Modeling interphase mass flux between PCE and permanganate in the
presence of surfactants
by
Mark Julian
Clarkson University
Modeling interphase mass flux between PCE and permanganate in the presence of
surfactants
A Thesis Proposal by
Mark Julian
Department of Chemical and Biomolecular Engineering
Mentor: Michelle Crimi
March 2011
2 Abstract
This proposal describes the modeling of perchloroethylene mass transfer with oxidation
by permanganate in the presence of sodium dioctyl sulfosuccinate. Through the use of an existing
reactive diffusion model, theoretical mass flux values based on experimentally obtained
parameters are to be calculated. It is hypothesized that the existing model can adequately describe
the reactive mass transfer in the presence of surfactant molecules through adjustment of the PCE
aqueous solubility limit, diffusion coefficient, and second order reaction rate constant. PCE
aqueous/surfactant solubility is to be found using batch mixing and gas chromatography analysis.
Diffusion coefficients for both PCE and permanganate are to be obtained using membrane
diffusion cells. Reaction rate constants will be found by conducting kinetic experiments at various
surfactant concentrations. The resulting mass flux values obtained from the theoretical model will
be compared to those found experimentally. Through this comparison it will be possible to
determine if the reactive diffusion model can adequately be extended to describe the mass transfer
associated with in situ chemical oxidation coupled with surfactant enhanced aquifer remediation.
3 Introduction
Both in situ chemical oxidation (ISCO) and surfactant enhanced aquifer remediation
(SEAR) are groundwater remediation techniques used to remove organic pollutants from the
environment. As with most “pump and treat” efforts, the aim of these practices is to utilize the
fundamental knowledge of aqueous chemistry to remove (in the case of surfactant solubilization)
or destroy (in the case of in situ oxidation) environmentally dangerous chemicals. The biggest
obstacle with such techniques is the fact that the organic pollutants are often immiscible and
much denser than the fluid used to treat them, making removal extremely difficult. Within recent
years, many efforts have been made to combine these two techniques in an attempt to
simultaneously remove and destroy organic pollutants from known source zones. While both
remediation methods have been studied extensively on their own, a new approach that combines
the two techniques is still in its infancy. Studies that combine oxidation and surfactant
enhancement have been conducted, and are mentioned in this work. However, there still remains
a significant amount of question as to how exactly the addition of surfactant molecules affect the
PCE mass flux into the aqueous phase, and if existing models can accurately describe these
processes. In order to begin to answer these questions, the commonalities between surfactant
chemistry, organic to aqueous phase dissolution, and oxidation processes must be understood.
The proposed research aims to further this understanding by considering the oxidation of
the contaminant perchloroethylene (PCE) by permanganate ions in the presence of the surfactant
sodium dioctyl sulfosuccinate (Aerosol OT). Specifically, the project attempts to determine how
the addition of surfactants alters the oxidation processes due to the increased dissolution rate of
PCE into the aqueous phase. To do so, reactive solutions at various surfactant and permanganate
concentrations will be studied. For each system analyzed, the molecular diffusion and aqueous
solubility of the PCE will be determined. In addition, the second order reaction rate constant for
the PCE-permanganate oxidation will be measured at various surfactant concentrations. Through
the use of these parameters and a model that analyzes simultaneous chemical reaction and
diffusion, a mass flux for the PCE into the aqueous phase can be determined. By comparing the
mass flux values from the model to those obtained experimentally, the validity of the model can
be assessed. These results will be useful for understanding ISCO and SEAR related processes,
and will provide fundamental data for the determination of optimum oxidant and surfactant
combinations.
