Applicability of the Local Equilibrium Assumption to Transport through

11 Applicability of the Local Equilibrium Assumption to Transport through Soils of Solutes Affected by

Ion

Exchange RONALD V. JAMES and JACOB RUBIN

Downloaded by UNIV LAVAL on July 14, 2016 | http://pubs.acs.org Publication Date: March 19, 1979 | doi: 10.1021/bk-1979-0093.ch011

Water Resources Division, U.S. Geological Survey, Menlo Park, CA 94025 In an attempt to deal w i t h such unwanted substances as r a d i o a c t i v e and chemical wastes, d i s p o s a l s i t e s are often used that are h y d r a u l i c a l l y connected w i t h usable water supplies v i a subsurface t r a n s p o r t routes. To manage these wastes e f f e c t i v e l y , i t i s d e s i r a b l e to have the c a p a b i l i t y of p r e d i c t i n g the course of s o l u t e t r a n s p o r t along these connecting routes. Subsurface solute t r a n s p o r t i s a f f e c t e d by hydrodynamic d i s p e r s i o n and by chemical r e a c t i o n s w i t h s o i l and rocks. The e f f e c t s of hydrodynamic d i s p e r s i o n have been e x t e n s i v e l y studied (_1, _2, _3, 4) . Chemical r e a c t i o n s i n v o l v i n g the s o l i d phase a f f e c t subsurface solute transport i n a way that depends on the r e a c t i o n rates r e l a t i v e to the water f l u x . I f the r e a c t i o n rate i s f a s t and the flow r a t e slow, then the l o c a l e q u i l i b r i u m assumpt i o n may be a p p l i c a b l e . I f the r e a c t i o n rate i s slow and the f l u x r e l a t i v e l y high, then r e a c t i o n k i n e t i c s c o n t r o l s the chemi s t r y and one cannot assume l o c a l e q u i l i b r i u m . Theoretical treatments f o r t r a n s p o r t of many kinds of r e a c t i v e solutes are a v a i l a b l e f o r both s i t u a t i o n s (5-10). I t i s o f t e n d e s i r a b l e , where a p p l i c a b l e , to use the l o c a l e q u i l i b r i u m assumption when p r e d i c t i n g the f a t e of subsurface s o l u t e s . Advantages of t h i s approach may include 1) data such as e q u i l i b r i u m constants are r e a d i l y a v a i l a b l e , as opposed to the l a c k of k i n e t i c data, and 2) f o r transport i n v o l v i n g ion exchange and adsorption, the mathematics f o r e q u i l i b r i u m systems are g e n e r a l l y simpler than f o r those c o n t r o l l e d by k i n e t i c s . To u t i l i z e f u l l y these advantages, i t i s h e l p f u l to know the flow rate below which the l o c a l e q u i l i b r i u m assumption i s a p p l i c a b l e for a given chemical system. Few i n d i c a t o r s are a v a i l a b l e which allow determination of that c r i t i c a l water f l u x . The work presented here addresses t h i s question f o r a laboratory ion-exchange system that i s r e l a t i v e l y simple. Effluent concentration h i s t o r i e s were obtained f o r calcium and c h l o r i d e ions during m i s c i b l e displacement of calcium c h l o r i d e s o l u t i o n s through v e r t i c a l columns c o n t a i n i n g homogeneous, repacked sandy s o i l that was water-saturated. Calcium self-exchange was the 0-8412-0479-9/79/47-093-225$05.00/0 This chapter not subject to U.S. copyright Published 1979 American Chemical Society Jenne; Chemical Modeling in Aqueous Systems ACS Symposium Series; American Chemical Society: Washington, DC, 1979.

