ASPECTS OF LIQUID AND VAPOR FLOW IN RETORTED OTT. SHAT.F.
by '
Glenn 0. Brown and David B. McWhorter :
Agricultural and Chemical Engineering
Colorado State University
Fort Collins, Colorado 80523 '
Presented at the 1986 Eastern Oil Shale Symposium,
Lexington, Kentucky
November 19-21, 1986
Notice
The development of the information in this document has been funded
wholly or in part by the United States Environmental Protection Agency under
Cooperative Agreement CR812225 to Colorado State University. It has been
subject to the Agency's peer and administrative review, arid it has been
approved for publication. j
-------
ASPECTS OF LIQUID AND VAPOR FLOW IN RETORTED OIL SHALE
by
Glenn 0. Brown and David B. McWhorter
Agricultural and Chemical Engineering
Colorado State University
Fort Collins, Colorado 80523
Abstract '
Reclamation and impact analysis of retorted oil shale piles will!
require prediction of water and solute transport rates over the
entire water content range down to and including the relatively
dry region. Experimental measurements of water transport
coefficients in relatively dry oil shales have brought forward
long-standing questions concerning the mechanics of combined
liquid/vapor flow. In an attempt to ensure proper interpretation1
of experimental data, a new analytical solution has been obtained
for combined liquid and vapor flow with solute transport. The
solution shows that the relative magnitudes of the separate
transport coefficients produce . many of the flow, features seen in
experimental data, and significant liquid transport can occur in
regions without apparent solute transport. This development is
new and represents a significant addition to understanding solute
transport. As such its methods and results can be applied to
other problems in multiple phase transport and to materials such
as high volume mining wastes and some hazardous waste disposal
sites. The paper shows that an earlier conclusion reached by the
authors, that a critical water content exists in retorted oil!
shale below which solute transport ceases, is unjustified.
Introduction
Research into the movement of water and
dissolved constituents in retorted oil shale has
been motivated by the adverse impacts that can be
anticipated if leachate is generated and left
uncontrolled in the large disposal piles planned.
Accurate prediction of water and solute movement
through disposal piles will require measurements
of both water and solute transport over the entire
range of solution contents, down to and including
the relatively dry region. Dry region transport
phenomena are of concern for two reasons. First,
retorting produces a shale that is oven dried, and
it is expected that only enough water to control
dust and aid compaction will be added before
placement, and much of that may rapidly be lost to
evaporation. Thus, the pile initial condition
before infiltration of precipitation will probably
be relatively dry, and leaching may be strongly
influenced by transport processes near the initial
condition. Second, an earlier study by Colder
Associates1 has proposed that piles be designed to
eliminate pile leachate by evaporating all excess
infiltration. Such evaporation would of course
require the portion of the pile near the
evaporating interface to remain quite dry.
In a previous study ;by the authors2'3 the
hydraulic and solute transport characteristics of
Lurgi retorted oil shale; were measured from'
saturation (0.49 cm3/cm3) ; down to almost zero.
solution content. These measurements were carried
out by use of a dual source; gamma ray system4'5'8
which can simultaneously measure water content and
solute concentrations during column sorption
experiments. The experiments showed, as expected
from earlier studies7'8'8'1*"11, that water vapor
diffusion becomes the dominant water transport
mechanism in the retorted material at low solution
contents. Empirical examination of the data
indicated that the liquid water and solute
transport ceased at 0.066 cm3/cm3 volumetric
solution content and that all water transport was
by vapor diffusion. :
As soon as these results were obtained,
questions arose as to the accuracy of the data
Interpretation. More specifically, it was
questioned if the solute transport observed was a
function more of the test conditions, and if
liquid transport actually ceased at a finite
solution content. Examination of the literature
showed that a considerable number of questions
remained about the interactions of combined
-------
liquid/vapor flow12'13'1* and unexplained solute
transport phenomena which occur near low water
contentsl*.
'.[n an attempt to ensure proper interpretation
of the experiment data a theoretical model of two-
phase water and solute transport in a pore was
developed. The model allows analytical numerical
solutions to the unsteady sorption of solution.
The model solutions are able to produce many of
the flow features seen in the experimental data.
From the results it can be concluded that the
relative magnitude of the separate liquid and
vapor transport coefficients produce the solute
profiles seen, and that significant liquid
transport can occur in a region without apparent
solute transport.
