A parametric transfer function methodology for analyzing reactive transport in nonuniform flow

We analyze reactive transport during in-situ bioremediation in a nonuniform flow field, involving multiple extraction and injection wells, by the method of transfer functions. Gamma distributions are used as parametric models of the transfer functions. Apparent parameters of classical transport models may be estimated from those of the gamma distributions by matching temporal moments. We demonstrate the method by application to measured data taken at a field experiment on bioremediation conducted in a multiple-well system in Oak Ridge, TN. Breakthrough curves (BTCs) of a conservative tracer (bromide) and a reactive compound (ethanol) are measured at multi-level sampling (MLS) wells and in extraction wells. The BTCs of both compounds are jointly analyzed to estimate the first-order degradation rate of ethanol. To quantify the tracer loss, we compare the approaches of using a scaling factor and a first-order decay term. Results show that by including a scaling factor both gamma distributions and inverse-Gaussian distributions (transfer functions according to the advection-dispersion equation) are suitable to approximate the transfer functions and estimate the reactive rate coefficients for both MLS and extraction wells. However, using a first-order decay term for tracer loss fails to describe the BTCs at the extraction well, which is affected by the nonuniform distribution of travel paths.     

Reactive solute transport in macroscopically homogeneous porous media: analytical solutions for the temporal moments

In this work, we investigate one-dimensional solute transport affected by rate-limited sorption, first-order mass transfer, and first-order transformation. Analytical expressions are obtained for the temporal moments of the solute in the solution phase. The effect of various rate coefficients on the temporal moments is examined. It was found that, in the presence of transformation reactions, the mean arrival time, and the spread and skewness of the breakthrough curves, are not monotonic functions of the rate coefficients. These solutions will be useful as a preliminary analysis tool for ascertaining the relative importance of various processes under given conditions. They may also be used to analyze the accuracy of various numerical techniques used for simulation of reactive transport.     

Prediction of mass-transfer coefficient for solute transport in porous media

Several previously reported laboratory studies related to transport of solutes through packed columns were utilized to develop predictive relationships for mass-transfer rate coefficient. The data were classified into two groups: those obtained under rate-limited mass transfer between mobile and immobile water regions (physical nonequilibrium conditions), and those derived from rate-limited mass transfer between instantaneous and slow sorption sites (sorption nonequilibrium conditions). The mass-transfer coefficient in all these studies was obtained by fitting breakthrough data to a transport model employing a first-order rate limitations with a "constant" mass-transfer coefficient, independent of flow conditions. This study demonstrated that the mass-transfer coefficient in these models is dependent on system parameters including pore-water velocity, length-scale, retardation coefficient, and particle or aggregate size. Predictive relationships were developed, through regression analysis, relating mass-transfer coefficient to residence time. The developed relationships adequately estimated previously reported field mass-transfer values. Successful simulations of field desorption data reported by Bahr [J. Contam. Hydrol. 4 (1989) 205] further demonstrate the potential applicability of the developed relationships.     

Characterizing pesticide sorption and degradation in microscale biopurification systems using column displacement experiments

Biopurification systems treating pesticide contaminated water are very efficient, however they operate as a black box. Processes inside the system are not yet characterized. To optimize the performance, knowledge of degradation and retention processes needs to be generated. Therefore, displacement experiments were carried out for four pesticides (isoproturon, bentazone, metalaxyl, linuron) in columns containing different organic mixtures. Bromide, isoproturon and bentazone breakthrough curves (BTCs) were well described using the convection-dispersion equation (CDE) and a first-order degradation kinetic approach. Metalaxyl and linuron BTCs were well described using the CDE model expanded with Monod-type kinetics. Freundlich sorption, first-order degradation and Monod kinetics coefficients were fitted to the BTCs. Fitted values of the distribution coefficient K_f,column were much lower than those determined from batch experiments. Based on mobility, pesticides were ranked as: bentazone > metalaxyl - isoproturon > linuron. Based on degradability, pesticides were ranked as: linuron > metalaxyl - isoproturon > bentazone.     

Transport of coliphage PRD1 in a surface flow constructed wetland.

