Equilibrium versus Nonequilibrium Treatment Modeling in the Optimal Design of Pump-and-Treat Groundwater Remediation Systems Karen L. Endres1; Alex Mayer2; and David W. Hand3 Abstract: The present work proposes that the incorporation of granular activated carbon 共GAC兲 treatment model that accounts for nonequilibrium adsorption into the optimal design of pump-and-treat systems will result in more realistic costs and better-engineered remediation systems. It was found that, when nonequilibrium GAC adsorption effects are considered, the predicted cost of optimal remediation strategies increases consistently when compared to costs obtained assuming equilibrium GAC adsorption, for a wide range of cleanup goals. This finding implies that when simpler equilibrium models are used for GAC adsorption, cleanup costs will be underestimated. GAC treatment costs are shown to be particularly sensitive to the degree of mass transfer limitations in the aquifer–contaminant system, especially when nonequilibrium GAC adsorption is accounted for. Time-varying pumping rates are shown to produce more efficient remediation solutions; the increase in efficiency is even more pronounced when nonequilibrium GAC adsorption is accounted for. Further results show that the optimal remediation designs can be significantly more efficient when the number of GAC adsorber units is selected through optimization. DOI: 10.1061/共ASCE兲0733-9372共2007兲133:8共809兲 CE Database subject headings: Ground-water management; Water treatment; Optimization; Mass transfer; Equilibrium.
Introduction Pump-and-treat 共PAT兲 technologies have become common for groundwater remediation. Optimization of these systems has primarily focused on design of the hydraulic components of the system; however, the treatment component of the remediation often comprises the majority of the total cost 共e.g., Culver and Shoemaker 1997; Culver and Shenk 1998; Aksoy and Culver 2000兲. A common treatment for removing dissolved organic contaminants is adsorption by granular activated carbon 共GAC兲. The primary expense associated with GAC treatment of contaminated groundwater is the cost of replacing the GAC as its capacity for removing contaminants is exhausted. The GAC treatment process typically has been modeled by assuming equilibrium between the contaminant in the aqueous and solid phases. The equilibrium assumption allows the use of simple algebraic models of GAC usage that depend on a limited number of GAC-contaminant properties. 1 Dept. of Civil and Environmental Engineering, Michigan Technological Univ., 1400 Townsend Dr., Houghton, MI 49931; formerly, Graduate Student. 2 Professor, Dept. of Geological and Mining Engineering and Sciences, Michigan Technological Univ., 1400 Townsend Dr., Houghton, MI 49931 共corresponding author兲. E-mail:
[email protected] 3 Professor, Dept. of Civil and Environmental Engineering, Michigan Technological Univ., 1400 Townsend Dr., Houghton, MI 49931. Note. Discussion open until January 1, 2008. Separate discussions must be submitted for individual papers. To extend the closing date by one month, a written request must be filed with the ASCE Managing Editor. The manuscript for this paper was submitted for review and possible publication on January 20, 2006; approved on January 8, 2007. This paper is part of the Journal of Environmental Engineering, Vol. 133, No. 8, August 1, 2007. ©ASCE, ISSN 0733-9372/2007/8-809–818/ $25.00.
However, it is well known that the process of adsorption onto GAC is complex and that mass transfer limitations can be significant 共e.g., Sontheimer et al. 1988兲. The use of equilibrium methods to predict carbon usage has been shown to be inadequate by several investigators 共e.g., Crittenden et al. 1987a,b; Hand et al. 1984, 1989, 1997; Jarvie et al. 2005兲. Crittenden et al. 共1987a兲 discussed the importance of considering mass transfer in performed GAC usage rate calculations. Hand et al. 共1989兲 showed that equilibrium calculations used to calculate GAC usage rates for dichloroethene and trichloroethene for several empty bed contact times 共EBCTs兲 significantly overpredicted the adsorption capacity as compared to GAC usage rates obtained from pilot plant data and mass transfer model predictions. They attributed the failure of the equilibrium model to predict the GAC usage rates to mass transfer limitations in the adsorption of contaminants to the carbon. Jarvie et al. 共2005兲 demonstrated that using an equilibrium approach to model GAC adsorption can greatly underestimate the rate of carbon usage by comparing models that account for nonequilibrium and equilibrium behavior. They modeled groundwater treatment scenarios with a range of chemical types and concentrations, EBCTs, and influent flow rates, target effluent concentrations, and background groundwater compositions containing natural organic matter. They found that the equilibrium approach underestimated carbon usage by a factor of 2–10 without the effects of natural organic matter and up to 20 times when natural organic matter was considered. Operational costs for a GAC groundwater treatment system are based primarily on the GAC usage rate, given that once breakthrough of the contaminant occurs in the treatment system, the GAC must be replaced. With equilibrium modeling of the GAC system, the replacement rate is based on the assumption that the entire adsorptive capacity of the GAC is exhausted at the time of breakthrough. Residence time in the adsorption unit does not need to be considered. However, when nonequilibrium processes are