226

CHEMICAL MODELING IN AQUEOUS SYSTEMS

only r e a c t i o n considered. The flow r a t e was kept constant during each experiment but i t had a l a r g e range of v a r i a t i o n among the experiments. Experimental 4 5

Calcium c h l o r i d e s o l u t i o n s (pH = 6.2) l a b e l e d w i t h C a or C 1 were d i s p l a c e d v e r t i c a l l y downward through columns of homo geneous, repacked, water-saturated sandy s o i l by a c h e m i c a l l y i d e n t i c a l s o l u t i o n l a b e l e d w i t h C 1 or C a , r e s p e c t i v e l y . Constant water f l u x e s , and s o l u t i o n a c t i v i t i e s of 1 to 15 μα/dm , were used. Calcium s o l u t i o n s were analyzed by t i t r a t i o n w i t h disodium dihydrogen ethylenediamine t e t r a a c e t a t e to a murexide end p o i n t (11). The a c t i v i t y of r a d i o a c t i v e l y l a b e l e d s o l u t i o n s was obtained by l i q u i d s c i n t i l l a t i o n techniques. Concentrations of adsorbed c a l c i u m were c a l c u l a t e d from isotope d i l u t i o n . The c o n c e n t r a t i o n of c a l c i u m c h l o r i d e i n the i n f l u e n t s o l u t i o n was 0.08 N. Because exchange of c a l c i u m f o r i t s e l f was the only chemical process a f f e c t i n g t r a n s p o r t , the c a l c i u m c h l o r i d e con c e n t r a t i o n remained constant a t 0.08 Ν throughout each experiment, both w i t h i n the column and i n the e f f l u e n t . The s o i l s employed i n t h i s study were D e l h i (11) and Oakley (12) sands, i n which most of the c l a y appears to be present as c o a t i n g s on the sand p a r t i c l e s . C h a r a c t e r i s t i c s of each s o i l are shown i n Table I . Oakley sand i s q u i t e a c i d i c . This can be ex p l a i n e d (13, p. 282-289) by the s u b s t a n t i a l aluminum component of the exchangeable c a t i o n s . 36

36

4 5


3

Table I C h a r a c t e r i s t i c s of Experimental S o i l s

Porosity Soil

3

3

(cm /cm )

Bulk density 3

(g/cm )

pH** ( i n 0.01 CaCl

Cation exchange capacity* M

(meq/g)

g

D e l h i sand

0.40

1.6

6.1

0.04

Oakley sand

0.34

1.8

4.2

0.02

^Exchange c a p a c i t y values are averages. S p e c i f i c values were obtained f o r each column and used as e x p l a i n e d i n the t e x t . ^ D e t e r m i n a t i o n of s o i l pH as described by Black (15). Each Oakley column was pre-leached, removing about h a l f the adsorbed aluminum. As subsequent l e a c h i n g continued d u r i n g r e p e t i t i v e experiments, exchangeable aluminum was removed and the e f f l u e n t pH increased. I n a l l of the i n d i v i d u a l experiments d e s c r i b e d here, the c o n c e n t r a t i o n of aluminum i n the e f f l u e n t

Jenne; Chemical Modeling in Aqueous Systems ACS Symposium Series; American Chemical Society: Washington, DC, 1979.

11.

JAMES

Local

AND RUBIN

Equilibrium

Assumption

227


5

was so low ( l e s s than 3 χ 10~ M) that the increases i n adsorbed calcium r e s u l t i n g from removal of aluminum were n e g l i g i b l e and the e f f l u e n t pH increased l e s s than 0.2 u n i t s . Therefore, i n each given experiment the c o n c e n t r a t i o n of adsorbed calcium was taken to be constant. The aluminum c o n c e n t r a t i o n was determined c o l o r i m e t r i c a l l y (14). D e l h i sand i s a n e a r l y n e u t r a l s o i l and no pH change occurred during the experiments using i t . The concen t r a t i o n of adsorbed calcium was determined f o r each column by g r a p h i c a l i n t e g r a t i o n of the area between p l o t t e d curves showing the calcium and c h l o r i d e e f f l u e n t h i s t o r i e s . The exchangeable calcium thus determined was used i n subsequent t h e o r e t i c a l pre d i c t i o n s that were compared w i t h the e m p i r i c a l r e s u l t s obtained from the corresponding columns. The columns used i n these s t u d i e s and the general experimen t a l techniques employed are described by James and Rubin (16). The s o i l columns were 10 cm long and 5 cm i n diameter. Special care was taken i n packing the columns to avoid r a d i a l and l o n g i t u d i n a l p a r t i c l e - s i z e segregation (17). F l u i d volumes i n the columns averaged 73 cm f o r Oakley sand and 81 cm f o r D e l h i sand. Results are presented as e f f l u e n t h i s t o r i e s i n which the r e l a t i v e e f f l u e n t c o n c e n t r a t i o n (REC) of l a b e l e d calcium i s p l o t t e d against time and i s given by: 3