A tracer study was conducted in a 3-ha surface flow constructed wetland to analyze transport performance of PRD1, an enteric virus model. The convection-dispersion equation (CDE), including a first-order reaction model, adequately simulated transport performance of PRD1 in the wetland under an average hydraulic loading rate of 82 mm/d. Convective velocity (v) and longitudinal dispersion coefficient (D) were estimated by modeling a conservative tracer (bromide) pulse through the wetland. Both PRD1 and bromide were simultaneously added to the entering secondary treated wastewater effluent. The mass of bromide and PRD1 recovered was 76 and 16%, respectively. The PRD1 decay rate was calculated to be 0.3/day. The findings of this study suggest that the CDE model and analytical moment equations represent a suitable option to characterize virus transport performance in surface flow constructed wetlands.     

Comparison of nonideal sorption formulations in modeling the transport of phthalate esters through packed soil columns

Sorption of dimethyl phthalate (DMP), diethyl phthalate (DEP) and dipropyl phthalate (DPP) to two soil materials that vary in organic matter content was investigated using miscible displacement experiments under saturated flow conditions. Generated breakthrough curves (BTCs) were inversely simulated using linear, equilibrium sorption (LE), nonlinear, equilibrium sorption (NL), linear, first-order nonequilibrium sorption (LFO), linear, radial diffusion (LRD), and nonlinear, first-order nonequilibrium sorption (NFO) models. The Akaike information criterion was utilized to determine the preferred model. The LE model could not adequately describe phthalate ester (PE) BTCs in higher organic matter soil or for more hydrophobic PEs. The LFO and LRD models adequately described the BTCs but a slight improvement in curve-fitting was gained in some cases when the NFO model was used. However, none of the models could properly describe the desorptive tail of DPP for the high organic matter soil. Transport of DPP through this soil was adequately predicted when degradation or sorption hysteresis was considered. Using the optimized parameter values along with values reported by others it was shown that the organic carbon distribution coefficient (K(oc)) of PEs correlates well with the octanol/water partition coefficient (K(ow)). Also, a strong relationship was found between the first-order sorption rate coefficient normalized to injection pulse size and compound residence time. A similar trend of timescale dependence was found for the rate parameter in the radial diffusion model. Results also revealed that the fraction of instantaneous sorption sites is dependent on K(ow) and appears to decrease with the increase in the sorption rate parameter.     

Evaluating the sensitivity of a subsurface multicomponent reactive transport model with respect to transport and reaction parameters.

The input variables for a numerical model of reactive solute transport in groundwater include both transport parameters, such as hydraulic conductivity and infiltration, and reaction parameters that describe the important chemical and biological processes in the system. These parameters are subject to uncertainty due to measurement error and due to the spatial variability of properties in the subsurface environment. This paper compares the relative effects of uncertainty in the transport and reaction parameters on the results of a solute transport model. This question is addressed by comparing the magnitudes of the local sensitivity coefficients for transport and reaction parameters. General sensitivity equations are presented for transport parameters, reaction parameters, and the initial (background) concentrations in the problem domain. Parameter sensitivity coefficients are then calculated for an example problem in which uranium(VI) hydrolysis species are transported through a two-dimensional domain with a spatially variable pattern of surface complexation sites. In this example, the reaction model includes equilibrium speciation reactions and mass transfer-limited non-electrostatic surface complexation reactions. The set of parameters to which the model is most sensitive includes the initial concentration of one of the surface sites, the formation constant (Kf) of one of the surface complexes and the hydraulic conductivity within the reactive zone. For this example problem, the sensitivity analysis demonstrates that transport and reaction parameters are equally important in terms of how their variability affects the model results.     

Effect of pore-water velocity on chemical nonequilibrium transport of Cd,Zn, and Pb in alluvial gravel columns