Fig. 1. Description of equilibrium and nonequilibrium 关after Weber 共1972兲兴 GAC models
considered, breakthrough can occur before the adsorptive capacity of the GAC is exhausted. The time to breakthrough depends on many factors, such as the influent contaminant concentration, the EBCT, the flow rate into the adsorption unit, and the contaminant treatment goal 共Crittenden et al. 1987a,b; Hand et al. 1984, 1989, 1997兲. The difference between the equilibrium and nonequilibrium conceptual approaches is illustrated in Fig. 1. For the equilibrium approach, the concentration “front” that moves along the GAC bed is vertical, such that the concentration behind the wave is equal to the influent concentration 共C0兲 and the concentration ahead of the front is zero. The entire length of the GAC is saturated 共meaning that the adsorptive capacity is completely exhausted兲 at the time when the effluent concentration 共Ce兲 from the GAC unit reaches the operating limit 共CL兲. The operating limit is the level at which the GAC must be replaced. This level usually corresponds to a treatment goal, such as the maximum contaminant level for drinking water. In contrast, with the nonequilibrium approach, the concentration front is “S” shaped. The leading 共downstream兲 edge of the front is significantly lower than the influent concentration. When the leading edge of the front reaches the end of the GAC bed and Ce → CL a fraction of the adsorptive capacity remains at the point when Ce → CL and the GAC must be replaced. The amount of adsorptive capacity remaining depends on several factors, such as the EBCT, the influent concentration, and the operating limit. The conceptual approaches presented in Fig. 1 demonstrate that the equilibrium approach can significantly overpredict the lifetime, and thus underpredict the replacement rate, of the GAC bed. The present work proposes that the incorporation of GAC adsorption treatment models that consider nonequilibrium adsorption effects will result in more realistic costs for and betterengineered remediation systems. First, we investigate the effect of using equilibrium and nonequilibrium based treatment models on the optimal design of PAT systems. We take a multiobjective optimization approach, where we determine the set of designs that simultaneously optimize cost and cleanup performance for a simple, hypothetical aquifer–contaminant system. We next examine the significance of aquifer–contaminant system mass transfer limitations on the optimal PAT design, using the nonequilibrium
approach for modeling the treatment system. It is expected that the optimal design will be sensitive to aquifer–contaminant mass transfer limitations, since the greater the degree of aquifercontaminant mass transfer limitations, the more severe the tailing will be in the extracted groundwater 共Aksoy and Culver 2000兲. With more severe tailing, the influent concentration to the treatment system will decrease, potentially resulting in less efficient use of the treatment system. Finally, we assess the importance of using variable pumping rates and variable number of absorber units in the optimal design of PAT systems, again using the nonequilibrium approach for modeling the treatment system. It is hypothesized that flexibility in these design parameters will result in improved utilization of the GAC, as compared to fixed values.
Methodology The goal of the computational framework is to determine optimal values of decision variables while satisfying multiple objectives and constraints. The framework, summarized in Fig. 2, includes objective functions and models for simulating groundwater flow and transport processes and the groundwater treatment process. The two objective functions are to minimize capital and operational costs and to minimize the contaminant mass remaining in the aquifer
Fig. 2. Schematic of computational framework
冋
New Nt
min f 1 = min a1New + a2NGAC +
冋 冉冕
min f 2 = min
1 M0
兺 兺 k=1 l=1
˙ 共a3QkHktl + a4M GACtl兲
C共x,t兲dV
⍀D
冊册
册 共1兲
at t = t f
共2兲
The terms in Eq. 共1兲 represent, in order of appearance, capital costs associated with pumping well installation, capital costs associated with the treatment system, operational costs associated with pumping, and operational costs associated with groundwater treatment by GAC. Eq. 共2兲 essentially represents the objective of maximizing cleanup performance, measured by the contaminant mass remaining in the aquifer, normalized by the initial contaminant mass. The decision variables are the pumping rates at fixedlocation extraction wells, Qk, and the number of GAC adsorption units, NGAC. The state variables are concentration, C, and hydraulic head, h, which is related to the total head, H. The constraints on the decision variables and state variables are 0 艋 Qk 艋 Qmax
