Environ. Sci. Technol. 2009, 43, 4113–4118
Development and Assessment of a Rate Equation for Chemical Transformations in Reactive Porous Media BY J.F. DEVLIN* Department of Geology, Lindley Hall, University of Kansas, 1475 Jayhawk Blvd., Lawrence, Kansas 66049
Received January 5, 2009. Revised manuscript received April 6, 2009. Accepted April 7, 2009.
With increasing interest in reactive porous media for groundwater remediation, such as granular iron, kinetic rate equations based on the Langmuir-Hinshelwood (L-H) assumptions have proven useful. Three parameters describe L-H kinetics: the two Langmuir sorption parameters, J and Cmax, and the first order rate constant, k. Unfortunately, the Cmax and k are lumped in the L-H equation, making it impossible to estimate their individual magnitudes. A re-examination of the theory underlying the L-H rate equation showed that L-H kinetics are not necessarily appropriate for packed reactive porous media experiments in columns or in the field. A more general rate equation was derived by accounting for changes in sorbed concentrations over time. The equation contains the Langmuir sorption parameter Cmax not lumped with the reaction rate constant, k, so it is possible to obtain unique estimates of J and Cmax and the rate constant, k. A sensitivity analysis suggested that this separation of variables can be achieved over a finite range of conditions applicable to granular iron media. The equation was demonstrated to be applicable, and the separation of variables possible, using the reduction of 4-chloronitrobenzene with Connelly granular iron as a test case.
Introduction The kinetics of organic compound degradation in the presence of granular iron is of scientific interest, as well as practical importance to those who design permeable reactive barriers for groundwater remediation. Consequently, the subject has received much attention over the past decade. The first kinetic studies yielded rate data that followed pseudofirst-order behavior (1, 2), or second-order behavior if the iron surface was varied between experiments (3). However, additional work revealed that reaction rates did not vary linearly with the aqueous reactant concentrations, Cw (4, 5), and therefore more sophisticated kinetic models were soon applied. The kinetics of sorption, desorption, and reaction ultimately control the observed transformation rates of reactants in heterogeneous media, and rate laws accounting for these processes have been presented (6). However, as discussed below, simplifying assumptions have almost always been required to apply these equations, including the assumptions of equilibrium sorption, and constant sorbed mass. * Corresponding author phone: 785-864-4994; fax: 785-864-5276; e-mail:
[email protected]. 10.1021/es900025r CCC: $40.75
Published on Web 04/27/2009
2009 American Chemical Society
The assumptions mentioned above lead to the following hyperbolic function (eq 1), which is similar in form to the Langmuir isotherm and the Michaelis-Menten microbial kinetic equation, and which was put forward as a suitable basis for the nonlinear dependence of rate on Cw, in granular iron experiments (4, 7-9) dCw kappFmCw )dt 1 + Cw J
(1)
where Cw is the aqueous concentration (M/L3), kapp is an apparent rate constant (T-1), Fm is the ratio of solid iron mass to solution volume (M/L3), J is a sorption parameter related to the affinity of a solute for the solid surface (L3/M), and t is time (T) (note: generalized units are given where M is mass or moles, L is length, and T is time). A reaction sequence such as that shown below is assumed in eq 1. k1
k2
k3
Γ + Cw a CΓ 98 PΓ {\} Γ + Pw k-1
(2)
k-3