This paper investigates the effects of pore-water velocity on chemical nonequilibrium during transport of Cd, Zn, and Pb through alluvial gravel columns. Three pore-water velocities ranging from 3 to 60 m/day were applied to triplicate columns for each metal. Model results for the symmetric breakthrough curves (BTCs) of tritium (3H2O) data suggest that physical nonequilibrium components were absent in the uniformly packed columns used in these studies. As a result, values of pore-water velocity and dispersion coefficient were estimated from fitting 3H2O BTCs to an equilibrium model. The BTCs of metals display long tailing, indicating presence of chemical nonequilibrium in the system, which was further supported by the decreased metal concentrations during flow interruption. The BTCs of the metals were analysed using a two-site model, and transport parameters were derived using the CXTFIT curve-fitting program. The model results indicate that the partitioning coefficient (beta), forward rate (k1), and backward rate (k2) are positively correlated with pore-water velocity (V); while the retardation factor (R), mass transfer coefficient ((omega), and ratio of k1/k2 are inversely correlated with V. There is no apparent relationship between the fraction of exchange sites at equilibrium (f) and V. The influence of Von k2 is much greater than on R, beta, omega, and k1. A one-order-of-magnitude change in V would cause a two-order-of-magnitude change in k2 while resulting in only a one order-of-magnitude change in R, beta, omega, and k1. The forward rates for the metals are found to be two to three orders-of-magnitude greater than the corresponding backward rate. However, the difference between the two rates reduces with increasing pore-water velocity. Model results also suggest that Cd and Zn behave similarly, while Pb is much more strongly sorbed. At input concentrations of about 4 mg/l and pore-water velocities of 3-60 m/day in the groundwater within alluvial gravel, this study suggests retardation factors of 26-289 for Cd, 24-255 for Zn, and 322-6377 for Pb.     

Use of tracer tests to investigate changes in flow and transport properties due to bioclogging of porous media

Tracer tests were conducted in three laboratory columns to study changes in the hydraulic properties of a porous medium due to bioclogging. About 30 breakthrough curves (BTCs) for each column were obtained. The BTCs were analyzed using analytical equilibrium and dual-porosity models, and estimates of the hydrodynamic dispersion and mass transfer coefficients were obtained by curve fitting. The change in transport properties developed in three stages: an initial phase (I) with no significant changes in transport properties, phase II with growth of biomass near the inlet of the columns causing changes in dispersivity, and phase III with added growth of micro-colonies deeper in the columns causing mass transfer of solutes from the water phase to the biophase. Tracer transport changed from being uniform to more non-uniform with increase in mass transfer of the tracer between the mobile phase and the immobile biomass. An increase in the bulk dispersivity value of up to one order of magnitude was observed. Numerical simulations suggest that local dispersivity values may be as much as 40 times higher in the more severe clogged areas inside the column. The bulk hydraulic conductivities of the columns decreased by up to three orders of magnitude. The hydraulic conductivity and dispersivity parameters were almost recovered after disinfection of the columns. Different models relating the changes of the hydraulic conductivity to the changes in the mobile porosity due to bioclogging were reviewed, and the micro-colony relation of Thullner et al. [Thullner, M., Zeyer, J., Kinzelbach, W., 2002. Influence of microbial growth on hydraulic properties of pore networks, Transport in Porous Media, 49, 99-122.] was found to best describe the relation between the bulk hydraulic parameters.     

Modelling of diffusion-limited retardation of contaminants in hydraulically and lithologically nonuniform media

A new reactive transport modelling approach and examples of its application are presented, dealing with the impact of sorption/desorption kinetics on the spreading of solutes, e.g. organic contaminants, in groundwater. Slow sorption/desorption is known from the literature to be strongly responsible for the retardation of organic contaminants. The modelling concept applied in this paper quantifies sorption/desorption kinetics by an intra-particle diffusion approach. According to this idea, solute uptake by or release from the aquifer material is modelled at small scale by a "slow" diffusion process where the diffusion coefficient is reduced as compared to the aqueous diffusion coefficient due to (i) the size and shape of intra-particle pores and (ii) retarded transport of solutes within intra-particle pores governed by a nonlinear sorption isotherm. This process-based concept has the advantage of requiring only measurable model parameters, thus avoiding fitting parameters like first-order rate coefficients.In addition, the approach presented here allows for modelling of slow sorption/desorption in lithologically nonuniform media. Therefore, it accounts for well-known experimental findings indicating that sorptive properties depend on (i) the grain size distribution of the aquifer material and (ii) the lithological composition (e.g. percentage of quartz, sandstone, limestone, etc.) of each grain size fraction. The small-scale physico-chemical model describing sorption/desorption is coupled to a large-scale model of groundwater flow and solute transport. Consequently, hydraulic heterogeneities may also be considered by the overall model. This coupling is regarded as an essential prerequisite for simulating field-scale scenarios which will be addressed by a forthcoming publication.This paper focuses on mathematical model formulation, implementation of the numerical code and lab-scale model applications highlighting the sorption and desorption behavior of an organic contaminant (Phenanthrene) with regard to three lithocomponents exhibiting different sorptive properties. In particular, it is shown that breakthrough curves (BTCs) for lithologically nonuniform media cannot be obtained via simple arithmetic averaging of breakthrough curves for lithologically uniform media. In addition, as no analytical solutions are available for model validation purposes, simulation results are compared to measurements from lab-scale column experiments. The model results indicate that the new code can be regarded as a valuable tool for predicting long-term contaminant uptake or release, which may last for several hundreds of years for some lithocomponents. In particular, breakthrough curves simulated by pure forward modelling reproduce experimental data much better than a calibrated standard first-order kinetics reactive transport model, thus indicating that the new approach is of high quality and may be advantageously used for supporting the design of remediation strategies at contaminated sites where some lithocomponents and/or grain size classes may provide a long-term pollutant source.     