for k = 1, . . . ,New
NGAC 艋 h 艌 hmin
共3兲
max NGAC
共4兲
over ⍀D
共5兲
Nt
tl = t f 兺 l=1
共6兲
Eq. 共3兲 limits the maximum pumping rate at each well location. Eq. 共4兲 limits the number of GAC adsorption units. Eq. 共5兲 effectively constrains the maximum drawdown in the aquifer. Finally, Eq. 共6兲 constrains the remediation horizon, or cleanup time. The subsurface flow and transport simulators used in this work are based on the two-dimensional steady-state flow equations and contaminant mass balance equations. The steady-state, confined groundwater flow equation for a nondeforming, saturated aquifer system is New
ⵜ共K · ⵜh兲 =
Q⬘k ␦共x − xk,y − y k兲 兺 k=1
共7兲
The hydraulic head, h, is related to the head that the pump in extraction well k must overcome to deliver water to the treatment system, H, by H = zgs − h + hl. Contaminant concentrations are determined by solving the contaminant mass balance equation, given by
C + ⵜ关v − 共D · ⵜC兲兴 = − t
兺k
Ck Q⬘␦共x − xk,y − y k兲 n k
共8兲
i j 兩v兩
共9兲
The pore velocity, v, is given by Darcy’s law as nv = − K ⵜ h
共nmCm兲 共nimCim兲 + + ⵜ关nmv − 共nmD · ⵜCm兲兴 t t =−
兺k Cm,kQ⬘k ␦共x − xk,y − yk兲 共nimCim兲 = ␣共Cm − Cim兲 t
共10兲
To represent mass transfer limitations in aquifer–contaminant systems, we modify Eq. 共8兲 by utilizing the dual domain concept. This concept considers the aquifer as partitioned into mobile and
共11兲
共12兲
A two-dimensional, finite difference approximation is used to solve the groundwater flow equation 关Eq. 共7兲兴 and a particletracking method to solve the mobile zone mass transport equation 关Eq. 共11兲兴. The numerical codes have been developed and validated by Maxwell 共1998兲. Additional background information pertaining to the development of this numerical simulator can be found in LaBolle et al. 共1996兲. The transport code of Maxwell 共1998兲 has been modified to include mobile–immobile mass exchange following the approach of Valocchi 共1985兲, where particle transfers between the mobile and immobile pore volumes are based on a normal probability distribution with a variance calculated from the first-order rate constant, ␣, and the fractional porosities, nm and nim. The modified code was validated agains analytical solutions for Eqs. 共11兲 and 共12兲 共van Genuchten and Alves 1982兲. Two approaches are taken to estimate the carbon usage rate, ˙ M GAC. The equilibrium approach relies on the assumption that the contaminant in the groundwater and GAC are in instantaneous equilibrium. The carbon utilization rate for the equilibrium approach is based on using a Freundlich isotherm to describe partitioning between the groundwater and GAC, or q = KABC1/n
共13兲
where the Freundlich isotherm constants, KAB and 1 / n, are particular to the groundwater–GAC–contaminant system. Given Eq. 共13兲, we can determine the GAC utilization rate for the equilibrium approach as ˙ M GAC = Qk
The hydrodynamic dispersion tensor, D, is defined as D = 共␣T兩v兩 + D*兲I + 共␣L − ␣T兲
immobile zones, such that the total contaminant concentration in the aquifer and the total porosity is divided into mobile and immobile pore volumes, as in nC = nmCm + Cimnim. The exchange of mass between the pore volumes is driven by the concentration gradient between the zones of mobile and immobile water. The origin of the conceptual model and its mathematical representation can be traced to Coats and Smith 共1964兲 and has been applied in the last 2 decades to simulate transport under natural and engineered field conditions 共e.g., Haggerty and Gorelick 1994; Feehley et al. 2000兲. In conceptual terms, the mobile–immobile domain approach allows for each grid block of the aquifer system to be considered as homogenous, while accounting for subgrid heterogeneity. Eq. 共8兲 is replaced with mass balance equations for the mobile and immobile pore volumes, as in
Ck KABC1/n k
共14兲
For the nonequilibrium approach, we use the pore and surface diffusion model 共PSDM兲 to simulate intraparticle mass transfer in fixed-bed GAC adsorption 共Crittenden et al. 1987a,b; Hand et al. 1984, 1989; Sontheimer et al. 1988, Jarvie et al. 2005兲. In the PSDM, the intraparticle mass transfer is modeled by assuming that diffusion occurs at the surface of the GAC particles and in water-filled pores within homogeneous particles and that the pore diffusion is retarded by equilibrium sorption within the pores. The
Fig. 3. Illustration and mathematical description of GAC adsorption processes
