SSC 107, Fall 2002 – Chapter 7 Page 7-1

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 H2S SO2

NH3 N2 N2O CxHy NO

SSC 107, Fall 2002 – Chapter 7 Page 7-2

An example of diffusion processes related to contaminants in the vadose zone

SSC 107, Fall 2002 – Chapter 7 Page 7-3

Mechanisms for transfer of gases 1. Mass flow in vapor phase

a. Barometric pressure changes b. Wind c. Irrigation or rain d. 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 phase b. 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. Da

g is the diffusion coefficient in air without soil and Cg is concentration of gas. What happens to Da

g for soil? Le is the effective length

L

Le

SSC 107, Fall 2002 – Chapter 7 Page 7-4

In soil, not air, the equation becomes

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

LL

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 a

gsg (not too good when a is small)

e

gag

e

g

LCD

tAm ∆−=

ee

LaALA =

SSC 107, Fall 2002 – Chapter 7 Page 7-5

Thus

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 ag2

10/3sg

φ Millington and Quirk

D a a aa

Dgs

b

ga= +

FHGIKJ

+

2 0 041003

100100

2 3

./

d i 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 slope of the Campbell water retention function. If b is not known, it can be estimated from the clay fraction given 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

TTDD 12

=

SSC 107, Fall 2002 – Chapter 7 Page 7-6

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

The SWC dependent model is the one by Moldrup et al. (1999)

0

0.5

Dgs/Dga

0 0.5 Soil-air content (a)

Penman

Millington & Quirk

SSC 107, Fall 2002 – Chapter 7 Page 7-7

An apparatus for measuring the diffusion coefficient for soil Diagram giving initial and boundary conditions X=0 X=-L X=-(L+a)

Soil Core

C=C0 t=0

C=C0 t≥0

C=Ci t=0

C=f(t) t›0

Diffusion Chamber

Soil Core

Diffusion Chamber

Sample Port

Slide - Horizontal

Position A

SSC 107, Fall 2002 – Chapter 7 Page 7-8

Method of measuring Ds

g in lab

down)(flow dz

dC D- = Atm - = J - gs

gg

g

mg - mass of gas Cg - 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( VLAD - =

dtdC

og

sgg

Rearranging

Diffusion Chamber

Sample Port

Soil Core

Position B

SSC 107, Fall 2002 – Chapter 7 Page 7-9

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.

SSC 107, Fall 2002 – Chapter 7 Page 7-10

0 -0.02 -0.04 ln [(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.

SSC 107, Fall 2002 – Chapter 7 Page 7-11

Diffusion equations With concentration on fluid basis Steady State

dzdC D - = J gs

gg

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 ∂∂∂

2g

2sg

g

zC D =

tCa

∂∂

∂∂

aD = D

zC D =

tC s

gm2

g2

mg

∂∂

∂∂

ssoil cm = D

2

m∴

SSC 107, Fall 2002 – Chapter 7 Page 7-12

With concentration on soil basis use Cm Steady State

dzdC D - = J m

mg

Cm - g gas/cm3 soil Transient State

2m

2

mm

zC D =

tC

∂∂

∂∂

soil cm soil cm g

ssoil cm =

s soil cmg

23

2

3

SSC 107, Fall 2002 – Chapter 7 Page 7-13

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 of gas 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 3

d is the liquid/soil partition coefficient. Substituting Sw from the equation above gives

Taking the derivatives of Sw and Ss with respect to time and substituting into the original equation gives

Dividing both sides by a gives

tS -

tS -

zC D =

tC a sw

2g

2sg

g

∂∂

∂∂

∂∂

∂∂

K / C = Sor / S K = C HvgwvwHg θθ

θρ vwdbs / S K = / S

K / C K = S Hgbds ρ

2g

2sg

g

H

bd

H

v

zC D =

tC

KK

Ka

∂∂

∂∂

ρ+

θ+

2g

2

mg

H

bd

H

v

zC D =

tC

aKK +

aK + 1

∂∂

∂∂

ρθ

SSC 107, Fall 2002 – Chapter 7 Page 7-14

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

2g

2mg

zC

RD =

tC

∂∂

∂∂

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

∂∂

∂∂

SSC 107, Fall 2002 – Chapter 7 Page 7-15

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∫∫

dzdCDJ gs

gg −=

)t,z(SzCD

tC

2

2g

mg

−∂∂

=∂∂

SSC 107, Fall 2002 – Chapter 7 Page 7-16

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 depth L = 0 or dCg/dz = 0. Substituting dCg/dz = 0 in equation determined after first integration gives

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

SSC 107, Fall 2002 – Chapter 7 Page 7-17

Oxygen profiles in soils as related to degree of biological activity and the soil gaseous

diffusion coefficient.

Curve Degree of Activity Dgs/Dg

a (liters/m3 day) 1 10 0.06 2 5 0.06 3 10 0.25 4 5 0.25

0

1

Soil Depth (m)

15

21 Oxygen (%)

12

3

4

SSC 107, Fall 2002 – Chapter 7 Page 7-18

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 solubility of 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.

SSC 107, Fall 2002 – Chapter 7 Page 7-19

Several figures related to aeration follow:

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

SSC 107, Fall 2002 – Chapter 7 Page 7-20

Above figure from Glinski and Stepniewski (1983)

SSC 107, Fall 2002 – Chapter 7 Page 7-21

Other Diffusion Processes -Flooded soil or sediments • O2 diffusion • NH3 volatilization • Solute diffusion A diagram showing diffusion processes in flooded soil (Reddy et al.)

SSC 107, Fall 2002 – Chapter 7 Page 7-22

- 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

Anoxic Zone

O2

NO3

N2O or N2

SSC 107, Fall 2002 – Chapter 7 Page 7-23

Diagrams concerning volatile organic chemical transport processes follow:

SSC 107, Fall 2002 – Chapter 7 Page 7-24

Water Vapor Movement

dzd D - = J v

vwvρ

ρv- vapor density in gaseous phase Dv - diffusion coefficient for water vapor in soil corrected for tortuosity (See book) Vapor density gradients caused by

1. Differences in matric potential and solute potential 2. Temperature differences

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

Water Potential Temperature (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 humidity at - 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 of water 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 the development of large vapor density gradients