Journal of Contaminant Hydrology, 1 (1987) 375-393 375 Elsevier Science Publishers B.V., Amsterdam - - Pr inted in The Nether lands

THE INFLUENCE OF MASS TRANSFER ON SOLUTE TRANSPORT IN COLUMN EXPERIMENTS WITH AN AGGREGATED SOIL

PAUL V. ROBERTS 1, MARK N. GOLTZ 1, R. SCOTT SUMMERS 1, JOHN C. CRITTENDEN 2 and PETER NKEDI-KIZZA 3

1Department of Civil Engineering, Stanford University, Stanford, CA 94305, U.S.A. 2 Department of Civil Engineering, Michigan Technological University, Houghton, MI 49931, U.S.A. 3 Soil Science Department, University of Florida, Gainesville, FL 32611, U.S.A.

(Received October 31, 1986; revised and accepted February 16, 1987)

ABSTRACT

Roberts, P.V., Goltz, M.N., Summers, R.S., Crit tenden, J.C. and Nkedi-Kizza, P., 1987. The influence of mass t ransfer on solute t ranspor t in column experiments with an aggregated soil. J. Contam. Hydrol., 1: 375-393.

The spreading of concentra t ion fronts in dynamic column experiments conducted with a porous, aggregated soil is analyzed by means of a previously documented t ranspor t model (DFPSDM) tha t accounts for longitudinal dispersion, external mass t ransfer in the boundary layer surrounding the aggregate particles, and diffusion in the intra-aggregate pores. The data are drawn from a previous report on the t ranspor t of t r i t ia ted water, chloride, and calcium ion in a column filled with Ione soil having an average aggregate particle diameter of 0.34 cm, at pore water velocities from 3 to 143 cm/h. The parameters for dispersion, external mass transfer, and internal diffusion were predicted for the experimental conditions by means of generalized correlations, independent of the column data. The predicted degree of solute front-spreading agreed well with the experimental observations. Consistent with the aggregate porosity of 45%, the tortuosity factor for in ternal pore diffusion was approximately equal to 2. Quant i ta t ive cri teria for the spreading influence of the three mechanisms are evaluated with respect to the column data. Hydrodynamic dispersion is thought to have governed the front shape in the experiments at low velocity, and in ternal pore diffusion is believed to have dominated at high velocity; the external mass t ransfer resis tance played a minor role under all conditions. A t ranspor t model such as DFPSDM is useful for in terpre t ing column data with regard to the mechanisms control l ing concentra t ion front dynamics, but care must be exercised to avoid confounding the effects of the re levant processes.

INTRODUCTION

Measuring and interpreting the concentration responses of laboratory col- umns filled with a porous medium of interest has frequently been used to gain insight into the processes governing solute transport in the medium, as well as to evaluate parameters needed for mathematical modeling. In dealing with solutes that sorb on the porous medium, it is necessary to consider the solute's

0169-7722/87/$03.50 © 1987 Elsevier Science Publishers B.V.

376

retardation, together with advection and dispersion (Bear, 1979). Local equili- brium between the pore water and the sorbing stat ionary phase is commonly assumed as an expedient in data interpretation and modeling (Bear, 1979), al though there is ample evidence of deviations from local equilibrium in some studies (van Genuchten and Wierenga, 1977). Deviations from local equili- brium have been characterized from two basic points of view: as a quasi- chemical kinetic rate process (Coats and Smith, 1964), or as a diffusion process (Rasmuson and Neretnieks, 1980). Kinetic limitation can be postulated based on knowledge of a specific, slow reaction step. Diffusive limitation can be anticipated where the medium is aggregated or stratified, such that the charac- teristic length scale of separation between the pore fluid and the sorption sites is appreciable, and the resulting time constant for diffusive mass transfer is not small compared to the advection time constant (Goltz and Roberts, 1986).

In principle, the effluent concentration response of a soil column following a defined input stimulus can be predicted if the medium and its interactions with the solute are well characterized. However, this goal has proven some- what elusive, because models have been lacking that account for all of the potentially relevant transport processes simultaneously. To alleviate this shortcoming, Crittenden et al. (1986) have developed a family of numerical models for computing solute transport in a quasi-homogeneous porous medium consisting of spherical porous aggregates packed into a one-dimensional, semi- infinite column. One of that family of models - - the dispersed flow, pore and surface diffusion model ( D F P S D M ) - - accounts for longitudinal dispersion, as well as external and internal mass transfer limitation. External mass transfer refers to the transport through the fluid boundary layer surrounding the particles, whereas the internal mass transfer is treated as the sum of diffusive transport in the internal pore fluid and along the surface of the internal pore walls, as illustrated in fig. 2 of Crittenden et al. (1986).