3

c -c REC = - S _ £ c -c oo ο

(1)

where c , c , and c are the concentrations of l a b e l e d calcium i n the s o i l at zero time, i n the d i s p l a c e d s o l u t i o n , and i n the e f f l u e n t at time t , r e s p e c t i v e l y . 0

œ

e

Theoretical One-dimensional t r a n s p o r t through s o i l s of calcium a f f e c t e d by e q u i l i b r i u m - c o n t r o l l e d self-exchange i s described by: Λ

ôc . St

P

ôs ïït

2

=

_ ôc ôP" "

ôc q

/ 0

v

( 2 )

ôz

where θ = p o r o s i t y of the medium, ρ = bulk d e n s i t y of the medium (g/cm ), ζ = d i s t a n c e from input end of column (cm), t = time (sec) , c = c o n c e n t r a t i o n of labeled s o l u t e i n the s o i l s o l u t i o n at ζ and t (meq/cm ), s = c o n c e n t r a t i o n of adsorbed s o l u t e at ζ and t (meq/g), D = hydrodynamic d i s p e r s i o n c o e f f i c i e n t (cm /sec), and q = water f l u x (cm/sec). The e q u i l i b r i u m r e l a t i o n i s 3

3

2

^4=1

(3)

cswhere the a s t e r i s k denotes unlabeled calcium.

Noting t h a t c+c*


228


i s the t o t a l c o n c e n t r a t i o n of calcium i n s o l u t i o n , C , and t h a t , w i t h the c o n c e n t r a t i o n of adsorbed calcium n e g l i g i b l y a f f e c t e d by the presence of aluminum, s+s* can be taken to be the exchange c a p a c i t y of the s o i l , λ, i t f o l l o w s from Equation 3 that t

For the c o n d i t i o n s used i n t h i s study C and λ are constant, t h e r e f o r e , p a r t i a l d i f f e r e n t i a t i o n of Equation 4 y i e l d s


t

as _ λ_

ôc

ôt " C

ôt

t

(5)

Combining Equation 5 w i t h Equation 2 gives the /Λ

_i_

Λ

^

. ÔC

~

2

Ô C

equation

ÔC

which i n t h i s study was solved n u m e r i c a l l y f o r e f f l u e n t concent r a t i o n s as a f u n c t i o n of time u s i n g the method and boundary c o n d i t i o n s of Rubin and James (7)· The numerical method i s needed because experimental columns of t h i s type are best represented as two consecutive l a y e r s : the s o i l column and the v o i d space of the apparatus passages, r e s p e c t i v e l y . The two-layer d e s c r i p t i o n i s necessary i n order to account f o r d i s p e r s i o n induced by the apparatus (16). The d i s p e r s i o n c o e f f i c i e n t s used at each flow r a t e are given i n Table I I . P o r o s i t y , bulk d e n s i t y , and water flow r a t e were determined independently of the e l u t i o n h i s t o r i e s , as described p r e v i o u s l y (16). The d i s p e r s i o n c o e f f i c i e n t s f o r the s o i l were obtained by t r i a l and e r r o r determination of the D values g i v i n g the best agreement between the p r e d i c t e d and the e m p i r i c a l c h l o r i d e e f f l u e n t h i s t o r i e s . The p r e d i c t e d h i s t o r i e s were obtained from the s o l u t i o n to Equation 2 w i t h ôs/ôt = 0 . To f i n d the d i s p e r s i o n c o e f f i c i e n t s f o r the end caps (16), a length of 0.2 cm and a p o r o s i t y of 1.0 were used. The water f l u x i n the end cap i s the same as that i n the s o i l . Results and D i s c u s s i o n For both s o i l s s t u d i e d , comparison of c a l c i u m - e f f l u e n t h i s t o r i e s p r e d i c t e d by the s o l u t i o n to Equation 6 w i t h those obtained from experimental columns gave good agreement only f o r the lowest flow r a t e s . For the three higher water f l u x e s , more apparent d i s p e r s i o n was observed than could be explained by p r e d i c t i o n s that assume l o c a l e q u i l i b r i u m . Examples of these comparisons are shown i n Figures 1 and 2.