PSDM further incorporates the following assumptions: 共1兲 plug– flow conditions exist in the GAC bed; 共2兲 a linear driving force describes the mass flux from the bulk flowing phase to the exterior surface of the adsorbent particle; 共3兲 intraparticle mass flux is described by surface and pore diffusion; and 共4兲 local adsorption equilibrium exists between the solute adsorbed onto the adsorbent particle and the intraaggregate stagnant fluid. A graphical depiction of the water–contaminant–GAC processes simulated by the PSDM is given in Fig. 3, along with mathematical descriptions of the mass flux from the bulk phase to the surface of the particle and the intraparticle mass flux. The GAC usage rate is related to the contaminant fluxes in the PSDM, described conceptually and mathematically in Fig. 3. Since the fluid phase mass transfer rate, k f , the pore and surface diffusivities, Ds and Dl, respectively, the adsorbent density, a, the intraparticle porosity, p, and the intraparticle porosity, p, all control the rate of adsorption onto the GAC, they also control the rate of contaminant breakthrough, and thus the GAC usage rate. For example, higher fluid phase mass transfer rates would produce behavior closer to equilibrium, more efficient GAC utilization, and lower GAC usage rates. In this work, we use a single contaminant and carbon type, such that the previously listed properties and other GAC–contaminant properties 共i.e., Freundlich isotherm constants兲 are held constant. We also fix the effluent treatment goal. Since these properties are held constant, the most significant ˙ variable factors controlling M GAC for the nonequilibrium PSDM approach are the influent concentration and the residence time in the GAC absorber unit 共EBCT兲. The influent concentration delivered to the GAC adsorber unit is controlled by the selected pumping rate and the mass transfer limitations in the aquifer materials. The EBCT is controlled by the geometry of the GAC adsorber unit, which is fixed in this work, but is also related to the selected pumping rate. In the PSDM, the differential equations describing transport in the bulk phase and fluxes to and inside the GAC particles are solved using radial and lateral collocation techniques. The radial collocation defines diffusion across the bed and the lateral defines the length of the bed, which gives the solutions to the space
derivatives. Time derivatives are solved using the DGEAR solution method. The GAC utilization rate for the nonequilibrium approach is calculated as an output of the PSDM through the following logic. When the effluent treatment goal is exceeded, the GAC bed is considered as exhausted, the mass of the GAC is added to the GAC usage, and the exhausted bed is replaced by a “fresh” bed. Obtaining optimal solutions to Eqs. 共1兲 and 共2兲 is a multiobjective problem, which is solved using a niched-Pareto genetic algorithm 共NPGA兲. The NPGA uses evolutionary methods to search for optimal design candidates based on a fitness evaluation of each candidate. The fitness is based on evaluating each candidate solution with respect to how many other solutions dominate the solution in a Pareto optimal sense. The NPGA also uses a “niching” operator to force the solutions to span the limits of the tradeoff curve. McKinney and Lin 共1994兲, Ritzel et al. 共1994兲, and Huang and Mayer 共1997兲 give detailed descriptions of the traditional GA selection, reproduction, and mutation operators and a general overview of the GA as applied to single-objective groundwater problems. Erickson et al. 共2002兲 give a complete description of the application of the NPGA to multiobjective groundwater remediation design. The output of the NPGA is a tradeoff curve of cost versus mass remaining. In this work, the NPGA is also used to find single-objective optimal solutions, where the objective function described in Eq. 共1兲 was considered, but Eq. 共2兲 is transformed into a constraint with a fixed target value of the mass remaining. This constraint is formulated as 1 M0
冉冕
⍀D
冊
C共x,t兲dV 艋 MR⬘
at t = t f
共15兲
The constraint was enforced by using a standard penalty approach 共Erickson et al. 2002兲.
Numerical Experiments The hypothetical aquifer used in this set of experiments is homogenous with respect to hydraulic conductivity. In some simulations, mass transfer limitations in the aquifer–contaminant system are considered. A schematic description of the hypothetical aquifer is given in Fig. 4. Each simulation begins with the development of a plume over a 500-day period. The plume emanates from a continuous source and is transported by groundwater flow, which is imposed by constant head boundary conditions on the west and east boundaries of the aquifer. At the end of the 500-day period, the source is removed and PAT remediation begins using a single extraction well. All groundwater from the extraction well is treated in the GAC system, unless the concentration in the extraction well falls below the treatment objective. The remediation continues for a 5,000-day period. The aquifer, contaminant, and treatment system parameters are given in Table 1. The hypothetical contaminant has properties similar to trichloroethylene, one of the most frequently found groundwater contaminants associated with hazardous waste disposal. The GAC properties are based on Calgon Filtrasorb 400, which is a commercially available GAC and is widely used in groundwater treatment systems. The parameters associated with the PSDM are given in Table 2. The coefficients associated with the cost objective function 关Eq. 共1兲兴 are indicated in Table 3. The well installation, pumping, and GAC unit cost coefficients are taken from Erickson et al.
Table 2. Parameters Used in PSDM Model Simulations Parameter
Fig. 4. Illustration hypothetical aquifer system. No specific concentration values are associated with plume contours.
