Numerical analysis of sorption and diffusion in soil micropores, macropores, and organic matter

Computers chem. Engng Vol.21, No. 5, pp, 489--499,1997

Pergamon PII: S0098-1354(96)00288-8

Copyright© 1996ElsevierScienceLtd Printedin GreatBritain.All rightsreserved 0098-1354/97 $17.00+0.00

Numerical analysis of sorption and diffusion in soil micropores, macropores, and organic matter Marco A. Arocha, Alan P. Jackman and Ben J. McCoy* Department of Chemical Engineering and Materials Science, University of California, Davis, CA 95616, U.S.A.

(Received 19 December 1995; revised 22 March 1996) Abstract

Sorption and diffusion of volatile organic compounds (VOCs) in soil are important processes in subsurface contamination, but are complicated by the heterogeneous nature of soil. The soil pore-size distribution is herein approximated as a biporous geometry of micropores and macropores. The microspheres embedded in macrospheres contain micropores. Voids between microspheres constitute the macropores. Soil organic matter is assumed present on the surface of microspheres. We develop numerical solutions for the transient diffusion processes with linear partitioning in soil organic matter and nonlinear adsorption in the biporous soil particles. The three coupled nonlinear partial differential equations describe the diffusion in micropores and organic matter in series with diffusion in macropores. The computation uses an implicit Crank-Nicolson method with a predictor-corrector scheme for fast and efficient calculations. The solutions are applied to toluene and water sorption--desorption experimental data obtained by gravimetric uptake measurements. The procedure allows the estimation of key parameters, including micropore diffusivities. Copyright © 1996 Elsevier Science Ltd

Introduction The sorption and diffusion of volatile organic compounds (VOCs) in soil are important processes in subsurface contamination, but are complicated by the complex and heterogeneous nature of soil. A typical soil may consist of clay grains containing micropores that are agglomerated into particles such that macropores comprise the voids between grains. The presence of soil organic matter (SOM) further adds to the complexity. The diffusion and sorption processes are significantly influenced by the geometrical make up of the soil. Mathematical models for transport in soils must necessarily account for these geometrical features, in addition to the sorption and diffusion processes. We have proposed a transport model based on soil agglomerates described as macrospheres composed of microspheres (Arocha et al., 1996). The voids between microspheres are macropores, and the embedded porous

*To whom all correspondence should be addressed: Ben J. McCoy, Department of Chemical Engineering and Materials Science, University of California, Davis, CA 95616, U.S.A. [email protected],916/752-6923 FAX 916n52-8058

microspheres contain micropores (Fig. 1). Here we extend the model by considering that soil organic matter (SOM) is present on microsphere surfaces. The model has its roots in the analysis of linear adsorption in biporous sorbents proposed by Ruckenstein et al. (1971). Chiou et al. (1988) reported that VOC absorption into SOM is linear, but that adsorption isotherms for VOCs on mineral surfaces are nonlinear (Chiou et al., 1983 and Chiou and Shoup, 1985). The more realistic nonlinear adsorption requires a numerical procedure to solve the coupled macropore and micropore equations (Arocha et al., 1996). In the present case three coupled diffusion equations--for VOC transport in micropores, macropores, and SOM--must be solved. Our analysis has features that may be useful in the numerical solution of similar embedded and coupled partial differential equations. For example, we describe a procedure to handle the singularities at the initial condition that are caused by the nonconstant coefficients in the differential equations. In addition, to fit the numerical solution to experimental data, it is essential that the computer code be optimized to reduce the time needed to obtain a solution. Our solution uses an implicit Crank-Nicolson method in conjunction with a predictor--corrector scheme for fast calculations.

489

M. A. AROCHAet al.

490

micmooms

batter

me,, ar,u macropores Fig. 1. Conceptual picture of a biporous soil aggregate with soil organic matter on microspheres. Model equations The conceptual model of VOC transport in soil is shown in Fig. 1, where we assume that (i) sorption takes place in a medium of macropores, micropores, and organic matter, (ii) microparticles are spherical and of

the same size, (iii) SOM covers a fraction of the microspheres, and (iv) local equilibrium applies. For adsorption in micropores and macropores and partitioning into organic matter, the governing differential equations and boundary conditions are shown in Table 1. The governing equation for the macropore zone (equa-

Table 1. Governing differential equations with initial and boundary conditions Macropore:

OCa+(1 s

OC,

Da~ff 0 ( ~ 8 C , ~

(1.1)

[OCil

[achl

adsorption

desorption

C, = Cb.,k #C/Or. = 0

r. = R. r. = 0

C. = 0 aC.lar, ffi 0

c.=c.

t=o

c.=c.

micropore:

(1.3)

oc,,_ 1 o {no

oc,

(1.2)

ei.~-+(1-.%)--~-- ? ~ri~i "'#~-r/J adsorption c~=c,

r~=R~

desorption C,=C,

aC/Or~ = 0 c, = C~o

r+ = 0 t=o

OCJOri = 0 c,=c~

(1.4)

organic matter: ac~

a~c~

(1.5)

absorption

desorption

c, = K,C.

