Analysis of radioactive cesium and cobalt adsorption to rocks using the two-site kinetic model equations

Journal of Contaminant Hydrology, 8 (1991) 43-69 43 Elsevier Science Publishers B.V., Amsterdam

Analysis adsorption

of radioactive cesium and cobalt to rocks using the two-site kinetic

model equations

Yoko Fuj ikawa and Masami Fukui

Research Reactor Institute, Kyoto University, Kumatori-Cho, Sennan-Gun, Osaka, 590-04, Japan

(Received March 20, 1990; revised and accepted January 14, 1991)

ABSTRACT

Fujikawa, Y. and Fukui M., 1991. Analysis of radioactive cesium and cobalt adsorption to rocks using the two-site kinetic model equations. J. Contam. Hydrol., 8: 43-69.

The present study shows that equations of the two-site kinetic model, which have been applied to trace-metal and pesticide adsorption to soil, could also be used to simulate diffusion- controlled adsorption behaviour of 6°Co and ~37Cs to rock powder (diameter less than 105 x 10 6 m), pieces (diameter 2 x 10 3m ~ 3 x l0 3m) and slabs (0.05 m x 0.05m x 0.01m) of chert, shale and granodiorite in saline water. The adsorption rate coefficient was dependent on sizes of rocks and rock pore structures but was independent of distribution coefficients. The desorption rate coefficient was dependent on sizes of rocks, rock pore structures and distribution coefficients. The difference in adsorption and desorption rate coefficients between rock powder, pieces, and slabs was zero to two orders of magnitude for ~37Cs and zero to three orders of magnitude for 6°Co.

Theoretical relationships between parameters of the two-site kinetic model and the Fickian diffusion model were also developed. Effective diffusion coefficients could be evaluated from the fitted rate parameter values of the two-site kinetic model. Observed characteristics of rate parameters agreed well with the theoretical equation.

Depedence of adsorption and desorption rate coefficients and equilibrium distribution coefficient of 137Cs and e°Co on solid to liquid ratios were insignificant for rock powder.

Applying rate parameter values obtained in this study to field conditions, it was concluded that kinetics of adsorption may often become important in predicting field-scale solute transport.

INTRODUCTION

Local equilibrium for pollutant distribution between groundwater and geologic media has been frequently assumed in studies on underground pollutant transport (Freeze and Cherry, 1979). However, recent studies have shown that the local equilibrium assumption (called LEA hereafter) is often inappropriate in actual geologic media (Friedman and Fried, 1979; Melynk

0169-7722/91/$03.50 © 1991--Elsevier Science Publishers B.V.

44 Y. FUJIKAWA AND M. FUKU|

et al., 1983; W ol f rum and Lang, 1985; Valocchi, 1985) when: (1) slow reactions like redox reac t ions are preva len t (Hodgson , 1960; Sposi to, 1981); a n d / o r (2) supply o f chemicals to adso rp t ion sites are ra te- l imited by diffusion processes such as in terpar t ic le diffusion (Fukui , 1978) or diffusion in rock matr ices (Yornstenfel t et al., 1982). Accord ing to Rubin (1983) and Valocchi (1985), it is no t possible to app ly L E A in solute t r anspor t models if adso rp t ion rates are

NOTATION

a~ radius of rock particle [m] c solute concentration in the liquid phase [unit/m 3]

co solute concentration in the liquid phase before the adsorption experiments were initiated [unit/m 3 ]

Ceq solute concentration in the liquid phase after adsorption equilibrium was attained [unit/m 3 ]

c~m solute concentration in the immobile phase [unit/m 3] Cm solute concentration in the mobile phase [unit/m 3] Da effective diffusion coefficient in rock pores [m 2/s]

f fraction of sorption sites which is in direct contact with mobile liquid Kd equilibrium distribution coefficient [m3/kg] K~ distribution coefficient of instantaneous adsorption [m3/kg]

K,m distribution coefficient in immobile region [m3/kg] Km distribution coefficient in mobile region [m3/kg] kl adsorption rate coefficient [s -~] k2 desorption rate coefficient [s J] L spatial scale of interest [m]

M mass of rock in each bottle [kg] Rim retardation constant in immobile phase [m3/kg]

r ratio of immobile solution to pore solution s total amount of solute adsorbed to the solid phase [unit/kg - rock]

s2 solute adsorbed at kinetic sites [unit/kg - rock] ill2 reaction half life [s]

u velocity of groundwater [m/s] V volume of liquid in each bottle [m 3]

Vim volume of immobile water [m 3] VL volume of liquid phase in the batch system other than pore solution [m 3] Vm volume of mobile water [m 3] Vp pore volume of rock [m 3]

V,o~ whole volume of a batch system [m 3] coefficient defined as 1 + M K ~ / V

at mass transfer coefficient [s -1] fl coefficient defined as - k2 - kl/0~

0~m fraction of the system filled with immobile water Or. fraction of the system filled with mobile water 0p porosity of rock p bulk density of rock defined in terms of rock volume [kg/m 3]

Pb bulk density defined in terms of volume of the system [kg/m 3]

CESIUM AND COBALT ADSORPTION TO RQCKS USING THE TWO-SITE KINETIC MODEL 4 5

"insufficiently fast" compared with the system's other processes that change solute concentration (e.g. a convection process). In order to judge the validity of LEA, it is necessary to examine adsorption time scales of the nuclides of interest.

The sorption of nuclides to geologic materials is governed by various kinetic reactions and processes such as ion-exchange, complex formation, redox reactions, degradation, diffusion, dissolution and precipitation which occur in and between solution and solid. Each of these processes is governed by different rate laws (Sparks, 1988): the ion-exchange reaction itself occurs within microseconds or milliseconds, while the dissolution and precipitation of minerals continues on the orders of years. Therefore, in order to simulate the kinetic adsorption accurately from the first millisecond to years, all rate- controlled reactions relevant to the sorption process should be incorporated into the model. This requires vast efforts and is almost impossible in complicated heterogeneous systems. In this study, we confined the scope of analysis to adsorption kinetics which was observed during the time range of a few months to a few years in batch adsorption systems using rock powder, pieces, and slabs as solid phases and saline water (sea water was used in this case) spiked with 6°Co and ~37Cs as a liquid phase.

The adsorption process was simulated using equations of the two-site kinetic model developed by Selim et al., (1976) and Cameron and Klute (1977). The model was originally used to simulate nonequilibrium sorption which constrains solute transport in soils. It was composed of first-order reversible kinetics combined with instantaneous adsorption. In our case, processes modeled by instantaneous adsorption are fast reactions such as adsorption to charged surfaces and ion-exchange. Those modeled by first- order reversible kinetics are principally diffusion of nuclides into rock pores as will be discussed in the "Results and Discussion" section. The magnitude of the kinetic parameters or the raction half-life, which were obtained through curve fitting, served as a useful criterion to judge the validity of LEA.