共2002兲. The GAC absorber unit costs are based on the purchase of a unit, excluding the carbon. The parameters used in the NPGA are as follows: population size= 50; tournament selection size= 2; niche radius= 0.5; probability of crossover= 0.9; and probability of mutation= 0.001. These parameter values were determined to give optimal performance in previous PAT optimization work by Erickson et al. 共2002兲. Four sets of numerical experiments were conducted. The purpose of the first set of experiments is to compare multiobjective optimal solutions obtained with the equilibrium and nonequilibrium GAC models. The solutions are obtained in the form of cost versus mass remaining tradeoff curves. In these experiments, the effects of the mobile–immobile mass exchange were not considered and the number of absorber units was fixed at one 共NGAC = 1兲. The second set of experiments was conducted to assess the effects of mass transfer limitations in the aquifer–contaminant system on optimal PAT designs, using the nonequilibrium GAC Table 1. Base Case Parameters for Flow, Transport, and Treatment Simulations Parameter
Symbol
Value
共a兲 Aquifer properties Total porosity 0.25 n 0.25 Mobile zone porositya nm 0.00 Immobile zone porositya nim Hydraulic conductivity 3.82⫻ 10−5 K 2.7⫻ 10−2 Background pore velocity v 10 Longitudinal dispersivity ␣L 2 Transverse dispersivity ␣T Mobile–immobile zone 0 ␣ exchange ratea
Value
Unit
Void fraction of the particle 0.667 共—兲 共unitless兲 Apparent density 6.50⫻ 105 g/m3 −4 Particle radius 4.2⫻ 10 m Length 3.00 m Weight of adsorbent in bed 50.0 kg Adsorber diameter 0.3 m Operating temperature 24.0 °C Number of radial collocation points 10 共—兲 Number of axial collocation points 50 共—兲 Number of axial elements 10 共—兲 Molecular weight of adsorbate 119.38 g/gmol m3 / gmol Molar volume of adsorbate 8.75⫻ 10−4 11.3 共mol/ g兲共L / mol兲1/n Freundlich KAB of adsorbate 0.78 共—兲 Freundlich exponent 1 / n of adsorbate Surface to pore diffusion flux ratio 4.0 共—兲 number Tortuosity constant of adsorbate 1.0 共—兲
treatment approach. These experiments were single objective experiments, where cost was minimized 关Eq. 共1兲兴 and the mass remaining was treated as a constraint 关Eq. 共15兲兴. A range of mobile–immobile zone mass exchange rates and mobile– immobile zone porosities were used to assess the sensitivity of the optimal solutions to the parameters controlling mobile–immobile zone exchange. The values of the parameters are given in Table 4 and were selected based on values in the literature 共Sardin et al. 1991; Haggerty and Gorelick 1995; Zhang and Brusseau 1999; Feehley et al. 2000兲. Decreasing values of ␣ and nm correspond to greater degrees of mass transfer limitations, and hence the expectation of greater tailing in the concentrations in the extraction well. The number of absorber units was fixed at one 共NGAC = 1兲. In the third set of experiments, the pumping rate was allowed to vary in each of ten 500-day periods. These experiments also were single objective experiments, where cost was minimized 关Eq. 共1兲兴 and the mass remaining was treated as a constraint 关Eq. 共15兲兴. As has been demonstrated by several authors 共e.g., Gorelick and Remson 1982; Chang et al. 1992; Yu et al. 1998兲, variable
Units Table 3. Base Case Values Used in Objective Function and Constraints 共—兲 共—兲 共—兲 m/s m/day m m day−1
共b兲 Groundwater treatment system properties 11.3 GAC adsorption coefficient 共mg/ g兲共L / mg兲1/n KAB GAC adsorption coefficient 0.78 共—兲 1/n 0.005 mg/L Effluent treatment goal C* a This parameter is varied in numerical experiments; the value given is for the base case.
Parameter
Symbol
Value
Units
共a兲 Cost coefficients in objective function 10,800 Dollars/well Well installation cost a1 coefficient Adsorber unit cost 1,000 Dollars/ a2 coefficient adsorber unit 1.05 Dollars/ m4 Pumping operation cost a3 coefficient Treatment cost coefficient 2.14 Dollars/g GAC a4 共b兲 Constraint values 250 Maximum extraction rate Qmax max 3 Maximum number of NGAC adsorption units Remediation horizon 5,000 tf 0.001 Maximum mass remaining MR⬘
m3 / day 共—兲 day 共—兲
Table 4. Parameter Values for Mobile–Immobile Zone Simulations Mobile–immobile zone exchange rate ␣ 共day−1兲 0.02 0.02 0.002 0.002 0.0002 0.0002
Mobile zone porosity, nm
Immobile zone porosity, nim
0.05 0.20 0.05 0.20 0.05 0.20
0.20 0.05 0.20 0.05 0.20 0.05
pumping rates can lead to more efficient solutions, since as a contaminant plume shrinks, pumping rates can be reduced while still capturing the plume. Variable pumping rates are expected to be even more significant when nonequilibrium GAC adsorption is considered, because it is hypothesized that flow rates can be optimized to feed as high a concentration stream as possible to the GAC adsorber. Higher influent concentrations should lead to more efficient use of the GAC capacity. The number of GAC units was fixed at one 共NGAC = 1兲 in these experiments. The fourth set of experiments considers the feasibility of simultaneous design of the pumping system and treatment system by considering the number of absorber units in series 共NGAC兲 as a decision variable. These experiments were conducted with the nonequilibrium PSDM model and with both the non-mass transfer limited aquifer system and the aquifer system incorporating mobile–immobile mass exchange. Using more than one absorber unit in series could lead to more efficient use of the GAC, since the successive units can manage the breakthrough concentrations from the preceding units, allowing for more of the capacity of the GAC to be utilized in the preceding units. The pumping rates are held constant over the remediation time period in these experiments.