where Γ represents the iron surface, CΓ is the concentration of reacting solute on sorption sites (M/M), PΓ is the concentration of product on sorption sites (M/M), Pw is concentration of product in solution (M/L3), and kx are rate constants for the forward (positive subscript) and reverse (negative subscript) directions. All sorption sites are assumed equivalent and reactive, and with sorption assumed to be at equilibrium, the second step in the reaction sequence (k2) is rate controlling (9-11). Prior to its use with granular iron systems, the hyperbolic function (eq 1) was found to be applicable to other surface reaction cases, such as industrial catalysis (9, 10, 12) and aspects of aquatic chemistry (6). Once again, reaction sequence 2 was assumed to apply (6, 10). Zepp and Wolfe (6) examined such a system by developing a model that accounted for sorption, desorption, and reaction kinetics. However, to apply their model to experimental data, they postulated that the reaction rates were rate controlling (i.e., sorption equilibrium was achieved quickly) and, “... as the reaction proceeds, the concentration of sorbed reactant reaches a steady state and dCΓ/dt becomes nearly zero” (ref 6 p 428). The practical outcome of their efforts is easily shown to be equivalent to eq 1 (see Supporting Information). Sorption of organics in porous media is currently recognized as a complex process involving different kinetic and equilibrium behaviors that depend on the sorbate and sorbent (13-16). Nevertheless, where sorption is followed by a chemical reaction, the notion that sorption occurs rapidly compared to the reaction has been widely accepted and applied with considerable success (4-9, 17). Therefore, though this assumption is doubtless a simplification of reality, it appears to be a useful one. The assumption that sorbed concentrations do not change in time (i.e., dCΓ/dt ) 0) is more questionable at first glance because aqueous concentrations are expected to change in time, and as they do the equilibrium assumption requires that the sorbed concentration must also change in time (unless the surface is fully saturated). This is certainly what must happen in batch tests of the kind frequently reported in granular iron studies (1, 2, 13, 18, 19). However, if the sorbed mass is a small fraction of the total, the assumption that dCΓ/dt ) 0 is easily justified because the term disappears in the mass balance, as explained more fully in the following section. The assumption is less easily justified when the solid surface area VOL. 43, NO. 11, 2009 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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available for sorption is large, and sorbed mass is not a negligible quantity, as might be the case in porous media packed columns or permeable reactive barriers. The objective of this work was to develop a new kinetic model based on an assumption of sorption equilibrium but in which dCΓ/dt is not assumed to be zero. Limited experience with the model has been presented, involving work with granular iron (20-23), and hence it was named the kinetic iron model (KIM). Here, for the first time, its theoretical basis is given, its utility in estimating sorption and reactivity parameters is evaluated and demonstrated. Although the model has so far only been applied to granular iron reactions, the following theoretical development suggests that it may have more general applicability. Theory. Equation Development. In the case of single, reactive site sorption, the following mass balance applies for a reactant Mtot ) MΓ + Mw
(3)
where Mtot is the total mass of reacting solute in the system, MΓ is the mass of reacting solute sorbed, and Mw is the mass of reacting solute in solution. If all the reactant is consumed by a first order reaction of the sorbed phase, dMtot ) -kMΓ dt
(4)
where k is the first-order rate constant for a reaction on the solid surface (T-1) Differentiating eq 3, and expressing both eqs 3 and 4 in terms of concentrations, the following equations result, dCw dCΓ dMtot )S + Vp dt dt dt
(5)
dMtot ) -kSCΓ dt
(6)
and
where Vp is the water-filled volume of the reaction vessel. S should represent the number of reactive sites on the solid. However, this quantity is difficult to determine accurately, so surrogate values are used instead. For example, solid surface area has been suggested as a surrogate in granular iron studies (3, 11). Iron masses and surface areas have been found to be linearly related, with a factor of about 1 m2/g often being representative of the commercial products (range is from about 0.4 to 1.5 m2/g1, 5, 11), so iron mass is another possible surrogate for these kinds of studies, and is used in this presentation. Combining eqs 5 and 6 -kSCΓ ) S