The DFPSDM is potentially well-suited for interpreting the results of column studies with aggregated soils, a possibility explored by Hutzler et al. (1986) by applying the model to interpret results of miscible displacement experiments with chloride ion, trichloroethylene, and bromoform in columns filled with a sandy loam. Hutzler et al. (1986) were able to simulate the results of their column experiments by optimizing the values of parameters to improve the fit. However, utilizing parameter values predicted from independent correlations, they could not account for the degree of front spreading observed in their column experiments, and were compelled to assume aggregation of the soil particles on a scale larger than was consistent with the physical properties of the porous medium.

It is the objective of this paper to apply the DFPSDM transport model to interpret the results of a set of miscible displacement experiments conducted with an aggregated soil, namely the data of Nkedi-Kizza et al. (1982), employing to the maximum extent parameter values predicted independent of the column data. Further, this paper aims to evaluate the relative contributions of lon- gitudinal dispersion, external mass transfer, and intra-aggregate diffusion to

377

the solute f ront -spreading in the soil co lumn experiments of Nkedi-Kizza et al. (1982).

SOIL COLUMN EXPERIMENTS

Nkedi-Kizza et al. (1982) presented results of soil co lumn experiments con- ducted over a wide range of veloci t ies for d isplacement of four radio-labelled solutes t h r o u g h a sa tu ra t ed porous medium consis t ing of an aggrega ted Oxisol. This paper considers the resul ts of seven of the experiments repor ted by Nkedi- Kizza et al. (1982): namely, the i r exper iments 1-3 (3H20); 8 and 9 (36C1); and 10 and 11 (45Ca), which represent a range of sorpt ion behavior .

The condi t ions are summarized in Table 1 of Nkedi-Kizza et al. (1982); only those values necessary for the model computa t ions are repeated here (Table 1), us ing nomenc l a tu r e cons is tent with Cr i t tenden et al. (1986) with sl ight modifi- cations. For all experiments, the average aggrega te d iameter was 0.34cm (range: 0.2 to 0.47 cm); it has been shown tha t nonun i fo rm porous media can be adequa te ly modeled as beds of equivalent uniform spheres in predic t ing the effects of in t ra -aggrega te diffusion (Rao e t al., 1982; Rasmuson, 1985) and dispersion (Han et al., 1985) on the dynamic behavior , if the range of par t ic le sizes does not exceed a fac tor of two. The bed diameter was 7.6 cm, giving a bed-to-part icle d iameter ra t io of g rea te r t han 16:1, which is usual ly ample to avoid serious wall effects if the co lumn is careful ly packed. The average pore water veloci t ies ranged from 3 to 143cm/h. Sorpt ion equil ibrium isotherms were repor ted by Nkedi-Kizza et al. (1982) to be linear, with d is t r ibut ion coefficients, K D, equal to 0.03, 0.44, and 1.37ml/g, respectively, for tr i t ium, chloride, and calcium. However , it mus t be noted tha t the sorpt ion equil ibrium for the ionic solutes could not be determined independent ly in ba tch equili- br ium experiments, because of compl ica t ions ar is ing from pH changes, and hence the KD values had to be eva lua ted by fi t t ing a por t ion of the co lumn data (Nkedi-Kizza et al., 1982). The input s t imulus in all experiments was in the form of a broad pulse, r ang ing in length, Ppv, from 2.87 to 5.32 pore volumes.

TABLE 1

Experimental conditions for the experiments of Nkedi-Kizza et al. (1982) analyzed in this work

Exp. Solute K D C O PB ~B L v PPv No. (cm a/g) (mM) (g/cm 3) ( ) (cm) (cm/h) ( - )

1 3H20 0.03 1.0 × 10 -7 1.05 0.40 10.1 15 2.87 2 3H20 0.03 1.0 × 10 -7 1.05 0.40 10.1 68 2.87 3 aH20 0.03 1.0 × 10 -7 1.05 0.40 10.1 143 2.87 8 ~C1 0.44 1.0 × 10 -6 1.01 0.41 6.0 3.0 5.32 9 ~6CI 0.44 1.0 × 10 _6 1.01 0.41 6.0 121 4.78

10 45Ca 1.37 1,0 × 10 _6 1.01 0.41 6.0 3.2 5.32 11 45Ca 1.37 1.0 × 10 _6 1.01 0.41 6.0 127 5.32

Data common to all experiments: T -~ 20°C; a A ~ 0.175cm; PA = 1.76g/cm~; ~A = 0.45; d B = 7.6cm,

378

TRANSPORT MODEL

The DFPSDM transport model proposed by Crittenden et al. (1986) accounts for advection and longitudinal dispersion of a sorbing solute in a uniform, semi-infinite porous medium under conditions of one-dimensional flow, and allows for mass transfer limitation in the external fluid boundary layer sur- rounding the solid grains as well as internal diffusion. A set of simultaneous, nonlinear partial differential equations is solved numerically by orthogonal collocation (Finlayson, 1980).

