aquifer material originating from the water saturated zone of a sandy, ..... We like to thank Dr. Phil Jardine, Oak Ridge National Labs, who was so kind to.
The modeling of Reactive Solute Transport with Sorption to Mobile and Immobile Sorbents Part II: Model Discussion and Numerical Simulation K. U. Totsche2*, P. Knabner1 and I. Kögel-Knabner2 1Institute for Applied Mathematics, University of Erlangen-Nürnberg, Martensstr. 3,
D-91058 Erlangen, Germany. 2Soil Science Group, University of Bochum, NA 6/134, D-44780 Bochum; *present address: Soil Physics Group, University of Bayreuth, D-95440 Bayreuth, Germany. ABSTRACT We analyze mathematically and numerically a model derived in Part I of this paper. The model deals with carrier influenced transport and besides advective and dispersive transport and equilibrium and nonequilibrium sorption takes into account both carrier facilitation and co-sorption. Guidelines are derived, whether the overall mobility is enhanced or reduced and various other properties of the model elucidated, in particular for varying carrier concentrations. We indicate how to modify existing numerical codes for the usual adsorption model and discuss simulations for experimental data sets. INTRODUCTION This paper is a sequel of Knabner et al. [1995], in the following referred to as part I. In part I based on a discussion of experimental findings, we have set up a mathematical model to describe the advective and dispersive transport of both a dissolved carrier, which undergoes possibly nonlinear equilibrium and nonequilibrium sorption to the soil, and the movement of a dissolved contaminant, which may get attached to the carrier, thus existing in solution in a free and a bound form, where both of these forms are also subject to all the transport mechanisms already mentioned. We always refer to the carrier as dissolved organic carbon (DOC) and to the contaminant as hydrophobic organic chemicals (HOC). In particular this model both takes into account facilitated transport of contaminants (co-transport) and the adsorption of the HOC attached to reactive carrier to the soil matrix (co-sorption), i. e. two competing mechanisms. A transformation in terms of total concentrations of contaminant and carrier simplifies the model, insofar the model now has the structure of the well-known equilibrium-nonequilibrium multiple site adsorption model, but with space and time dependent, implicitly defined isotherms and rate functions. They are called effective, as they combine the competing mechanisms. The effective isotherm resulting for linear isotherms has already been pointed out in part I. This part II assumes the knowledge of the basic notation and equations of part I. Its aim is to analyze the model mathematically and numerically, which in particular includes a comparison with the experimental data discussed in part I. The mathematical analysis is not directed towards closed form (analytical) solutions: These solutions are not available in the general case, but in some special cases quite obvious: If the carrier concentration is constant and the 1
isotherm Π describing the HOC-DOC formation, is linear, then the model is just a scaled standard equilibrium-nonequilibrium multiple-site-adsorption model. This means, that if only linear isotherms and rate functions appear, we can apply, depending on spatial domain and boundary conditions, the whole range of closed form solutions, which have been developed for decades. These solutions are wellknown in soil science (see e.g. Kreft and Zuber [1978] or van Genuchten and Alves [1982]), such that a repetition is unnecessary. We rather want to elucidate qualitative and quantitative properties of the solution, also in the nonlinear case, without knowing the solutions explicitly. In particular we are concerned how to detect whether there is enhanced or reduced mobility. We can derive certain parameters which answer this question only from the knowledge of the participating isotherms. Nevertheless also numerical simulation is necessary to compare quantitatively