Zh = 8

c, : r , c .

aCdaz. : o c,=c~

z, = 0

acJaz, = o c,=c~

t=o

(1.6)

Freundlich isotherms at maeropore and micropore fluid-solid interfaces: (1.7) C~,=K.,C,

(1.8)

Numerical analysis of sorption and diffusion tion (1.1) Table 1) includes terms for accumulation in the gas and at the solid surfaces, diffusion in the macropore volume, and diffusional fluxes into micropores and organic matter. The governing equation for the micropore zone (equation (1.3) Table 1) accounts for accumulation in the gas phase and at the surfaces, and for diffusion into the micropore volume. An additional equation (equation (1.5) Table 1) accounts for accumulation, diffusion, and linear partitioning into SOM. In these equations, Sh and S~ represent the microsphere surface area (per volume of microsphere) covered and not covered by organic matter, respectively. The equilibrium adsorption isotherms, equations (1.7) and (1.8) Table 1, link concentrations in the gas phase to surface concentrations, In the present development the Freundlich isotherm describes the local equilibrium adsorption on mineral surfaces up to a relative partial pressure of 0.6 (Arocha et al., 1996). Sorption in SOM is by linear partitioning. At equilibrium (7. = 6'.® = C bulk, C i --- Ci~ = Cbulk, and Ch = Ch~ = KhC.® = K~Cb.lk where Cb.tk is the external (bulk) concentration. Equilibrium sorption experiments (Arocha et al., 1996; Lin et al,, 1994) show that the mass of the sorbate in the gas phase is always below 1% of the sorbed mass, thus e.(OCffOt)<<(l-¢.)OC,./Ot, and ~ ( O C , l O 0 < < ( 1 - ~z~)OCffOt. These approximations simplify equations (l.1) and (1.3) Table 1. The equations (1.1)-(1.6) Table 1 are rendered dimensionless with the variables c~, c. ch, x, y, and z

491

defined in the Nomenclature. The governing equations take the form shown in Table 2, where four dimensionless groups, a;, ah, fli, and fib, are defined in the Nomenclature. The outer microsphere surface (cm2/cm3 macrosphere) can be expressed as Srz=r/4crR~= (4/3).rrR/3 (1 - e,) .47rR/2= -----~--/, 3(1 - ~ , ) where 71is the number of microspheres per macrosphere volume. The micropore surface covered with organic matter is 3(1 - e.) S~= - f and the uncovered surface fraction is

e,

S~=

3(1 - ~.)

ei

( l - J ) ; thus Sr = Si + Sh. Here f is the

fraction of the microsphere outer surface covered with organic matter, 0 < - f -< 1. Thus/3~ and flh can be defined as shown in the Nomenclature section. The dimensionless groups (Bh]Oth)n a and (B]ai)n.ln i correspond to similar groups that are reported in earlier works (Ruckenstein et al., 1971; Arocha et al., 1996). The apparent diffusivities, Da~ptco~ and D~.app~c,., are defined at the constant equilibrium gas-phase concentration, C,,~ and C~, and are thus constant values, i.e., independent of concentration. The SOM effective diffusivity, Dh,.n, is independent of concentration since SOM sorption equilibrium is described by a linear isotherm. The transformations U(x,O)= xc.(x,O) and W(x,y,O)=

Table 2. Governing differential equations with initial and boundary conditions in dimensionless variables macropore:

o¢o ¢,_.I I o ( Olo

(2.1) Or I.,.=~

OZ ]:=,J

adsorption c,= 1

x= 1

desorption ca=O

OcJOx = 0

x =0

OcJOx = 0

ca = Cao micropore:

O= 0

Ca = Cao

Oc,

1

(2.3)

O { 2 8c"~

adsorption ci = c~

y= 1

desorption ci = q

8c/ay = 0

y =0

OcilOy = 0

C i = Cio

O = 0

C i = Cio

organic matter: Och 0¥h

(2.4)

(2.5)

sorption ch = c~

z= 1

desorption Ch = C~

Och/Oz = 0

z =0

OcdOz = 0

= Cho

0 = 0

Ch = Cho

Ch

(2.2)

(2.6)

492

M. A. AROCHAet al.

yc~(x,y,O) change the spherical operator into a rectangular operator and the equations for macropores and micropores, respectively, are simplified to

and micropores are therefore replaced by the small values,

U~=l × lO-nSx~and W~j=l X lO-'ty~, ij=l,2 ....

x

/

(1)

The initial condition for SOM does not need this approximation since SOM equations are linear and regular at the initial condition.