In the present work, the equations were solved on a microcomputer (IBM- PC/AT) using 4 radial (i.e., intra-aggregate) and 18 axial collocation points. The accuracy of the computational procedure, including the absence of numeri- cal dispersion and instability, was verified as follows: the results of the or- thogonal collocation computations were compared with the results obtained from analytical solutions that are available for simplified cases in which only one spreading mechanism (i.e., dispersion, external mass transfer limitation, or internal diffusion limitation) is operative; these tests were performed under conditions chosen to conform to the range of spreading behavior represented in Fig. 1.

SPREADING CRITERIA

Crittenden et al. (1986) suggested that criteria can be established for compar- ing the relative contributions of longitudinal dispersion, external mass trans- fer resistance, and internal diffusion resistance to the spreading of the con- centration wave during transport through the porous medium. Based on a

comparison of analytical solutions for limiting cases, Crittenden et al. (1986) proposed the following equivalence criteria:

(Pe.D) -1 = [3(1 + 1/Dc)2St] -~ = [15(1 + 1/DG)2E, MT] ~ (1)

where SHD = (PenD) -1 represents the spreading contribution of longitudinal hydrodynamic dispersion; SEM T = [3(1 + 1/Da) St] -1 represents the contribu- tion of the external mass transfer resistance; and SIMT = [15(1 + 1/DG) 2 EIMT] 1 represents the contribution of the intra-aggregate diffusional resistance.

In eq. (1), PeHD is the dimensionless Peclet Number, characterizing longi- tudinal dispersion in the inter-aggregate pores, i.e. the mobile domain, and is defined as:

PeHD = vL/EnD (la)

where v is the average interstitial velocity in the inter-aggregate pores (L/t); L is the column length (L); and EHD is the longitudinal hydrodynamic dispersion coefficient (L2/t). Pent) can be understood as the ratio of advective to dispersive transport.

Da in eqn. (1) is the solute distribution ratio, which corresponds to the ratio of the mass of solute within the aggregate grains (including the mass in the intra-aggregate pore fluid as well as that sorbed) to the mass in the inter-

379

aggregate solution, evaluated at equilibrium with the influent concentration:

DG = (1 - ~B)(SAC0 + PAqE) (lb) ~BC0

and e~ is the fraction of the bed volume consisting of inter-aggregate pores; eA is the fraction of the aggregate volume consisting of internal pores; PA is the aggregate density, i.e., the mass per unit volume of total aggregate (M/L3); qE is the mass of solute sorbed per unit mass of soil, in equilibrium with C o (M/M); and Co is the influent concentration (M/L3).

St in eqn. (1) is the dimensionless Stanton Number, which is the ratio of the rate of external mass transfer into the aggregates to the rate of advective transport through the porous medium

St = (1 - ~B)kEMTL/(~BVaA) (lc)

where kEM w is the mass transfer coefficient for the external boundary layer surrounding the aggregate (L/t); and aA is the aggregate grain radius (L).

EIM T in eqn. (1) is the intra-aggregate diffusion modulus, the ratio of the rate of diffusional mass transfer within the aggregate to the advection rate in the inter-aggregate macropores:

EIM w = LDpDG/Va2A (ld)

where Dp is the intra-aggregate pore diffusion coefficient (L2/t):

De = DL/~p (le)

DL is the aqueous diffusivity of the solute (L2/t), and re is the tortuosity factor (dimensionless), which accounts for the irregularity of the intra-aggregate pore structure. Equation (ld) accounts for only the pore diffusion contribution to intra-aggregate transport; surface diffusion is neglected in the present work on the grounds that pore diffusion is expected to dominate internal transport where the sorptive interactions are relatively weak owing to low specific surface area (Komiyama and Smith, 1974) as in the systems treated here.

Equality of the members of eqn. (1) implies equal contributions to spreading of the effluent concentrat ion response from dispersion, (PeHD)-I; external mass transfer resistance, [3(1 + 1/DG)2St]-'; and internal mass transfer resistance, [15(1 + 1/DG)2EIMT]-I; respectively. This conclusion, reached by Crittenden et al. (1986) by inspection of asymptotic solutions corresponding to relatively sharp fronts (e.g., PeHD > 40), is limited because of the restriction that the column be long relative to the length of the front, a condition not always satisfied in soil column experiments. However, the conclusions are consistent with the more general analysis based on comparing the influence of dispersion, first-order rate limitation, and diffusion into spherical aggregates, respective- ly, upon the second time moment of a column's effluent concentration history (Valocchi, 1985; Goltz, 1986; Parker and Valocchi, 1986).