As discussed by Rao et al., (1980), van Genuchten and Dalton (1986), and Parker and Valocchi (1986), Fickian diffusion coefficients can be converted to rate coefficients of the two-region mass transfer model and vice-versa in the case of aggregates which have well defined geometrical shape. Based on their study, theoretical relationships between the Fickian diffusion model and the two-site kinetic model were developed. Rate parameters of the two-site kinetic model could be related to intrinsic parameters including diffusion coefficients, particle size, and distribution coefficients.

MATERIALS AND METHODS

For batch sorption experiments, rocks were cut into 0.05 m x 0.05 m x 0.01m slabs, crushed into 2 x 10-3m ~ 3 x 10-3m pieces, and powder

46 Y. F U J I K A W A A N D M. F U K U I

with diameter less than 105 x 10-6m. The rock slabs were thoroughly washed twice for five minutes with distilled water. The rocks were two metamorphosed cherts (designated as rocks A and B hereafter), a slightly metamorphosed shale (rock C), a granodiotie (rock D), and two metamorphosed shales (rocks E and F). All rocks were collected in a coastal area in Fukui prefecture, Japan. Sea water sampled in Wakasa bay, Fukui prefecture, was filtered through membrane filters of 0.45 × 10-6m pore size and was spiked with 137Cs (2.2 x 109Bq/m 3) amd 6°Co (1.5 x 109Bq/m3). Concen- trations of major cationic species in the sea water were 8.6 x 10 °, 9.9 x 10 ', 3.8 x 10 I, and 3.0 x 10 I kg/m 3 for Na, Mg, K and Ca, respectively. The pH of the sea water was initially 8.0 and afterwards varied from 7.3 to 8.1 depending on the rock sample soaked. Mass concentrations of cobalt and cesium in the traced solution were 5.8 x 10-7kg/m 3 and 6.9 x 10 7kg/m3, respectively.

Batch experiments were carried out under aerobic conditions by adding spiked sea water to each rock fraction in a glass bottle. The bottles were allowed to stand for three months to three years in a dark place at 15°C. All samples were prepared in duplicate. Solid to liquid ratios were 680 kg/m 3 for rock slabs, 103 kg/m 3 for rock pieces. Three different solid to liquid ratios, 25 kg/m 3, 50 kg/m 3 and 10 kg/m 3, were applied in the case of rock powder. A Ge detector was used for radioactivity assays of 3 x 10-6m 3 supernatant solutions periodically sampled from liquid phase. Liquid samples were returned to bottles each time after measurements.

MODEL AND PARAMETER ESTIMATION

Two-site kinetic model

Basic assumptions of the model are: (1) different soil constituents react with solutes at different rates, some rapidly and other slowly; and (2) adsorption to certain sites (defined as kinetic sites) is rate limited by slow processes such as solute diffusion and therefore adsorption to such sites appears slow, while adsorption to the other sites (defined as instantaneous adsorption sites) appears fast. The standard of "fast" and "slow" in classification of adsorption rate is relative rather than absolute. It depends on the kinds of adsorption reactions considered. In the two-site kinetic model "fast" reactions are approximated by instantaneous adsorption term. A model by Karickhoff and Morrison (1985) represents "fast" reactions as another kinetic reaction term. In fact, such fast adsorption was observed in experimental measurements of our study as initial rapid decrease of liquid phase concentration before the effect of slower adsorption becomes significant. However, since our present interest is in "slow" adsorption, we applied the two-site model here to

CESIUM AND COBALT ADSORPTION TO ROCKS USING THE TWO-SITE KINETIC MODEL 47

minimize numbers of parameter as far as possible, in view of Occam's razor (Akaike, 1976).

The two-site kinetic model was first developed by Selim et al., (1976), Fukui and Katsurayama (1976), and Cameron and Klute (1977) to simulate nonequilibrium sorption that constraints trace material transport in soils and has been applied by many authors (de Camargo et al., 1979; Rao et al., 1979; Nkedi-Kizza et al., 1984). Although the model is empirical, it describes typical adsorption behaviour that shows a very quick decrease in solute concentration in the early stage followed by a slow reaction step (Mckenzie, 1970; Wolfrum and Lang, 1985).

Our approach was to apply the model equations to adsorption kinetics of radioactive cesium and cobalt to rocks. Conceivable rate-limiting steps of adsorption are as folows: (1) diffusion of nuclides within rock matrices (Neretnieks, 1980); (2) exchange adsorption to certain minerals (e.g. manganese dioxides) that accompanies redox reaction (Burns, 1976); (3) formation of weathering products which serve as the source of adsorption sites during the batch experiments (Walton et al., 1984). Diffusion of nuclides within the matrix of the rock is considered to be the principal rate limiting step in the observed adsorption to rocks, as discussed in the following section. Here diffusion in rock matrices was represented by first-order reversible kinetics, whereas diffusion in rock matrices is usually represented by Fick's second law (Tang et al., 1981; Tornstenfelt et al., 1982). The two-site kinetic model is mathematically simpler and can be incorporated into migration models more easily in comparison to the Fickian diffusion model (Brusseau and Rao, 1989). For example, complicated boundary conditions governed by rock geometry is not necessary in applying the two-site kinetic model.

Mathematical descriptions of sorption dynamics, as contained in the two- site kinetic model, are shown below:

S - - S 2 ~- K d e ( l )

d s 2 / d l = k 1 V c / M - - k 2 s 2 (2)

c = c o - M s / V (3)

Here s[unit/kg - rock], and s2[unit/kg - rock] are the total amount of solute adsorbed to solid phase and the solute fraction adsorbed at kinetic sites respectively, c[unit/m 3] is the solute concentration in the liquid phase, kl Is 1] and k 2 [s- 1] are adsorption and desorption rate coefficients each, and K~[m 3/kg] is a distribution coefficient of instantaneous adsorption. Experimental variables Co, M and V are radionuclide concentration in liquid before the adsorption experiments were initiated [unit/m3], mass of rock [kg], and volume of liquid


[m3], in each bottle, respectively. Initial conditions are as follows:

q,=0+l = c0/(1 + MK~/V) (4)

s = Kdco/(l + MKd/V) (5)

s2<,=o+) = 0 (6)

In above equations, initial distribution of solute between solid and liquid phase as a result of adsorption to instantaneous adsorption sites is assumed (Fujikawa and Fukui, 1990a). The number of unknown parameters can be reduced to two from three by using experimentally measured values of equilibrium distribution coefficient Kd [m 3/kg]. Namely, after equilibrium was attained:

s = [K~ + k, V/(k2M)]ceq (7)

