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The influence of mass transfer on solute transport in column experiments with an aggregated soil
Journal of Contaminant Hydrology, 1 (1987) 375-393
375
Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands
THE I N F L U E N C E OF MASS T R A N S F E R ON S O L U T E T R A N S P O R T IN COLUMN E X P E R I M E N T S WITH AN A G G R E G A T E D 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. 2Department of Civil Engineering, Michigan Technological University, Houghton, M I 49931, U.S.A. 3Soil 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., Crittenden, J.C. and Nkedi-Kizza, P., 1987. The influence of mass transfer on solute t r a n s p o r t in column experiments with an aggregated soil. J. Contam. Hydrol., 1: 375-393. The spreading of concentration fronts in dynamic column experiments conducted with a porous, aggregated soil is analyzed by means of a previously documented transport model (DFPSDM) t h a t accounts for longitudinal dispersion, external mass transfer 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 r a n s p o r t of tritiated 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 internal pore diffusion was approximately equal to 2. Quantitative criteria for the spreading influence of the three mechanisms are evaluated with respect to the column data. Hydrodynamic dispersion is t h o u g h t to have governed the front shape in the experiments at low velocity, and internal pore diffusion is believed to have dominated at high velocity; the external mass transfer resistance played a minor role under all conditions. A transport model such as DFPSDM is useful for interpreting column data with regard to the mechanisms controlling concentration front dynamics, but care must be exercised to avoid confounding the effects of the relevant processes.
INTRODUCTION
Measuring and interpreting the concentration responses of laboratory columns 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 t h a t sorb on the porous medium, it is necessary to consider the solute's
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© 1987 Elsevier Science Publishers B.V.
376 retardation, together with advection and dispersion (Bear, 1979). Local equilibrium between the pore water and the sorbing stationary phase is commonly assumed as an expedient in data interpretation and modeling (Bear, 1979), although there is ample evidence of deviations from local equilibrium in some studies (van Genuchten and Wierenga, 1977). Deviations from local equilibrium have been characterized from two basic points of view: as a quasichemical 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 characteristic 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 somewhat 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, semiinfinite column. One of t h a t 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 longitudinal dispersion, external mass transfer, and intra-aggregate diffusion to
377 the solute f r o n t - s p r e a d i n g in the soil c o l u m n e x p e r i m e n t s of Nkedi-Kizza et al. (1982). SOIL COLUMN EXPERIMENTS Nkedi-Kizza et al. (1982) p r e s e n t e d results of soil c o l u m n experiments cond u c t e d over a wide r a n g e of velocities for d i s p l a c e m e n t of four radio-labelled solutes t h r o u g h a s a t u r a t e d p o r o u s m e d i u m c o n s i s t i n g of an a g g r e g a t e d Oxisol. This p a p e r considers the results of seven of the experiments r e p o r t e d by NkediKizza et al. (1982): namely, t h e i r e x p e r i m e n t s 1-3 (3H20); 8 and 9 (36C1); a n d 10 and 11 (45Ca), w h i c h r e p r e s e n t a r a n g e of s o r p t i o n behavior. The c o n d i t i o n s are s u m m a r i z e d in Table 1 of Nkedi-Kizza et al. (1982); only t h o s e v a l u e s n e c e s s a r y for the model c o m p u t a t i o n s are r e p e a t e d here (Table 1), u s i n g n o m e n c l a t u r e c o n s i s t e n t with C r i t t e n d e n et al. (1986) with slight modifications. F o r all experiments, the a v e r a g e a g g r e g a t e d i a m e t e r was 0.34cm (range: 0.2 to 0.47 cm); it has been s h o w n t h a t n o n u n i f o r m p o r o u s media c a n be a d e q u a t e l y modeled as beds of e q u i v a l e n t u n i f o r m spheres in p r e d i c t i n g the effects of i n t r a - a g g r e g a t e diffusion (Rao e t al., 1982; R a s m u s o n , 1985) and dispersion ( H a n et al., 1985) on the d y n a m i c behavior, if the r a n g e of particle sizes does n o t exceed a f a c t o r of two. The bed d i a m e t e r was 7.6 cm, giving a bed-to-particle d i a m e t e r r a t i o of g r e a t e r t h a n 16:1, w h i c h is u s u a l l y ample to avoid serious wall effects if the c o l u m n is carefully packed. The a v e r a g e pore w a t e r velocities r a n g e d from 3 to 143cm/h. S o r p t i o n equilibrium i s o t h e r m s were r e p o r t e d by Nkedi-Kizza et al. (1982) to be linear, with d i s t r i b u t i o n coefficients, K D, equal to 0.03, 0.44, and 1.37ml/g, respectively, for tritium, chloride, a n d calcium. However, it m u s t be n o t e d t h a t the s o r p t i o n equilibrium for the ionic solutes could n o t be d e t e r m i n e d i n d e p e n d e n t l y in b a t c h equilib r i u m experiments, b e c a u s e of c o m p l i c a t i o n s arising from pH changes, and h e n c e the KD v a l u e s h a d to be e v a l u a t e d by fitting a p o r t i o n of the c o l u m n d a t a (Nkedi-Kizza et al., 1982). The i n p u t stimulus in all experiments was in the form of a b r o a d pulse, r a n g i n g 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. No.