Background
The Agency for Toxic Substances and Disease Registry ranked perchloroethylene (PCE)
rd
33 in it’s 2007 listing of hazardous substances (ATSDR 2007). Today, PCE continues to be one
of many volatile organic compounds labeled as a “contaminant of concern.” In addition to the
adverse health effects associated with PCE, chlorinated solvents are considered to be particularly
dangerous to the environment, and are often noted as dense non-aqueous phase liquids (DNAPLs)
because of their relatively high density and immiscibility with water. DNAPLs pose a unique
challenge for classic “pump and treat” remediation techniques. Due to their low aqueous
solubility, high density, and low viscosity, DNAPLs tend to sink through the vadose and saturated
groundwater zones. This often results in DNAPL pools, or residuals, above confining layers and
in areas with low soil permeability. These pools are highly undesirable since the residual DNAPL
aqueous solubility tends to be high enough to allow contamination of aquifers at a dangerous
level, yet low enough to avoid dissolution in water used for remediation purposes (Johnson and
Pankow 1992). As a response to the failure of classic pump and treat methods to overcome this
dilemma, in situ chemical oxidation (ISCO) is one technical approach that has been developed
and refined over the past twenty years.
Through the use of an oxidative permanganate solution as a remediation fluid,
degradation of the DNAPL can be achieved through reaction within the groundwater. The
4 oxidation reaction between PCE and the aqueous phase permanganate ion can be summarized in
Equation 1.
!
!
3C! Cl! (l) + 4MnO!
! (aq) + 4H! O(l) → 4MnO! (s) + 6CO! (g) + 8H (aq) + 12Cl (aq)
(3.5 < pH < 12)
Such a reaction is desirable within the aquifer as the pollutant is oxidized to carbon dioxide gas at
pH values greater than 3. Through kinetic studies of the reaction, it has been concluded that the
oxidation is second order overall and first order with respect to both PCE and permanganate (Yan
and Schwartz 1999). ISCO field studies and laboratory tests have been studied extensively with
the realization that dissolution mass transfer from the DNAPL pool to the reactive aqueous phase
is both enhanced by and a parallel process to the chemical oxidation (Schnarr et al. 1998).
Consequently, one of the greatest limitations to successful ISCO implementation is the limited
mass transfer of the PCE to the aqueous phase where oxidation occurs. Thus, improvements to in
situ oxidation technology may be possible through the understanding and enhancement of
DNAPL mass transfer (Petri et al. 2008).
The mass transfer of PCE from the organic to aqueous phase can be described through
the stagnant-film model developed by Sherwood in 1975 (Heiderscheidt 2005). Through this
approach the mass flux expression can be represented as a linear driving force model (Equation
2).
! = !(!! − !)
Eq. 1 Eq. 2 Alternatively, the mass flux at the interface of the stagnant film can be calculated according to
Fick’s first law (Equation 3).
!"
! = −! !"
!!!
Where J is the mass flux of the DNAPL, K is the mass transfer coefficient, D is the DNAPL’s
aqueous diffusion coefficient, C is the concentration of the DNAPL in the aqueous phase, Cs is
the aqueous solubility of the DNAPL in the aqueous phase, and x is the distance from the
DNAPL/aqueous phase interphase (Siegrist et al. 2001). A conceptual model has been proposed
(Urynowicz and Siegrist 2005) to describe the concentrations of both the PCE and permanganate
ion across a “stagnant film boundary layer.” This model is shown in Figure 1 in which the origin
is set as the liquid/liquid phase interphase.
Figure 1 Theoretical Concentration Profiles across the Stagnant Film Reactive Boundary Layer
While the term “stagnant film boundary layer” is used in the literature, it should not be confused
with Sherwood’s conceptual stagnant film or the concentration boundary layer associated with
fluid flow (Chrysikopoulos et al. 2003). In addition, the term “stagnant film” is somewhat
misleading, as convective transport arising from density changes will cause slight motion
(Cussler 1997). These convective currents are however ignored in the proposed analysis. Thus, in
order to avoid any misconception with Sherwood’s stagnant film or the boundary layers
associated with flow, this region will be referred to as the “diffusive layer.”
5 Eq. 3 Through a mass balance across a differential shell within the diffusive layer, the
following system of nonlinear equations can be derived to model both the simultaneous diffusion
and oxidation processes (Cussler 1997) for PCE (Equation 4)
!!