Here Gq is the equilibrium concentration in the liquid phase. Comparing eq. 7 with the following general definition of Kd;

S = KdCeq ( 8 )

a relation between Kd, k, and k 2 c a n be obtained:

Kd = K d - k, V/(k2M ) (9)

Analytical solutions of model equations are:

C/Co = [(f + k2)exp (fit) - k2]/(ef) (10)

s _ V ( l ( f l + k 2 ) e a ' - k2) c o M - 0of (11)

s2 _ V {'1 (1 + M K / J V ) [ ( f + k2)exp ( f i t ) - k2]) Co M ~, -- ~fl } (12)

where

= 1 + M K d / V

f = - k 2 - kl/~

(13)

(14)

Here we introduce another parameter, fi/2[s], which is called a reaction half- life (Melynk et al., 1983) and defined as the time required for liquid concentration to reach (%=0+1 - Ceq) /2" The reaction half-life is expected to serve as a criterion to judge the validity of LEA by comparing it with groundwater resident time (Rubin, 1983; Valocchi, 1985). Here q,=0+)is defined in eq. 4 and % is the equilibrium concentration. The equation of fi/2 derived from the analytical solutions of the two-site kinetic model is:

tl/2 = In 2/(k2 + k~/~) (15)


Experimental data of c/c o were fitted with eq. 10. the nonlinear least square method (NLS) of Marquardt (1963) was applied to evaluate two unknown parameter values. Optimized parameter values of k~ and k 2 with 95% asymptotic confidence range, which reflects the deviation of optimized curve from fitted experimental data, could be obtained. Optimized values of Kd and ill2 could be estimated by substituting obtained kl and k 2 values into eqs. 9 and 15. However, NLS was also applied to experimental data for unknown values of K,~ and tl/2 in order to obtain 95% asymptotic confidence range of t~/2 and K~. Before the fitting, eq. 10 was rewritten in terms of Kd and t~,2 as follows:

C/C o 1 + MK~/V 1 + ~IKd/V exp ( - In 2/tl/2) +

1 + MK /V

(16)

Several earlier data points which represent rapid adsorption were eliminated from each experimental data sets, in order to avoid estimated kinetic parameters from being greater influenced by initial rapid adsorption kinetics. As discussed previously, such rapid adsorption kinetics was approximated by instantaneous adsorption term because our principal interest is to estimate rate constants of slower adsorption. Number of data to be excluded was determined by applying NLS several times to experimental data sets with different starting points. Obtained best-fit rate parameters varied depending on the starting point: higher for earlier points, then take constant values for a while, and become lower for later points. It was considered that higher values of rate parameters were caused by initial rapid kinetics, lower values were caused by equilibrium phase data points. Consequently, earliest of the data pooints that gave constant rate coefficients were chosen as a starting point. As a result, approximately 2% of the time for each system to reach transient equilibrium observed during three years of laboratory experiments was eliminated.

One of the optimized fits by the two-site kinetic model is shown in Fig. 1. The concentration of ~37Cs in the liquid phase is expressed as relative concentration c/c o. A curve that fits with the initial phase of rapid concentration decrease is also shown in the figure. It is evident that the initial phase is governed by much faster rate laws compared with the later phase.

Relationship between parameters of two-site kinetic model, two-region mass transfer model, and physical parameters

Mass transfer into rock pores investigated in this study is comparable to mass transfer into intra-aggregate pores of aggregated soils, which is often

5 0 Y, FUJIKAWA AND M. FUKUI

10

_ °9 f " o.o ( J

0.7

.~ 0.6

~ 0.5 E ~ 0.4 o ~ 0.3

-~ 0.2

~ 0.1 r'r

- - Best- f i t

..... R0pid- phose

o Pieces D

o """- ......... ~ 0 - -

o z 10s 4x'io5 6 1o5 8x'1o5 T i m e i s )

Fig. 1. Optimized fit of L37Cs adsorpt ion behavior to rock pieces D. K d = 0.2[m-~/kg], Kd = 8,54 × 10 3[m3/kg], k I = 2.06 x 10 5[s i], and k 2 = 1.04 × 10 6[s-I] for the optimized fit; K a = 0.2[m3/kg], K~ = 1.58 x 10 3[m3/kg], k I = 5.03 x 10 5[s 1], and k 2 = 2.46 × I0 6[s-l] for the K~ = 0[m3/kg]

case.

modeled using the first-order-type, two-region mass transfer model (van Genuchten and Wierenga, 1976). The model is proved to be mathematically equivalent to the two-site kinetic model in non-dimensional form (Nkedi- Kizza et al., 1984). On the other hand, Villermaux (1981), van Genuchten and Dalton (1986), and Nicoud and Schweich (1989) reported that diffusion coefficients can be converted to rate coefficients of the first-order-type, two- region mass transfer model and vice-versa when diffusion into particles which have well defined geometrical shape (e.g. spheres) is considered. Consequently, parameters of the two-site kinetic model and the Fickian diffusion model for rocks with well-defined geometrical shapes can be related together and rate parameter values obtained through fitting can be compared with diffusion coefficients.

Relationships between rate parameters of the two-site kinetic model and effective coefficients obtained in batch experiments using spherical rocks are as follows (See Appendix for the development):

k I

k 2 =

where

Rim =

150imOa/(Oma ) ( 1 7 )

15Da/(Rima ) (18)

1 + pb(1 -- f)K~mlOim (19)

Here 0m and 0~m are the fractions of the system filled with mobile and immobile water respectively, D, is an effective dispersion coefficient in rock pore [m2/s], a S is a radius of a rock particle [m], Pb is the bulk density [kg/m3], f is the fraction of sorption sites which is in direct contact with the mobile


l i q u i d , Kim is a distribution coefficient in immobile region [m 3/kg], and Rim is a retardation constant of immobile phase. Designating the whole volume of a batch system, volume of mobile water, and volume of stagnant water as gtotal[m3], Vm[m3], and Vim[m3], respectively:

0 m = r m / gtota 1 (20 )

Oim = Vim / gtota I (21 )

Pb = M/Vtotal (22)

Note that Pb is defined based on volume of batch system, not on rock volume. Equations 17 and 18 indicate that parameter kl of the two-site kinetic

model is governed by Fickian diffusion coefficients but not by distribution coefficients of solutes, while k 2 is goverened by both of them, as far as the principal rate-limiting step being the diffusion to rock matrix.