Examination of table 2 of Valocchi (1985), after considering differences in

380

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nomenclature, reveals that for a model which incorporates the mechanisms of hydrodynamic dispersion and intra-aggregate diffusion, if the terms SHD and SIM T a r e equal, each mechanism makes an equal contribution to the second time moment. A further comparison can be drawn by noting that the external mass transfer resistance incorporated in the DFPSDM model (Crittenden et al., 1986) is tantamount to a first-order rate limitation. Examination of table 2 of Valocchi (1985) also shows that for a model which incorporates the mechanisms of hydrodynamic dispersion and a first-order rate limitation, if the terms SHD and SEM T a r e equal, each mechanism makes an equal contribution to the second time moment. As a more general corollary, it can be stated that for equal values of SHD, SIMT, and SEMT, each of the three mechanisms (hydrodynamic dispersion, intra-aggregate diffusional resistance, and external mass transfer resistance) contributes equally to the second temporal moment of a column's effluent concentrat ion history. Significantly, these conclusions are independent of the front length relative to the column length, i.e., independent of the effective Peclet number.

Parker and Valocchi (1986) performed an analysis similar to the discussion above, whereby they used the formulae in table 2 of Valocchi (1985) as a point of departure to derive equivalent model parameters which yielded equal second temporal moments. The model parameter equivalence criteria derived by Parker and Valocchi (1986), based on temporal moment equivalence, are direct- ly analogous to the equal spreading criteria proposed by Crittenden et al. (1986), but Parker and Valocchi's analysis is more general, in that there is no restriction on Peclet number. On the other hand, the DFPSDM model of Critten- den et al. (1986) permits the calculation of the full concentration response.

It is not surprising that mass transfer limitations associated with sorption in porous aggregates affect the spreading in a manner similar to longitudinal dispersion, analogous to the increase in apparent dispersion owing to slow interchange with fluid trapped in stagnant pockets (Aris, 1959). The degree of spreading increases with increasing values of the members in eqn. (1). For example, as the hydrodynamic dispersion coefficient, EMD, increases, (Pe~D) 1 increases, leading to increased spreading. A decrease in the external mass transfer coefficient, kEMT, or the pore diffusion coefficient, Dp, leads to a propor- t ionate increase in the corresponding spreading parameter, because of the slower communication with the intra-aggregate domain.

Parker and Valocchi (1986) have explored the concept of equivalent effects of longitudinal dispersion and spherical diffusion on spreading of concentra- tion responses, but the effects of external mass transfer were not included. To illustrate the effect of the spreading criteria, including that for the external resistance, on the shape of the concentration fronts, several examples are shown in Fig. 1; the examples correspond to spreading criteria values, S, equal to (a) 0.01; (b) 0.03; (c) 0.07; and (d) 0.7. The conditions of column dimensions, flow rate, and pulse width are consistent with the experimental conditions used by Nkedi-Kizza et al. (1982), specifically their Experiment 3, but the values of the spreading parameters are chosen arbitrarily to illustrate the effects of spreading caused by the different phenomena.

382

Each curve in Fig.1 is computed considering only one of the three spreading phenomena, and neglecting the other two. The shapes of the concentration responses are virtually identical for small values of the spreading parameter, S = 0.01 and 0.03 (PeHD ~ 100 and 33), as shown in Figs. 1A and lB. However, for S = 0.07 (PeHD = 14), minor differences arise, and for S = 0.7 (PeHD = 1.4) there are major differences in the forms of the concentration responses corre- sponding to the different spreading mechanisms (Figs. 1C and 1D). Hydro- dynamic dispersion causes more spreading at the leading edge of the response, whereas the internal diffusion limitation elicits more extended tailing; the effect of external mass transfer limitation is generally intermediate between those of dispersion and internal mass transfer limitation. These differences in the detailed form of the concentration responses arise despite the fact that the second temporal moments of the three simulations at a given value of S are by definition identically equal. The qualitative differences noted here would ap- pear relatively more pronounced if a narrower pulse width had been chosen.

Given the similarity of the forms of the concentration responses when S < 0.1 (PeHD > 10) - - illustrated in Figs. 1A, 1B, and 1C - - it is not easy to identify the mechanism of spreading from column data based on curve shape alone. This is consistent with the proposal that the effects of mass transfer limitation can be simulated by adjustment of the hydrodynamic dispersion coefficient (Passioura, 1971; Rao et al., 1980; DeSmedt and Wierenga, 1984; Valocchi, 1985), at least where the degree of spreading is small to moderate. However, the latter adaptation of lumping the effects of mass transfer resis- tance into the dispersion parameter should be avoided if possible, and can lead to severe distortions of the concentration response where the degree of spread- ing is large, i.e., S > 0.1 (PeHD < 10).