J A M E S AND RUBIN

Local

Equilibrium

Assumption


Table I I V a r i a t i o n w i t h Water F l u x of S o i l and Apparatus D i s p e r s i o n C o e f f i c i e n t s A.

D e l h i Sand D

V

(cm /sec)

(cm /sec)

q 2

(cm/sec) 1.7 1.8 1.7 1.8

χ χ χ χ

lCT lCT lCT lCT

5

5.0 2.7 3.5 3.5

4

3

2

B.

χ χ χ χ

ier 10r ΙΟ lCT

e

5

-4 3

4 3 7 6

χ χ χ χ

lCT lCT lCT lCT

6

5

5

4

Oakley Sand D

q

x χ χ χ

2

(cm /sec)

2

(cm /sec)

(cm/sec) 1.7 1.7 1.7 1.7

2

5

lCT lCT 1CT 10-

4

3

2

3.0 6.6 1.0 2.0

χ χ χ χ

lCT lCT 10~ 10~

6

5

4

3

4 3 7 6

χ χ χ χ

lCT ΙΟ" lCT lCT

6

5

5

4

*The s u b s c r i p t A i n d i c a t e s the d i s p e r s i o n c o e f f i c i e n t used f o r the l a y e r which d e s c r i b e s the d i s p e r s i o n induced by the apparatus.





Figure

2.

Elution

histories for Delhi and Oakley

5

sands with water flux of 1.7 χ 10~


cm/sec

CO

bo

>

CHEMICAL MODELING ΓΝ AQUEOUS SYSTEMS

232

The m o l e c u l a r - d i f f u s i o n c o e f f i c i e n t f o r calcium i n d i l u t e aqueous s o l u t i o n s of calcium c h l o r i d e , D ^ i s about 1 χ 10" cm /sec (18,p.700). The molecular-diffusion c o e f f i c i e n t i n the s o i l , D , c a n be estimated by the r e l a t i o n D ~ ^ / T , where Τ i s the t o r t u o s i t y , assumed here t o be about 1.4. The estimated m o l e c u l a r - d i f f u s i o n c o e f f i c i e n t of calcium f o r the s o i l s employed i n t h i s study i s , t h e r e f o r e , about 3 χ 1(Γ cm /sec. Comparing t h i s value w i t h the d i s p e r s i o n c o e f f i c i e n t s i n Table I I , one finds that a t the lowest flow r a t e studied f o r each s o i l the c o e f f i c i e n t s are s i m i l a r , but a t higher f l u x e s they d i f f e r by orders of magnitude. Thus, when the hydrodynamic d i s p e r s i o n c o e f f i c i e n t i s n e a r l y the same as the estimated m o l e c u l a r - d i f f u s i o n c o e f f i c i e n t i n the s o i l , the e q u i l i b r i u m assumption a p p l i e s . When the d i s persion c o e f f i c i e n t i s s i g n i f i c a n t l y l a r g e r , the l o c a l e q u i l i b r i u m assumption does not apply and a k i n e t i c s - b a s e d model presumably i s indicated. In order to f u r t h e r s u b s t a n t i a t e t h i s c o n c l u s i o n , i t i s of i n t e r e s t to compare i t w i t h the p r e d i c t i o n obtained from a simple t h e o r e t i c a l model. Glueckauf's well-known t r a n s p o r t model (19, p. 449-453), supplemented by the more modern concept of hydrodynamic d i s p e r s i o n , i s w e l l s u i t e d f o r t h i s purpose. The model simulates d i s p e r s i o n - a f f e c t e d s o l u t e t r a n s p o r t w i t h i o n exchange for which d i f f u s i o n processes are r a t e l i m i t i n g . I n h i s develop ment, Glueckauf assumes: 1) exchange takes place i n porous s p e r i c a l p a r t i c l e s w i t h radius r and p o r o s i t y θ ; 2) w i t h i n these p a r t i c l e s , water and s o l u t e movement occur only by molecu l a r d i f f u s i o n ; 3) the r e a c t i o n rate i s determined by the rates of f i l m and i n t r a p a r t i c l e d i f f u s i o n , which are p r o p o r t i o n a l t o the d i f f e r e n c e s between the t h e o r e t i c a l e q u i l i b r i u m and a c t u a l concentrations of s o l u t i o n and adsorbed s o l u t e s , r e s p e c t i v e l y ; and 4) the r e l e v a n t e q u i l i b r i u m r e l a t i o n s are described by Equation 4. Using the modified Glueckauf model described above, and em p l o y i n g h i s s i m p l i f y i n g mathematical assumptions (the correctness of which has been confirmed by the authors w i t h the a i d o f numer i c a l methods), one obtains 5