Physical measurements of some of the above-mentioned parameters, for example 0~m, 0m and f, are difficult. Although estimated values of D a and as of rock can be found in the literature quite often, values of 0ira, 0m and f a r e rarely found. However, if Kim is assumed to be equal to K d and porosity of the rock (accordingly pore volume of rock Vp) is known, uppermost values of kl and lowermost values of k2 can be derived from the data on diffusion without knowledge of 0 m and 0ira.

(1) Uppermost values of k~ can be evaluated as follows: Assuming that the solution out of rock pores is "mobile":

Om >~ VL/Vtot,, (23)

0ira ~ Vp / Vtota I (24 )

where VL is the volume of liquid phase in the batch system other than pore solution [m3], Vp is the pore volume of rock [m3]. Therefore uppermost value of k~ can be derived from eq. 17 as follows:

k, ~ 15D, Vp/(VLa~) (25)

(2) Lowermost values of k 2 can be evaluated as follows: Assuming that the fraction of sorption sites in contact with "immobile"

solution is less than the fraction of immobile solution in pore solutions:

1 - f <~ VtotOim/V p (26)

Substituting eq. 26 into eq. 18, we obtain the lowermost value of k2:

k2 >1 15D,/[(1 + MKim/Vp)a~] (27)

Using eqs. 25 and 27, Da can be estimated as follows when values of kl and kz are given:

Va~kl/(15Vp) ~ O a ~ (1 -+- MKim/Vp)a~k2/15 (28)

k~

b

~

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1

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1.

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l0

8 X

l0

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C E S I U M A N D C O B A L T A D S O R P T I O N T O R O C K S U S I N G T H E T W O - S I T E K I N E T I C M O D E L 53

r~ r,i

~ x

• r , r - i x

.~ +1~_ ,,6 x

,m'- ,,.., ,._; x

~ +t ~

• ~ M x

+t ~

("4 ~ ¢ q t'-,I ( :~

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+1~ +1~ +1~ +l~ +iS ~ x ~ x ~ x ~ x ~ x

~ x ~ x ~ x ~ x ~ x

~t ~ x ~ x ~ x ~ x

~ ~ ~ ~ ~ ~ t..q ~ ¢ ~ ~ ,..,..; x ,,.6 x ~ x ~ x ..,4 x

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0.2

1.5

±

0.5

x 10

-3

x 10

-1

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x

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X

10

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X

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0.

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3

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±

21

x 10

-7

5.0

+

0.6

x 10

-9

7.2

±

5.5

X

|0 -

I°

3.6

___

1.3

x 10

.9

2.5

+

0.3

x 10

-7

1.8

+

0.2

xl0

7

1.9

±

0.2

xl0

7

1.4

4- 0

.3

xl0

7

2.4

+

0.3

xl0

7

8.5

+

0.9

x 10

-8

Kd/

Kd

g t~

/2[s

] g

slab

s pi

eces

p

ow

der

sl

abs

piec

es

po

wd

er

A"

4.0

+ 0.

2 5.

5 +

2.5

4.4

_+

1.2

6.5

_+ 2

.5

3.8

+ 2.

0 4.

9 4-

0.

83

× l0

3

X 1

0 -2

X

10

2 ×

10+

6 ×

10+

6 ×

10+

5

B b

1.

3 +

2.0

8.0

___

2.3

2.3

4-

0.8

1.2

___

0.50

5.

1 ___

__ 1.

7 4.

1 ±

0.70

x

10

3 x

10

3 x

10 -2

×

10

+7

×

10 +

6 ×

10 +

5

C c

5.

6 +

3.7

5.3

___

15

1.9

_ 0.

6 1.

9 4-

1.

1 1.

8 +

12

3.1

+ 0.

51

× 10

-1

× 10

-1

X 1

0 -2

×

10 +

8 X

10

+6

X 1

0 +5

D d

8.

8 +

4.6

3.1

+ 0.

5 2.

0 +

0.8

6.1

__+

2.6

8.6

+ 1.

4 3.

7 +

0.89

×

10

-3

× 10

3

X 1

0 2

x 10

+6

× 10

+5

X 1

0 +5

E e

1.

1 +_

_ 2.

6 1.

8 4-

1.

8 5.

3 4-

1.

2 31

.8

____

_ 3.1

4.

1 __

4.3

6.

5 +

0.94

x

10

I x

10

3 x

10

2 ×

10+

7 ×

10+

6 ×

10+

5

F f

2.3

_+ 2

.3

9.5

_+ 4

.3

7.5

_+ 2

.0

8.5

_+

1.8

4.8

_+ 2

.2

2.2

+ 0.

31

× 10

2

x 10

-3

× 10

-3

× 10

+6

x 10

+6

× 10

+5

~M

etam

orp

ho

sed

ch

ert;

bm

etam

orp

ho

sed

ch

ert;

Csl

ight

ly m

etam

orp

ho

sed

sh

ale;

dg

ran

od

iori

te;

emet

amo

rph

ose

d

shal

e; f

met

amo

rph

ose

d

shal

e.

gE

stim

ated

par

amet

er

val

ues

are

sh

ow

n

wit

h i

ts a

sym

pto

tic

95

%

con

fid

ence

ran

ge.

A

ll t

he

val

ues

wer

e ro

un

ded

to

tw

o

sig

nif

ican

t fi

gure

s.

hA m

ark

'-

' m

ean

s th

at a

deq

uat

e p

aram

eter

w

as n

ot

ob

tain

ed

du

e to

sh

ort

age

of

exp

erim

enta

l d

ata.

'So

lid

to

liqu

id

rati

os

wer

e 68

0 k

g/m

3,

103

kg

/m 3

, 10

0 k

g/m

3 f

or

rock

sl

abs,

pie

ces,

an

d

po

wd

er,

resp

ecti

vel

y.

56 Y. F U J I K A W A A N D M. F U K U ¿

RESULTS AND DISCUSSION

Dependence o f parameter values on kinds and sizes o f rocks

Values ofk~, k2, K~/Kd and t~/2 obtained by curve-fitting and of Kd obtained experimentally are shown in Tables 1 and 2 for 6°Co and ~37Cs. Estimation of Kd were made by the following mass-balance equation:

K a = V(c o - Ceq)/(Mceq ) (29)

Values of both k~ and k 2 w e r e smaller for larger sizes of rocks except for rock D. The difference in rate parameters between the three sizes of rocks varied from zero to two orders in the case of 137Cs, and zero to three orders in the case of 6°C0. As shown in Tables 1 and 2, both k~ and k 2 decreased with sizes of rocks, which is consistent with relationships expressed in eqs. 17 and 18, except for rock D. Values of k, and k 2 or rock D were smaller for powder than for pieces. Since rock D was more breakable than other rocks, a large number of fissures in rock pieces D induced crushing seems to have caused faster adsorption-desorption rates.