Reaction-enhanced mass transfer and transport from non-aqueous phase liquid source zones

The objective of the following research is to theoretically quantify the enhancement of interphase mass transfer of dissolved non-aqueous phase liquid (NAPL) compounds from the non-aqueous phase to the aqueous phase and the enhancement of dispersive mass transport from a NAPL zone due to destruction of dissolved NAPL compounds. For relatively slow reaction rates, such as for permanganate and perchloroethene (PCE), local-scale mass transfer enhancement is expected to be small. Dispersive mass transport with reaction from a horizontal NAPL zone can be quantified using equations derived for a mathematically equivalent falling film reactor system. In contrast to local-scale interphase mass transfer, dispersive mass transport from NAPL zones may be significantly increased by reaction. Enhancement factors due to destruction of the NAPL compound(s) are mainly dependent on NAPL solubility and oxidant concentration and to a lesser extent on reaction rate, stoichiometry, and transverse dispersion coefficients. Higher NAPL solubility and/or lower oxidant concentration reduces the maximum expected enhancement factor. Reaction enhancement factors for mass transport from NAPL zones are expected to be in the range of 5-50 for permanganate and chlorinated solvents. Theoretical results suggest that assuming instantaneous reaction rates may be appropriate for dispersive mass transport from NAPL zones.     

A mass transfer based method for measuring the reaction coefficients of a photocatalyst

We have developed a new method, the mass transfer based (MTB) method, for measuring the Langmuir–Hinshelwood (L–H) rate form reaction coefficients of photocatalysts. The conventional method for determining the reaction coefficients disregards the effect of mass transfer on the reaction surface by designing and controlling a reaction process to be reaction-limited. In contrast, the new MTB method takes the mass transfer effect into account by using a computational fluid dynamics (CFD) method. The reaction coefficients can be regressed by the measured reaction rates and the calculated VOC concentrations in the air adjacent to the reaction surface. Thus, by using the new method, the reaction coefficient of a reaction process can be accurately determined even if it is not reaction-limited. This is very important in cases where it is difficult to realize reaction-limited processes, such as photocatalytic oxidation of VOCs with strong UV radiation intensity. The relative error of the regressed reaction coefficients obtained by the new method is analyzed. To illustrate, we apply this method to measuring the reaction coefficients of TiO₂ photocatalytic decomposing formaldehyde. This method is very useful in determining the reaction coefficients of the photocatalytic oxidation of various VOCs simultaneously.     

Second-order modeling of arsenite transport in soils

Rate limited processes including kinetic adsorption-desorption can greatly impact the fate and behavior of toxic arsenic compounds in heterogeneous soils. In this study, miscible displacement column experiments were carried out to investigate the extent of reactivity during transport of arsenite in soils. Arsenite breakthrough curves (BTCs) of Olivier and Windsor soils exhibited strong retardation with diffusive effluent fronts followed by slow release or tailing during leaching. Such behavior is indicative of the dominance of kinetic retention reactions for arsenite transport in the soil columns. Sharp decrease or increase in arsenite concentration in response to flow interruptions (stop-flow) further verified that non-equilibrium conditions are dominant. After some 40-60 pore volumes of continued leaching, 30-70% of the applied arsenite was retained by the soil in the columns. Furthermore, continued arsenite slow release for months was evident by the high levels of residual arsenite concentrations observed during leaching. In contrast, arsenite transport in a reference sand material exhibited no retention where complete mass recovery in the effluent solution was attained. A second-order model (SOM) which accounts for equilibrium, reversible, and irreversible retention mechanisms was utilized to describe arsenite transport results from the soil columns. Based on inverse and predictive modeling results, the SOM model successfully depicted arsenite BTCs from several soil columns. Based on inverse and predictive modeling results, a second-order model which accounts for kinetic reversible and irreversible reactions is recommended for describing arsenite transport in soils.     