Phenomena of Interest
In this paper two experimentally observed
phenomena will be examined. These phenomena have
been either unexplained or explained by thresholds
of transport processes which have not been
rigorously proven to exist.
Critical Water Content
Several experimenters have observed or
speculated on the existence of a "critical" water
content below which liquid conduction of solute
ceases. From sorption or solute diffusion in dry
media, Grismer et al.6, McWhorter and Brown2'3,
Rose1", and Porter et al.ls speculated that the
.liquid phase becomes discontinuous at low water
contents. From sorption at high solution
concents, Krupp et al.16, Laryea et al.17, and
Robin et al.18 postulated the existence of an
immobile layer of water near the solid surface,
while Van Schaik and Kemper19, Smiles and
Gardiner20, and Bond et al.21 concluded that
significant anion exclusion effects were
occurring.
Most of the above conclusions were based on
unsteady infiltration of solution. Typical
results of such an experiment are taken from
McWhorter and Brown2 and are shown in Fig. 1. In
this experiment Nal solution was injected into a
completely dry column. Water and l" were tracked
by dual source gamma ray. After significant
inflow the water was observed to develop a
gradually varying profile, but the solute was
restricted to water contents above 0.06
(cm3/cm3).
Solute Concentration Humps
Smiles et al.1* observed, as shown in Fig. 2.
that, on infiltrating soil with a low initial
solution content, there was not perfect piston-
like displacement of the resident fluid, as would
be expected. A small increase or hump in the
resident solute concentration was formed ahead of
the wetting front where the solute concentration
was increased above the initial. This hump could
not be explained, as no single convectlve or
dispersive process can explain an increase in
solute concentration in a region of increasing
solution content. The same hump can be seen in
the data but was not noted by Smiles and
Phillip22.
Likewise McWhorter and Brown2'3 observed
large Increases in the invading solute
concentration behind the wetting front:. They
attributed the hump to evaporation.
The theoretical model;, which is detailed in
the Appendix, will be applied to determine if these'
two phenome a can be explained by the two-phase
transport processes without the aid of transport,
thresholds such as a critical water content. :
Method of Analysis
The phenomena of interest will be explained
by appeal to a theoretical solution of liquid and
vapor flow with solute transport in a porous media
of known hydraulic properties. To obtain the
theoretical solution new equations of water and!
solute transport have been developed. The
derivation of these transport equations is,
secondary to the purpqse of ' this paper.
Therefore, only the assumptions and final resultsL
will be presented here, while a brief derivation
is presented in the Appendix.
Model Assumptions
i
The developed model assumes one-dimensional
horizontal adsorption of !liquid solute into a
nearly dry column of porous media as typical with
most laboratory experiments of interest here.'
Solution is transported by gradients of hydraulic
pressure, but it is assumed that the water
characteristic can be used to transform Darcy's
conductivity to a diffusivity where the liquid
flow is driven by gradients .of water content. The
vapor transport is assumed to be driven by vapor
diffusion but again it is ^assumed that the vapor
adsorption isotherm can be used to transform the
driving gradient to the [liquid water content.
These well established ! transformations are
routinely used by investigators in this field.
The liquid and vapor phases are assumed to be in
instantaneous local equilibrium; i.e., where.
kinetic effects of evaporation and condensation
are negligible. ' ;
Solutes are present in both the initial
solution and the invading solution, but at
different concentrations. ; Solute is transported
only in the liquid phase. The solute is not
adsorbed by the solid phase and moves with the
mean velocity of the liquid phase. This last
assumption implies miscible displacement; i.e.
the initial and invading - solutes are mixed by
only small-scale dispersive processes.
column has a uniform, low initial
content. At thd start and during the
The
solution
remainder of the adsorption the column inlet is
maintained at a constant ^solution content. The
invading solution enters the column with a
constant concentration. , This is a slightly
different boundary condition than used in some
experiments . 2 ' s
Diffustvtties , '
Using the concept of liquid and vapor
diffusivities it can be istated that the total
water flux expressed as liquid volume is
or (la)
- ' V<>
-------
o
U
X (cm)
Figure 1. Solution Content (9) and Relative Invading Solute Concentration
(C/CQ) Versus Column Position (x), After Various Times. These
Results Are from an Unsteady, Constant-Flux Experiment Performed
by McWhorter and Brown2 ,
-------
i
i—i—i—i—i—i
i—i—i—i—i—i—i—i—i
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
x/f
Figure 2. Solution Content (0) and Relative Resident Solute Concentration
(c/cn) Versus Relative Column Position (x/t1/2). These Results
Are From an Unsteady Constant Concentration Experiment Performed
by Smiles et al.1* (Curves Are Best Fit to Reported Data)
-------
where
8.