,=,

Implicit Scheme

[

OCh

(2.1 ')

The implicit finite-difference equations are as follows where n = l, 2 ..... for macropores (see Appendix A):

(2.3')

p(i)UT+~+q(i)UT~++~]+r(i)UT~=s(i) i=2,3...,ix-3,

(2)

q(l)U~2*l+r(1)U~3+]=s(l)-p(1)U~ ÷l i=1,

(3)

p(ix - 2) UTah+)2+ q(ix - 2)U7-*', = s(/x - 2) - r(ix - 2 )/-Pa*' The same transformation is applied to initial and boundary conditions. The parameters in the original governing differential equations are thus reduced to ai, ct,, fli, fib, na, and ni. Three coupled diffusion equations (2.1'), (2.3'), and (2.5) Table 2 are to be solved, two of them with concentration-dependent coefficients, (U/ x) j-"` and (W/y) ~-",, which act mathematically as diffusivities. When no = n~ = l, the isotherms for the mineral surfaces are linear, and the sorption-diffusion coupled equations reduce to a linear system. The finite-difference discretization of the partial differential equations and their boundary conditions is described in the Appendix. Figure 2 shows the finitedifference grid points for the diffusion model. The grid points reflect how the macropore diffusion is in series with diffusion in the micropores and organic matter. For the sorption case the equations for macropores and micropores (2.1') and (2.3') are singular at the initial condition, O =0. The initial conditions for macropores

i=ix ( ~

0 •

0 •

i = i x - 2. For micropores:

pl(id3W~f' + q 1(ij)W~'~f+~,+rl(ij)W~if+~2=sl(ij) i=1, 2 . . . ; j = 2 , 3 ..... j x - 3 ,

i = 1 , 2 . . . . . /x;j= 1,

(7)

(i=2, 3 ..... ix;j=jx).

(8)

= s l ( i j x - 2) - rl(ijx - 2)wT~f'

0 •

0 •

0 •

0 •

0

0

0

0

0

0

0

0

0

0

0 •

0 •

0 •

0 •

0 •

0 •

i=2

O

Q

0

o

o

o

i=1

0 •

0 •

0 •

0 •

0 •

0 •

x

i=l;j=jx, p l ( i j x - 2)wTq+~_2+ql(ijx- 2) wTy,

0

O i, grid point i ~ ~ 0 k, gdd point 10T ~' • j, grid point Fig. 2. Finite-difference grid points for the diffusion model.

(6)

p l ( 1 j x - 2)W~t_2+ql(1jx- 2)W7~~_,=sl (1 j x - 2)

0

•

(5)

ql(i,1)Vc~h;t +rl(i,1)W~,;'=sl(i,1) -pl(i,1)W~.~'

<~.,-

/, •

(4)

•

•

•

•

D

))

%.

"~ "-

Numerical analysis of sorption and diffusion For organic matter:

and desorption with an explicit scheme on a VAX 6000-410 OS VMS computer as compared with several minutes for the implicit scheme. The explicit finitedifference formulas for macropores, micropores, and SOM are

• .+l • ~.1 i +r2(i,k)c~,~l.2=s2(i,k) p2(l,k)c~, k +q2(t,k)ci•k÷ i= 1, 2 . . . . ; k=2, 3 .... k x - 3, [p2(i, 1) + q2(i, 1)]c,".,+' + r2(i, 1)c7,~' = s2(i, 1) i = 1 , 2 .... ; k = l ,

493

(9)

(lO)

l n+l p2( 1,kx - 2)c nl•+k_ 2+ q2( 1,kx - 2)ci.~_, = s2( 1,kx - 2)

i= l; k=kx,

(11)

• n+l • p2(l,kx - 2)ci,<_ 2+ q2(i,k.x2)ci, ~_ i= s2(i,k) n+l

•

- r2(i,kx- 2)ci"~I

(14) i=2, 3 . . . . ; k=kx.