dCΓ dCw + Vp dt dt
(7)
The first term on the right side of eq 7 describes changing sorbed concentration with time. If this is assumed to be equal to zero, i.e., if it is assumed that either sorption is in steady state, or that the mass sorbed is very small compared to the mass in solution, dCΓ/dt ) 0, and it is easily shown that dCw ) -kFmCΓ dt
(8)
where Fm is equal to S/Vp. Substituting the Langmuir isotherm expression for CΓ, CmaxJCw CΓ ) 1 + JCw 4114
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where Cmax is the capacity of the solid surface to sorb the solute of interest (M/M), and an expression equivalent to eq 1 is obtained kCmaxFmCw kCmaxFmJCw dCw ))dt 1 + JCw 1 + Cw J
(10)
where kCmax is a lumped parameter given as kapp in eq 1. This model is hereafter referred to as the Langmuir-Hinshelwood (L-H) model. If dCΓ/dt is not assumed to be zero, then it can be obtained by differentiating eq 9,
(
)
(11)
)
(12)
dCw CmaxJ dCΓ ) dt (1 + JCw)2 dt Substituting eqs 9 and 11 into 7 -kS
(
dCw CmaxJ dCw CmaxJCw )S + Vp 2 1 + JCw dt dt (1 + JCw)
Simplifying, rate )
kCmaxFmCw dCw )dt CmaxFm 1 + Cw + J 1 + JCw
(13)
The values of Cw and Fm are fixed in the experimental design. The magnitude of the rate term is determined from the experimental data. The unknowns in the equation are therefore k, J, and Cmax. Equation 13 is the aforementioned KIM, and differs from eq 1 by the fractional term in the denominator. Further inspection reveals that this term is likely to be of importance when either Cmax or Fm (or both) are appropriately large. This makes intuitive sense since increasing the solid surface concentration (Fm) or surface capacity for sorption (Cmax) leads to a larger fraction of total reactant mass in the sorbed phase and a greater likelihood that dCΓ/dt * 0. Relationship to Previously Reported Rate Equations. Equation 13 is a general form of other models reported in the literature for reactions involving granular iron. For example, assuming a small value of Cw, the expression collapses to
(
)
dCw kCmaxFm Cw ) -kobsCw )dt 1 + CmaxFm J
(14)
which is a pseudofirst-order expression, if Fm is constant, and where kobs is the apparent rate constant estimated by fitting data to a first-order-kinetic model. When Fm or Cmax is small, the second term in the denominator of eq 14 vanishes and dCw ) -(kJCmax)FmCw ) -k2ndFmCw dt
(15)
Equation 15 is the familiar pseudosecond-order expression that has been applied to granular iron systems in which it was shown that rate depended on both Fm and Cw (3, 7). The expression is pseudosecond-order because in addition to the time dependency of Cw, the term Fm may also change slowly in time (5, 24). When the reactant concentration is very high, Cw dominates the denominator in the KIM and the expression reduces to a zero order kinetic equation. In cases where CmaxFm/(1/J + Cw) , (1/J + Cw), eq 13 collapses to the L-H model dCw kCmaxFmCw )dt 1 + Cw J
(16)
FIGURE 1. Normalized sensitivities as a function of aqueous reactant concentrations for the case where Gm ) 4500 g/L and k, Cmax, and J are in the range of values considered typical for work with granular iron. Sensitivities for k and Cmax are an order of magnitude greater than those for J. (A) Sensitivity to k, (note J ) 0.01 µM-1 and Cmax ) 0.1 µmol g-1). (B) Sensitivity to Cmax (note J ) 0.01 µM-1 and k ) 0.1 min-1). (C) Sensitivity to J (note k ) 0.1 min-1 and Cmax ) 0.1 µmol g-1). (D) Broken lines indicate D(rate)/ Dk*k and solid lines indicate D(rate)/ DCmax*Cmax as functions of concentration for k )0.1 min-1, Cmax ) 0.01 µmol/g, and J ) 0.01 µM-1 (lightweight lines read from right axis) and k ) 1 min-1 Cmax ) 1 µmol/g, and J ) 0.01 µM-1 (heavy weighted lines read from left axis). In none of the cases of eqs 14, 15, or 16 can the contributions of k and Cmax be separated. Therefore, observed rates might be dominated by electron-transfer (high k and low Cmax) or sorption (low k and high Cmax) in any particular case, and the experimenter cannot easily make this distinction. However, under