Values of Kd were smallest for slabs, and increased in the order of powder and pieces except for rock C. For rock C, values of Kd were smallest for slabs, and increased in the order of pieces and powder. The smallest K o values of slabs are probably because dead-end rock pores, which cause decrease of effective surface areas, are more abundant in slabs than in pieces and powder. The reason why Kd of rock pieces were greater than those of powder for most rocks and not for rock C is not elucidated.

The value of K~/K d represents a ratio of the rapidly adsorbed nuclide to the total nuclide adsorbed. As shown in the tables, values of K~/K d were smaller for rock slabs compared with powder and pieces, probably because the contribution of diffusion-controlled slow adsorption to whole adsorption becomes more important for larger rocks. However, values of K~/Kd of pieces were not always smaller than those of powder. It was considered that microfis- sures and cracks of rock pieces induced during crushing increased the fraction of fast adsorption sites. This means that not only the size but also the number of fissures in the rock and extent of weathering influences K~/Ka. Therefore, K~/Kd is expected to be smaller for larger, less-fissured, and less-weathered rocks and apparent adsorption of nuclides to such rocks may be represented by simple first-order reversible kinetics with no instantaneous component.

Coefficients of variation (c.v.) of k~ between different kinds of rocks were greater for slabs and pieces (86% and 194% for ~37Cs, 91% and 125% for 6°Co) compared with powder (43% for '37Cs, 51% for 6°Co) for both 6°Co and ~37Cs. This may be interpreted in terms of eq. 17, which shows that k~ is a


function of an effective diffusion coefficient Da, values of which is determined by rock pore structures according to Wadden and Katsube (1982). The observed variations in k~ probably reflects the different microscopic rock pore structures which exists in slabs and pieces but are destroyed in the course of making rock powder.

On the other hand, k2 did not vary much between different kinds of rocks for rock powder (c.v. were 52% for 137Cs, 34% for 6°Co) as well as for slabs (20% for 137Cs, 77% 6°Co) but varied a lot for pieces (131% for 137Cs, 219% for 6°Co). The variation of k 2 values could not be explained by the difference in microscopic rock pore structures as in the case of k~. This may be explained by combined effects of different Kd and Da values of different rocks, as expressed in eq. 18.

Rate-limiting steps of adsorption

Rate-limiting steps of the kinetic process investigated here may be inferred by comparing the estimated parameter values with the rate coefficients of known fundamental steps such as redox reaction and diffusion. For example, redox exchange adsorption of cobalt to manganese dioxide minerals is suspected to be occurring in the experiment of our study (Fujikawa and Fukui, 1991). Although data on rate coefficients of the redox kinetics of cobalt were not found in the literature, rate coefficients of redox kinetics obtained for arsenite oxidation by manganese oxides is on the order of 10 2 to 10 -4 [S -I] according to Oscarson et al. (1983a, b). Effective diffusion coefficients reported ranged from 2.5 × 10 -14 to 1.6 × 10 -12 [m2/s] for granitic rocks (Bradbury et al., 1982; Skagius and Neretnieks, 1982). The values of effective diffusion coefficients can be converted to uppermost kl values using eq. 25. Here, porosity of rock was assumed to be 0.75%, which is the value reported for granitic and metamorphosed rocks by Chidanken (1970). Comparison of the above-mentioned parameter values with those in Tables 1 and 2 obtained by direct fitting of experimental results by the two-site kinetic model is given in Table 3. Rate parameters of redox kinetics were greater than k~ values of the two-site kinetic model. On the other hand, k~ values obtained by conversion of diffusion coefficients were almost comparable with those obtained by fitting for powder, and were smaller than those obtained by fitting for pieces. Therefore, it may be inferred that diffusion of nuclides into rock is the probably rate-limiting step for the adsorption process considered in this study.

Another conceivable rate-limiting step of adsorption is formation of weathered materials that sorb nuclides, which is expected to continue for orders of months to years (Sparks, 1988). According to Walton et al., (1984), precipitation of iron-oxyhydroxides that sorb nuclides are known to continue

TA

BL

E

3

Co

mp

aris

on

of

rate

coe

ffic

ient

s a

kl o

f ro

ck p

ow

der

[s

~]

kl o

f ro

ck p

iece

s [s

- l]

k~ (D

a) b

k~

(fi

t) c

k~ (f

it) c

k~

(D

a) b

k~

(fit

) ¢

kl (

fit)

~ 13

7Cs

60C

0 13

7 CS

6°C

0

red

ox

rat

e d

[s-~l

A e

1.

7 X

1

0 -

71

2.

5 ×

1

0 -

6

1.5

× I0

6

7.1

× 10

9n

3.

9 ×

10

7 2.

9 ×

10

7 B

f 8.

1 ×

10 -6

1.

9 X

1

0 -

6

_k

2.3

× 10

-7

C g

3.

8 ×

l0

6 2.

6 ×

10

6 4.

8 ×

10

7 8.

5 ×

10

8 D

h

1.1

× l0

5m

8.

3 ×

10

6 2.

3 ×

10

6 1.

1 ×

10

10o

2.1

× 10

-5

1.6

× 10

6

E i

4.5

× 10

-6

1.0

x 10

-6

6.0

× 10

-7

3.0

× 10

-7

F j

5.9

× 10

6

4.4

×

10

-6

8.

4 ×

1

0 -

7

2.2

× 10

7

10

2 ~

10

4

~AI1

par

amet

ers

are

exp

ress

ed i

n m

, kg

an

d s

uni

ts.

hU

pp

erm

ost

k~

valu

es e

stim

ated

fro

m e

ffec

tive

dif

fusi

on

coe

ffic

ient

s re

po

rted

in

Sk

agiu

s an

d N

eret

nie

ks

(198

2) u

sin

g e

q. 2

5. C

on

stan

ts u

sed

to

ca

lcul

ate

k~ w

ere

Vp

= M

O/p

=

0.00

4 ×

0.00

75/2

600,

V

=

4 ×

10 -5

, as

=

5 ×

10

5 fo

r ro

ck p

ow

der

, V

p =

MO

/p =

0.

01

× 0.

0075

/260

0,

V =

9.