Biodegradation of hydrocarbons vapors: Comparison of laboratory studies and field investigations in the vadose zone at the emplaced fuel source experiment, Airbase Vaerløse, Denmark

The natural attenuation of volatile organic compounds (VOCs) in the unsaturated zone can only be predicted when information about microbial biodegradation rates and kinetics are known. This study aimed at determining first-order rate coefficients for the aerobic biodegradation of 13 volatile petroleum hydrocarbons which were artificially emplaced as a liquid mixture during a field experiment in an unsaturated sandy soil. Apparent first-order biodegradation rate coefficients were estimated by comparing the spatial evolution of the resulting vapor plumes to an analytical reactive transport model. Two independent reactive numerical model approaches have been used to simulate the diffusive migration of VOC vapors and to estimate degradation rate coefficients. Supplementary laboratory column and microcosm experiments were performed with the sandy soil at room temperature under aerobic conditions. First-order kinetics adequately matched the lab column profiles for most of the compounds. Consistent compound-specific apparent first-order rate coefficients were obtained by the three models and the lab column experiment, except for benzene. Laboratory microcosm experiments lacked of sensitivity for slowly degrading compounds and underestimated degradation rates by up to a factor of 5. Addition of NH3 vapor was shown to increase the degradation rates for some VOCs in the laboratory microcosms. All field models suggested a significantly higher degradation rate for benzene than the rates measured in the lab, suggesting that the field microbial community was superior in developing benzene degrading activity.     

Successive identification of biodegradation rates for multiple sequentially reactive contaminants in groundwater.

At the field scale, the biodegradation rate is usually estimated from analytical solutions to single species transport with first-order reactions, using measured data as input. Because many contaminants, e.g., chlorinated solvents, are degraded in a sequential pattern, with degradation products further reacting to produce new species, it is of great interest to quantify the transformation rate of every reaction. The conventional inverse solutions for identifying the transformation rates are limited to single species problems. In the present study, we propose a successive optimization approach to identify the biodegradation rate for each species by using a previously developed analytical solution to multi-species first-order reactive transport using data obtained at the field scale. By specifying a link between analytical solutions to sequentially reactive transport problems and optimization methods and assuming constant transport parameters (velocity, dispersivities, and retardation factors), the first-order transformation rates are optimized successively from parent species to its daughter species.     

Modeling in-situ uranium(VI) bioreduction by sulfate-reducing bacteria

We present a travel-time based reactive transport model to simulate an in-situ bioremediation experiment for demonstrating enhanced bioreduction of uranium(VI). The model considers aquatic equilibrium chemistry of uranium and other groundwater constituents, uranium sorption and precipitation, and the microbial reduction of nitrate, sulfate and U(VI). Kinetic sorption/desorption of U(VI) is characterized by mass transfer between stagnant micro-pores and mobile flow zones. The model describes the succession of terminal electron accepting processes and the growth and decay of sulfate-reducing bacteria, concurrent with the enzymatic reduction of aqueous U(VI) species. The effective U(VI) reduction rate and sorption site distributions are determined by fitting the model simulation to an in-situ experiment at Oak Ridge, TN. Results show that (1) the presence of nitrate inhibits U(VI) reduction at the site; (2) the fitted effective rate of in-situ U(VI) reduction is much smaller than the values reported for laboratory experiments; (3) U(VI) sorption/desorption, which affects U(VI) bioavailability at the site, is strongly controlled by kinetics; (4) both pH and bicarbonate concentration significantly influence the sorption/desorption of U(VI), which therefore cannot be characterized by empirical isotherms; and (5) calcium-uranyl-carbonate complexes significantly influence the model performance of U(VI) reduction.     