Solute
1 2 3 8 9 10 11
3H20 3H20 aH20 ~C1 ~6CI 45Ca 45Ca
KD
CO
PB
(g/cm3)
L (cm)
PPv
(mM)
~B ( )
v
(cma/g)
(cm/h)
(-)
0.03 0.03 0.03 0.44 0.44 1.37 1.37
1.0 1.0 1.0 1.0 1.0 1,0 1.0
1.05 1.05 1.05 1.01 1.01 1.01 1.01
0.40 0.40 0.40 0.41 0.41 0.41 0.41
10.1 10.1 10.1 6.0 6.0 6.0 6.0
15 68 143 3.0 121 3.2 127
2.87 2.87 2.87 5.32 4.78 5.32 5.32
× × × × × × ×
10-7 10-7 10-7 10-6 10_6 10_6 10_6
Data common to all experiments: T -~ 20°C; aA ~ 0.175cm; PA = 1.76g/cm~; ~A = 0.45; dB = 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 surrounding 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 (IBMPC/AT) using 4 radial (i.e., intra-aggregate) and 18 axial collocation points. The accuracy of the computational procedure, including the absence of numerical dispersion and instability, was verified as follows: the results of the orthogonal 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 comparing the relative contributions of longitudinal dispersion, external mass transfer resistance, and internal diffusion resistance to the spreading of the concentration 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 contribution of the external mass transfer resistance; and SIMT = [15(1 + 1/DG)2EIMT] 1 represents the contribution of the intra-aggregate diffusional resistance. In eq. (1), PeHD is the dimensionless Peclet Number, characterizing longitudinal 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) ~BC0
(lb)
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 Co (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 kEMw is the mass transfer coefficient for the external boundary layer surrounding the aggregate (L/t); and aA is the aggregate grain radius (L). EIMT 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 concentration 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, respectively, 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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381
nomenclature, reveals that for a model which incorporates the mechanisms of hydrodynamic dispersion and intra-aggregate diffusion, if the terms SHD and SIMT 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 SEMT 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 concentration 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 directly 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 DFPSDMmodel of Crittenden 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, o r the pore diffusion coefficient, Dp, leads to a proportionate 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 concentration 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 corresponding to the different spreading mechanisms (Figs. 1C and 1D). Hydrodynamic 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 appear 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 resistance into the dispersion parameter should be avoided if possible, and can lead to severe distortions of the concentration response where the degree of spreading 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 hydrodynamic dispersion coefficient, EHD, the external mass transfer coefficient, k E M T , and the pore diffusion coefficient for intra-aggregate mass transfer, D e. The latter three parameters - - E H D , 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-dependent (Nkedi-Kizza et al., 1982, 1984), and much too complex to permit prediction by means of generalized correlations. Quantification of the partitioning by interpretation of column experiments conducted at low velocity is likely to furnish a more accurate estimate than any correlation, and a more representative 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 :
(2)
1.09DLeB(Re)I/3(Sc) '/3
where R e is the dimensionless Reynolds Number based on the grain diameter and the average interstitial velocity, R e = 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 Reynolds 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/DL
:
0.67
EHD/DL = (0.67 + 0.5Pe~) EHD/DL =
1.8PeMD
PeMD
<
1
(3a)
1 < PeMD < 260
(3b)
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 recommended 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 tritiated water was assumed to be approximately that of deuterium hydroxide (Reid and Sherwood, 1958, p. 288) and adjusted for temperature according to the ratio T/g, where T is the absolute temperature (K)
3H20
3H20 3H20 a~Cl ~C1 45Ca
4SCa
1
2 3 8 9 10
11
10 ~ 10 -5 10 -s 10 5 10 -5
i0 -s
0.79 × 10 -s
2.8 × x × x × ×
2.8 2.8 2.03 2.03 0.79
DL (cm2/s)
1270
357 357 357 493 493 1270
Sc ( )
curves
1.24
0.147 0.658 1.39 0.029 1.18 0.031
(-)
Re
V a l u e s Of DG: 3 H 2 0 , D G = 0.74; ~C1, D G = 1.71; 45Ca, D o = 4.0.
Solute
Exp. No.
Model parameters for predicting breakthrough
TABLE 2
5.98 9.85 1.27 2.78 9.56 1.51 5.17
× × × x × × ×
kEMT (cm/s) 10 4 10 -4 10 3 10 4 10 4 10 -4 10 _4 2.23 × 10 _2
10 -3 10 -2 10 -4 10 -2 10 -4
1570
× × x × ×
9.82 2.50 2.63 2.12 3.31
1.64 × 10 -3
EHD (cm2/s)
235 496 14.4 582 39.5
52.5
PeMD (-) I0
x × × × ×
10 - s 10 -6 10 -6 10 -6 10 -7 7.9 x 10 -7
2.8 2.8 2.0 2.0 7.9
2.8 × 10 -6
zp =
Dp(cm2/s) 2
× × × × ×
10 - s 10 -~ 10 _5 l 0 -5 10 6 4.0 × 10 ~
1.4 1.4 1.0 1.0 4.0
1.4 x 10 -5
Zp =
385 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 CONCENTRATIONRESPONSES The predicted responses, computed with the DFPSDM under the assumptions 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 concentration 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 = EA1
(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 SIMw a r e given for each experiment, with values for SIMw corresponding to ~p values of 2 and 10. For experiments 1 and 8, the value of SHD exceeds that of SIMTby a factor of 2 to 3; in all other experiments, SIMT 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 1.1 10.90.8-
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Fig. 2. Predicted and observed effluent concentration responses for tritiated 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).