- total equivalent liquid
volumetric flux,
DVW - vapor diffusivity,
C^ - concentration of water in
solution,
dp^/dff - slope of vapor adsorption
isotherm,
D^(0) - liquid diffusivity,
t> — solution content,
Dt(*) - total water diffusivity, and
x - horizontal position.
As noted both D, and D are functions of
With Eq. (la) it is clear tKat the total water
Note that the vapor diffusivity is greatest at the
lower solution contents,1 while the liquid
diffusivity dominates 'as solution content
increases. Nowhere in the range of interest is
either transport coefficient zero. The total
diffusivity has the characteristic double maximum
of most porous media. these functions were
obtained from a theoretical model for a set of
plate capillaries, but for this analysis any set
of values could be used as long as they had the
same general shape. A different set of transport
coefficients would only change the numerical
results, not the general conclusions.
From the Appendix, the general equation for
the water flow is !
A M
2 dA
where
A - x/t
dA '•"t dA
1/2
(2)
and
flow is the sum of the two components , bulk liquid
flow and vapor diffusion. The fraction of water
transport in either phase at a particular position
is simply the ratio of the phase diffusivity to
the total.
Assumed Diffusivities
Values of the individual diffusivities must
be assumed to achieve a flow solution. Figure 3
presents the assumed values . In the figure the
horizontal axis is the solution volumetric
content, 6, over the range of interest 0.02 to
0.10, while the vertical axis is the diffusivity.
0.0005
t - time from start of sorption.
Equation (2) is subject to |
ff(A-O) - ff ; a constant, and
0(A«>) - g • a constant.
(3)
(4)
The use of the transformed Boltzman coordinate,
A=x/t1/2 is possible due to the boundary
conditions used, and "normalizes" the solution.
That is, it allows the solution content profiles
from different times to be collapsed onto one
another. For ease of reference, A can "
0.02
Figure 3. Assumed Values of Liquid (D^), Vapor (D ), and Total (D )
Diffusivities. Note That Vapor Has a Maximum at Low Solution
Contents; The Liquid, at High Solution Contents; and the Total,
a Minimum at an Intermediate Content
-------
thought of as a horizontal scale which stretches
uniformaly (like a rubber band) with time.
The evaporation and condensation along the
column can be determined by the relation:
d MS} A d£_
1 dfl 2 J dA ' (:>)
where e - the phase transfer, and
A(0) - transport function.
The transport function is calculated during
the solution procedure and is a function of the
diffusivities and boundary conditions. The phase
transfer variable e is positive for condensation
and negative for evaporation.
The solute concentration along the column can
be determined by solution of the following
differential equation:
C — inlet concentration of invading
solute, and
!
C -• initial ; concentration of
resident solute.
The left-hand side of Eq. (6) represents
phase transfer and convective effects, while the
right-hand side represents dispersion.
Results and Discussion
CCA).
[A(«> - f ]
subj ect to :
C(X-O) - C
C(A— ) - C
and
where
C(A) — solute concentration,
D (0) - coefficient
dispersion,
of
, (6)
(7)
(8)
solute
Flow Solution
Figure 4 presents the solution of Eq. (2) for ,
solution content profiles i for the initial and
boundary conditions given i -by Eqs . (7 ) and ( 8 ) .
In Fig. 4 the vertical ax:is is solution content!
and the horizontal axis the normalized distance,
x/t1/2. As could be expected the solution content
falls from the boundary 'condition at the inlet ,
(x/t1/2 - 0) , to the initial solution content at
about x/t1/2 - 0.07. 'As in all sorption
experiments where vapor transport plays a role, a,
"vapor nose" elongates the p'rofile. This nose is,
due to the secondary diffusivity naTimum at the
low solution contents as Ishown in Fig. 3. This
theoretical profile is similar and consistent with
experimentally observed profiles.