INPUT PARAMETERS

All required input parameters for model predictions can be obtained from the direct measurements of Nkedi-Kizza et al. (1982), without calibration using their column data, except the sorption distribution coefficient, KD, the hy- drodynamic dispersion coefficient, EHD, the external mass transfer coefficient, kEMT, and the pore diffusion coefficient for intra-aggregate mass transfer, D e. The latter three parameters - - EHD , kEMT, and D p - - are evaluated in this section employing independent, generalized correlations, as described below, and the values are summarized in Table 2. The values ~. "~D cannot be predicted in like manner, however, and we resort to using the values estimated by Nkedi-Kizza et al. (1982), obtained by interpretation of a portion of the column data (Table 1). It must be recognized that sorption equilibria for binding of ionic solutes to charged natural solids such as oxisols are strongly pH-depen- dent (Nkedi-Kizza et al., 1982, 1984), and much too complex to permit prediction by means of generalized correlations. Quantification of the partit ioning by interpretation of column experiments conducted at low velocity is likely to furnish a more accurate estimate than any correlation, and a more represen- tative estimate than direct measurement in batch experiments.

383

The external mass transfer coefficient, kEMT, is evaluated according to the correlation of Wilson and Geankoplis (1966) for Darcy flow in packed beds of uniform spheres, identical to the procedure recommended by Crittenden et al. (1986):

kEM w : 1.09DLeB(Re)I/3(Sc) '/3 (2)

where Re is the dimensionless Reynolds Number based on the grain diameter and the average interstit ial velocity, Re = 2aGv/v, Sc is the dimensionless Schmidt Number, Sc = V/DL, and v is the kinematic viscosity of water, v = ~ 0.01 cm2/s at 20°C. Equation (2) is a semi-empirical relation for external mass transfer in packed beds, based on boundary layer theory and calibrated in the range 1.6 × 10 -3 < (eB Re) < 55 by Wilson and Geankoplis (1966), who reported the precision to be + 6.7%; however, greater deviations must be expected for nonuniform, nonspherical grains (Roberts et al., 1985). The Rey- nolds Number values for the experiments considered here are in the range from 0.03 to 1.4, well within the scope of validity of eqn. (2).

The longitudinal hydrodynamic dispersion coefficient, EHD, is evaluated from a set of generalized correlations that account for the combined effects of molecular diffusion and hydrodynamic dispersion (Fried, 1975; Bear, 1979). The specific relationships used were inferred from fig. 7.4 of Bear (1979):

EHD/D L : 0.67 PeMD < 1 (3a)

EHD/DL = (0.67 + 0.5Pe~) 1 < PeMD < 260 (3b)

EHD/DL = 1.8PeMD 260 < PeMD < 105 (3C)

and depend on the value of the Peclet Number for molecular diffusion, PeMD : 2ac V/DL = Re" Sc. This procedure differs slightly from that recom- mended by Crittenden et al. (1986), who proposed that eqn. (3b) be used in the intermediate range of PeMD , but offered no suggestion for lower or higher values of PeMD. The constant of 0.67 in eqn. (3a) is adopted in accordance with capillary network models for flow in porous media (Saffman, 1960; Fried and Combarnous, 1971); the upper limit for eqn. (3c), which corresponds to the limit of validity of Darcy flow at Re > 10, is chosen in accord with fig. 2.4.2 of Fried (1975). In the experiments considered here, the values of PeMD w e r e in the range 15 to 1500; at the upper end of that range, the difference between eqn. (3b) and (3c) amounts to 20%. The values of EHD range from 2.6 × 10 4 to 2.2 × 10 -2 cm2/s, dependent mainly on the velocity (Table 2).

The effective intra-aggregate pore diffusion coefficient is estimated as a range of values using eqn. (le), assuming a feasible range of tortuosity values, 2 < zp < 10, as recommended by Satterfield (1980) and Cussler (1984) for porous sorbents and catalysts possessing substantial internal, interconnected porosity. The values of the aqueous diffusivity, DL, were estimated as follows." The diffusivity for tr i t iated water was assumed to be approximately that of deuterium hydroxide (Reid and Sherwood, 1958, p. 288) and adjusted for tem- perature according to the ratio T/g, where T is the absolute temperature (K)

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and # is the dynamic viscosity (M/Lt). The diffusivities for Ca 2÷ and C1- were taken from the compilation of aqueous ionic diffusion coefficients in Cussler (1984, p. 147).

PREDICTION OF CONCENTRATION RESPONSES

The predicted responses, computed with the DFPSDM under the assump- tions outlined above and using the data in Table 1 and the predicted dispersion and mass transfer rate parameters in Table 2, are shown in Figs. 2, 3, and 4 for Experiments 1-3 (tritiated water), 8 and 9 (chloride), and 10-11 (calcium), respectively. The graphs show dimensionless concentration, normalized to the influent concentrat ion during the pulse, versus the number of pore volumes eluted. For each experiment, two predictions are shown, corresponding to the high (solid curve) and low (broken curve) end of feasible values of the effective pore diffusion coefficients, Dp. In each case, the high value of D e is calculated assuming ~p = 2, and the low value of Dp assumes Tp - ~ - 10.

