SSC 107, Fall 2002 – Chapter 7
Page 7-1
Chapter 7 - Gas Flow
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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 NO
CxHy
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.
b.
c.
d.
Barometric pressure changes
Wind
Irrigation or rain
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)
mg = = - a dCg
Jg
Dg
At
dz
Fick's law assumes that gases are dilute and that there is equimolar counter diffusion of each gas.
a
Dg is the diffusion coefficient in air without soil and C g is concentration of gas.
What happens to Dag for soil?
Le
L
Le is the effective length
SSC 107, Fall 2002 – Chapter 7
Page 7-4
In soil, not air, the equation becomes
mg
∆Cg
= − Dag
Aet
Le
where Ae is the effective pore area. The soil-air content, a, is
a=
vol gas Ae Le
=
total vol A L
Ae =
or
aAL
Le
Thus
mg = - a LDag ∆Cg
At
Le Le
Multiplying top & bottom by L gives Jg = - a
 L
Let D = a 

 Le 
s
g
Thus Jg = - Dsg
L2 Dag ∆Cg
Le2 L
2
Dag
∆ Cg
or Jg = - Dsg dCg
L
dz
s
Dg 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
 L
 
 Le 
2
is called the tortuosity factor
For some cases Dsg = 0.66 a Dag (not too good when a is small)
SSC 107, Fall 2002 – Chapter 7
Page 7-5
Thus
2
2
L
cm soil
=
0.66
 
2
cm air
 Le 
Penman
Other Relationships
s
3/2
a
Marshall
Dg = a Dg
s
4/3
a
Millington
Dg = γ a Dg
Currie
Dg = a Dg
µ
s
a
10/3
a
a
D = 2 Dg
φ
s
g
Dsg
=
d
3
2a100
Millington and Quirk
+ 0.04a100
iFGH a a IJK
2+3/ b
D ag
Moldrup et al. (1999)
100
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
 T2 
DT 2 = DT1 
 T1 
n
Where T is in degrees Kelvin. The value of n has been found to be about 1.72-1.75.
SSC 107, Fall 2002 – Chapter 7
Page 7-6
0.
5
Dgs/Dga
Penman
Millington &
Quirk
0
0
Soil-air content (a)
0.5
Two relationships of soil gas diffusion coefficient to soil-air content.
The SWC dependent model is the one by Moldrup et al. (1999)
SSC 107, Fall 2002 – Chapter 7
Page 7-7
An apparatus for measuring the diffusion coefficient for soil
Diagram giving initial and boundary conditions
C=C0
t≥0
X=0
Soil
Core
X=-L
C=C0
t=0
C=Ci
t=0
C=f(t) t›0
Diffusion
Chamber
X=-(L+a)
Position A
Soil
Core
Slide - Horizontal
Diffusion Chamber
Sample Port
SSC 107, Fall 2002 – Chapter 7
Page 7-8
Position B
Soil
Core
Diffusion Chamber
Sample Port
Method of measuring Dsg in lab
- Jg = -
dC
mg
= - Dsg g
At
dz
(flow down)
mg - mass of gas
Cg - concentration in chamber
-
dmg
dC
= - Dsg A g
dt
dz
V - volume of chamber
mg = Cg V
s
dC
D A dCg
∴ g =+ g
dt
V dz
Co - Concentration of tracer gas
dCg Cg - Co
≅
dz
-L
at the top of the core
L - Length of soil core
s
g
dCg
D A
=(Cg - Co )
dt
VL
Rearranging
SSC 107, Fall 2002 – Chapter 7
Page 7-9
 Dsg A 
dCg
=-
 dt and integrating
Cg - Co
 VL 
cg
∫
ci
t
 Ds A 
dCg
= -  g  ∫ dt
Cg - Co
 VL  o
Where Ci is initial concentration in chamber at t=0
 Dsg A  t
Cg
ln (Cg − C 0 ) Ci = - 
t0
 VL 
 Ds A 
ln (Cg - Co ) - ln (Ci - Co ) = -  g  t
 VL 
 Cg - Co   Dsg A 
ln 
t
=-
 Ci - Co   VL 
If Ci = 0
 C - C   Ds A 
ln  o g  = -  g  t
 Co   VL 
dCg Cg - Co
≠
dz
-L
Also, equation does not take into account change in storage of gas in soil core.
Equation not valid for small times because
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
Time (hours)
1.0
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
Diffusion equations
With concentration on fluid basis
Steady State
s
J g = - Dg
dCg
dz
Cg is concentration on fluid basis and units are g gas/cm3 soil air
3
g gas
s g gas/ cm air
=
D
g
2
cm soil
cm soil s
∴ Dsg =
3
cm air
cm soil s
Transient State
∆ storage = ∆ flux
a
∂ Cg
∂J
=- g
∂t
dz
a
2
∂ Cg
∂ Cg
= Dsg
∂t
∂z 2
2
∂ Cg
∂ Cg
= Dm
∂t
∂z 2
∴ Dm = cm
2
soil
s
s
Dg
Dm =
a
Page 7-11
SSC 107, Fall 2002 – Chapter 7
With concentration on soil basis use Cm
Steady State
J g = - Dm
dCm
dz
Cm - g gas/cm3 soil
Transient State
2
∂ Cm
∂ Cm
= Dm
∂t
∂z 2
2
g
g
cm soil
=
3
3
2
s
cm soil cm soil
cm soil s
Page 7-12
SSC 107, Fall 2002 – Chapter 7
Page 7-13
Dissolution of gas in soil water and adsorption on soil
2
∂ Cg
∂
s ∂ Cg ∂ Sw
a
= Dg
- Ss
2
∂t
∂z
∂t
∂t
where Sw (g gas / cm3 soil) is amount of gas dissolved in water and Ss (g gas / cm3 soil) is amount of
gas adsorbed to soil solids.
The relationships between gas phase and water phase concentrations are
Cg = K H Sw / θv or Sw = Cg θv / K H
where K H 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
Ss / ρb = Kd Sw / θv
where Kd (cm3 water /g soil) is the liquid/soil partition coefficient. Substituting Sw from the
equation above gives
Ss = Kd ρb Cg / K H
Taking the derivatives of S w and S s with respect to time and substituting into the original equation
gives
2