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°
•
~ 3A
o
J
I
2
4
r
6
I
B
10
PORE VOLUMES
1.1 1 0.9
0.8
3B
0.7
0.6 0.5 0.40.30.20.1 - t 0 0
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 s p e r s i o n at the s c a l e of i n d i v i d u a l grains and pores (Freeze and Cherry, 1979). F o r example, H a n et al. (1985) o b s e r v e d t h a t the l o n g i t u d i n a l d i s p e r s i o n coefficient w a s a p p r o x i m a t e l y doubled by t h e addition of o n l y 10 v o l u m e p e r c e n t o f differently sized particles. The a b o v e r a t i o n a l i z a t i o n , w h i l e tedious, serves to i l l u s t r a t e h o w i n a c c u r a c y in e s t i m a t i n g o n e of t h e s p r e a d i n g effects c a n i n t e r a c t to c o n f o u n d t h e e s t i m a t e o f a n o t h e r o f t h e parameters, if i n v e s t i g a t o r s are t e m p t e d 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-fitting. F r o m t h e v a l u e s in Table 3, it is a p p a r e n t that t h e e x t e r n a l m a s s transfer r e s i s t a n c e in n o c a s e exerts t h e p r e d o m i n a n t i n f l u e n c e o n spreading, and in m o s t c a s e s is n e a r l y an order o f m a g n i t u d e less i m p o r t a n t t h a n at least o n e o f t h e o t h e r factors. Clearly, t h e n e g l e c t o f t h e e x t e r n a l m a s s transfer r e s i s t a n c e
388 1.1 b -
1 0.9 0.8 0.7 0.6 0.5 0.4 0.30.2 0.1 0 2
4
6
8
10
12
14.
PORE VOLUMES
1.1
1 o.g 0.8 0.7 O.S 0.5 0.4 0,3 0.2 0,1 D
i
2
i
i
4
i
i
6
i
~
i
S
10
i
~
12
i
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). in t h e data a n a l y s i s w a s n o t a s e r i o u s o m i s s i o n in t h e w o r k of N k e d i - K i z z a et al. (1982). N e v e r t h e l e s s , the spreading p a r a m e t e r v a l u e s for the different mecha n i s m s in a g i v e n e x p e r i m e n t do n o t differ by m o r e t h a n o n e to t w o orders o f m a g n i t u d e , and it is surely beneficial to e m p l o y for data i n t e r p r e t a t i o n a m o d e l c a p a b l e o f a c c o u n t i n g for all t h r e e processes. SENSITIVITY ANALYSIS T h e r e l a t i v e effects o f t h e three m e c h a n i s m s u p o n spreading m a y be qualitatively a s c e r t a i n e d by m e a n s o f a s e n s i t i v i t y analysis. F i g u r e s 5 t h r o u g h 7 s h o w the impact, o n b r e a k t h r o u g h curves, o f i n d e p e n d e n t l y v a r y i n g SHD, SrMT, and SEMT o v e r t w o orders o f m a g n i t u d e . The base c a s e for t h e a n a l y s i s u s e s t h e
389 TABLE 3 Summary of predominant influence on spreading Exp. No.
V (cm/h)
DG
SEMT
SHD
Dominant mechanism(s)
~MT
~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
1.71
0.053
0.008
0.017
0.087
Dispersion and internal diffusion
1.71
0.105
0.097
0.71
3.51
Internal diffusion
3.97
0.062
0.026
0.075
0.382
Internal diffusion and dispersion
3.97
0.104
0.30
3.0
8
3.0
9
121
10
3.2
11
127
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 i n T a b l e 2) f o 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 c o e f f i c i e n t is c h o s e n a s 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 f o r SHD, SIMT, a n d SEMT 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 f e c 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 SIMT a n d SEMT c o n s t a n t . A n a l o g o u s l y , F i g s . 6 a n d 7 s h o w t h e e f f e c t s o f p e r t u r b i n g SIMT 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 i n e a c h d i r e c t i o n f r o m t h e i r 1
]
/ 0.6
4"
SHD = (~.019
IZq
~
1:1.2
0.1
x ~ 0
20
'
TIME
(MINUTES
40
al 60
~
Fig. 5. Sensitivity analysis, showing the effect of varying the spreading contribution of hydrodynamic dispersion (SRD) while holding the other terms constant at ~MT -- 0.17 and S~MT = 0.022.
390
Z
0.9
ar
S,~T
=
1.7
•
0,6
/~
SIMT = ~).056
~ 0 ~
0.4
~
0.2
r~
0.2,
0.1
o4--
,
,
0
,
20
40
TIME
60
(MINUTES)
Fig. 6. Sensitivity analysis, showing the effect of varying the spreading contribution of intraaggregate diffusional resistance (SIMT)while holding the other terms constant at S~D = 0.055 and S~Mw = 0.022. 1
z 0 ,Y PZ LU o Z
F
0.9 0.8 0.7
g
O.S
U) Cn W .J Z 0 m Z W
0.5 0.4
"I"
SE~-r
=
0.22
~>
Se~T
=
0. 0 6 7
rl
SEMT
=
0.022
/%
SE,~
=
0.00"74
0.3 0.2
°-'tJ 0
i-0
i
i
2O
40 TIME
SO
(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 SIMT = 0.17.