(12)

These coupled matrix equations must be solved simultaneously for each time step. Since the coefficients of the tridiagonal matrix are defined at n + l (i.e., a time in the future) a predictioncorrection scheme is necessary. Predicted values of the independent variables are obtained using an Adams formula (Hornbeck, 1975), originally developed for solving ordinary differential equations• The formula used here is accurate within order (AO) 3, exceeding the accuracy of the Crank-Nicolson scheme, which has an error of order (A O)-'. For the prediction of the macropore gas-phase transformed variables we obtain

~+'=~+ao ~ \ 7 ~

.),

,+77 7g .

(13)

Similar equations were developed to predict dependent variables W~,j+l and c;'~I. Since three levels of information (i.e., n, n - 1, and n - 2) are required at the start, we need three values in the past of the dependent variables before being able to predict a new value. The set of equations is therefore solved with an explicit scheme at least three time steps before applying the Crank-Nicoison predictor--corrector technique. The explicit scheme requires only one value in the past of the dependent variable, which for the starting point is provided by the initial condition.

Explicit scheme Though the simplest finite-difference approach for solving partial differential equations, explicit schemes often are slowed by severe restrictions in the time step. Several hours of computational time were required for modeling the experimental data for long-term sorption

~;'=wT,.+

ao~, { ~ \'-",/

7)t t

- 2WT,'j+WTij+,), (15)

~

-c,.,+ ~ Az-

c , 5 _ , - 2cD+c,D+,

.

(16)

These formulas require only one value in the past of a dependent variable. The finite-difference representations for boundary and initial conditions are the same as for the implicit scheme. The explicit scheme so developed is accurate within an error of order (AO, zlx 2, Ay 2, liz2), i.e., first-order in time and second-order in space, in contrast to the implicit scheme, which is second-order in both time and space. This scheme was used as the starter for a few time steps, and then the more accurate and efficient implicit approach was implemented. The fractional uptake is expressed as the ratio of the mass sorbed at certain time, M,, and the mass sorbed at equilibrium, M~,

M]M~=(M.+MI+Mh)I(M,,~+M~+Mh~).

(17)

The concentration profiles at each time can be volumeaveraged to calculate MIMe. A more efficient method is based on the sorbate mass diffusional flux into the macrosphere volume:

MIMe= [ (OcJOx)x=,dOI S (OcJOx)~=,dO. 0

(18)

0

The computation is stopped when M / M s =0.9999, i.e., when equilibrium is essentially reached. The numerical computations were validated by confirming limiting cases of the model. For values of -

-

M~®

= - - n o and ah approaching zero the diffusion and ah

partitioning processes in SOM are negligible and the

494

M. A. A R O C H A et al.

model reduces to the microsphere-in-macrosphere bipor-

g,=

/3,

ous model (Arocha et al., 1996). For - - = - - n o and Ma= ah ah approaching zero and n o = n~ = 1 the governing equations reduce to the linear model of Ruckenstein et al. (1971). The Freundlich exponents are responsible for the asymmetry in sorption and desorption curves (Crank, 1975). For values of n o e l and n : ¢ l the highest asymmetry is observed. When no = n~=l (linear isotherm) the desorption curve is a mirror image of the sorption curve and no asymmetry is observed (Ruckenstein et al., 197 I). Once we have calculated the concentration profiles for at least three time steps, the values are transferred to the implicit technique. We then calculate the predicted values with the Adams formulas. An iterative loop corrects the values by comparing the predicted values against the calculated one. Initial predicted values are replaced by the newly calculated ones and so on. Iterations are stopped when the absolute difference between predicted and corrected values is less than 10 -5 . Usually, the number of iterations necessary for a tolerance of 10 -5 on the absolute difference between predicted and calculated values was no more than two. The implicit Crank-Nicolson predictor-corrector scheme so developed is accurate within an error of second-order in time and space, (AO, Ax, Ay, Az) 2. Stability of the program was investigated numerically by varying the time step. For the range of parameters used in this work, stability was ensured by starting the calculations with a dimensionless time step, dO = 1.0 × 10 -4, increasing up to dO = 1.0 as the calculations progressed. The values of the dimensionless spatial increments are Ax = Ay = Az =1/20. The overall algorithm produced a very efficient and accurate computational scheme requiring no more than 5 minutes of