! ! !!
!! !
− 3!!! !! = 0
Eq. 4 and for the permanganate ion, (Equation 5)
! ! !!
!!
− 4!!! !! = 0
!! !
-
Eq. 5 where the subscripts A and B refer to PCE and MnO4 respectfully. In this model both DA and DB
are diffusion coefficients, CA and CB denote species concentrations at a distance x from the liquidliquid phase interphase, and k denotes the second order reaction rate constant for the PCE/MnO4oxidation. Boundary conditions for this system of equations include:
!! 0 = !! ,
!!!
!"
! = 0, !! ! = !! ,
!!!
!"
0 = 0.
The choice for the first boundary condition is made with the assumption that the PCE is in
thermodynamic equilibrium with the aqueous phase; making the concentration of the PCE at the
phase interface it’s aqueous solubility. The assumptions used for the second and fourth boundary
conditions are made to ensure that a constant PCE concentration is maintained once the bulk
aqueous phase is reached (this value is typically zero), and that no permanganate enters the
DNAPL phase. For the third boundary condition, the concentration of the permanganate at the
end of the diffusive layer is simply set to its bulk aqueous phase concentration, as illustrated in
Figure 1.
These governing equations and their associated boundary conditions were implemented
by Reitsma and Dai (2001) to determine the enhancement of DNAPL interphase mass transfer
caused by increased concentration profiles during chemical oxidation. While their theoretical
results predict little mass flux enhancement, they recognize that verification of their prediction
requires a value for the diffusive layer thickness,δ. The proposed research, while different from
Reitsma and Dais’ work, will suffer from the same level of uncertainty, as the diffusive layer will
be estimated numerically. However, the theoretical PCE mass flux in this proposed research
(obtained from the model’s prediction of the instantaneous change in concentration at the
interface) will be compared to experimentally obtained values of the mass flux. Thus, verification
of the model will be possible in the proposed research.
Through two-dimensional porous media flow modeling performed by Chrysikopoulos et
al. (2003), the concentration boundary layer thickness was approximated as a simple function of
the hydrodynamic dispersion coefficient (Dx), horizontal distance along the DNAPL pool (z), and
interstitial fluid velocity above the DNAPL pool (Uz). The approximation can be summarized in
Equation 6.
!! ≈ 4
!! !
!!
!
!
This concentration boundary layer thickness is defined as the vertical distance from the DNAPLwater interface where the aqueous-phase concentration of the DNAPL has depleted to 1% of the
saturation concentration, Cs, and is somewhat analogous to the diffusive layer thickness. This
value is expected to be just a few centimeters thick under typical groundwater conditions
(Chrysikopoulos et al. 2003). Yet this approximation, as mentioned before, is not valid for the
stagnant case studied in this work. In the diffusive layer being studied in this project, the
horizontal distance, z, along the DNAPL pool is of no concern, and the velocity above the pool is
zero, which would cause an isolated singularity in the approximation above. Yet, the
approximation provides useful insight into the relationship between boundary layer thickness and
dispersion. As the dispersion coefficient increases, the boundary layer thickness will increase.
Analogously, as the diffusion coefficient of the DNAPL increases, its diffusive layer thickness
should increase. Therefore, should the diffusion coefficient of the DNAPL be increased
6 Eq. 6 sufficiently, the diffusive layer in which the DNAPL is oxidized should increase, resulting in a
more efficient use of the aqueous permanganate solution.