Next, solving Eq. (5) for the evaporation and'
condensation of water vapor along the column
provides the data plotted! in Fig. 5. ]in Fig. 5
the vertical axis is !the normalized phase
0.02
0.04
x/f'/2 (cm/s'/2)
0.06
0.08
Figure 4. Calculated Solution Content ($) Versus Relative Position
(x/t1/2)
-------
0.5
0.4 -
0.3 -
0.2 -
0.1 -
-0.1 -
-0.2 -
-0.3 -
-0.4 -
-0.5
T
T
0.02
0.04
A= x/t'/2(cm/s'/2)
—I—
0.06
O.OB
Figure 5. Calculated Evaporation and Condensation (e) Versus Relative
Position (x/t1'2) |
transfer, e, with condensation positive and
evaporation negative. Again the horizontal axis
is Che normalized distance x/t1/2. The point
labeled A^, where e is zero at about 0.03 cm/s1'2,
separates the regions of evaporation and
condensation. In consideration of solute
transport it can be stated that the evaporation
behind \^ will tend to increase the solute
concentration, while the condensation ahead of the
point will tend to reduce solute concentrations.
The maximum evaporation rate occurs at about
0.021 cm/s1'2, while the maximum condensation
occurs at about 0.044 cm/s1'2.
Finally, Eq. (6) must be solved to obtain the
solute concentration along the column. For
illustrative reasons Eq. (6) is solved assuming
that the dispersive term on the right-hand side is
zero. This allows the clear presentation of the
convective and evaporation effects without the
blurring caused by dispersion. Figure 6 presents
these results. In Fig. 6 the vertical axis is the
normalized solute contents C/C and C/C . The
invad.ing solution has only advanced to a position.
labeled A , of about 0.018 cm/s1/2. From the
inlet the invading solution concentration, C/C
rises from its injection value, 1, to a maximum o?
about 1.5 at A . This increase in concentration
is due to both evaporation and convective effects.
Possibly of greater interest is the
concentration of the resident fluid, C/C . While
condensation at the far end ahead of An dilutes
the resident solution, as could be expected,
convective and evaporative effects between A
and Ag have the net effect of first increasing
the concentration above 1 and then reducing it to
about 0.8. If dispersion were added, these
concentration profiles would be blurred by the
smoothing effect of the dispersion, but the trend
would still be present. I
Explanation of Phenomena of Interest
We can now explain the two phenomena of
interest. First re-examine Fig. 6. Looking only
at the invading solute concentration profile
(McUhorter and Brown2'3) it could be argued that a
"critical" water content occurred at position A
which (from the solution content profile, Fig. H)
would be estimated to have a value of about 0.06.
Thus it would be concluded that liquid diffusivity
dropped to zero at that point. But this, of
course, would be wrong. From Fig. 3 it can be
seen that the liquid diffusivity is about four
times the vapor value at that solution content.
The solute is limited to the region behind A
simply because that position is as far ai
convection will transport it during a given
infiltration time. Also note that A is not an
explicit function of the evaporation The
evaporation front. A is well ahead of A and
AS Is also slishtly behind the position o!'
maximum evaporation. '
Now examine the resident solution content
profile. The increase in' solute concentration
ahead of Ag is similar to the unexplained
observation of Smiles et al.l<. Thus their data
can be explained as the effects of combined liquid
and vapor flow which they did not consider.
-------
•a
o
0.08
Figure 6.
Calculated Solute Concentration (C/C and C/C ) Versus Relative
o n |
Position (x/t'/2)
Finally note that in this case the minimum
resident solution content occurs not where
condensation is a maximum but instead where the
-phases transfer is zero. This point demonstrates
clearly that the solute concentration is a
function of both convection and phase transfer,
and that at some positions convection alone can
produce concentration changes.
Conclusions
The development of the information in this
document has been funded wholly or in part by the
United States Environmental Protection Agency
under Cooperative Agreement CR812225 to Colorado
State University. It has been subject to.the
Agency's peer and administrative review, and it
has been approved for publication.