CPU time to calculate the sorption history curve of an 18-day experiment. Results Experiments conducted in the microbalance described by Shonnard et al. (1993) and Arocha et al. (1996) yielded measurements of M/M=. Sorption history curves for toluene and water on Yolo silt loam are shown in Figs 3 and 4, respectively. To ensure the curves finally reached equilibrium, these data cover a longer time period than most experiments reported in the adsorption literature. As smaller water molecules are able to penetrate the micropores faster, equilibrium was achieved in under 3 h whereas toluene required up to 14 days. Solid lines represent a fit of the present model to the experiments by choosing parameters t~j, Oth, fli, and flh that minimize the sum of squares of deviations between experimental and computed values of fractional uptake, M/M=. The values of the Freundlich exponents, n a and nl, were obtained in the previous study (Arocha et al., 1996) of Yolo silt loam with lower SOM. Results of desorption experiments and numerical analysis are available in Arocha (1995). The model satisfactorily portrays the features of the uptake histories for toluene and water. Values of model parameters are supplied in Table 3. Macropore apparent diffusion coefficients for toluene, DQ~r~ = 1.5 X l0 -s cm2/s, obtained by curve fitting were the same as obtained in the previous analysis that did not include SOM diffusion and partitioning (Arocha et al., 1996). This indicates that the higher SOM content in the current sample has negligible effect on the uptake and diffusion attributed to macropores. For water we obtained DQ.a~ = 1.6 × l0 -s cm2/s. Multiplying D,.=pp by macropore retardation factors, 255 and 2729 for toluene

12

12

8

2

i .......... 10 .7 1 0 4

0

10 4

04

10 "~ 10"2

i

0

5

10 "1 10 0

101

10 2

I

0 time, days

15

20

Fig. 3. Toluene sorption history curve for Yolo siltloam; solid linesrepresentthe curve fitwith the sorption-diffusionmodel.

Numerical analysis of sorption and diffusion

495

15

15

" -'.... "

1

i

,

10

!

i., o

i i

,

|

L_

/ i

t

0 10 ~

C

0

..... 1 0 "3 1 0 "2 1 0 "

10 o

i

i

5O

I00

~ 10'

I 0 2

103

10'

I

150

time, minutes Fig. 4. Water vapor sorption history curve for Yolo silt loam; solid lines represent the curve fit with the sorption-diffusion model.

and water, respectively, gives Down = 3 . 8 x 10 -3 and 4.4 x 10 -2 cm2/s. For diffusion in micropores and SOM, the diffusion time constants, D ~ / ~ and Dh.~nlo~, are simply related to the fitting parameters ai and ah reported in Table 3. If we assume the thickness 8 o f the S O M patches is 1/zm, then Dh.af is o f the order 10 -~2 cm2/s, comparable to the

diffusivity of VOCs in polymeric materials such as polyvinyl acetate (Kokes and Long, 1953). If R i = 1 p,m and the retardation factors are 241 and 4277 for toluene and water, respectively, then Di.eu = 7 . 2 × 1 0 -t3 and 1.3 x 10 -8 cm2/s. The larger retardation for water is due to its stronger adsorption, while the larger diffusivity is due to its smaller molecular size.

Table 3. Experimental and simulation parameters for sorption history curves of toluene and water on Yolo silt loam Parameter Experimental

Units

PIPo Cbulk T R, Ri

(mg/cm 3 gas) (°C) (cm) (/zm)

~ Kinetics

D.~/R~ D~x~ D~

Di~plR~

(l/s) (cm:/s) (cm%) (I/s)

Di.al~

(cm2/s)

3.0

Di.eff Dh.c~t82 Dh,dr a~

(cm2/s) (l/s) (cm2/s)

7.2 × 10 -13 7.5 × 10 -4 8.0 X 10 -~2 1.4 X 10 -~ 3.75 X 10 -2 0.397 0.427 0.26 1.2 11.5 41 11 48

orh

Equilibrium

Toluene 0.16 0.022 25 0.0272 1.0 0.22 0.26 0.02 1.5 × 10 -5 3.8 x 10 -3 3.0 × 10 -7

no ni (/~/a~)n~

(fl¢3oOnJn ~ total sorption % in maempoms % in SOM % in micropoms

(mg/g)

X

1 0 -15

Water 0.042 0.00097 25 0.0272 1.0 0.22 0.26 .021 1.6 × 10-5 4.4 × 10-2 3.0 x 10-4 3.0 × 10-'2 1.3 × 10-s 3.0 × 10-4 3.0 X 10-'2 1.4 X 10-5 1.4 x 10-2 0.400 0.400 0.400 1.40 13.1 36 14 50