θ v K dρ b  ∂ Cg
s ∂ Cg
a
=
+
+
D
g

KH
K H  ∂t
∂z 2

Dividing both sides by a gives
2

K d ρ b  ∂ Cg
θv
∂ Cg
1
+
+
=
D
m
 aK
aK H  ∂t
∂z 2
H

SSC 107, Fall 2002 – Chapter 7
Page 7-14
or
∂ Cg Dm ∂ 2 Cg
=
R ∂z 2
∂t
where Dm = Dsg / a and R is the retardation coefficient given by
R =1+
Kρ
θv
+ d b
aK H aK H
Consumption or Production of Gases
- O2 consumed
- CO2 produced
- some gases adsorbed
- some gases react
2
∂ Cg
s ∂ Cg
a
= Dg
+ r g (z, t)
∂t
∂z 2
r g (z, t) is sink or source terms to take consumption or production into account
SSC 107, Fall 2002 – Chapter 7
Page 7-15
A steady-state solution for gas diffusion and consumption
Jg = − D sg
dCg
dz
For O2
∂Cg
∂ 2 Cg
= Dm 2 − S( z, t )
∂t
∂z
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,
∂ Cg
=0
∂t
2
d Cg
= α,
Dm
2
dz
Integrate once with respect to z
∫d C
2
dz
g
2
dz =
∫
α
Dm
dz,
dCg α
=
z + c1
dz Dm
where c1 is a constant of integration.
2
d Cg α
=
2
dz
Dm
SSC 107, Fall 2002 – Chapter 7
Page 7-16
Integrate again with respect to z
∫ ddzC
g
dz =
∫
α
Dm
∫
zdz + c1 dz
to give
Cg =
α
2 Dm
2
z + c1 z + c 2
where c2 is an additional constant of integration.
At z = 0, Cg = Co. Therefore, Co = C2, and
Cg =
α
2 Dm
2
z + c1 z + Co
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
0=
α
Dm
c1 = -
L + c1
α
L
Dm
α 2 αL
z + Co
Cg =
z 2 Dm
Dm
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
Dgs/Dga
Degree of Activity
(liters/m3 day)
10
5
10
5
1
2
3
4
0.06
0.06
0.25
0.25
0
Soil
Depth
(m)
1
2
3
4
1
1
5
Oxygen (%)
2
1
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
Jg =
It
n * F′A
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
s
Dg 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
Above figure from Glinski and Stepniewski (1983)
Page 7-20
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"
O2
Anoxic
Zone
NO3
N2O or
N2
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
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
J wv = - Dv
d ρv
dz
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)
15
18
20
21
22
23
24
25
30
35
-0.1 bar (-9.8 kPa)
12.83 x 10-6
15.37 x 10-6
17.30 x 10-6
18.34 x 10-6
19.43 x 10-6
20.58 x 10-6
21.78 x 10-6
23.05 x 10-6
30.38 x 10-6
39.63 x 10-6
-15 bars (-1500 kPa)
12.70 x 10-6
15.22 x 10-6
17.13 x 10-6
18.16 x 10-6
19.24 x 10-6
20.37 x 10-6
21.56 x 10-6
22.82 x 10-6
30.08 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