base c a s e v a l u e s , h o l d i n g t h e o t h e r t w o p a r a m e t e r s c o n s t a n t . C o m p a r i n g t h e three figures v i s u a l l y , it is a p p a r e n t that t h e 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 e n s i t i v e to v a r i a t i o n in SIMT,n e x t m o s t s e n s i t i v e to SHD, and least s e n s i t i v e to SEM T •
This r a n k order o f s e n s i t i v i t y c o n f o r m s to e x p e c t a t i o n s formed a c c o r d i n g to the r e l a t i v e v a l u e s o f t h e i n d i v i d u a l spreading p a r a m e t e r v a l u e s in t h e base case. It is a p p a r e n t t h a t t h e effluent r e s p o n s e is r e l a t i v e l y i n s e n s i t i v e to changing t h e v a l u e o f a g i v e n s p r e a d i n g term so l o n g as t h e r e l e v a n t v a l u e is e x c e e d e d
391
by the term corresponding to one of the other mechanisms. For example, increasing SEMTdoes not significantly influence the response until SEMTexceeds 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 importance 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 latter 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 following proportionality relations for homogeneous beds of porous, spherical aggregates (PeMD >> 1)
SHD ~: aAL-' SEMT
OC
D2n213a2j3v2/3L G~ A
1
~: DL2/3a~3v213L 1
S~Mw oc D~eA1DLI~pa2AvL-1 oc ~AIDL-"cpa2AVL I
(5)
(DG ~ 1)
(6a)
(DG >> 1)
(6b)
(DG ~ 1)
(7a)
I)
(7b)
(DG >~
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 critical parameters for longitudinal hydrodynamic dispersion, external mass transfer, 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 characteristic spreading parameters, hydrodynamic dispersion dominated 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 partitioning 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 t o govern frontspreading 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 external 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 predictions and experimental data may be fortuitous. Application to a less uniform porous medium would require a different approach for estimating the dispersion 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 warnings 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 University; 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 preparation was supported in part by the R.S. Kerr Environmental Research Laboratory of the U.S. Environmental Protection Agency under EPACR-808851. REFERENCES Aris, R., 1959.The longitudinal diffusioncoefficientin flowthrough a tube with stagnant pockets. Chem. Eng. Sci., 11: 194-198, Bear, J., 1979. Hydraulics of Groundwater. American Elsevier Publishing Co., New York, NY. Coats, K.H. and Smith, B.D., 1964.Dead-endpore volumeand dispersion in porous media. Soc. Pet. Eng. J., 4: 73-84.
393 Crittenden, J.C., Hutzler, N.J., Geyer, D.G., Oravitz, J.L. and Friedman, G., 1986. Transport of organic compounds with saturated groundwater flow: model development and parameter sensitivity. Water Resour. Res., 22:271 284. Cussler, E.L., 1984. Dlffilmon: Mass Transfer in Fluid Systems• Cambridge University Press, Cambridge, Great Britain. DeSmedt, F. and Wierenga, P.J., 1984. Solute transfer through columns of glass beads. Water Resour. Res., 20: 225-232. Finlayson, B.A., 1980. Nonlinear Analysis in Chemical Engineering. McGraw-Hill, New York, NY. Freeze, R.A. and Cherry, J.A., 1979. Groundwater. Prentice-Hall, Englewood Cliffs, NJ. Fried, J.J., 1975. Groundwater Pollution. Elsevier, Amsterdam, 330 pp. Fried, J.J. and Combarnous, M.A., 1971. Dispersion in Porous Media. In: V.T. Chow (Editor), Advances in Hydroscience. Academic Press, New York, NY, Vol. 7, pp. 169-282. Goltz, M.N., 1986. Three-dimensional analytical modeling of diffusion-limited solute transport. Doctoral Diss., Department of Civil Engineering, Stanford University, Stanford, CA. Goltz, M.N. and Roberts, P.V., 1986. Interpreting organic solute transport data from a field experiment using physical nonequilibrium models. J. Contam. Hydrol., 1:77 93. Han, N-W., Bhakta, J. and Carbonell, R.G., 1985. Longitudinal and lateral dispersion in packed beds: effect of column length and particle size distribution. AIChE J., 31:277 288. Hutzler, N.J., Crittenden, J.C., Gierke, J.S., and Johnson, A.S., 1986. Transport of organic compounds with saturated groundwater flow: Experimental results. Water Resour. Res., 22: 285-295. Komiyama, H. and Smith, J.M., 1974. Surface diffusion in liquid-filled pores. AIChE J., 20: 111(~1117. Nkedi-Kizza, P., Rao, P.S.C., Jessup, R.E. and Davidson, J.M., 1982. Ion exchange and diffusive mass transfer during miscible displacement through an aggregated oxisol. Soil Sci. Soc. Am. J., 46: 471~176. Nkedi-Kizza, P., Biggar, J.W., Selim, H.M., van Genuchten, M.Th., Wierenga, P.J., Davidson, J.M. and Nielsen, D.R., 1984. On the equivalence of two conceptual models for describing ion exchange during transport through an aggregated Oxisol. Water Resour. Res., 20: 1123-1130. Parker, J.C. and Valocchi, A.J., 1986. Constraints on the validity of equilibrium and first-order kinetic transport models in structured soils. Water Resour. Res., 22: 399-407. Passioura, J.B., 1971. Hydrodynamic dispersion in aggregated media, 1, Theory. Soil Sci., 111: 339-344. Rao, P.S.C., Rolston, D.E., Jessup, R.S. and Davidson, J.M., 1980. Solute transport in aggregated porous media: theoretical and experimental evaluation. Soil Sci. Soc. Am. J., 44: 1139-1146. Rao, P.S.C., Jessup, R.E. and Addiscott, T.M., 1982. Experimental and theoretical aspects of solute diffusion in spherical and nonspherical aggregates. Soil. Sci., 133: 342-349. Rasmuson, A., 1985. The effect of particles of variable size, shape, and properties on the dynamics of fixed