

Soil + Pesticide Pesticide

AnoxicZone

O2

NO3

N2O orN2


Diagrams concerning volatile organic chemical transport processes follow:


Water Vapor Movement

dzd

D - = J vvwv

ρ

ρv- vapor density in gaseous phaseDv - diffusion coefficient for water vapor in soil corrected for tortuosity (See book)

Vapor density gradients caused by

1. Differences in matric potential and solute potential2. Temperature differences

Vapor density, Dv, in grams of vapor per cubic cm of pore space (g/cm3) at various temperatures andat two soil-water potentials.

Water PotentialTemperature (C)-0.1 bar (-9.8 kPa) -15 bars (-1500 kPa)

15 12.83 x 10-6 12.70 x 10-6

18 15.37 x 10-6 15.22 x 10-6

20 17.30 x 10-6 17.13 x 10-6

21 18.34 x 10-6 18.16 x 10-6

22 19.43 x 10-6 19.24 x 10-6

23 20.58 x 10-6 20.37 x 10-6

24 21.78 x 10-6 21.56 x 10-6

25 23.05 x 10-6 22.82 x 10-6

30 30.38 x 10-6 30.08 x 10-6

35 39.63 x 10-6 39.23 x 10-6

at - 0.1 bars have 100% relative humidityat - 15 bars have 98.98% relative humidity

... Differences in water potential will have little effect on vapor transport

Temperature differences have the much larger effect, but still little difference in effects ofwater potential over the range between - 0.1 and - 15 bars at different temperatures

Appreciable vapor phase water flow will occur in the field surface soil due to thedevelopment of large vapor density gradients

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Chapter 7 - Gas Flowlawr.ucdavis.edu/classes/ssc107/SSC107Syllabus/chapter7-00.pdf · Chapter 7 - Gas Flow ... w is amount of gas dissolved in water and S (g gas / cm3 soil) s is