Solute Transport

1
Solute Transport

• There are three basic mechanisms that affect
  solute transport in soils
• Mass Flow
• Molecular Diffusion
• Hydrodynamic Dispersion (Dispersivity)

2
The solute flux (Js) is essentially controlled by
the various mechanisms of liquid flux (Jl) and
gaseous flux (Jg). Js Jl Jg The solute liquid
flux consists of that due to liquid convection
(Jlc), dispersion (Jlh), and diffusion (Jld),
such that Js Jlc Jlh Jld Jg where for
non-volatile chemicals Jg
0
3
Jlc JwCl where Jw -K(?)?H, and Cl
concentration of the solute in solution Jlh
-Dlh?C/?x Dlh dispersion coefficient Jld
-Dld?C/?x Dld diffusion coefficient Jdh
-De?C/?x De combined dispersivity
coefficient Jg -Dg?C/?x Dg gaseous
diffusion coefficient
4
Time Dependent Solute Flux
Just as for transient state water flow, time
dependent solute flux is derived from the
conservation of mass theory.
?z
Js
Js
?y
?x
?CT/?t - ?Js/?x,y,x r where r internal
reactions
5
Horizontal Solute Flux ?CT/?t -?Js/?x
r where Js JwCl - De?C/?x - Dg ?C/?x and ?CT/?t
-?/?x (JwCl - De?C/?x -Dg?C/?x) - r
NOTE Solute flux due to dispersivity and gaseous
diffusion are normally small in comparison to the
mass transport due to liquid water flow.
6
Inert, Non-Volatile, Non-Adsorbing Solute
Inert Not chemically reactive r
0 Non-Volatile Remains in solution Jg 0 If
the solute concentration is low and water flow is
due to gravity alone Js Jw -K(?)?H Average
velocity (v) Jw/?
7
Average Residence Time (time to breakthrough) tb
L/v L?/Jw where ?H 1 and Jw K(?) such
that tb L?/K(?) and the distance L L tbK(?)/?
8
Inert, Non-Volatile, Adsorbing Solute
In this instance, solute transport is delayed due
to the adsorption process v v/(1k) where k
is the adsorption coefficient (selectivity
constant). Hence, tb ((1k)L?)/K(?) and L
tbK(?)/((1k)(?))
9

• How do we determine the value of k?
• If the solute is cationic and the adsorption
  mechanism is non-specific we can apply a cation
  exchange equation to determine the equilibrium
  constant k.
• If the adsorption mechanism is specific we can
  develop an adsorption isotherm to determine k
  (simplest form is linear).