Results The results of the first set of experiments, where multiobjective optimal solutions 共tradeoff curves兲 obtained with the equilibrium and nonequilibrium GAC models are compared, are shown in
Fig. 5. Cost versus mass remaining tradeoff curves for equilibrium and nonequilibrium GAC models
Fig. 6. GAC usage rate 共gm GAC used/volume of water treated兲 versus mass remaining for equilibrium and nonequilibrium GAC models
Fig. 5. Each point in the tradeoff curve represents a Pareto optimal design obtained with either the equilibrium or nonequilibrium GAC model. As expected, for both approaches, the remediation costs increase as the level of mass remaining decreases. The costs obtained with the nonequilibrium approach are higher than those obtained with the equilibrium approach, with the greatest differences occurring at the lowest levels of mass remaining. The pumping and treatment capital costs are constant for all of the designs included in the tradeoff curve results, since the number of wells and adsorber units are fixed for each design. For a given mass remaining target, the pumping rates, and hence the pumping operating costs, are similar for the equilibrium and nonequilibrium GAC model results. The average difference in pumping rates and costs between the equilibrium and nonequilibrium GAC model results is 7%. Since the pumping and treatment capital costs and the pumping operating costs are similar, the major factor contributing to the difference between the equilibrium and nonequilibrium costs is the treatment operating cost, which is directly related to GAC usage. The difference in GAC usage between the equilibrium and nonequilibrium approaches is emphasized in Fig. 6. Fig. 6 is a plot of the volume of water treated per mass of GAC used, which is a measure of treatment efficiency, as a function of mass remaining. The results in Fig. 6 show that the treatment efficiency predicted by the nonequilibrium approach is significantly less 共lower volumes of water treated per mass of GAC usage兲 than for the equilibrium approach. Furthermore, the treatment efficiency for the nonequilibrium model results decreases sharply as mass remaining decreases, whereas the treatment efficiency remains nearly constant for the equilibrium model results. The differences in GAC usage 共and hence the remediation costs兲 between the two modeling approaches can be explained as follows. Fig. 7 shows time-varying concentrations obtained at the pumping well for selected Pareto optimal solutions, corresponding to various levels of normalized mass removal at the end of remediation. For the low mass remaining targets, the concentrations in the extracted groundwater 共and hence the influent concentrations to the GAC adsorber unit兲 decrease sharply as the bulk of the contaminant has been removed. For example, for the MR⬘ = 0.2% design, the influent concentrations drop by two or-
Fig. 7. Concentration at extraction well versus time for range of cleanup performances, measured as mass remaining 共MR⬘兲
ders of magnitude in the latter 3,000 days of the remediation period. Further details of this particular 共MR⬘ = 0.2% 兲 design are given in Table 5. As suggested previously, the pumping and treatment capital costs and pumping operating costs are similar for both of the model approaches. The major difference is found in the GAC usage and corresponding treatment operating cost, where the GAC usage is significantly lower and the treatment costs are significantly higher for the nonequilibrium model results. The lower GAC usage is explained by the fact that, when the nonequilibrium effects in carbon adsorption are considered, lower influent concentrations translate into less efficient use of the GAC per mass of contaminant removed. The nonequilibrium model accounts for this lower efficiency, while the equilibrium model does not. The next set of results pertains to the numerical experiments where mass transfer limitations in the aquifer–contaminant system are accounted for, using the mobile–immobile mass exchange modeling approach for groundwater contaminant transport. Fig. 8 shows the costs for optimal designs obtained using single objective optimization, where the mass remaining target was fixed at MR⬘ = 0.1%. The costs obtained for ␣ = 0 day−1 and nm = n = 0.25 correspond to costs obtained with no mass transfer limitations. The costs for the mobile zone porosity of nm = 0.2 increase slightly as ␣ decreases, and are similar to the costs obtained with no mass transfer limitations. However, for the lower mobile zone porosity, the total costs increase sharply as ␣ decreases, due to sharp increases in treatment costs. For all of these solutions, except for the ␣ = 0.002 day−1 and nm = 0.05 case, the pumping rates were similar, as indicated by the similar pumping operational
Fig. 8. Treatment and pumping costs for range of mobile–immobile zone exchange rates and for high and low mobile zone porosities. “T Op” corresponds to treatment operational costs; “T Cap” corresponds to treatment capital costs; “P Op” corresponds to pumping operational costs; and “P Cap” corresponds to pumping capital costs.