A fundamental hypothesis of the proposed research is that an increase in DNAPL
diffusion can be accomplished through the use of surfactant-enhanced solubilization. Surfactant
molecules have been shown to increase both the solubility and mass dissolution rate of organic
molecules into the aqueous phase (Grimberg et al. 1995). Theoretically this will provide a higher
concentration gradient to drive mass transfer, as well as a thicker diffusive layer within which
oxidation of the DNAPL can occur. In a similar fashion to DNAPL aqueous phase mass transfer,
the surfactant-enhanced model follows the same linear driving force model described by
Sherwood in 1975 (Mayer et al. 1999; Grimberg et al. 1999). Similarly, the mass flux can be
described with an effective diffusion coefficient, where the dissolution occurs across a distance
referred to as the “hydrodynamic boundary layer.” Thus, the diffusion coefficient and diffusive
layer studied in the proposed research will be a combination of both reaction driven and
surfactant-enhanced contributions. Theoretically, the presence of the surfactant molecules should
result in higher PCE interphase mass flux values.
This modeling approach differs from the more common use of Gilland-Sherwood
correlations, in which the mass transfer coefficient is correlated through the Sherwood number as
a function of Reynolds number, Schmidt number, and porous media grain characteristics. While
these methods have been used successfully to model DNAPL mass transfer to the aqueous phase
(Powers et al. 1994), and in the presence of surfactant solutions (Mayer et. al. 1999), such a
correlation is not appropriate for the stagnant nature of these experiments. However, once
baseline data concerning the chemical oxidation processes in the presence of surfactants is
established in this proposed research, developments of Gilland-Sherwood correlations in flowthrough column studies could be a logical progression of this project.
The feasibility of combining surfactant-enhanced solubilization and in situ chemical
oxidation has been studied in laboratory scale column reactors. Findings from continuous stir
batch reactors showed that the combination of potassium permanganate and surfactant molecules
at the critical micelle concentration (CMC) significantly enhanced DNAPL removal (Tsai et al.
2009). The critical micelle concentration is an important surfactant concentration to consider, as
micellar transport of DNAPL to the aqueous phase is a key mechanism in DNAPL dissolution
(Mayer et al. 1999). In addition, the oxidation processes seem to be altered, as higher pseudofirst-order reaction rate constants have been observed with increasing surfactant concentration (Li
and Hanlie 2008). The kinetic studies conducted by Li and Hanlie (2008) also showed that
experimental data at higher surfactant concentrations became more and more non-linear. This
behavior was thought to be attributed to quick permanganate consumption in response to
increased DNAPL dissolution rates. Thus, it will be important to consider how surfactant
molecules change both the diffusion coefficient and the reaction rate constant.
Methodology
The proposed research is to be conducted under the advisement of professor Michelle
Crimi through The Institute for a Sustainable Environment. In the study, a total of 9
permanganate/surfactant concentration combinations will be analyzed. Permanganate
concentrations of 0.0316 M, 0.0158 M, and 0.0040 M were chosen to represent a range of typical
concentrations used in ISCO implementation. At each permanganate concentration, Aerosol OT
concentrations of 10,000 mg/L, 500 mg/L, and 0 mg/L will be studied to compare mass flux
values in the absence of surfactants to those at various surfactant levels leading up to it’s aqueous
solubility limit. For each of the nine possible permanganate/surfactant combinations, various
parameters must be experimentally determined. The PCE aqueous solubility, the reactant
diffusion coefficients (both PCE and MnO4-), and the second order reaction rate constant must be
determined for each system in order to model the PCE interphase mass flux.
7 Solubility Experimets
Since PCE aqueous solubility in the presence of surfactant molecules is assumed to be
independent of MnO4- concentration, only three values of CS will be required for the various
surfactant concentrations studied. In order to determine aqueous
solubility, 5mL of PCE will be mixed with 5mL of aqueous
surfactant solution (10,000 mg/L, 500 mg/L, and 0 mg/L). After
the settling of the two liquid phases, gas chromatography
analysis of the aqueous phase will allow for the calculation of the
PCE solubility limit at each surfactant concentration. These
values will be incorporated into the model as the PCE
concentration at the liquid-liquid interphase.