7 ×

10

5, a

~ =

0.00

1 fo

r ro

ck p

iece

s. V

alu

es o

f D

a ar

e gi

ven

in t

he t

able

. ~

Val

ues

of

kl e

stim

ated

dir

ectl

y by

fit

ting

ex

per

imen

tal

resu

lts

wit

h t

he t

wo

-sit

e ki

neti

c m

od

el.

d R

ate

coef

fici

ent

of

red

ox

kin

etic

s o

f ar

sen

ite

ox

idat

ion

by

man

gan

ese

ox

ides

(O

scar

son

et

al.,

19

83a,

b)

. M

etam

orp

ho

sed

che

rt;

fmet

amo

rph

ose

d c

hert

; g s

ligh

tly

met

amo

rph

ose

d

shal

e; h

gra

no

dio

rite

; ~

met

amo

rph

ose

d s

hale

; J m

etam

orp

ho

sed

sh

ale.

kA

mar

k -

m

ean

s th

at a

deq

uat

e p

aram

eter

was

no

t o

bta

ined

du

e to

sh

ort

age

of

exp

erim

enta

l d

ata.

~

Up

per

mo

st k

~ va

lues

est

imat

ed f

rom

eq.

25

wh

en D

a =

1.6

× 10

~2

an

d a

~ =

5 ×

10 -5

. m

Up

per

mo

st k

j va

lues

est

imat

ed f

rom

eq.

25

wh

en D

~ =

2.5

x 10

J4

an

d a

s =

5 ×

10 -5

. n

Up

per

mo

st k

l va

lues

est

imat

ed f

rom

eq.

25

wh

en D

a =

1.6

× 10

12

an

d a

~ =

1 x

10

-3

.

°Up

per

mo

st k

l va

lues

est

imat

ed f

rom

eq.

25

wh

en D

a =

2.5

× 10

-~4

and

as

= 1

× 10

3.


for a few months. Although such precipitation may have played an important role in our experimental system, the effect was not considered here because the rate of precipitation was not measured in the present study. It is also possible that the process of weathering such as clay formation is as important as diffusion in rate limiting the whole adsorption process in the period longer than considered in this study. Incorporation of weathering kinetics into the adsorption model needs further investigation.

Estimation of effective diffusion coefficient

Lower and upper limit values of effective diffusion coefficients were estimated using eq. 28 for rock pieces and powder from values given in Tables 1 and 2 assuming: (1) diffusion is the principal rate limiting step of the process; and (2) the shape of each particle of rock pieces and rock powder is spherical. Results are given in Table 4. The lower and the upper limit values were in close agreement with each other. This suggests that (1) assumptions (eqs. 23, 24 and 25) made in the process of deriving eqs. 25 and 27 were valid; and (2) rate parameter values of the two-site model obtained by fitting could be interpreted in terms of the Fickian diffusion model. The latter agrees with the discussion in the previous paragraph that diffusion of nuclides into the rock is the principal rate limiting step. Also, values of effective diffusion coefficients in Table 4 obtained for powder were comparable with those obtained by Skagius and Neretnieks (1982) and Bradbury et al. (1982), while those for pieces were much greater. This may be due to crakcs in pieces induced during crushing.

Dependence of parameters on solid to liquid ratios

Parameters k~, k 2 and K d of rock powders under three different solid to liquid ratios, 25kg/m 3, 50kg/m 3, and 100kg/m 3, are shown in Tables 5 and 6. Dependence of kl, k2 and Kd on solid to liquid ratios were not detected at the confidence level of 95% for both 6°Co and 137Cs. However, rewriting theoretical equations of kl and k2 expressed in terms of Da, 0m and 0ira into equations expressed in terms of experimental variables, both k~ and k 2 were shown to depend on solid to liquid ratios in the following manner. Namely, substituting eqs. 20 and 21 into eqs. 17 and 18:

k, ~- 15VimOa/(Vma~) (30)

k2 = 15Da/{[1 + pb(1 -- f)K~m Vtota~/V~m]a~} (31)

Here Vim, Vm and Vtota~ can be represented using experimental variables as follows:

Vim = rMOp/p (32)

V m = V L -~- (1 - - r)MOp/p ( 3 3 )

TA

BL

E 4

Eff

ecti

ve

dif

fusi

on

co

effi

cien

ts e

stim

ated

fro

m k

~ an

d k

2 v

alu

es o

f th

e tw

o-s

ite

kin

etic

mo

del

137C

s 6O

Co

po

wd

er

pie

ces

po

wd

er

pie

ces

D.

(lo

w) a

D

, (h

igh

) b

D.

(lo

w)

a D

a (h

igh

) b

Da

(lo

w)

a D

a (h

igh

) b

D, (

low

)"

D, (

hig

h) b

[m

2/s]

[m

2/s]

[m

2/s]

[m

2/s]

[m

2/s]

[m

2/s]

[m

2/s]

[m

2/s]

A c

1.

5 ×

10

-12

1.7

x 10

-12

8.8

×

10

I1

1.1

×

10

1o

8.7

×

10

~3

9.1

×

10 -1

3 6.

6 ×

10

11

6.

9 ×

10

II

B

d 4.

8 ×

10

-12

4.9

×

10 -1

2 _i

_i

1.

1 ×

10

-~2

1.2

×

10

12

5.3

×

10

11

5.4

×

10

Jl

C e

2.

2 ×

10

-12

2.7

x 10

12

1.

1 ×

10

lO

1.

7 ×

10

Io

1.

5 x

10

12

1.6

X

l0 -

12

1.9

×

10

11

4.0

x 10

11

D

r 4.

9 ×

10 -1

2 5.

2 ×

10 -1

2 4.

7 x

10

9 4.

9 ×

10 -9

1.

4 ×

10

-12

1.4

×

10 -1

2 3.

6 ×

10

1o

3.

6 ×

10

io

E

g

2.6

× 10

-12

2.9

×

10 -1

2 1.

4 ×

10

1o

1.

9 ×

10

1o

6.

0 X

10

-13

6.4

×

10

13

6.7

×

10

11

6.7

×

10

I1

F h

3.5

×

10 -1

2 3.

9 x

10

12

1.9

×

10

1o

2.3

×

10 -1

° 2.

6 ×

10

12

2.

6 ×

10 -1

2 5.

1 ×

10

11

5.

2 ×

10

11

aLo

wer

lim

it o

f ef

fect

ive

coef

fici

ent

(Da)

val

ues

[m

2/s]

est

imat

ed f

rom

eq

. 30

. P

aram

eter

val

ues

use

d f

or

the

esti

mat

ion

are

0

=

0.0

75

%,

as =

5

×

10

5m f

or

po

wd

er,

as

=

0.00

1 m

fo

r p

iece

s, s

oli

d t

o l

iqu

id r

atio

is

100

[kg/

m3]

.

bU

pp

er l

imit

of

effe

ctiv

e co

effi

cien

t (D

a) v

alu

es [

m2/

s] e

stim

ated

fro

m e

q.

30.