beds. Chem. Eng. Sci., 40: 621429. Rasmuson, A. and Neretnieks, I., 1980. Exact solution of a model for diffusion in particles and longitudinal dispersion in packed beds. AIChE J., 26: 686~90. Reid, R.C. and Sherwood, T.K., 1958. The Properties of Gases and Liquids. McGraw-Hill, New York, NY. Roberts, P.V., Cornel, P. and Summers, R.S., 1985. External mass-transfer rate in fixed bed adsorption. J. Environ. Eng., ASCE, 111: 891-905. Saffman, P.G., 1960. Dispersion due to molecular diffusion and macroscopic mixing in flow through a network of capillaries. J. Fluid Mech., 7:194 208. Satterfield, C.N., 1980. Homogeneous Catalysis in Practice. McGraw-Hill Book Co., New York, NY. Valocchi, A.J., 1985. Validity of the local equilibrium assumption for modeling sorbing solute transport through homogeneous soils. Water Resour. Res., 21: 808-820. Van Genuchten, M.Th. and Wierenga, P.J., 1977. Mass transfer studies in sorbing porous media: II. experimental evaluation with tritium. Soil Sci. Soc. Am. J., 41: 272-278. Wakao, N. and Smith, J.M., 1962. Diffusion in catalyst pellets. Chem. Eng. Sci., 17: 825-834. Wilson, E.J. and Geankoplis, C.J., 1966. Liquid mass transfer at very low Reynolds Numbers in packed beds. Ind. Eng. Chem. Fund., 5: 9-12.
375
Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands
THE I N F L U E N C E OF MASS T R A N S F E R ON S O L U T E T R A N S P O R T IN COLUMN E X P E R I M E N T S WITH AN A G G R E G A T E D 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. 2Department of Civil Engineering, Michigan Technological University, Houghton, M I 49931, U.S.A. 3Soil 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., Crittenden, J.C. and Nkedi-Kizza, P., 1987. The influence of mass transfer on solute t r a n s p o r t in column experiments with an aggregated soil. J. Contam. Hydrol., 1: 375-393. The spreading of concentration fronts in dynamic column experiments conducted with a porous, aggregated soil is analyzed by means of a previously documented transport model (DFPSDM) t h a t accounts for longitudinal dispersion, external mass transfer 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 r a n s p o r t of tritiated 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 internal pore diffusion was approximately equal to 2. Quantitative criteria for the spreading influence of the three mechanisms are evaluated with respect to the column data. Hydrodynamic dispersion is t h o u g h t to have governed the front shape in the experiments at low velocity, and internal pore diffusion is believed to have dominated at high velocity; the external mass transfer resistance played a minor role under all conditions. A transport model such as DFPSDM is useful for interpreting column data with regard to the mechanisms controlling concentration front dynamics, but care must be exercised to avoid confounding the effects of the relevant processes.
INTRODUCTION
Measuring and interpreting the concentration responses of laboratory columns 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 t h a t sorb on the porous medium, it is necessary to consider the solute's
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© 1987 Elsevier Science Publishers B.V.
376 retardation, together with advection and dispersion (Bear, 1979). Local equilibrium between the pore water and the sorbing stationary phase is commonly assumed as an expedient in data interpretation and modeling (Bear, 1979), although there is ample evidence of deviations from local equilibrium in some studies (van Genuchten and Wierenga, 1977). Deviations from local equilibrium have been characterized from two basic points of view: as a quasichemical 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 characteristic 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 somewhat 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, semiinfinite column. One of t h a t 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 longitudinal dispersion, external mass transfer, and intra-aggregate diffusion to
377 the solute f r o n t - s p r e a d i n g in the soil c o l u m n e x p e r i m e n t s of Nkedi-Kizza et al. (1982). SOIL COLUMN EXPERIMENTS Nkedi-Kizza et al. (1982) p r e s e n t e d results of soil c o l u m n experiments cond u c t e d over a wide r a n g e of velocities for d i s p l a c e m e n t of four radio-labelled solutes t h r o u g h a s a t u r a t e d p o r o u s m e d i u m c o n s i s t i n g of an a g g r e g a t e d Oxisol. This p a p e r considers the results of seven of the experiments r e p o r t e d by NkediKizza et al. (1982): namely, t h e i r e x p e r i m e n t s 1-3 (3H20); 8 and 9 (36C1); a n d 10 and 11 (45Ca), w h i c h r e p r e s e n t a r a n g e of s o r p t i o n behavior. The c o n d i t i o n s are s u m m a r i z e d in Table 1 of Nkedi-Kizza et al. (1982); only t h o s e v a l u e s n e c e s s a r y for the model c o m p u t a t i o n s are r e p e a t e d here (Table 1), u s i n g n o m e n c l a t u r e c o n s i s t e n t with C r i t t e n d e n et al. (1986) with slight modifications. F o r all experiments, the a v e r a g e a g g r e g a t e d i a m e t e r was 0.34cm (range: 0.2 to 0.47 cm); it has been s h o w n t h a t n o n u n i f o r m p o r o u s media c a n be a d e q u a t e l y modeled as beds of e q u i v a l e n t u n i f o r m spheres in p r e d i c t i n g the effects of i n t r a - a g g r e g a t e diffusion (Rao e t al., 1982; R a s m u s o n , 1985) and dispersion ( H a n et al., 1985) on the d y n a m i c behavior, if the r a n g e of particle sizes does n o t exceed a f a c t o r of two. The bed d i a m e t e r was 7.6 cm, giving a bed-to-particle d i a m e t e r r a t i o of g r e a t e r t h a n 16:1, w h i c h is u s u a l