m

5

2

D

mg

m


6

2

ρ

which i s s i m i l a r to Equation 6 w i t h D replaced by the e f f e c t i v e d i s p e r s i o n c o e f f i c i e n t D, which i s given by: D = D + F + F Ρ

1

f

(8)

when Fp and F f describe the c o n t r i b u t i o n s of i n t r a p a r t i c l e and f i l m d i f f u s i o n , r e s p e c t i v e l y , and are expressed by


11.

Local

J A M E S AND RUBIN

Equilibrium

Ο.ΙΟρλο^Γ F

233

Assumption

2

ρ ~ D ,θ (6C +ρλ) ρ t

2

(9) 0.133 ρ λ g r.2 F, = f D ^(9C +pX) (l+70qr) 2

2

2

2

m

t

As q decreases, the terms F and F f become small compared to D, and Equation 7 becomes more l i k e Equation 6. This i n d i c a t e s that the l o c a l e q u i l i b r i u m assumption i s a p p l i c a b l e when q i s s u f f i c i e n t l y small. Using the r e l a t i o n s h i p Downloaded by UNIV LAVAL on July 14, 2016 | http://pubs.acs.org Publication Date: March 19, 1979 | doi: 10.1021/bk-1979-0093.ch011

p

D = D + ccq ms

(10)

where α i s an e m p i r i c a l constant (jL) , one f i n d s that when the d i s p e r s i o n c o e f f i c i e n t i s n e a r l y equal t o the m o l e c u l a r - d i f f u s i o n c o e f f i c i e n t i n the s o i l , aq i s much smaller than D , g i v i n g mg

D ms

q « M

α

D «Θ πι^υ =— — αΤ

/1 ι \ (11)

I n e q u a l i t y 11 was s u b s t i t u t e d i n t o Equation 8, together w i t h reasonable values o f other parameters and 0.1 cm < α < 0.2 cm as α was found to be i n t h i s study. This leads t o the c o n c l u s i o n t h a t , f o r systems i n which r i s l e s s than 0.1 cm, the l o c a l e q u i l ibrium assumption i s a p p l i c a b l e ( i . e . , q i s s u f f i c i e n t l y small) when D i s n e a r l y equal to D as observed i n the experiments. I n s o i l s i n which the exchanging p a r t i c l e s are not s p h e r i c a l , r would represent approximately the mean d i f f u s i o n path w i t h i n c l a y aggregates or w i t h i n c l a y coatings on coarse p a r t i c l e s . For s o i l s without appreciable c l a y aggregation, the e x p e r i mental r e s u l t s and t h e o r e t i c a l a n a l y s i s described here i n d i c a t e that when d i f f u s i o n i s r a t e - l i m i t i n g , the r a t i o of the hydrody namic d i s p e r s i o n c o e f f i c i e n t t o the estimated s o i l s e l f - d i f f u s i o n c o e f f i c i e n t may be a u s e f u l i n d i c a t o r of the a p p l i c a b i l i t y of the l o c a l e q u i l i b r i u m assumption. For r e a c t i n g s o l u t e s i n l a b oratory columns such as those used i n t h i s study, systems w i t h r a t i o s near u n i t y can be modeled using e q u i l i b r i u m chemistry. The experimental r e s u l t s i n d i c a t e that when the r a t i o i s consid e r a b l y l a r g e r than one, another r e l a t i o n s h i p , presumably one i n v o l v i n g k i n e t i c s , must be used. m g