Par

amet

er v

alu

es u

sed

fo

r th

e es

tim

atio

n a

re

0 =

0

.07

5%

, as

=

5 ×

10

5m

fo

r p

ow

der

, as

=

0.

001

m f

or

pie

ces,

so

lid

to

liq

uid

rat

io i

s 10

0 [k

g/m

3].

CM

etam

orp

ho

sed

ch

ert;

dm

etam

orp

ho

sed

ch

ert;

esl

igh

tly

me

tam

orp

ho

sed

sh

ale;

fg

ran

od

iori

te;

gm

etam

orp

ho

sed

sh

ale;

hm

etam

orp

ho

sed

sh

ale.

~A

mar

k -

m

ean

s th

at a

deq

uat

e p

aram

eter

was

no

t o

bta

ined

du

e to

sh

ort

age

of

exp

erim

enta

l d

ata.

,<

).

Z

TA

BL

E

5

Par

amet

ers

of 1

37C

s ad

sorp

tion

to

roc

k p

owd

er

wit

h

diff

eren

t so

lid

to l

iqui

d ra

tios

25

kg/

m 3

, 50

kg/

m 3

, 10

0 k

g/m

3

Kd[

kg/m

~1

kl[

s i]

g k2

[s

,]g

S/I

rati

o S/

! ra

tio

s/1

rati

o 10

0 50

25

10

0 50

25

10

0 50

25

A"

2.7

2.2

1.9

2.5

_ 0.

7 8.

6 __

_ 3.

8 8.

8 _

24.8

1.

1 _+

0.3

1.

3 _+

0.4

3.

7 _

7.3

X 1

0 .2

X

10

2 X

10

2 X

l0

-6

X

10 -7

X

10

-7

X 1

0 -6

X

10

-6

X 1

0 -6

B b

1.

8 2.

2 2.

1 8.

1 __

1.

5 7.

1 __

+ 1.

1 4.

6 __

+ 1.

4 4.

7 __

+ 0.

8 7.

3 +

1.1

1.1

_+ 0

.3

X 10

-I

X 10

I

x 10

-1

X 10

6

X 10

-6

X 10

-6

X 10

7

X 10

7

X

10

-6

C ~

2.

5 2.

7 2.

2 3.

8 __

_ 0.

5 8.

1 __

_ 9.

4 2.

9 __

_ 2.

1 1.

8 +_

_ 0.

2 1.

2 +

1.2

7.9

+ 4.

3 x

10 -2

x

10

2 x

10 -2

x

10

6 x

10

7 X

10

-6

X l

0 6

X 1

0 6

X 1

0 6

D ~

1.

9 2.

0 2.

1 8.

3 +

2.0

8.9

_+

1.6

7.5

+ 1.

6 4.

6 +

1.1

1.0

_+ 0

.2

1.7

+ 0.

3 xl

0 I

xl0

' xl

0 -1

xl

0 6

xl0

6 xl

0 6

xl0

7 xl

0 6

xl0

6

E ¢

5.8

6.1

7.8

4.5

+ 0.

8 2.

8 __

_ 0.

7 2.

9 +

1.0

8.5

+__

1.5

1.0

___

0.2

2.0

+ 0.

6 X

10 -2

X

10

2 X

10

2 X

10

6 X

10 -6

X

10 -6

X

10

7 x

10 -6

X

10

6

F f

5.8

6.9

4.7

5.9

-t- 0

.7

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Vtota I = V L -~- m/p ( 3 4 )

where r and 0p are ratio of immobile solution to pore solution and porosity of rock, and p is bulk density defined in terms of rock volume, respectively. Substitution of eqs. 32, 33 and 34 into eqs. 30 and 31 gives:

k, = 15rOpOa/{[pVe/m + (1 - r)Op]a~} (35)

k2 = 15D,/{[1 + pb(1 --f)K~m(pVL/M + 1)/(rOp)]a~} (36)

Solid to liquid ratio of batch solution (M/V) can be approximated by M~ V c because usually V -- VL in batch systems. Equations 35 and 36 indicate that, theoretically, rate coefficients increase as solid to liquid ratios (M/VL) increase. A mechanism that causes the deviation of the fitted k~ and k2 values from the theoretical is not elucidated at the present state. Possible explanations are that D, in the numerator and/or r in the denominator of eqs. 35 and 36 decreased as solid to liquid ratio (M/VL) increased. Thus the effect of higher solid to liquid ratios, namely, smaller V/ML values in the denominator of eqs. 35 and 36, was masked.

Extrapolation of parameter values to field conditions

Parameter values for large volume of rocks such as those in the field may be estimated by extrapolating the values for smaller pieces of rocks obtained by fitting the experimental data. As discussed in the previous section, if the shape of the rock body could be approximated as spherical, theoretical basis of such extrapolation is provided by Villermaaux (1981), van Genuchten and Dalton (1986), and Nicoud and Schweich (1989). However, it should be noted that diffusion-controlled adsorption to larger volumes of rock usually takes longer than to smaller rock, and phenomena such as weathering which were not important in shorter time scales may become important factors concerning the adsorption behaviour of nuclides. Simple extrapolation of parameter values obtained for small rocks may become inadequate in such cases.

If the effect of weathering is insignificant, estimation of parameter kl, k2 and tl/2 c a n be made as follows. Assuming that values of r, f a n d D, estimated from batch-experiments is the same with those in field, kl and k2 can be extrapolated to those for field using eqs. 35 and 36. fi/2 can be evaluated from eq. 15. Solid to liquid ratios, rock diameter and K d need to be adjusted to appropriate values.

Equations of r and f a r e obtained by rewriting eqs. 35 and 36 as follows:

r = k~a~(VL/M + Op/p)/[Op/p(15D, + k~a~)] (37)

f = 1 + [1 - 15D,/(a~k2)]rOp/[pbK~m(Vep/M + 1)] (38)

Values of r were estimated by substituting into eq. 37, values of k~ and k2 from

6 4 Y. FUJIKAWA AND M. FUKUI

Tables 1 and 2, values of O a from Table 4, corresponding experimental values of M! VL and as, and 0.0075 for 0p (Chidanken, 1970). Values of f were similarly evaluated by substituting into eq. 38 values o f r estimated before and Kd values from Table 1 and 2. As for Da, mean upper and lower limit values from Table 4 were used.