496

M. A. AROCHA et al.

Nomenclature

D~..a= e.~D/~,micropore effective diffusivity, cm2/s D i.app=

D~,eff

ei~,+Sj./~.~n~: ~, micropore apparent diffu-

Co= macropore gas-phase concentration, mg/cm3 gas C,,o= macropore gas-phase concentration at zero time, mg/cm ~ gas Co.= macropore gas-phase concentration at equilibrium, mg/cm3 gas

Co- C~o

c°= C~-Coo' gas-phase macropore dimensionless concentration 3

=

e~ !' C.~dr.,

macropore volume-average con-

centration mg/cm3 macrosphere

C~k= outside macrosphere (bulk) concentration, mg/ cm 3 gas C°~a,= saturated bulk concentration, mg/cm3 gas Cn= concentration of VOC dissolved in SOM, mg/ cm 3 SOM C : micropore gas-phase concentration, mg/cm3 gas C~0= micropore gas-phase concentration at zero time, mg/cm3 gas C~= micropore gas-phase concentration at equilibrium, mg/cm~ gas cn=

C~- C~,,

Cn~ - Cn,' dimensionless concentration of VOC in SOM

C,- c~o

c~= - - ,

C~ - C~o

gas-phase micropore dimensionless

concentration

average concentration mg/cm~ macrosphere.

Ch= organic matter sorbed concentration, mg/cm3 organic matter

[(Mt/Mo~ )~,p- (Mt/Moo ).a]2 COD= 1 -

.,~=' i=l

. coeffi-

[(MtlM c~)~p - (M t/M ~ )©~p~,.r]2

cient of determination ~.~,,= macropore sorbed concentration, mg/cm2 (7~= micropore sorbed concentration, mg/cm2 C,.= ~, S J ( 1 - co) macropore sorbed concentration. mg/cm~ solid C,= ~,~SJ(I - e~), micropore sorbed concentration. mg/cm3 solid C~= gas phase concentration at equilibrium, mg/cm~ macrosphere D.= maeropore diffusivity, cm2/s Do~= e,,D,/%, macropore effective diffusivity, cm2/s D..~=

D.~ e o + S ~ / . n . ~ . - " macropore apparent diffusivity, cm2/s

D.~¢.=

D,u~ eo+S~/~no~C, , macropore apparent diffusiv-

ity evaluated at Co=, cm2/s D~= micropore diffusivity, cm~/s

sivity, cm2/s D~.,~ e~.~+S~j./~..nz~Fi , micropore apparent diffusivity evaluated at C~, cm2/s effective organic matter diffusivity, cm2/s fraction of microsphere outer surface cover with SOM. ~',,/C~", macropore Freundtich equilibrium partition coefficient, (mg/cm2)/(mg/cm3 pore) Hi= Cs,/C", micropore Freundlich equilibrium partition coefficient, (mg/cm2)/(mg/cm3 pore) '/ g : C,/C~~, macropore Freundlich equilibrium partition coefficient, (mg/cm3 solid)/(mg/cm3 pore) K : C,/~', micropore Freundlich equilibrium partition coefficient, (mg/cm 3 solid)/(mg/cm3 pore) '/ g~= CJC., SOM linear partition coefficient. (mg/cm 3 solid)/(mg/cm~ gas) M,= total uptake at time t, mg sorbate/cm~ macrosphere total uptake at equilibrium, mg sorbate/cm3 M~= macrosphere n = Freundlich exponent for overall (micro- plus macropore) isotherm h i m Freundlich exponent for micropore isotherm no= Freundlich exponent for macropore isotherm P= partial pressure, ton" I~= saturation vapor pressure, tort pipo= relative partial pressure, dimensionless r = radial coordinate, cm eo= macrosphere radius, cm e : microsphere radius, cm so= specific macropore surface area, cm2/cm s macrosphere S~= specific micropore surface area for single microsphere, cm2/cm 3 microsphere s : specific microsphere surface area covered by organic matter, cm2/cm3 microsphere S T = S~ + Sh, total specific micropore surface area t = time, s U= xc#, dimensionless variable W= yc~, dimensionless variable X = rJR~, dimensionless macrosphere radial position y= r/R~. dimensionless microsphere radial position Z = zh/8, dimensionless SOM distance coordinate

Dhxff f=

R~Dh,¢[e.o+(l-e,,)K,,n,,C~']

[Oh,~/82]

~= R~D,,c[e,,+(1 - e,,)K~noC~-'] = [O,,~lR~]c," Oll=

~= sD,.:~ oo.,~, overall average porosity, cm 3 pomtcm3 macrosphere e.= macropor¢ average porosity, c m ~ pore/eras mac-

rosphcre

Numerical analysis of sorption and diffusion ~,= micropore zone average porosity, cm3 pore/cm3 macrosphere ~o= micropore average porosity for single microsphere, cm~pore/cm3 microsphere ,5= thickness of organic matter deposits