This analysis has shown that, in relatively
dry retorted oil shale, proper interpretation of
experimental solute transport data requires
careful theoretical considerations. The vapor
transport of water, while small when compared to
liquid flow, nevertheless induces convective
transport processes in the liquid phase which can
increase or dilute the concentration of solutes.
Without a clear understanding of the effects of
evaporation and condensation on solute convection
it is easy to misinterpret experimental data.
Additional analyses of liquid and vapor flow
with solute transport have been performed. These
included the constant concentration with
dispersion, the constant flux boundary with and
without dispersion, and a case with actual
measured oil shale liquid and vapor diffusivities.
Results of these analyses will be published at a
later date.
References
1. Colder Associates, 1983. Movement of Water.
through a Processed Oil Shale Pile. Report
to AMOCO/Rio Blanco Dili Shale Co., March.
HcWhorter, D.B., and Brown, G.O. , 1985.
Adsorption and Flow of ] Water in Nearly Dry
Lurgi Retorted Oil Shali. Unpublished report :
prepared for Standard!Oil Company, AMOCO
Research Center, Colorado State University,
February. :
McUhorter, D.B., and Brown, G.O., 1985.
Liquid and Vapor Transport Coefficients for
Retorted Oil Shale. Proceedings of the 18th
Oil Shale Symposium. Colorado School of
Mines Press, Colden, Colorado.
-------
4. Grlsmer, M.E., 1984. Water and Salt Movement
In Relatively Dry Soils. Ph.D. Dissertation,
Department of Agricultural and Chemical
Engineering, Colorado State University, Fort
Collins, Colorado.
5. Grismer, M.E., McWhorter, D.B., and Klute,
A., 1986. Determination of Diffusivity and
Hydraulic Conductivity in Soils at Low Water
Contents from Nondestructive Transient Flow
Observations. Soil Science, 141:10-19.
6. Grismer, M.E., McWhorter, D.B., and Klute,
A., 1986. Monitoring Water and Salt Movement
in Soils at Low Solution Contents. Soil
Science, 141:163-171.
7. Jackson, R.D., 1964a. Water Vapor Diffusion
in Relatively Dry Soil: I. Theoretical
Considerations and Sorption Experiments.
Soil Sci. Soc. Am. Proc. 28:172-176.
8. Jackson, R.D., 1964b. Water Vapor Diffusion
in Relatively Dry Soil: II. Desorption
Experiments. Soil Sci. Soc. Am. Proc.
28:464-466.
9.
Jackson, R.D.,
in Relatively
Experiments.
28:467-470.
1964c. Water Vapor Diffusion
Dry Soil: III. Steady-State
Soil Sci. Soc. Am. Proc.
10. Rose, D.A., 1963a. Water Movement in Porous
Materials: I. Isothermal Vapor Transfer.
Brit. J. App. Phys. 14:256-262.
11. Rose, D.A., 1963b. Water Movement in Porous
Materials: II. The Separation of the
Components of Water Movement. Brit. J. App.
Phys. 14:491-496.
12. Cass, A., Cambell, G.S., and Jones, T.L.,
1984. Enhancement of Thermal Water Vapor
Diffusion in soil. Soil Sci. Soc. Am. J.
48:25-32.
13. Jurry, W.A., and Lety, J. Jr., 1979. Water
Vapor Movement in Soil: Reconciliation of
Theory and Experiment. Soil Sci. Soc. Am. J.
43:823-827.
14. Smiles, D.E., Phillip, J.R., Knight, J.H.,
and Elrick, D.E., 1978. Hydrodynamic
Disperion During Adsorption of Water by Soil.
Soil Sci. Soc. Am. J. 44:229-234.
15. Porter, L.K., Kemper, W. D., Jackson, R.D.,
and Stewart, B.A., 1960. Chloride Diffusion
in Soils as Influenced by Moisture Content.
Soil Sci. Soc. Am. Proc. 24:460-463.
16. Krupp. H.K., Biggar, J.W., and Nielsen, D.R.,
1972. Relative Flow Rates of Salt and Water
in Soil. Soil Sci. Soc. Am. Proc 36-412-
417.
17. Laryea, K.B., Elick, D.E., and Robin, M.J.L.,
1982. Hydrodynamic Dispersion Involving
Cationic Adsorption During Unsaturated,
Transient Water Flow in Soil. Soil Sci. Soc.