l y ample to avoid serious wall effects if the c o l u m n is carefully packed. The a v e r a g e pore w a t e r velocities r a n g e d from 3 to 143cm/h. S o r p t i o n equilibrium i s o t h e r m s were r e p o r t e d by Nkedi-Kizza et al. (1982) to be linear, with d i s t r i b u t i o n coefficients, K D, equal to 0.03, 0.44, and 1.37ml/g, respectively, for tritium, chloride, a n d calcium. However, it m u s t be n o t e d t h a t the s o r p t i o n equilibrium for the ionic solutes could n o t be d e t e r m i n e d i n d e p e n d e n t l y in b a t c h equilib r i u m experiments, b e c a u s e of c o m p l i c a t i o n s arising from pH changes, and h e n c e the KD v a l u e s h a d to be e v a l u a t e d by fitting a p o r t i o n of the c o l u m n d a t a (Nkedi-Kizza et al., 1982). The i n p u t stimulus in all experiments was in the form of a b r o a d pulse, r a n g i n g 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. No.
Solute
1 2 3 8 9 10 11
3H20 3H20 aH20 ~C1 ~6CI 45Ca 45Ca
KD
CO
PB
(g/cm3)
L (cm)
PPv
(mM)
~B ( )
v
(cma/g)
(cm/h)
(-)
0.03 0.03 0.03 0.44 0.44 1.37 1.37
1.0 1.0 1.0 1.0 1.0 1,0 1.0
1.05 1.05 1.05 1.01 1.01 1.01 1.01
0.40 0.40 0.40 0.41 0.41 0.41 0.41
10.1 10.1 10.1 6.0 6.0 6.0 6.0
15 68 143 3.0 121 3.2 127
2.87 2.87 2.87 5.32 4.78 5.32 5.32
× × × × × × ×
10-7 10-7 10-7 10-6 10_6 10_6 10_6
Data common to all experiments: T -~ 20°C; aA ~ 0.175cm; PA = 1.76g/cm~; ~A = 0.45; dB = 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 surrounding 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 (IBMPC/AT) using 4 radial (i.e., intra-aggregate) and 18 axial collocation points. The accuracy of the computational procedure, including the absence of numerical dispersion and instability, was verified as follows: the results of the orthogonal 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 comparing the relative contributions of longitudinal dispersion, external mass transfer resistance, and internal diffusion resistance to the spreading of the concentration 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 contribution of the external mass transfer resistance; and SIMT = [15(1 + 1/DG)2EIMT] 1 represents the contribution of the intra-aggregate diffusional resistance. In eq. (1), PeHD is the dimensionless Peclet Number, characterizing longitudinal 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) ~BC0
(lb)
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 Co (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 kEMw is the mass transfer coefficient for the external boundary layer surrounding the aggregate (L/t); and aA is the aggregate grain radius (L). EIMT 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 concentration 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, respectively, 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
o~ o'~
m
0
N
n
T
d
~_
o~
.~
O
0.
~ l
l
~
o
l
d
l o
l o
l d
l o
N011~IN30N00
l d
l O
~
n
l
n
~
d
d
d
d
d
u
n
,
o
u
~
o
o
d
o
d
d
H
~ ~
3 ~ 114~73~
o
d
d
d
d
d
d
d
d
d
i
d
d
d
o
c,
d
d
d
d
b.-~ ~
~
381
nomenclature, reveals that for a model which incorporates the mechanisms of hydrodynamic dispersion and intra-aggregate diffusion, if the terms SHD and SIMT 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 SEMT 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 concentration 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 directly 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 DFPSDMmodel of Crittenden 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, o r the pore diffusion coefficient, Dp, leads to a proportionate 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 concentration 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 corresponding to the different spreading mechanisms (Figs. 1C and 1D). Hydrodynamic 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 appear 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 resistance into the dispersion parameter should be avoided if possible, and can lead to severe distortions of the concentration response where the degree of spreading 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 hydrodynamic dispersion coefficient, EHD, the external mass transfer coefficient, k E M T , and the pore diffusion coefficient for intra-aggregate mass transfer, D e. The latter three parameters - - E H D , 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-dependent (Nkedi-Kizza et al., 1982, 1984), and much too complex to permit prediction by means of generalized correlations. Quantification of the partitioning by interpretation of column experiments conducted at low velocity is likely to furnish a more accurate estimate than any correlation, and a more representative 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 :
(2)
1.09DLeB(Re)I/3(Sc) '/3
where R e is the dimensionless Reynolds Number based on the grain diameter and the average interstitial velocity, R e = 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 Reynolds 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/DL
:
0.67
EHD/DL = (0.67 + 0.5Pe~) EHD/DL =
1.8PeMD
PeMD
<
1
(3a)
1 < PeMD < 260
(3b)
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 recommended 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 tritiated water was assumed to be approximately that of deuterium hydroxide (Reid and Sherwood, 1958, p. 288) and adjusted for temperature according to the ratio T/g, where T is the absolute temperature (K)
3H20
3H20 3H20 a~Cl ~C1 45Ca
4SCa
1
2 3 8 9 10
11
10 ~ 10 -5 10 -s 10 5 10 -5
i0 -s
0.79 × 10 -s
2.8 × x × x × ×
2.8 2.8 2.03 2.03 0.79
DL (cm2/s)
1270
357 357 357 493 493 1270
Sc ( )
curves
1.24
0.147 0.658 1.39 0.029 1.18 0.031
(-)
Re
V a l u e s Of DG: 3 H 2 0 , D G = 0.74; ~C1, D G = 1.71; 45Ca, D o = 4.0.