497

Appendix A For the macropores, the dimensionless gas-phase transformed (2.1') is discretized by the implicit Crank-Nicolson scheme,

4

7/= (1 - G)/~ ¢r~ (volume of microparticles/unit macrosphere volume)/unit microsphere volume (i.e., number of microspheresper macrosphere) z.= macropore network tortuosity factor ~= micropore network tortuosity factor O= tD,~ppl~,,IR~, dimensionlesstime .....

.÷l+

f OCh Acknowledgements

The authors thank the California Integrated Waste Management Board, Gibson Environmental, Inc., the National Institute of Environmental Health Science (grant ESO 4699-06), and the Center for Ecological Health Research for financial support.

References

Arocha, M. A., Jackman, A. P. and McCoy, B. J. (1996) Adsorption Kinetics of Toluene on Soil Agglomerates: Soil as a Biporous Sorbent, Environ. Sci. Technol., 30, 1500. Arocha, M. A. (1995) Sorption, Diffusion and Immobilization of Volatile Organic Compounds in Soil, UC Davis Dissertation. Chiou, C. T., Kile, D. E. and Malcolm, R. L. (1988) Sorption of Vapors of Some Organic Liquids on Soil Humic Acid and its Relation to Partitioning of Organic Compounds in Soil Organic Matter. Environ. Sci. Technol. 22, 298-303. Chiou, C. T., Porter, P. E. and Schmedding, D. W. (1983) Partitioning Equilibria of Nonionic Organic Compounds between Soil Organic Matter and Water. Environ. Sci. Technol. 17, 227. Chiou, C. T. and Shoup, T. D. (1985) Soil Sorption of organic Vapors and Effects of Humidity on Sorptive Mechanism and Capacity. Environ. Sci. Technol. 19, 1196. Crauk, J. (1975) The Mathematics of Diffusion, Oxford University Press. Hornbeck, R. W. (1975) Numerical Methods, pp. 196-202. Prentice-Hall, Inc., Englewood Cliffs, NJ. Kokes, R. J. and Long, R. A. (1953) Diffusion of organic Vapors into Polyvinyl Acetate. J. Am. Chem. Soc. 75, 6142. Lin, T. E, Little, J. C. and Nazaroff, W. W. (1994) Transport and Sorption of Volatile Organic Compounds and Water Vapor within Dry Soil Grains. Environ. Sci. Technol. 28, 322. Ruckenstein, E., Vaidyanathan, A. S. and Youngquist, G. R. (1971) Sorption by Solids with Bidisperse Pore Structures. Chem. Eng. Sci. 26, ! 305. Shonnard, D. R., Bell, R. L. and Jackman, A. P. (1993) Effects of Nonlinear Sorption on the Diffusion of Benzene and Dichloromethane from Air-Dry Soils.. Environ. Sci. Technol. 27, 457.

(A1)

~l.=,jj,

Here, n and i are the time and spatial coordinate indexes, respectively. For the first-order time derivative the finitedifference approximation with a central difference representation accurate to within an error of order (AO/2)~ is

aU [~]

,,+,,2_~ ,

'*'_

,

(A2)

A#

After the time index is omitted, the finite-differenceequations for the spatial derivatives accurate within an error of order h2 (here h is the step change in the independentvariable) are given by

[ o~u] _ v,_,- 2v,+v,+,

(A3)

ox2 ] -

~j~

[ OchI

(A4)

Ay

=c"'"-Ch' .....

0z j ,.~.~

(A5)

Az

Defining

~'*~= ~ AO[ ( -U ) ' - " ] '+' and #"=21A0[( U ) ......]i" (A6) we reexpress the finite-differenceequation for the macropores

as

M. A. AROCHA et al.

498

p l ( i j ) = - ~+',,

(A20)

ql(ij)=(1 + 2~++~),

(A21)

rl(id') = - ~j++'~,

(A22)

sl(id)=~l'~/j+~÷,(W~j+~'j÷~+W.:~÷~) n=l,2 .....

(A23)

~:',+ l+---~- ~÷'---M

-

{

Thus, the difference equation for micropores is

pl(ij)w~7' +ql(ij)~7÷', +rl(i,j)~7+~=sl(ij).