In all cases, the data are fairly well bracketed by the two sets of predictions. In most instances (Figs. 2B, 2C, 3B, 4A, and 4B) the data agree better with the prediction corresponding to the high end of the feasible Dp range than with that for the low Dp assumption. The high end of the range of Dp estimates corresponds to Tp = 2.0; this value is consistent with the measured internal porosity of the aggregate grains, eA = 0.45, according to a commonly accepted model for predicting the tortuosity (Wakao and Smith, 1962):

~r = EA 1 (4)

which gives Zp = 2.2 for the aggregate grains considered here. The two exceptions where the data are better matched by the prediction

assuming the low end of the Dp range are experiments 1 and 8 (Figs. 2A and 3A), both conducted at relatively low velocities. From comparison of the spreading parameter values (Table 3), it can be seen that hydrodynamic dispersion is likely to have dominated the spreading of the concentration fronts in those two experiments. In Table 3, values of SHD , SEMI, and SIM w a re given for each experiment, with values for SIM w corresponding to ~p values of 2 and 10. For experiments 1 and 8, the value of SHD exceeds that of SIMT by a factor of 2 to 3; in all other experiments, SIM T exceeds SHD, and internal mass transfer is likely to have dominated.

Hence, the better matching of the data from experiments 1 and 8 using the lower value of Dp, i.e. where Tp = 10, is likely to be an artifact of confounding with the dispersion mechanism. If the value of the dispersion coefficient, EnD, were underestimated by eqns. (3), the predicted response would be biased toward less spreading, which could be largely compensated by choice of a lower Dp value. Indeed, it is probable that the estimates of EHD obtained from eqns. (3) are negatively biased throughout this work, as the aggregated soil used in the experiments considered here was certainly less than ideally homogeneous; differences among the hydraulic conductivities along individual flowpaths are likely to have contributed to dispersion-like spreading beyond that arising

386

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0 2 4

PORE VOLUM r q

Fig. 2. Predicted and observed effluent concentra t ion responses for t r i t ia ted water. (2A) v = 15.1 cm/h (Exp. 1); (2B) v = 67.8 cm/h (Exp. 2); (2C) v = 142.8 cm/h (Exp. 3). Curves represent DFPSDM predictions for high (solid) and low (dashed) end of Dp range; open squares are the data of Nkedi-Kizza et al. (1982).

o

1.1

o 0 O °

• ~

3A

J I r I

2 4 6 B 10

PORE VOLUMES

387

1.1

1

0 . 9

0.8

0 .7

0.6

0.5

0 . 4 -

0 . 3 -

0 . 2 -

0.1 - t

0 0

3B

2 4 6 8 10 12 14

PORE VOLUMES

Fig. 3. Predicted and observed effluent concentration responses for chloride. (3A) v = 3.0 cm/h (Exp. 8); (3B) v = 121.2 cm/h (Exp. 9). Curves represent DFPSDM predictions for high (solid) and low (dashed) end of Dp range; open squares are the data of Nkedi-Kizza et al. (1982).

from m e c h a n i c a l d i spers ion at the sca le of indiv idual grains and pores (Freeze and Cherry, 1979). For example , H a n et al. (1985) observed that the long i tud ina l d i spers ion coeff ic ient w a s approx imate ly doubled by the addi t ion of o n l y 10 v o l u m e percent o f di f ferent ly s ized partic les . The above ra t iona l i za t ion , wh i l e tedious , serves to i l lus trate h o w i n a c c u r a c y in e s t imat ing one of the s p r e a d i n g effects can in terac t to c o n f o u n d the e s t imate o f a n o t h e r o f the parameters , if i n v e s t i g a t o r s are tempted to draw c o n c l u s i o n s regarding process m e c h a n i s m s by m e a n s of curve-f itt ing.

From the v a l u e s in Table 3, it is apparent that the ex terna l mass transfer re s i s tance in no case exerts the p r e d o m i n a n t in f luence o n spreading, and in m o s t cases is near ly an order o f m a g n i t u d e less important t h a n at least o n e o f the o ther factors . Clearly , the neg l ec t o f the ex terna l mass transfer re s i s tance

388

1.1

1

0 . 9

0 .8

0 .7

0 .6

0 .5

0 . 4

0 . 3 -

0 .2

0.1

0

b -

2 4 6 8 10 12 14.

PORE VOLUMES

i i i i i i ~ i i ~ i

2 4 6 S 10 12 14

PORE VOLUMES

Fig. 4. Predicted and observed effluent concentration histories for calcium. (4A) v = 3,2 cm/h (Exp. 10); (4B) v = 127.5 cm/h (Exp. 11). Curves represent DFPSDM predictions for high (solid) and low (dashed) end of/~z range; open squares are the data of Nkedi-Kizza et al. (1982).