Abstract Miscible displacement of calcium-chloride solutions through water-saturated laboratory soil columns was studied for a wide range of constant water-flow rates. Calcium- and chloride-effluent



234


histories were obtained. Calcium self-exchange was the only reaction considered. Calcium-effluent histories were compared with predictions from a one-dimensional solute-transport model, assuming local chemical equilibrium. Good agreement between the predictions and the data was obtained for the slowest flow rate studied, but not for the higher fluxes. Thus, the local equilib rium assumption applies when the ratio of the hydrodynamic dispersion coefficient to the estimated molecular-diffusion coefficient is near unity. This conclusion is further substan tiated by comparison with the results of a theoretical analysis using the relatively simple transport model for solutes affected by ion exchange that has been developed by Glueckauf (19). It is suggested that the data obtained for the higher water fluxes that yield a coefficient ratio much greater than unity cannot be described by assuming local equilibrium, but must be modeled using another relationship, presumably one involving kinetics. Literature Cited 1. 2. 3. 4. 5. 6. 7.

8. 9. 10. 11.

Perkins, Τ. Κ., and Johnston, O. C . , A review of diffusion and dispersion in porous media, Soc. Petrol. Eng. J., 228 (March), 70-84 (1963). Pfannkuch, H. O., Contribution A l'Etude des Deplacement de Fluides Miscibles dans un Milieu Poreux, Paris Instit. Franc. due Petr. Rev., 18 (2), 215-270 (1963). Nunge, R. J., and G i l l , W. Ν., Mechanisms affecting disper sion and miscible displacement, Indust. Eng. Chem., 61 (9), 33-49 (1969). Bear, J., "Dynamics of Fluids in Porous Media", 764 p., American Elsevier Publishing Company, Inc., New York, 1972. Cameron, D. R., and Klute, Α., Convective-dispersive solute transport with a combined equilibrium and kinetic adsorption model, Water Resour. Res., 13 (1), 183-188 (1977). Skopp, J., and Warrick, A. W., A two-phase model for the miscible displacement of reactive solutes in soils, Soil Sci. Soc. Am. Proc., 38 (4), 545-550 (1974). Rubin, J., and James, R. V., Dispersion-affected transport of reacting solutes in saturated porous media: Galerkin method applied to equilibrium-controlled exchange in unidirectional steady water flow, Water Resour. Res., 9 (5), 1332-1336 (1973). Reiniger, P., "Movement and Exchange of Sodium and Calcium in Calcareous and Gypseous Soils", Ph.D. Thesis, Hebrew Univ., Jerusalem, 1970. Lai, S.-H., and Jurinak, J . J., Numerical approximation of cation exchange in miscible displacement through soil columns, Soil Sci. Soc. Am. Proc., 35 (6), 894-899 (1971). Peterson, Ε. Ε . , "Chemical Reaction Analysis", 276 p., Prentice-Hall, Inc., New Jersey, 1965. Cole, R. C., Gardner, R. Α., Koehler, L. F., Anderson, A. C., Bartholomew, O. F . , and Retzer, J . L., Soil Survey of the


11.

12. 13.


14.

15. 16. 17. 18. 19.

JAMES AND RUBIN

Local Equilibrium Assumption

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Applicability of the Local Equilibrium Assumption to Transport through

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