10
Whereas the effect of adsorption is to delay the
time to breakthrough, dispersivity imparts a
spreading or tailing effect as a result of
variable flow velocities and liquid
diffusion. Non-Adsorbing Solute ?CT/?t
-?JwCl/?x ?/?x De?Cl/?x where v Jw/? and D
De/? and since there is no adsorption ?CT/?t
?/?t(?bCa?Cl) ??/?t(Cl) -v?Cl/?x D?2Cl/?x2
0
11
Adsorbing Solute ?/?t(?bCa?Cl) -v?Cl/?x
D?2Cl/?x2 If we assume a linear adsorption
isotherm ?Ca/?t k ?Cl?t such
that (1k?b/?)?Cl/?t -v?Cl/?x D?2Cl/?x2
Defining a new varialble r (1k?b/?) r?Cl/?t
-v?Cl/?x D?2Cl/?x2
12
Methods of Study
Breakthrough Curves A steady water flow regime
is established in a soil column, and at some time
(t0), solute is suddenly added to the input.
t0
By analyzing the effluent at various times (tgtt0,
the shape of the curve showing the appearance of
solute at the outflow is characterized.
L
?
13
Pore Volume A pore volume is defined as the
amount of water which must be added in order to
move salt a given distance L (length of the
soil column). 1PV L? Piston Flow Piston
flow refers to the approximate model which
neglects all but the mass water flow component of
the solute flux. In this model, when a slug of
water is added, all the solute suddenly appears
at the distance L.
14
Breakthrough Time (tb) The breakthrough time is
defined as the time after which salt is added to
the inlet and it first appears at the outlet.
According to the piston flow model, tb L/v
L?/Jw
Piston Flow Model
1.0
The spreading effect is due to liquid diffusion
and dispersion (dispersivity).
c/c0
Actual Breakthrough
tb
15
This oversimplification applies primarily to
inert non-adsorbing solutes. Follow-up solute
free leaching of the soil column would typically
result in a more or less mirror image of solute
discharge.
Leaching Begins
1.0
c/c0
Average time to breakthrough
tb (or PV of solution)
16
The effect of adsorption on breakthough is to
delay the time of solute appearance at the column
outlet.
NonAdsorbing Solute
Adsorbing Solute
c/c0
time lag shift
tb
t
tb
17
In the case of a linear adsorption isotherm ?bCa
k?Cl (amount adsorbed) k (amount in solution)
where k may be taken as the average slope of
the breakthrough curve. Hence, the effective
velocity (v) of the solute is reduced to v
v/(1k) and tb L/v (1k)L?/K(?)
18
Depending on the reversibility of the
adsorption process, column leaching may take on a
different character.
Leaching Begins
Somewhat Irreversible
c/c0
Strongly Irreversible
tb
t
19
Example Calculation
Calculate the breakthrough time for a front of
solute moving through a vertical soil column, 100
cm in length of saturated conductivity Ks 5
cm/hr, ?s 0.50, and inflow pressure head of 10
cm.
b 10 cm
A 100 cm2
L 100 cm
K 5 cm/h
? 0.50
Js
20
In the absence of rapid and turbulent water flow,
for an inert non-adsorbing solute Js
Jw Hence, Jw -Ks?H where H1 0 and H2 10 cm
100 cm. Jw -5 cm/h (110 cm/100 cm) -5.5
cm/h v Jw/? (5.5 cm/hr)/0.50 11 cm/h tb
L/v 100 cm/11 cm/h 9.09 h
21
Example Calculation
Next, consider that there is a 1 mm diameter
wormhole extending through the column and through
which water flows according to Poiseuilles Law.
1 mm diameter wormhole
b 10 cm
A 100 cm2
L 100 cm
K 5 cm/h
? 0.50
Js
22
The liquid volume flow rate through the wormhole
is given by Q ((pR4/8?)?lg)(bL/L) 0.265
cm3/s and the average velocity is v Q/pR2 33.7
cm/s Hence, tb 2.97 s
Structural voids clearly have a dominant effect
on water and solute flow if fully conducting.
23
Effects of Cation Exchange on Breakthrough
Characteristics
Cation exchange is, in effect, an adsorption
process. As such, it can be qualitatively
described by observing what happens when a
multi-electrolyte solution is moving through a
porous medium. Consider the following scenario
Na2SO4
CaCl2
Ca Na Na Ca Na Na
Na
Na Na Na Na Na Na
Na
SO4 Cl2 Cl2 SO4 SO4 SO4
SO4 SO4 SO4 SO4 SO4 SO4
CaCl2
Na2SO4
24
2NaX Ca ? CaX 2Na
1.0
SO4
Na
C/C0
Cl
Ca
t
tb L/v
25
For more complicated systems we must also
consider the effects of diffusion and dispersion
which would tend to flatten the breakthrough
characteristics and to contribute to a
tailing-off effect as well. Similar effects are
observed in aggregated soil where water tends to
flow through restricted channels by-passing part
of the porous medium. Solutes then move into and
out of these stagnant regions only by
diffusion, thus creating a tailing effect.
26
1.0
Compacted
C/C0
Aggregated
t
27
Profile Distribution Measurements
This technique generally uses a sectional soil
column where the solute is initially applied to
the top of an initially dry soil column at a rate
ltKs. For a non-adsorbing solute, C/C0 1.0 in
each section down to the wetting front.
Wetting Front
C0
C
1.0
C
C
C/C0
C
C
Wetting Front
C
Z
28
For an adsorbing solute the distribution profile
is quite different. Here the amount on the
adsorbed phase is initially high at the point of
inflow, then decreases with depth at a rate
dependent on the distribution coefficient k.
The greater the distribution coefficient the more
strongly sorbed, the higher the amount on the
sorbed phase in the initial sections of the soil
column and the more rapid the decline in amount
adsorbed with depth.
k1gtk2gtk3
k2
Ca (µg/g)
k3
k1
Z
29
Upon leaching, the long-term effect is to more
evenly distribute the solute throughout the
entire soil column. The rate at which one can
attain this effect also is dependent upon the
distribution coefficient. Characteristically, a
weakly sorbed solute will move downward in a
wave-like fashion rapidly attaining a
more-or-less uniform distribution with depth.
For more strongly sorbed solutes, the wave will
move much more slowly with a prominent peak of
maximum concentration at depth over a much longer
time period.
30
k2 gtgtgt k1
Ca (µg/g)
C0
C0
C1
C2
C1
C3
Z
k2
k1
C2