conditions in which the KIM (eq 13) applies, it is possible to uniquely determine values of k and Cmax. A small value of Fm is one possible cause of the KIM collapsing to the L-H equation, and this eventuality is expected in standard batch tests (small amount of solid in a large volume of water). So, the application of eqs 14-16, is justifiable in most cases involving batch tests, and the KIM is not likely to be of assistance. However, eqs 14-16 are less likely to be applicable to columns or reactive barriers packed with reactive media, such as granular iron, because in these cases Fm can be a large number. In these cases, the generality of eq 13 is likely to be required. Sensitivity and Application. As discussed above, three parameters, not directly measurable, are required to calculate reaction rates with eq 13: J, Cmax, and k. The sensitivity of eq 13 to each of these parameters was assessed, beginning with the evaluation of the derivative of eq 13 with respect to each. rate ∂(rate) ) ∂k k
(
( )
∂(rate) rate ) ∂Cmax Cmax
(
1 + Cw J CmaxFm 1 + Cw + J 1 + JCw
∂(rate) rate ) ∂J CmaxFm 1 + Cw + J 1 + JCw
)(
)
CmaxFmCw 1 + J2 (1 + JCw)2
(17)
(18)
) (19)
Substituting eq 13 into eq 17, it can be seen that k cancels and ∂(rate)/∂k is independent of k. A similar substitution into eq 18 shows that ∂(rate)/∂Cmax is independent of Cmax when Cmax is small; at high values of Cmax the derivative tends to zero and sensitivity is lost. In the case of parameter J, for a specified set of conditions (constant k, Cmax, Fm and Cw) it is easily verified that, like Cmax, sensitivity is independent of J when the parameter is small, and tends to 0 when the parameter is large. In order to graphically assess sensitivity as a function of aqueous reactant concentration, the derivatives were normalized by multiplication with their respective parameters k, J and Cmax, and plotted against concentration for conditions representative of granular iron column tests (Fm ) 4500 g/L, Figure 1). A similar analysis for conditions representative of batch tests is presented in the Supporting Information. Inspection of the plots reveals that the concentration dependent sensitivities of k and Cmax are very similar in magnitude and functionality (Figures 1A and B), whereas the functionality of J is quite different, and the magnitude is considerably less (Figure 1C). This finding indicates that J is less influential in determining the reaction rate than the other two parameters, but that it can be uniquely estimated for aqueous reactant concentrations between about 10 and 1000 µM (see Figure 1C). The similarities in the shapes of the ∂(rate)/ ∂k*k and ∂(rate)/ ∂Cmax* Cmax vs. concentration curves raises the possibility that these parameters are well correlated and therefore difficult to uniquely determine. However, closer examination of the shapes of these curves reveals differences that make the separation of these variables possible. Comparing eqs 17 and 18, it is seen that the difference in functionality (i.e., curve shapes) depends on the second bracketed term in eq 18, and more particularly on the middle fraction in the denominator of that term. When the middle fraction is small, the functionality of ∂(rate)/ ∂k and ∂(rate)/ VOL. 43, NO. 11, 2009 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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FIGURE 2. (A) Sensitivity curves for k and Cmax under the conditions of the 4ClNB column experiment. The lack of overlap between the two curves (broken and solid lines) between 10 and 200 µM suggest the variables are separable with the KIM. The efficacy of the separation is demonstrated in (B) where for a constant value of kCmax, (the L-H lumped parameter ) 1.7 × 10-1 µmoles g-1 min-1) different curves result when the individual values of k and Cmax are varied. The best fit regression line occurs where k ) 4.5 min-1 and Cmax ) 3.8 × 10-2 µmoles g-1 and J ) 2.9 × 10-2 µM-1. Additional lines with varying k and Cmax are shown to illustrate the sensitivity. J is fixed at 2.9 × 10-2 µM-1 in all cases. The graph (B) is modified from (23) and data points are shown with ( 15% error bars.