Assuming that M] VL and rock diameter to be 2.5 x 105[kg/m 3] and 5 m in the field, and field Kd values are the same as those of rock pieces observed in batch experiments, parameter values are extrapolated as follows:

(1) kl ranged from 7 x 10 -1° to 8 x 10 -8 for 137Cs and 5 x 10 -t° to 9 x 10 -9 for 6°Co

(2) k2 ranged from 9 x 10 -12 to 1 x 10 -z° for 137Cs and 1 x 10 -13 to

1 x 10 -II for6°Co (3) t,/2 ranged from 5 x 109 to 7 x 101°[s] for 137CS and 6 x 10 '° to

7 x 1012[s] for 6°Co The above evaluation includes assumptions that r,f, Da and Kd are the same

between pieces in batch experiment and rocks in field which is larger and less fractured, and higher in solid to liquid ratios. Although characteristics of r and f are less well known, both D a and K a are expected to take smaller values in the field due to dead-end pores in rocks. In such case, kt may be smaller and k2 may be larger than the above estimation, but the response of t~/2 depends on the deviation of actual D, and Ka values from the assumed values.

Validity of LEA

According to Rubin (1983) and Valocchi (1985), it is not possible to apply LEA in solute transport models if adsorption rates are "insufficiently fast" compared with the system's other processes that change solute concentration (e.g. a convection process). Whether the adsorption rate is sufficiently fast or not, can be decided by comparing groundwater resident time in the spatial scale of interest with the reaction half-life of the nuclide. Melynk et al. (1983) also reported that for the equilibrium to be attained:

L > lOOu/kt (39)

where L is a spatial scale of interest, u is the groundwater velocity [m/s]. If we assume the reference distance, the distance of interest, to be one hundred meter, and using the groundwater velocity 2.08 x 10-6[m/s] which is reported at CRNL test site (Melynk et al., 1983), resident time of 4.8 × 107[s] is obtained. The reaction half-life obtained in the previous section is longer than the groundwater resident time and therefore adsorption is "insufficiently fast" compared with groundwater flow rates for the distance of interest. Applying kt values in the previous section to eq. 39, L > 2 × 103 to 3 × 105[m] for


137Cs and 2 x 104 to 4 x 105[m] for 6°Co for LEA to be valid. It implies that LEA is not always valid in field-scale transport problems and that kinetics of adsorption often needs to be included into the migration model.

CONCLUSIONS

The diffusion-controlled adsorption of 6°Co and 137Cs to various sizes of rocks could be simulated by the two-site kinetic model. Rate-limiting steps of the observed adsorption was identified as diffusion by comparing estimated kl values with rate parameters of known fundamental steps.

Both kt and k2 were generally smaller for greater sizes of rocks. The difference in adsorption and desorption rate coefficients between rock powder, pieces, and slabs was zero to two orders of magnitude for 137Cs and zero to three orders of magnitude for 6°C0.

The contribution of instantaneous adsorption to whole adsorption tended to be smaller for larger, less fissured and less weathered rocks. Therefore, diffusion-controlled adsorption to such rocks may be simulated by a model where all sorption is constrained by simple first-order reversible kinetics.

Equilibrium distribution coefficients were smaller for slabs for all the rocks tested. It is probably because dead-end pores are more abundant in slabs than in pieces and powder.

Theoretical relationships between effective diffusion coefficients, sizes of rocks, distribution coefficients, and adsorption and desorption rate coefficients of the two-site kinetic model were developed assuming a spherical geometry for the rocks.

Characteristics of rate parameters observed experimentally could be interpreted well by the theoretical relationships. For example, the adsorption rate coefficient kl was dependent on sizes of rocks and rock pore structures but was independent of distribution coefficients. The desorption rate coefficient k2 was dependent on sizes of rocks, rock pore structures, and distribution coefficients.

Using the relationships, effective diffusion coefficients could be evaluated from the fitted rate parameter values of the two-site kinetic model.

Adsorption and desorption rate coefficients and equilibrium distribution coefficients of ~37Cs and 6°C0 for rock powder were independent of solid to liquid ratios of batch experiments at the confidence level of 95%. Nevertheless, theoretical equations indicate that a correlation exists between those parameters. There is a possibility that the dependence of D a or r on solid to liquid ratios weakened the correlation.

Also, kl, k2 and tt/2 values obtained for smaller pieces of rocks can be extrapolated to larger ones provided the rocks are spherical and other parameters (Da and Kd) are constant.

66 Y. FUJIKAWA AND M, FUKUI

Comparing the extrapolated kl, k2 and tl/2 values with groundwater velocity and resident time reported at a field site, it was concluded that kinetics of adsorption may often become important in solute transport at the field scale.

A P P E N D I X

(1) Conversion o f the f irs t order type two-region mass transfer model into the two-site kinetic model

Mass transfer equations of the first-order type, two-region mass transfer model shown in van Genuchten and Wierenga (1976) are as follows:

dcim - - 0 ~ t ( ¢ m - - e i m ) dt 0ira Rim - - __

s m

Sim

where

Rim -=

gmCm

gimCim

(40)

(41)

(42)

1 + p b ( 1 - - f ) K i m / O i m ( 4 3 )

Here Cm, eim and at are solute concentration is mobile phase [unit/m3], solute concentration in immobile phase [unit/m3], and mass transfer coefficients Is I], respectively. Equations of the two-site kinetic model are:

ds 2 Vktc - k2s2 (44)

dt M

s - s 2 = K~c (45)

Relation between variables of the two models are assumed as follows:

c = Cm (46)

S - - S~ = f s m (47)

M s 2 = Vimeim -4- VtotalPb(1 - - f ) S i m (48)

Representing experimental variables of the two-site kinetic model by parameters of the two-region mass-transfer model:

M = Pb Vtotal (49)

V = V m ( 5 0 )

Substituting eqs. 47, 49 and 50 into eq. 48:

s2 = OimRimCim/Pb (51)

Equations 45, 49, 50 and 51 were substituted into the following model


equa t ion o f the two-site kinetic model:

ds2/dt = kl V e / M -- k2s2 (52)

Thus kinetic equa t ion of tim was obta ined as follows.

dcim _ Omklcm _ OimRimk2ci m (53) 0ira Rim dt

C o m p a r i n g the r igh t -hand sides o f eq. 53 and eq. 40, k~ and k2 can be expressed by a t as follows:

kl = 0 ~ t / 0 m (54)

k: = o~t/(OimRim ) (55)

(2) Relationship between the two-site model and the Fickian diffusion model

The relat ionship between mass t ransfer coefficient a t and effective diffusion coefficient O a for spherical particles o f radii as is given by Vil lermaux (1981) and van Genuch t en and Da l ton (1986) as follows:

O~ t = 15OaOim/(a 2) (56)

Subst i tu t ion o f eq. 56 into eqs. 54 and 55 gives:

k I = 150imOa/(Oma~) (57)

k 2 = 15O,/(Rima~) (58)

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Documents

Analysis of radioactive cesium and cobalt adsorption to rocks using the two-site kinetic model equations