Am. J. 46:667-671.
18. Robin, M.J.L., Laryea, K.£., and Elick, D.E.,
1983. Hydrodynamic ; Dispersion During
Adsorption of Water by Soil. J. Hydrol
65:333-348.
19. Van Schaik, J.C., and Kemper, W.D., 1966.
Chloride Diffusion in Clay-Water Systems.
Soil Sci. Soc. Am. Proc. !30:22-25.
20. Smiles, D.E., and Gardiner, B.N., 1982.
Hydrodynamic Dispersion During Unsteady,
Unsaturated Water Flow in Clay Soil. Soil
Sci. Soc. Am. J. 46:9-14.
21. Bond, W.J., Gardiner, B.N., and Smiles, D.E., ,
1982. Constant-Flux '' Adsorption of a
Tritiated Calcium Chloride Solution by a Clay
Soil with Anion Exclusion. Soil Sci. Soc.
Am. J. 46:1133-1137. I
22. Smiles, D.E., and Phillip, J.R.,. 1978.
Solute Transport During Adsorption of Water
by Soil: Laboratory Studies and Their
Practical Implications. Soil Sci. Soc. Am.
J. 42:537-544. ;
23. McWhorter, D.B., 1971. Infiltration Affected
by Flow of Air. Hydrology Papers, Colorado
State University, Fort Co'llins, Colorado.
24. Phillip, J.R., 1973. On Solving the
Unsaturated Flow Equation: I. The Flux-
Concentration Relation. I Soil Sci. 116:328-
335.
25. Phillip, J.R., and Knight, J.H., 1974. On
Solving the Unsaturated Flow Equation: III
New Quasi-Analytical Technique. Soil Sci.
117:1-13.
Appendix
i
Theoretical Development (
i
This appendix presents a development of the
equations of water, vapor, arid solute transport in
porous media. This development has been kept as
brief as possible and is limited to the case of
constant concentration adsqrption of solution.
Solutions for desorptlon and constant flux
sorption have been obtained by similar methods,
but are beyond the scope of this paper. A
complete development of this theory will be
published at a later date. ,
This development is new and represents a
significant addition to 'understanding solute
transport. As such its methods and results can be
applied to other problems in multiple phase
transport.
For one-dimensional sorption of water vapor
the mass flux of vapor, F , is given by:
F - -D
v dff
as.
3x
(A-l)
where
DV - vapor diffusivity,
p — vapor density, •
8 - volumetric water content, and
x — horizontal distance.
-------
The vapor diffusivity is a function of the
free space molecular diffusion coefficient, the
tortuosity, the porosity, and the solution
content. The ratio dp/d.8 is used to transform
the driving gradient for diffusion from vapor
concentration to water content. Its use assumes
that a unique isothermal equilibrium exists
between the vapor and liquid phases. Continuity
of vapor phase on a control volume of porous media
yields:
•fe
-D E
3x v dfl 3x
(A-2)
wheres
4> — porosity,
E — mass transfer to the liquid phase, and
t - time.
The source term, E, in general is assumed to be a
function of x and t.
For one-dimensional horizontal sorption of
liquid solution, the mass flux of water, F. is
given by the relation:
ft a
F2 - «t C* - -Cu D* fx" ' (A'3)
where
q. — liquid volumetric flux,
C — water concentration in solution, and
D^ - liquid diffusivity.
The liquid diffusivity is related to the hydraulic
conductivity and is a strong function of solution
content. Continuity of liquid phase on a control
volume of porous media yields:
- -C + E
(A-4)
Adding Eqs. (A-2) and (A-4) yields the total
water continuity:
Note in Eq. (A-5) the source terms have cancelled.
Using typical vapor adsorption isotherms, it is
simple to show that 3/3t(^-0)p « 3/3t(C 6) , and
thus, the first term can be dropped. Likewise for
dilute solutions, it can be shown that C =
constant — 1. With these assumptions Eq. (A-5)
becomes :
3JL_3_ (v +
3t 3x v C &B
3x
(A-6)
Now a combined diffusivity, D , is defined as:
V V
t ~ C dB !L '
w
Equations (A-6) and
equation of water flow:
3 d '3 fr. o 9 ,
8t - 3x (Dt 3x> ' °r
^ - - 3- q
3t. 3x qt '
(A-7)
(A-7) yield a simple
(A-8a)
(A-8b)
where qc is the total' water volunetric flux
expressed as solution. For constant concentration
sorption, the initial and boundary conditions are:
ff(«,t) - 6 , and
9(0,t) - So .