Transport of particulate material and dissolved tracer in a highly permeable porous medium: comparison of the transfer parameters

We are experimentally studying, by means of short-pulse injection, the transport and deposition kinetics of suspended particles (silts of the order of 10 microm) in a highly permeable medium consisting of a column of gravel. In our experiments, the breakthrough curves (BTCs) are well described by analytical solutions of a convection/dispersion model with first-order deposition kinetics. All the transport parameters calculated by the model for both particles and dissolved tracer depend on the flow rate. We demonstrate the existence of a critical flow rate, determined experimentally, beyond which the transfer time for the particles is longer than that for the tracer. This phenomenon is unusual in comparison with the results available in the literature. The increase in transfer time of particles in comparison to tracer leads us to assume a purely mechanical phenomenon, that is, collision between particles and grains of the medium with instantaneous reset in motion when the flow rate is sufficient to avoid settling. Thanks to the polydispersivity of the injected suspension and the control of grain size at the outlet, it can also be determined that the coarser particles are recovered before the finer particles, as expected when one considers the size-exclusion effect.     

Effect of water content on transient nonequilibrium NAPL-gas mass transfer during soil vapor extraction

The effect of water content on the volatilization of nonaqueous phase liquid (NAPL) in unsaturated soils was characterized by one-dimensional venting experiments conducted to evaluate the lumped mass transfer coefficient. An empirical correlation based upon the modified Sherwood number, Peclet number, and normalized mean grain size was used to estimate initial lumped mass transfer coefficients over a range of water content. The effects of water content on the soil vapor extraction SVE process have been investigated through experimentation and mathematical modeling. The experimental results indicated that a rate-limited NAPL-gas mass transfer occurred in water-wet soils. A severe mass transfer limitation was observed at 61.0% water saturation where the normalized effluent gas concentrations fell below 1.0 almost immediately, declined exponentially from the initiation of venting, and showed long tailing. This result was attributed to the reduction of interfacial area between the NAPL and mobile gas phases due to the increased water content. A transient mathematical model describing the change of the lumped mass transfer coefficient was used. Simulations showed that the nonequilibrium mass transfer process could be characterized by the exponent beta, a parameter which described the reduction of the specific area available for NAPL volatilization. The nonequilibrium mass transfer limitations were controlled by the soil mean grain size and pore gas velocity, were well described by beta values below 1.0 at low water saturation, and were well predicted with beta values greater than 1.0 at high water saturation.     

Assessment of intrinsic bioremediation of a coal-tar-affected aquifer using Two-dimensional reactive transport and Biogeochemical mass balance approaches.

Expedited site characterization and groundwater monitoring using direct-push technology and conventional monitoring wells were conducted at a former manufactured gas plant site. Biogeochemical data and heterotrophic plate counts support the presence of microbially mediated remediation. By superimposing solutions of a two-dimensional reactive transport analytical model, first-order degradation rate coefficients ((day-1) ) of various compounds for the dissolved-phase plume were estimated (i.e., benzene [0.0084], naphthalene [0.0058], and acenaphthene [0.0011]). The total mass transformed by aerobic respiration, nitrate reduction, and sulfate reduction around the free-phase coal-tar dense-nonaqueous-phase-liquid region and in the plume was estimated to be approximately 4.5 kg/y using a biogeochemical mass-balance approach. The total mass transformed using the degradation rate coefficients was estimated to be approximately 3.6 kg/y. Results showed that a simple two-dimensional analytical model and a biochemical mass balance with geochemical data from expedited site characterization can be useful for rapid estimation of mass-transformation rates.     

A method for estimating distributions of mass transfer rate coefficients with application to purging and batch experiments

Mass transfer between aquifer material and groundwater is often modeled as first-order rate-limited sorption or diffusive exchange between mobile zones and immobile zones with idealized geometries. Recent improvements in experimental techniques and advances in our understanding of pore-scale heterogeneity demonstrate that two (or even a few) rate coefficients are insufficient in many cases. Here, we investigate a piece-wise linear model for a continuous distribution of rate coefficients, that has several advantages over previously used ‘statistical' distribution models (with functional form from gamma or lognormal PDF's): (1) distributions of arbitrary, even bimodal, shapes can be represented; (2) linear estimation methods can be applied to determine the distribution from experimental data; (3) the uncertainty in the distribution can be determined for each of its sections; and (4) the relationship between the time scales of available data and those of estimatable mass transfer processes can be investigated. A statistical model refinement algorithm is presented that reduces the number of parameters (sections of the piece-wise linear model) to the admissible minimum. We show that purging experiments allow estimation of a wider zone of the rate distribution than do batch experiments, and hence will provide predictions that are accurate over a wider range of time scales. Finally, in an application to TCE gas-purging desorption data, the piece-wise linear rate-distribution model has a higher probability of being adequate than those using a gamma or lognormal distribution or a single rate coefficient.