Solute
Exp. No.
Model parameters for predicting breakthrough
TABLE 2
5.98 9.85 1.27 2.78 9.56 1.51 5.17
× × × x × × ×
kEMT (cm/s) 10 4 10 -4 10 3 10 4 10 4 10 -4 10 _4 2.23 × 10 _2
10 -3 10 -2 10 -4 10 -2 10 -4
1570
× × x × ×
9.82 2.50 2.63 2.12 3.31
1.64 × 10 -3
EHD (cm2/s)
235 496 14.4 582 39.5
52.5
PeMD (-) I0
x × × × ×
10 - s 10 -6 10 -6 10 -6 10 -7 7.9 x 10 -7
2.8 2.8 2.0 2.0 7.9
2.8 × 10 -6
zp =
Dp(cm2/s) 2
× × × × ×
10 - s 10 -~ 10 _5 l 0 -5 10 6 4.0 × 10 ~
1.4 1.4 1.0 1.0 4.0
1.4 x 10 -5
Zp =
385 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 CONCENTRATIONRESPONSES The predicted responses, computed with the DFPSDM under the assumptions 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 concentration 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 = EA1
(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 SIMw a r e given for each experiment, with values for SIMw corresponding to ~p values of 2 and 10. For experiments 1 and 8, the value of SHD exceeds that of SIMTby a factor of 2 to 3; in all other experiments, SIMT 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 1.1 10.90.8-
2A
O,7O.S* 0.5-
0.3 0.2 O.
2
0
4 PORE VOLUMES
1.1 1
.
.
.
.
.
.
.
.
0.9
os
/,~
o.~
2B
0.6
0.4
0
-
f
0
..
2
4
PORE VOLUMES 1.1 I .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0.9-
o.oI
/
o,~ o, ~
/j //
~,\
/1
ii
o.o~
o,j
0.,i
o,t -
v 0
2c
,.~
'l
It
'~k ~
,
,
2
4
PORE VOLUMr q
Fig. 2. Predicted and observed effluent concentration responses for tritiated 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).
387 1.1
o 0 O
°
•
~ 3A
o
J
I
2
4
r
6
I
B
10
PORE VOLUMES
1.1 1 0.9
0.8
3B
0.7
0.6 0.5 0.40.30.20.1 - t 0 0
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 s p e r s i o n at the s c a l e of i n d i v i d u a l grains and pores (Freeze and Cherry, 1979). F o r example, H a n et al. (1985) o b s e r v e d t h a t the l o n g i t u d i n a l d i s p e r s i o n coefficient w a s a p p r o x i m a t e l y doubled by t h e addition of o n l y 10 v o l u m e p e r c e n t o f differently sized particles. The a b o v e r a t i o n a l i z a t i o n , w h i l e tedious, serves to i l l u s t r a t e h o w i n a c c u r a c y in e s t i m a t i n g o n e of t h e s p r e a d i n g effects c a n i n t e r a c t to c o n f o u n d t h e e s t i m a t e o f a n o t h e r o f t h e parameters, if i n v e s t i g a t o r s are t e m p t e d 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-fitting. F r o m t h e v a l u e s in Table 3, it is a p p a r e n t that t h e e x t e r n a l m a s s transfer r e s i s t a n c e in n o c a s e exerts t h e p r e d o m i n a n t i n f l u e n c e o n spreading, and in m o s t c a s e s is n e a r l y an order o f m a g n i t u d e less i m p o r t a n t t h a n at least o n e o f t h e o t h e r factors. Clearly, t h e n e g l e c t o f t h e e x t e r n a l m a s s transfer r e s i s t a n c e
388 1.1 b -
1 0.9 0.8 0.7 0.6 0.5 0.4 0.30.2 0.1 0 2
4
6
8
10
12
14.