costs. For the ␣ = 0.002 day−1 and nm = 0.05 case, the pumping rate is almost doubled, as are the pumping operational costs. For this highly mass transfer limited case, the concentration in the groundwater extracted from the aquifer is, on average, much lower than in the cases with either higher ␣ or nm. Thus, a higher pumping rate is required to reach the mass remaining target. For the remaining cases 共␣ = 0.0002 day−1 and nm = 0.05兲, no feasible solution was obtained, meaning that the mass remaining constraint could not be met without exceeding the maximum allowed pumping rate. Fig. 9 shows a profile of normalized mass remaining in the aquifer for the case where ␣ = 0.002 day−1 and nm = 0.05, along with a mass remaining profile for the no mass transfer limitation case for reference. These results show that, for this severely mass transfer limited case, excessive tailing in the mass removal 共low concentrations兲 occurs due to the slow release of contaminant mass from the immobile zone during pumping. The tailing in mass removal results in the delivery of low concentration water
Table 5. Comparison of Nonequilibrium and Equilibrium Results for Mass Remaining Target Level of 0.2% Nonequilibrium Equilibrium Pumping capital cost Pumping operating cost Treatment capital cost Treatment operating cost Total cost Pumping rate GAC usage rate Average± standard deviation of log10 共influent concentration兲
10,800 5,875 1,000 74,340 92,015 115 1.52⫻ 10−3 1.80± 0.49
10,800 6,227 1,000 50,289 68,316 126 5.54⫻ 10−3 1.73± 0.61
Units Dollars Dollars Dollars Dollars Dollars m3 / day m3 / g mg/L
Fig. 9. Contaminant mass in mobile and immobile zones and total contaminant mass versus time for mobile–immobile zone exchange rate ␣ = 0.002 day−1 and mobile zone porosity nm = 0.05. Contaminant versus time for nonmass transfer limited case is provided for reference.
Fig. 10. Treatment and pumping costs for case where pumping rates are fixed 共“Fixed Q”兲 or are allowed to vary 共“Variable Q”兲, for range of mobile–immobile zone exchange rates and low mobile zone porosities. “T Op” corresponds to treatment operational costs; “T Cap” corresponds to treatment capital costs; “P Op” corresponds to pumping operational costs; and “P Cap” corresponds to pumping capital costs.
to the treatment system, inefficient use of the GAC, and higher treatment costs. For the infeasible case, corresponding to ␣ = 0.0002 day−1 and nm = 0.05, the mass release from the immobile zone is considerably slower. The result is extreme tailing, such that the mass remaining target cannot be achieved within the remediation horizon of 5,000 days, even at the maximum pumping rate. The impacts of allowing for variable pumping rates are show in Fig. 10. The results show that, as expected 共e.g., Gorelick and Remson 1982; Chang et al. 1992兲, varying the pumping rate produces more efficient solutions. For all of the solutions shown in Fig. 10, the pumping rates tended to decrease through time, starting with a pumping rate near the maximum 共250 m3 / day兲 and ending with a pumping rate that was sufficient to capture the plume 共20– 50 m3 / day兲. The increase in efficiency for variable pumping rates is especially pronounced for the case with more severe mass transfer limitations 共␣ = 0.002 day−1 and nm = 0.05兲. This result is due to the fact that, after about 1,000 days of pumping, the mobile fraction of contaminant present at the beginning of remediation has been largely removed. At this point, the remaining contaminant mass transfers very slowly from the immobile zone to the mobile zone and is present only at low concentrations in the mobile zone. Low pumping rates during this phase are much more efficient, since they lead to greater use of the GAC capacity. In Fig. 11, costs are shown for the optimal designs obtained with the number of adsorption units in series as a decision variable. For the case with no mass transfer limitations, the maximum number of three adsorption units was selected. The total GAC treatment costs 共capital and operating兲 for the case where the number of adsorption units is included as a decision variable are less than those for the case where the number of units is fixed. This result is due primarily to the fact the capital costs of the adsorption unit are relatively small compared to the GAC replacement costs. The outcome of lower costs being obtained when the number of adsorption units in series is taken as a decision variable implies that, when the design of the treatment system is optimized
Fig. 11. Treatment and pumping costs for cases where number of adsorber units was and was not considered as decision variable for nonmass transfer limited and mass transfer limited systems. Mass transfer limited system has mobile–immobile zone exchange rate of ␣ = 0.002 day−1 and mobile zone porosity of nm = 0.05. “T Op” corresponds to treatment operational costs; “T Cap” corresponds to treatment capital costs; “P Op” corresponds to pumping operational costs; and “P Cap” corresponds to pumping capital costs.
simultaneously with the design of the pumping system, more efficient solutions can be found. However, for the mass transfer limited case 共␣ = 0.002 day−1 and nm = 0.05兲, there is no difference in the designs obtained when the number of adsorption units in series is or is not a decision variable. This result is explained by the excessive tailing 共see Fig. 9兲 and consequently very low influent concentration to the GAC treatment system for the mass transfer limited case. Even when multiple units in series are considered, the efficiency of the GAC usage is not improved.