(A24)

The boundary conditions for the sorption and desorption cases are given by _ ~:,

-

~' ~ ~

fleri

W~.~=0 (i= 1,2,.. dj= 1),

ax ~

(A7)

Ay

We shift the i index by one (i.e. i + l ) in (A7) and define the coefficients of the tridiagonal matrix,

P(0= A : '

(A8)

(A25)

W~j~=0 (j=jx',x= 1) i= 1,2 .....

(A26a)

W~j~= U~ (i=2,3 ..... /x;x>0) n= 1,2 .....

(A26b)

Xi

Note that there are two representations for the boundary condition at j=jx (or y= 1) ((A26a),(A26b)) since equation (A26a) is singular at i=l (x=O). As I,l~/.~"l and W'/~l=U~+Jlx~are known quantities from the boundary conditions, the first and the last equations in the tridiagonal matrix are modified,

ql(i,1)W~.~%rl(i,1)14,~:.~'=sl(i,1)-pl(i I)W~:~" i=1,2 ..... / x j = l ,

(A27)

pl(1 j x - 2)W~,!_2+ql(ldx - 2)W/~ ~_,= s l ( l j x - 2) i= 1 j=jx, (A28)

p l ( i j x - 2)VC':~_2+ql(idx - 2)W~,f_z, r(i)= Ax2,

(A10)

s(i) = RHS of equation (A7).

(A 11)

=sl(ijx - 2) - rl(ijx - 2)W~+t(i= 2,3 ..... /x;j=jx), in the above equations n= l, 2 ..... ~. For soil organic matter we can define the group

The finite-difference equation for macropores becomes the tridiagonal matrix,

p(i)~'+~ +q(i)U~++~+r(i)~'~=s(i) n=l,2 .....

(A12)

The discretization for the sorption case boundary conditions leads to U l = 0 n= 1,2 .....

(A13)

U~x= 1.0 n= 1,2 .....

(A14)

AOah

(A15)

U~=0 n= 1,2 .....

(A16)

As U'~÷' is known ((A13)) the first equation (i=1) in the tridiagonal matrix is modified as

q(1)U'~+i+r(1)U'~÷i=s(1)-p(1)U'~+j i=I.

(AI7)

(A30)

o-= 2AzZ. In contrast to micropore and macropore equations, the o- group is independent of time and space (i.e., constant) because the SOM diffusion equation is linear• The coefficients of the resulting tridiagona] matrix are defined as

p2 = (i,k) = - o-,

(A31)

q2(i,k)=(1 + 2o-),

(A32)

r2(i,k) = - o',

(A33)

s2(i,k)=d/.k+~'~-2~+.+d/.,÷2) n = l , 2 .....

(A34)

and for the desorption step U, =0 n = 1,2 .....

(A29)

Therefore, the finite-difference equation for the organic matter solid-phase concentration is

• +1 ' I÷l ' I÷l __ • Similarly, /.,r~ ÷' is known ((Al4)), and the last equation (i.e., p2(t,k)~k +q2(t,k)~.~.÷j+r20,k)~.,.z-s20,k) i=ix-2) in the tridiagonalmatrix is modified to give

i,k= 1,2, ..;n= 1,2 .....

p(/x - 2)U'~2 + q(/x - 2)U7~+', =s(ix- 2) - r(/x - 2)/flu+'

i=ix-2.

(AI8)

A similar procedure for the micropores leads to the groups

c~.,=ci.2 (k= 1) i= 1,2 .... n= 1,2 .....

(A36)

ci.v=0 (kfkx) i= 1 n= 1,2 .....

(A37a)

Ui

c ~ = - - (k=kx) i=2,3 ..... /x n= 1,2 ..... ~j+l= ~

\ ~

and ~.t= ~

k ~

The coefficients of the tridiagonat matrix are

(A35)

The boundary conditions for sorption and desorption are

(A19)

(A37b)

X~

Since c~a and c~.~, are known by the boundary conditions, the first and last equations in (A35) are modified as

Numerical analysis of sorption and diffusion

[p2(i,l)+q2(i,l)]~' +r2(i,l)<~'=s2(i,l) i=l,2 .....

(A38)

p2(1 ,k.x- 2)~.~ ~_2+ q2(l,kx - 2 ) ~ ' _ ~=s2(l,kx - 2) i= 1, (A39)

499

p2(i,k.x- 2)d~'~_2+q2(i,k.r- 2 ) ~ _ , = s 2 ( i , k ) - r 2 ( i , k x - 2)~%' i=2,3 .....

(A40)

Thomas's algorithm was used to solve all the tridiagonal ezluations.