columns with different influent concentrations to steady state, assuming pseudofirst-order kinetics, and calculating the corresponding rate constant by fitting breakthrough curves to the reactive transport equation (25). Since the influent concentrations varied between columns, the sorbed concentrations were different from test to test making the KIM analysis possible (see Supporting Information). Ideally, a profile of steady state concentrations along a column length could supply a complete data set consisting of apparent rates as a function of concentration, if first-order kinetics applies. However, in practice, interferences with reaction products or oxide accumulations may affect rates down-column and over time. By conducting a series of experiments with short columns and different Co values, early time kinetic behavior in the presence of high Fe/Vp, with minimal reaction product interferences, can be examined as a function of reactant concentration, approaching an ideal data set (see also Supporting Information). Accordingly, the experiments were designed so that, in general, effluent concentrations were between 25 and 30% of the influent concentrations (Co), and the competing effects of reaction products, were minimal. The k and Cmax sensitivity curves were recalculated for the conditions of the experiment and it is seen that the differences in the curves are notable, suggesting that the k and Cmax parameters should be separable (Figure 2A). For this work, the three parameters were evaluated in two ways: first, the parameters were determined using a simplex optimization algorithm, and second they were estimated by well-known nonlinear regression methods (26, 27). The former method is based on the geometrical manipulation of a polygon, or simplex, with v vertices (v - 1 is the number of parameters to be fit) to explore a residual sum of squares surface and locate the global minimum (28). The latter method is more traditionally used for inverse problems, and its application to this data set is summarized below. In this work, the error function to be minimized was defined as, E(∆k, ∆Cmax, ∆J) )
∂Cmax become identical, and k and Cmax cannot be separated. This scenario corresponds to the KIM collapsing to the L-H model. To illustrate that the separation of k and Cmax is possible with experimental data, a data set obtained from column experiments involving Connelly iron and 4-chloronitrobenzene (4ClNB) is presented (Figure 2). Details of the experimental procedures are given elsewhere (23) and summarized in the Supporting Information. The transformation rates (dCw/dt) were estimated by running the
∑ [rate
model i
- rateiexp]2
(20)
i
where 0< i < n, and n is the number of experimental data points. As previously pointed out (27), a necessary condition for E(∆k, ∆Cmax, ∆J) to be minimized is that the partial derivatives of E(∆k, ∆Cmax, ∆J) with respect to ∆k, ∆Cmax, and ∆J go to zero. This was achieved by solving the following equation for ∆k ′, ∆C ′max ∆J ′, which are the estimated corrections necessary to revise the parameter estimates for the desired minimization,
TABLE 1. Parameter Estimates and Uncertainties from the Data Presented in Figure 2 As Estimated by the Simplex Optimization Algorithm and Nonlinear Regression for the KIM and L-H Modelsa KIM
L-H uncertainty range
parameter
simplex fit
b
NLR Fit
minimum
maximum
uncertainty range simplex fit
b
NLR Fit
minimum
maximum
-1
k (min ) 4.3 4.5 0.7 10.3 Cmax (µmoles g-1) 4.0 × 10-2 3.8 × 10-2 2.3 × 10-2 6.4 × 10-2 J (µM-1) 3.0 × 10-2 2.9 × 10-2 1.3 × 10-2 5.8 × 10-2 6.2 × 10-3 6.2 × 10-3 5.2 × 10-3 7.8 × 10-3 kCmax (µmols g-1 min-1) 1.7 × 10-1 1.7 × 10-1 2.3 × 10-1 2.3 × 10-1 2.0 × 10-1 2.6 × 10-1 residual sum of squares 2.99 × 104 2.90 × 104 3.44 × 104 3.44 × 104 a Uncertainty Range refers to the range of calculated values one standard deviation from the averages of the best fit Monte Carlo estimates (see text). Differences in the fitted estimates between methods are insignificant based on the estimated errors. b Nonlinear regression.
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[
∑ (U ) ∑U U ∑U U
2
ki
i
ki
Cmaxi
i
ji
i
ki
∑U U ∑ (U ∑U U ki
Cmaxi
i
)2
Ji
i
ji
[
Ji
ki
i
Cmaxi
i
∑U U ∑U U ∑ (U )
Cmaxi
i
Cmaxi
2
Ji
i
∑ U (rate ∑ U (rate ∑ U (rate ki
][
model i
]
∆k ′ ∆C ′max ) ∆J′
- rateiexp)
i
Cmaxi
model i
- rateiexp)
i
Ji
i
model i
nonreactive sorption to better understand the nature of the surface. When sorption and reaction parameters can be uniquely determined, it also becomes possible to assess the contributions of these two processes to reaction rates under different environmental conditions in situ, in benchtop experiments. This can lead to bench scale observations and insights concerning the nature, proportion and longevity of reactive sites and could complement spectroscopic and other microscale investigations that may require some disturbance of the reactive medium.