Dffusion Experiments
Determination of PCE and permanganate diffusion
coefficients will require the use of membrane diffusion cells. As
shown in Figure 2, these cells consist of two compartments (A
and B), separated by a membrane (D). Both compartments
contain stir bars (R and S) that are actuated by a revolving
magnet (M). In both compartments the aqueous surfactant
solution will be present at each of the three concentration levels
being studied. In the lower compartment PCE will be added at
it’s solubility limit for the given surfactant mixture. After 24
hours, PCE concentration in the two compartments will be
analyzed by gas chromatography. The same will be done for
permanganate, where it’s initial concentration in the bottom
compartment will be 0.0316 M. This procedure will be repeated
for statistical accuracy. With this information, a diffusion
coefficient can be calculated from Equation 7 (Mills et al. 1968).
!
! = !" log
Figure 2 Stokes Diffusion Cell
!
!
!!"##"$
!!!"#
!!"##"$ !!!"#
Here, the diffusion coefficient, D, is a function of the PCE
concentration at the top and bottom at the initial time and at some point, t. The constant, β,
depends on cell characteristics. This value will be found by calibrating the diffusion cell with an
aqueous urea solution, since the diffusion coefficient is already known. With this experimental
apparatus, accuracy of up to 0.2% can be obtained (Cussler 1997). The diffusion coefficients of
both PCE and permanganate will be used at the various surfactant concentrations (10000 mg/L,
500 mg/L, and 0 mg/L) as model parameters, thus
requiring a diffusion coefficient value for each of the two
reactants at each surfactant concentration.
Kinetic Experiments
In order to determine second order reaction rate
constants for the PCE/permanganate oxidation in the
presence of surfactant molecules, kinetic studies must be
performed in a stirred batch reactor. A setup similar to Yan
and Schwartz (1999) will be employed, and can be seen in
Figure 3. A kinetic study will be performed for each of the
9 permanganate/surfactant solution combinations being
considered. For each mixture, the initial PCE
concentration will be it’s aqueous solubility limit, and the
permanganate will be the reagent in excess. Triplicate
studies of each mixture are to be performed. Once PCE
Figure 3 Stirred Batch Reactor Setup
8 Eq. 7 concentration data have been obtained at various times via gas chromatography, a plot of
ln([PCE]/[PCE]0) vs. time can be generated. The slope obtained from a linear regression analysis
will give, kobs, a pseudo-first-order reaction rate constant. By dividing this by the initial value of
[MnO4-], a second order reaction rate constant can be obtained. This method is justified given the
excess of permanganate being used, and is similar to the methods used by both Yan and Schwartz
(1999) and Li and Hanlie (2008).
Modeling With Experimental Parameters
Once all experimental parameters have been obtained, the reactive diffusion model can
be solved for the various permanganate/surfactant combinations. The system of non-linear
ordinary differential equations will be solved using the Runge-Kutta method offered by Matlab’s
ode45 algorithm. In solving the equations, the code is able to easily incorporate the PCE
solubility concentration and zero permanganate flux boundary conditions at the interphase.
However, values for the initial PCE concentration gradient and permanganate interphase
concentration must be fit manually. These values are guessed until PCE concentration levels off
to zero, and permanganate reaches its bulk concentration at a distance “δ” from the interphase.
The value for delta must also be obtained by iterative guessing. Should a value for delta be too
small, PCE will drop to zero in a linear fashion. If a value for delta is guessed too high, there will
be an oscillatory output, showing that the model is breaking down as it tries to solve for
concentrations past the “diffusive layer.” Thus, a small initial value for delta will be guessed and
increased incrementally until a delta value is found that allows for both smooth depletion of PCE
and restoration of permanganate to its bulk concentration. Converting Equations 4 and 5 and their
associated boundary conditions to dimensionless form will be done to simplify the graphical
analysis. Once the model has been solved, the derivative of PCE concentration with respect to
space at the interphase will be multiplied by the diffusion coefficient to obtain an interphase mass
flux. These mass flux values will be compared to those obtained experimentally.