Flow Solution Method
(A-9)
(A-10)
(A-ll)
McWhorter23 and Phillip24 showed that
Eq. (A-8b) could be solved by a method of
fractional flow where the flux at any *(x,t) is
defined as a fraction, F(0,t), of the influx at
x-0, or: i
qt(x,t)
qo(t)|,
(A-12).
where q (t) is the inlet volumetric flux.
Phillip and Knight25 proved that, for constant
concentration boundary condition (Eq. A-ll), F was
not a function of time. Thus, an exact solution
for' the flow can be found through a semi-
analytical calculation of F(0). For this boundary
condition the inlet flux is'given by:
qo(t) - S/t1/2, ; (A-13)'
where S is the soil sorptivity constant
calculated from the fractional analysis.
With use of Eqs. (A-12) and (A-13), Eq. (A-
8b) is simplified further to:
3t
_ s_ a_
1/2 3x
(A-14)
For the boundary conditions of Eqs. (A-9),
(A-10), and (A-ll) the Boltzman transformation
will ease the solution of Eq. (A-14). A transform
coordinate is defined as: :
A - x/t
1/2
(A-15)
Making this substitution into Eqs. (A-9), (A-10),
(A-ll), and (A-14) yields a differential equation
of flow:
-fdT-s5I^> •
which is equivalent to:
A AS. d_ d£
" 2 dA ~ dA t dA '
Eq. (A-16) is subject to:
$ (•*>) — 9 , and
(A-16a)
(A-16b)|
(A-17)
(A-18)
With Eq. (A-16) and' the fractional flow
analysis of F(0), the flow profile at any time and
position can be determined for given diffusivity
functions D and D; or D .
Of importance to this appendix is the.
determination of the seepage velocity, v(0.), of a
particular solution content] 6,. Using Eq^ (A-14)
and the definition of substantial derivatives, it
can be shown that: '
f
(A-19)
10
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Liquid Phase Analysis
With the total flow analysis, the water
transport in either phase, the evaporation and the
condensation can be determined. First, since all
fluxes (liquid, vapor, and total) are driven by
dS/Bx, they can be related by the ratio of
diffusivities:
With Eqs. (A-12) and (A-13):
TN
,-V2 „
where
(A-20)
(A-21a)
(A-21b)
A(0) is a transport function equal to
Returning to the liquid continuity, Eq.
Applying the chain rule to Eq. (A-29) and'
regrouping produce: i H '
2
s dA'
Now it can be noted that the first term can be
replaced with phase transfer term e(0) as given'
by Eq. (A-24):
eC + (A
si - si (»Ds
(A-33)
This is the basic equation of solute transport
when water is transported! in vapor and liquid
phases, subject to the boundary conditions of Eqs
(A-30) and (A-31). :
(A-4)
Again, assuming C^- 1, making the Boltzman
transform, and substituting Eq. (A-21b) yields:
(A-22)
A d0 dA
2 dA + dA - E (X'C) «=
Here E' is E/C^. Now note that the left-hand
side is only a function of A. It can be stated
from inspection that:
E'(x,t) -
(A-23)
Substituting Eq. (A-23) into (A-22) yields a
relationship for the evaporation/condensation:
e^ ~ - 2 dA
Solute Transport
dA
(A-24)
With the known flow conditions it is now
possible to predict the solute transport.
Assuming solute is transported by convection and
Fickian dispersion in the liquid phase a control
volume analysis yields the equation of solute
motion:
at 3x
subject to:
C(x,0) - C
C(o,t) - C
o
C(co,t) - C
(en
(SD
(A-25)
(A-26)
(A-27)
(A-28)
whens C is the solute concentration, and D is
the coefficient of solute dispersion. Againf the
Boltzman transform is applied and q. is replaced
by Eq. (A-21b) to yield:
^A dflfi d_ _ d_ dC
2 dA + dA (CA) dA <•* s dA
subject to:
C(o) - C
o
C(») - C
and
(A-29)
(A-30)
(A-31)
11
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