- rateiexp)
]
(21)
where, Uk ) ∂rate/∂k from eq 17, UCmax ) ∂rate/∂Cmax from eq 18, and UJ ) ∂rate/∂J from eq 19. Equation 21 was solved iteratively until the solution converged on values of k, Cmax, and J such that the residual sum of squares between iterations changed less than 0.1%. Uncertainties in the parameter estimates, reported as errors in Table 1, were obtained by performing a Monte Carlo analysis in which the above equations were solved for the best fit parameters in 1000 realizations (29) (see Supporting Information). The data sets for these realizations were generated randomly from the experimental data in Figure 2, assuming a ( 15% uncertainty on each data point (Table 1). The two parameter estimation methods returned nearly identical parameter estimates (Table 1). The estimates were also similar in magnitude to those reported by Marietta and Devlin (23) who used an approximate two-step linearization scheme, instead of nonlinear regression, to solve the inverse problem (k ) 14 min-1, Cmax ) 1.7 × 10-2 µmol/g, J ) 8.4 × 10-3 µM-1). The uncertainties on the nonlinear regression estimates were found to be quite high (∼ (53% to (88%), which is due to both the uncertainty assigned to the data points ((15%), and the substantial correlation between k and Cmax that was noted earlier. Nevertheless, the data appear to define ranges of the three KIM parameters that clearly indicate the dominance of reaction rate (k) on the observed kinetics over sorption (Cmax) in this case. It should be noted that a Langmuir-Hinshelwood (L-H) style model can describe the data in Figure 2B reasonably well. Best fit parameters were calculated in a fashion similar to that described above, adapted for the L-H model (Table 1, see also Supporting Information). However, while the L-H model is restricted to providing estimates of the lumped parameter kCmax and J, the KIM can provide individual estimates of k, Cmax, and J. For a constant value of kCmax ) 1.7 × 10-1 µmoles g-1 min-1, it was found that different values of k and Cmax produced very different curves, using the KIM equation. This further illustrates the separability of the parameters (Figure 2B). Advantages and Implications. Equation 13, the KIM, is noteworthy because it is more general than the standard L-H-based expressions (for example, eq 1), and has a stronger theoretical basis for application to reactive porous media in columns, soils, or aquifers, than L-H based models, which are more suitable for batch type settings. The KIM assumes sorption equilibrium and is therefore likely to be applicable only in situations where sorption occurs quickly compared to reaction. The applicability is also limited to data sets that are not too noisy, because k and Cmax exhibit similar sensitivities. Nevertheless, data with error as much as (15% were as shown here to be suitable for parameter estimation. As pointed out above, the KIM can be used, in appropriately designed experiments, to uniquely determine a surface rate constant, k, and two Langmuir sorption isotherm parameters, J, and Cmax, for reactive sites in a porous medium. These could be compared to their counterparts describing total or
Acknowledgments The NSERC/Motorola/ETI Industrial Research Chair in Groundwater Remediation, NSERC, CRESTech, and NSF under Grant No. 0134545 are acknowledged for funding this work. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. Dr. Robert Gillham, Dr. Carl McElwee, Dr. Erping Bi, Cory Repta, Mary Morkin, Janet Patchen, Melissa Marietta, Bei Huang are acknowledged for their comments on, and contributions to this manuscript.
Supporting Information Available Equivalency of a sorption kinetic based model and the LH model is demonstrated; sensitivity of the KIM parameters in batch tests is presented; experimental methods for acquiring the data in Figure 2; justification of using column data to assemble data sets like that in Figure 2; details of the parameter estimation calculations; details of the Monte Carlo exercise. This material is available free of charge via the Internet at http://pubs.acs.org.
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