Experimentally Determining Mass Flux Values
In order to obtain mass flux values, reactive systems will be studied for each of the 9
permanganate/surfactant concentration combinations through the setup shown in Figure 4. In this
setup a long thin reactive column will contain a known volume of PCE, above which a layer of
permanganate/surfactant solution will rest. The volume of aqueous solution will be sufficiently
large compared to that of the PCE to allow for a constant bulk permanganate concentration. It is
essential to maintain the bulk permanganate concentration since it is assumed to be constant
beyond the “diffusive layer” where the reaction takes place. By observing changes in the
thickness of the PCE layer over time, estimate values for PCE mass flux can be obtained. Since
the beaker is of constant cross section, Equation 8 can be used to describe the mass flux of PCE.
!=
! !! !!!
!
Here, the fluid density, ρ, can be multiplied by the change in PCE thickness, (h0-ht), over a time
interval, t, to obtain an estimate of the mass flux value. By taking various height measurements at
different times, average flux values for a particular permanganate and surfactant concentration
can be calculated. If the theoretical differential equations and their associated boundary
conditions are truly capable of determining interphase mass flux during chemical oxidation, these
experimental values should be comparable to those obtained from the model.
9 Eq. 8 Figure 4 Experimental Setup for Mass Flux Analysis
10 Preliminary Results
Preliminary modeling of solutions where the surfactant concentration is 0 mg/L has been
performed. This is possible because model parameters for the reaction in the absence of
surfactants can readily be obtained from the literature. Such an analysis is beneficial as it
illustrates the proposal’s modeling approach, while providing insight into how the mass flux
values should change with varying permanganate bulk concentration. Model parameters were
obtained from Reitsma and Dai (2001) and are
Table 1—Model Parameters
summarized in Table 1.
k
0.041 L/(mol s)
In order to solve the model, a diffusive layer
Cb
0.0316 M
thickness of 0.01 cm was initially guessed. This value was
DA
9.4 X 10-6 cm2/s
then increased up to a value of 0.3 cm, where the PCE was
DB
17 X 10-6 cm2/s
then able to level off to a steady value at the end of the
Cs
0.000905 M
diffusive layer. Diffusive values significantly higher than
0.3 cm resulted in oscillatory outputs, illustrating the
breakdown of the model. This progression can be seen in Figures 5-7, where the “dimensionless
distance” goes from the liquid-liquid interface to the end of the diffusive layer, δ. The PCE (blue
curve) and permanganate (red curve) concentration profiles are scaled to dimensionless
concentration. A value of 1 for the PCE represents its max aqueous solubility, while a value of 1
for the permanganate represents its bulk aqueous concentration.
Figure 5 Diffusive layer = 0.01 cm
Figure 7 Diffusive layer = 3 cm
Figure 6 Diffusive layer = 0.3 cm
With a diffusive layer thickness equal
to 0.3 cm, concentration profiles consistent
with the theoretical model shown in Figure 1
were obtained. Given this thickness, the mass
flux of the PCE at the liquid-liquid interphase
was found to be 2.63 X 10-8 g/(cm2 s).
By changing the bulk permanganate
concentration to 0.0158 M, a diffusive layer
thickness of 0.37 cm was obtained,
corresponding to a mass flux value of 1.77 X
10-8 g/(cm2 s). These results indicate that in the
absence of surfactants, a decrease in
permanganate concentration will allow the
PCE to diffuse further into the aqueous
solution. However, it will do so at a slower
11 rate since the reaction rate decreases at lower permanganate concentrations. Such a relationship is
important to consider for in situ applications. Should the bulk permanganate concentration be set
too low, then the PCE will be able to diffuse well into the aqueous phase, causing the remediation
fluid to spread the DNAPL pollutant further downgradient from the source zone. Yet if the bulk
permanganate concentration is set too high, then rapid oxidation will occur relative to PCE
aqueous dissolution, resulting in much of the permanganate in the aqueous phase being wasted.