PORE VOLUMES
1.1
1 o.g 0.8 0.7 O.S 0.5 0.4 0,3 0.2 0,1 D
i
2
i
i
4
i
i
6
i
~
i
S
10
i
~
12
i
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). in t h e data a n a l y s i s w a s n o t a s e r i o u s o m i s s i o n in t h e w o r k of N k e d i - K i z z a et al. (1982). N e v e r t h e l e s s , the spreading p a r a m e t e r v a l u e s for the different mecha n i s m s in a g i v e n e x p e r i m e n t do n o t differ by m o r e t h a n o n e to t w o orders o f m a g n i t u d e , and it is surely beneficial to e m p l o y for data i n t e r p r e t a t i o n a m o d e l c a p a b l e o f a c c o u n t i n g for all t h r e e processes. SENSITIVITY ANALYSIS T h e r e l a t i v e effects o f t h e three m e c h a n i s m s u p o n spreading m a y be qualitatively a s c e r t a i n e d by m e a n s o f a s e n s i t i v i t y analysis. F i g u r e s 5 t h r o u g h 7 s h o w the impact, o n b r e a k t h r o u g h curves, o f i n d e p e n d e n t l y v a r y i n g SHD, SrMT, and SEMT o v e r t w o orders o f m a g n i t u d e . The base c a s e for t h e a n a l y s i s u s e s t h e
389 TABLE 3 Summary of predominant influence on spreading Exp. No.
V (cm/h)
DG
SEMT
SHD
Dominant mechanism(s)
~MT
~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
1.71
0.053
0.008
0.017
0.087
Dispersion and internal diffusion
1.71
0.105
0.097
0.71
3.51
Internal diffusion
3.97
0.062
0.026
0.075
0.382
Internal diffusion and dispersion
3.97
0.104
0.30
3.0
8
3.0
9
121
10
3.2
11
127
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 i n T a b l e 2) f o 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 c o e f f i c i e n t is c h o s e n a s 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 f o r SHD, SIMT, a n d SEMT 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 f e c 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 SIMT a n d SEMT c o n s t a n t . A n a l o g o u s l y , F i g s . 6 a n d 7 s h o w t h e e f f e c t s o f p e r t u r b i n g SIMT 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 i n e a c h d i r e c t i o n f r o m t h e i r 1
]
/ 0.6
4"
SHD = (~.019
IZq
~
1:1.2
0.1
x ~ 0
20
'
TIME
(MINUTES
40
al 60
~
Fig. 5. Sensitivity analysis, showing the effect of varying the spreading contribution of hydrodynamic dispersion (SRD) while holding the other terms constant at ~MT -- 0.17 and S~MT = 0.022.
390
Z
0.9
ar
S,~T
=
1.7
•
0,6
/~
SIMT = ~).056
~ 0 ~
0.4
~
0.2
r~
0.2,
0.1
o4--
,
,
0
,
20
40
TIME
60
(MINUTES)
Fig. 6. Sensitivity analysis, showing the effect of varying the spreading contribution of intraaggregate diffusional resistance (SIMT)while holding the other terms constant at S~D = 0.055 and S~Mw = 0.022. 1
z 0 ,Y PZ LU o Z
F
0.9 0.8 0.7
g
O.S
U) Cn W .J Z 0 m Z W
0.5 0.4
"I"
SE~-r
=
0.22
~>
Se~T
=
0. 0 6 7
rl
SEMT
=
0.022
/%
SE,~
=
0.00"74
0.3 0.2
°-'tJ 0
i-0
i
i
2O
40 TIME
SO
(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 SIMT = 0.17.
base c a s e v a l u e s , h o l d i n g t h e o t h e r t w o p a r a m e t e r s c o n s t a n t . C o m p a r i n g t h e three figures v i s u a l l y , it is a p p a r e n t that t h e 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 e n s i t i v e to v a r i a t i o n in SIMT,n e x t m o s t s e n s i t i v e to SHD, and least s e n s i t i v e to SEM T •
This r a n k order o f s e n s i t i v i t y c o n f o r m s to e x p e c t a t i o n s formed a c c o r d i n g to the r e l a t i v e v a l u e s o f t h e i n d i v i d u a l spreading p a r a m e t e r v a l u e s in t h e base case. It is a p p a r e n t t h a t t h e effluent r e s p o n s e is r e l a t i v e l y i n s e n s i t i v e to changing t h e v a l u e o f a g i v e n s p r e a d i n g term so l o n g as t h e r e l e v a n t v a l u e is e x c e e d e d
391
by the term corresponding to one of the other mechanisms. For example, increasing SEMTdoes not significantly influence the response until SEMTexceeds 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 importance 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 latter 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 following proportionality relations for homogeneous beds of porous, spherical aggregates (PeMD >> 1)
SHD ~: aAL-' SEMT
OC
D2n213a2j3v2/3L G~ A
1
~: DL2/3a~3v213L 1
S~Mw oc D~eA1DLI~pa2AvL-1 oc ~AIDL-"cpa2AVL I
(5)
(DG ~ 1)
(6a)
(DG >> 1)
(6b)
(DG ~ 1)
(7a)
I)
(7b)
(DG >~
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 critical parameters for longitudinal hydrodynamic dispersion, external mass transfer, 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 characteristic spreading parameters, hydrodynamic dispersion dominated 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 partitioning 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 t o govern frontspreading 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 external 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 predictions and experimental data may be fortuitous. Application to a less uniform porous medium would require a different approach for estimating the dispersion 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 warnings 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 University; 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 preparation was supported in part by the R.S. Kerr Environmental Research Laboratory of the U.S. Environmental Protection Agency under EPACR-808851. REFERENCES Aris, R., 1959.The longitudinal diffusioncoefficientin flowthrough a tube with stagnant pockets. Chem. Eng. Sci., 11: 194-198, Bear, J., 1979. Hydraulics of Groundwater. American Elsevier Publishing Co., New York, NY. Coats, K.H. and Smith, B.D., 1964.Dead-endpore volumeand dispersion in porous media. Soc. Pet. Eng. J., 4: 73-84.
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