Conclusions In this work, a GAC adsorption treatment model that accounts for nonequilibrium sorption has been incorporated into a PAT optimization framework. It was found that, when nonequilibrium GAC adsorption effects are considered, the predicted cost of optimal remediation strategies is consistently higher than when an equilibrium approach is used. This finding implies that when simpler equilibrium models are used for GAC adsorption, cleanup costs will be underestimated. In the case of the hypothetical aquifer– contaminant–treatment system investigated here, the equilibrium approach underestimated costs as much as 30%, when compared to the nonequilibrium approach. The difference between the two approaches is greatest when the aquifer cleanup target is stricter. Aquifer–contaminant system mass transfer limitations are modeled in this work by applying the mobile–immobile mass transfer concept. It was found that, when aquifer-contaminant system mass transfer limitations are accounted for, the cost of remediation increases, especially for lower mobile zone volume fractions and mobile–immobile zone mass transfer rates. It is an expected result that pumping costs would increase as aquifer– contaminant mass transfer limitations increase, since more pumping would be required to remove the “tail” of the contaminant mass under mass transfer limited conditions. However, it is only through the use of the nonequilibrium GAC adsorption model that we can see that treatment costs also tend to increase as the degree of aquifer-contaminant mass transfer limitation increases. This
result is significant because treatment costs generally comprise the majority of the overall PAT remediation cost. This work also shows that flexibility in PAT design parameters can produce more efficient designs. First, allowing for timevarying pumping rates produces more efficient remediation solutions. This is also not an unexpected result, but the results show that the increase in efficiency is even more pronounced when nonequilibrium GAC adsorption is accounted for. When nonequilibrium adsorption effects are considered, the pumping rates can be “tuned” to optimize the GAC usage rate, which is especially important during the period when contaminant mass tailing occurs. Second, we have extended the optimal remediation design to include configuration of the GAC adsorption system along with configuration of the pumping system, by including the number of absorber units as a design variable. The results show that the optimal remediation designs can be significantly less expensive when the number of adsorber units is selected through optimization.
˙ M GAC M0 m New NGAC max NGAC
⫽ ⫽ ⫽ ⫽ ⫽ ⫽
Nt ⫽ n nint nm Qk Q⬘k
⫽ ⫽ ⫽ ⫽ ⫽
Qmax ⫽ q ⫽ tf ⫽ tl ⫽
Acknowledgments This research was supported by NSF Grant No. BES-0083112, USEPA Grant No. CR826614-01-0, and the Statistical and Applied Mathematical Sciences Institute, Research Triangle Park, N.C.
Notation The following symbols are used in this paper: a1 ⫽ cost coefficient associated with extraction well installation 共$/well兲; a2 ⫽ cost coefficient associated with treatment system installation 共$/adsorber unit兲; a3 ⫽ cost coefficient associated with pumping operating cost 共$ / L4兲; a4 ⫽ cost coefficient associated with groundwater treatment operating costs 共$/M兲; C ⫽ contaminant concentration in aquifer, as function of location and time 关M / L3兴; Ck ⫽ aqueous concentration removed from well k 关M / L3兴; D ⫽ hydrodynamic dispersion tensor 关L2 / T兴; D* ⫽ molecular diffusivity 关L2 / T兴; f 1 ⫽ total cost 共$兲; f 2 ⫽ normalized mass remaining in aquifer 共—兲; H ⫽ total head, or head that pump must overcome to deliver water to treatment system 共L兲; Hk ⫽ total head, or head that pump in extraction well k must overcome to deliver water to treatment system 共L兲; h ⫽ hydraulic head 共L兲; hl ⫽ estimated head loss in treatment train 共L兲; hmin ⫽ minimum head allowed over model domain 共L兲; I ⫽ unit tensor 共—兲; im ⫽ subscript referring to immobile domain 共—兲; K ⫽ hydraulic conductivity tensor 关L/T兴; KAB ⫽ partitioning Freundlich isotherm constant 关共mol/ M兲 ⫻共L3 / mol兲1/n兴; k ⫽ well index 共—兲; l ⫽ time index 共—兲;
V v zgs ␣
⫽ ⫽ ⫽ ⫽
␣L ⫽ ␣T ␦ ⍀D 1/n
⫽ ⫽ ⫽ ⫽
carbon usage rate 关M/T兴; initial contaminant mass 共M兲; subscript referring to mobile domain 共—兲; number of active extraction wells 共—兲; number of GAC adsorption units 共—兲; maximum number of GAC adsorption units in series 共—兲; number of time steps within remediation horizon 共—兲; porosity 共—兲; fractional porosity for immobile domain 共—兲; fractional porosity for mobile domain 共—兲; pumping rate at well k 关L3 / T兴; extraction rate per unit aquifer volume from well k located at xk and y k 关L3 / T兴; maximum, individual pumping rate 关L3 / T兴; concentration on GAC 共mass contaminant/mass GAC兲 共—兲; time period over which remediation occurs 关T兴; incremental time period used to evaluate PAT operational costs 关T兴; volume 共L3兲; pore velocity 关L/T兴; ground surface elevation 关L兴; first-order rate constant controlling rate of exchange between mobile and immobile domains 关1/T兴; effective longitudinal dispersivity coefficient 关L2 / T兴; effective transverse dispersivity coefficient 关L2 / T兴; Dirac delta function 关—兴; global domain 关—兴; and exponential Freundlich isotherm constant 共—兲.
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