1 . 1

1

o .g

0 .8

0 .7

O.S

0 .5

0 .4

0 ,3

0 .2

0,1

D

in the data ana lys i s w a s no t a ser ious o m i s s i o n in the w o r k of Nked i -Kizza et al. (1982). N e v e r t h e l e s s , the spreading parameter v a l u e s for the different mech- a n i s m s in a g iven exper iment do not differ by more than o n e to t w o orders o f magn i tude , and it is sure ly benef ic ia l to employ for data in terpre ta t ion a mode l capable o f a c c o u n t i n g for all three processes .

SENSITIVITY ANALYSIS

The re la t ive effects o f the three m e c h a n i s m s upon spreading m a y be qualita- t ive ly a s c e r t a i n e d by m e a n s o f a sens i t iv i ty analys i s . F igures 5 t h r o u g h 7 s h o w the impact , on b r e a k t h r o u g h curves , o f independent ly v a r y i n g SHD, SrMT, and SEMT over two orders o f magni tude . The base case for the ana lys i s uses the

389

TABLE 3

Summary of predominant influence on spreading

Exp. V DG SHD SEMT ~MT Dominant No. (cm/h) mechanism(s)

~p = 2 vp = 10

1 15 0.74 0.039 0.005 0.016 0 .082 Dispersion and internal diffusion

2 68 0.74 0.052 0.013 0.074 0.37 Internal diffusion and dispersion

3 143 0.74 0.055 0.022 0.16 0.78 Internal diffusion

8 3.0 1.71 0.053 0.008 0.017 0 .087 Dispersion and internal diffusion

9 121 1.71 0.105 0.097 0.71 3.51 Internal diffusion

10 3.2 3.97 0.062 0.026 0.075 0 .382 Internal diffusion and dispersion

11 127 3.97 0.104 0.30 3.0 15.2 Internal diffusion

p r e d i c t e d p a r a m e t e r v a l u e s ( l i s t e d in T a b l e 2) fo r e x p e r i m e n t 3 o f N k e d i - K i z z a e t al . (1982), w i t h t h e e x c e p t i o n t h a t t h e i n t e r n a l d i f f u s i o n coe f f i c i en t is c h o s e n as Dp = 1.27 × 10-5cm2/s , o b t a i n e d u s i n g eqn. ( l e ) w i t h a t o r t u o s i t y re = 2.2

f r o m eqn. (4). T h e b a s e c a s e v a l u e s for SHD, SIMT, a n d SEM T a r e , t h e r e f o r e , 0.055, 0.17, a n d 0.022, r e s p e c t i v e l y . F i g u r e 5 s h o w s t h e e f fec t o f v a r y i n g SHD o v e r a r a n g e b e t w e e n 0.1 a n d 10.0 t i m e s i t s b a s e c a s e v a l u e , w h i l e h o l d i n g SIM T a n d SEM T c o n s t a n t . A n a l o g o u s l y , F igs . 6 a n d 7 s h o w t h e e f fec t s o f p e r t u r b i n g SIM T

a n d SEMT, r e s p e c t i v e l y , b y o n e o r d e r o f m a g n i t u d e in e a c h d i r e c t i o n f rom t h e i r

1 ]

/ 0.6 4" SHD = (~.019

IZq

~ 1:1.2

0.1 x ~ al

0 2 0 ' 4 0 6 0

TIME ( MINUTES ~

Fig. 5. Sensitivity analysis, showing the effect of varying the spreading contribution of hydro- dynamic dispersion (SRD) while holding the other terms constant at ~MT -- 0.17 and S~M T = 0.022.

390

Z 0 . 9 a r S , ~ T = 1 . 7

r ~

• 0 , 6 / ~ SIMT = ~).056

~ 0 .4 0 ~ 0.2,

~ 0 .2

0.1

o 4 - - , , , 0 2 0 4 0 6 0

TIME (MINUTES)

Fig. 6. Sensitivity analysis, showing the effect of varying the spreading contribution of intra- aggregate diffusional resistance (SIMT) while holding the other terms constant at S~D = 0.055 and S~M w = 0.022.

z 0

,Y P- Z LU o Z g U) Cn W .J Z 0

m Z W

1

0 . 9

0 . 8

0 . 7

O . S

0 . 5

0 . 4

0 . 3

0 . 2

°-'tJ 0 i - - 0

F i i

2O

"I" SE~-r = 0.22

~> Se~T = 0. 067

rl SEMT = 0.022

/% SE,~ = 0.00"74

40 S O

TIME (MINUTES)

Fig. 7. Sensitivity analysis, showing the effect of varying the spreading contribution of external mass transfer resistance (SEMT) while holding the other terms constant at SHD = 0.055 and SIM T = 0.17.