While the model is able to mathematically quantify these changes in the absence of
surfactants, experimental parameters for the reaction in the presence of surfactant molecules is
still necessary. It is hypothesized that the addition of Aerosol OT will change both the diffusion
coefficients and reaction rate in a way that allows for increased PCE interphase mass flux. With
these parameters and the reactive diffusion model, the interaction between varying surfactant and
permanganate concentrations and their effect on the interphase mass flux will be better
understood.
12 Timeline
The proposed timeline begins with solubility and kinetic studies, allowing for extensive
sensitivity analysis on the diffusion coefficients to be conducted over summer break. This leaves
the diffusion cell experiments, mass flux experiments, and final thesis writing for the fall
semester.
Solubility tests are expected to take a day of laboratory work. By beginning with this
simple test, a familiarization with gas chromatography analysis can be obtained early on. Since
more time will be required for the kinetic studies, the remainder of the spring semester will be
devoted to this. These initial findings will be incorporated in a poster presentation for the spring
2011 SURE conference, where the mathematical model and kinetic studies will be presented.
With these preliminary model parameters, a sensitivity analysis on the diffusion
coefficient can be conducted for the systems of interest over summer break. This will provide
good insight into how changes in the diffusion coefficient actually change the theoretical mass
flux values, given the other model parameters. Should the change be negligible when deviating D
within an order of magnitude (liquid diffusion coefficients are not expected to deviate much more
than this), the diffusion coefficient experiments will not be necessary. This analysis will be
crucial, so that if time is to be spent on the diffusion experiments, that time can be justified
mathematically.
Also, the portion of the thesis that concerns the kinetic and solubility studies can be
written over the summer break. This order of work will leave the diffusion cell experimentation
(pending the results of the sensitivity analysis) and the mass flux studies for the fall semester. By
splitting the experimental sections this way, complete data analysis can be accomplished mid-way
through the fall semester. Diffusion coefficient experiments are expected to take up to a month.
This will devote the rest of the fall to experimental validation of the modeled flux values, model
analysis, and final thesis writing.
Should unexpected difficulty be encountered with the solubility and kinetic studies this
spring, the diffusion coefficients could be approximated with correlations found in the literature
rather than be obtained experimentally. Although this may reduce the accuracy of the mass flux
values obtained from the model, the experimental mass flux values will be available for
comparison. Thus, the project accommodates room for error, with the possible alternative that the
diffusion coefficients be estimated, and their experimental determination be a task for future
work.
13 Nomenclature
Note: parameters are explained in terms of fundamental quantities mass (M), length (L), time (T),
and moles (mol)
Cs
Aqueous solubility concentration of PCE [mol/L3] or [M/L3]
C, CA, CB
Concentration, of PCE, of permanganate [mol/L3] or [M/L3]
Cb
Bulk aqueous permanganate concentration [mol/L3] or [M/L3]
C0bottom
Concentration at time 0 in the bottom compartment [mol/L3] or [M/L3]
C0top
Concentration at time 0 in the top compartment [mol/L3] or [M/L3]
D, DA, DB
Diffusion coefficient, of PCE, of permanganate [L2/T]
Dx
Hydrodynamic dispersion coefficient [L2/T]
h0
Height of PCE layer at time 0 [L]
ht
Height of PCE layer at time t [L]
J
Mass flux [mol/(L2T)], [M/(L2T)]
K
Mass transfer coefficient [L/T]
k
Second order reaction rate constant [L3/(mol T)]
kobs
Observed pseudo-first-order reaction rate constant [T-1]
t
Time [T]
Uz
Groundwater velocity above DNAPL [L/T]
x
Distance perpendicular to DNAPL/water interface [L]
β
Diffusion cell constant [L-2]
δ
Diffusive layer thickness [L]
δc
Concentration boundary layer thickness [L]
ρ
Density of PCE [M/L3]
14 References
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15 Siegrist, R.L, Urynowicz, M.A., West, O.R., Crimi, M.L., Lowe, K.S., 2001. Principles and
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