base case va lues , ho ld ing the o ther two parameters cons tant . Compar ing the three f igures v i sua l ly , it is apparent that the c o n c e n t r a t i o n r e s p o n s e is m o s t s ens i t i ve to v a r i a t i o n in SIMT, next m o s t s ens i t ive to SHD, and least s ens i t ive to S E M T •

This rank order o f s ens i t i v i ty conforms to e x p e c t a t i o n s formed accord ing to the re la t ive v a l u e s o f the ind iv idual spreading parameter va lues in the base case. It is apparent that the effluent r e sponse is re la t ive ly i n s e n s i t i v e to chang- ing the v a l u e o f a g iven spreading term so l o n g as the r e l e v a n t va lue is exceeded

391

by the term corresponding to one of the other mechanisms. For example, increasing SEM T does not significantly influence the response until SEMT exceeds both Stow and SHD (Fig. 7).

The following generalizations are proposed regarding the conditions for dominance of the spreading behavior by the individual mechanisms. Dispersion tends to dominate when the velocity, v, and the partition parameter, DG, are small. Both mass transfer resistances, external and internal, increase in impor- tance relative to dispersion as the partition parameter increases. Mass transfer resistances also become more important with increased velocity. Between the external and internal mass transfer resistances, the former tends to become more important than the lat ter at low velocities, whereas a larger aggregate particle size leads to a predominance of the internal resistance, as does a high value of the tortuosity. Increasing the bed length, L, influences all of the spreading parameters equally. These conclusions are apparent from the follow- ing proportionality relations for homogeneous beds of porous, spherical aggregates

SHD ~: aAL- ' (PeMD >> 1) (5)

D2n213a2j3v2/3L 1 (DG ~ 1) (6a) SEMT OC G ~ A

~: DL2/3a~3v213L 1 (DG >> 1) (6b)

S~M w oc D~eA1DLI~pa2AvL-1 (DG ~ 1) (7a)

oc ~AIDL-"cpa2AVL I (DG >~ I) (7b)

It is difficult to envision a porous medium in a real subsurface environment that would fulfill the conditions for external mass transfer dominance of spreading; such an environment would have to consist of a highly uniform, fine-grained medium, with little aggregation or layering, and a gradient large enough to generate a high velocity despite the small grain size.

SUMMARY AND CONCLUSION

It has been shown that the spreading of the concentration front of a sorbing solute during transport in a soil column packed with uniform, internally porous, aggregated particles can be successfully predicted with the DFPSDM. Previously published submodels were used to estimate independently the criti- cal parameters for longitudinal hydrodynamic dispersion, external mass trans- fer, and internal diffusion, and the model predictions were compared to previously published data for three solutes over two orders of magnitude of flow rate. The pore diffusion mechanism was found adequate to model the internal transport, without invoking surface diffusion. Judging from the values of the characterist ic spreading parameters, hydrodynamic dispersion domi- nated the front-spreading behavior at the low end of the velocity range (v = 3cm/h), and internal diffusion predominated at higher velocities (v 1> 120cm/h).

392

Interpretation of the functionalities implicit in the governing dimensionless groups leads to the following conclusions regarding dominance by the several mechanisms:

(1) dispersion is likely to dominate for heterogeneous, fine-grained media, especially where the pore water velocity, the scale of aggregation, and the parti t ioning into the aggregate domain are small;

(2) internal diffusion will tend to dominate in media in which the partition parameter and the scale of aggregation are large, especially where the internal transport is hindered by low internal porosity and substantial tortuosity or constriction of the internal pores, and where the pore water velocity is large; and

(3) the external mass transfer limitation is unlikely to govern front- spreading behavior under the conditions encountered in most soil column experiments, and certainly not in real subsurface environments, where heterogeneity, aggregation, and layering are the norm. Nonetheless, the exter- nal mass transfer resistance may contribute to spreading in a secondary role.

Despite the apparent success of this exercise in a posteriori prediction over a substantial range of conditions, the good agreement between model predic- tions and experimental data may be fortuitous. Application to a less uniform porous medium would require a different approach for estimating the disper- sion coefficient, or possibly an entirely different (e.g., stochastic) approach to modeling dispersion. The methodology needs to be verified by application to a wider range of media, solutes, and other conditions, especially at the lower velocities typical of groundwater movement (v = 0.001-1cm/h). These warn- ings notwithstanding, this example strengthens our conviction that the dynamic behavior of solute transport in soils is amenable to prediction, at least where conditions are carefully chosen and controlled.

ACKNOWLEDGMENTS

The computations were conducted in association with CE 372, a graduate course in the Environmental and Water Studies program at Stanford Univer- sity; the participating students, who are too numerous to mention individually, helped inspire this paper. William Ball, Suresh Rao, and Albert Valocchi read and criticized the manuscript before its submission. The manuscript prepara- tion was supported in part by the R.S. Kerr Environmental Research Labo- ratory of the U.S. Environmental Protection Agency under EPACR-808851.

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