with experimental data. We indicate how existing codes for the simulation of the usual adsorption model have to be extended and discuss some simulation results. MODEL PROPERTIES Properties of the effective isotherms and rate functions Our aim is to discuss the properties of effective isotherms and rate functions starting from general, i.e. in particular nonlinear isotherms ΨHf, ΨHb, Π and rate functions gHf, gHb in the original formulation (part I, Eq. 3 - 6 and 9). To make our reasoning correct, several properties of ΨHf etc. are necessary, which will be introduced later on and are fulfilled by all commonly used isotherms and rate functions (see e.g. van Duijn and Knabner [1992a]). In particular we can allow for the Freundlich isotherm with exponent less than 1. Same effects of carrier on equilibrium and nonequilibrium reactions Concerning the nonequilibrium reactions, we assume that the corresponding equilibria are given by isotherms, i.e. there are functions ϕHf, ϕHb such that
(
)
(
)
(1a)
g Hb (C Hb , S Hb ) = 0 ⇔ S Hb = ϕ Hb (C Hb )
(1b)
g Hf C Hf , S Hf = 0 ⇔ S Hf = ϕ Hf C Hf
Obviously this is the case if one uses the following explicit form:
(
)
( ( )
g Hf C Hf , S Hf = k Hf ϕ Hf C Hf − S Hf
)
(2)
with a rate parameter k Hf ≥ 0, and analogously for gHb. The effective rate functions fHf, fHb preserve the property (1a,b) with the following isotherms
2
f Hf ( x , t , C H , S Hf ) = 0 ⇔ S Hf = χ Hf ( x , t , CH ) = ϕ Hf ( G ( CD ( x , t ), CH )) f Hb ( x , t , CH , S Hb ) = 0 ⇔ S Hb = χ Hb ( x , t , C H ) = ϕ Hb ( CH − G ( CD ( x , t ), CH ))
(3a)
(3b)
Furthermore also the explicit form (2) is preserved with the isotherm defined in (3a,b). We observe that the isotherms χ Hf , χ Hb have the same structure as the constituents of the effective isotherm Ψ (defined in part I, Eq.(12)) describing the equilibrium adsorption process. This means that we can restrict the following discussion to the equilibrium adsorption process: Due to (3a,b) the nonequilibrium process will have the same properties discussed below, modified by the delays of a nonequilibrium process. Therefore we now turn to the properties of the effective isotherm. Limit cases We assume that Π(0) = 0, Π(C ) > 0 for C > 0, Π ′(C ) ≥ 0 for C ≥ 0
(4)
(Π ′(0) = ∞ is allowed ) and the same properties for ΨHf, ΨHb. Here ⋅′ denotes the derivative. If a function depends on several variables, we will write the differential explicitly. First we look at the limit cases of the model: For CD ( x , t ) → 0 (for fixed (x,t ) and C H (x , t )) we expect the model to reduce to the adsorption model only dealing with sorption to the soil. In fact, then by (part I, Eq. 10) CHf ( x , t ) → CH
(5)
and thus CHb ( x , t ) → 0, i.e. ρ Ψ( x , t , C H (x , t )) → ΨHf
ρ ΨH
ΨHf ( C H ( x , t ))
(6)
Thus locally at (x,t) there is no reduction in sorption of CHf = CH to soil, and there is no sorption of CHb to soil, as CHb = 0. This discussion holds also true, if the effective isotherm has the form of (35) in part I related to the modeling assumption of Jiang and Corapcioglu [1993], and the sorption of the carrier is in equilibrium, i.e. S D = ΨD (C D ) . For CD ( x , t ) → ∞ (for fixed (x,t)), we find by (part I, eq.10)
3
CHf ( x , t ) → 0
(7)
and thus CHb ( x , t ) → CH ( x , t ), i.e. Ψ( x , t , C H (x , t )) → ρ ΨHb
ρ ΨH
ΨHb ( C H ( x , t ))
(8)
Thus locally at (x,t) there is no reduction in sorption of CHb = CH to soil, and there is no sorption of CHf to soil, as CHf = 0. In particular, if there is no sorption of
C Hb (ρ ΨHb ΨHb = 0), the total concentration CH behaves as an inert solute. This was also observed by numerical simulations by Lafrance et al. [1989] for the special case considered there. If we restrict to the case of a linear isotherm Π (see (25)) the reasoning above can be sharpened by substituting CD → 0 by K HD CD → 0 and analogously for CD → ∞ , reflecting the fact, that not only the amount of carrier, but also its sorption capacity influences the process. Considering the alternative form of the effective isotherm, (35) from part I, then the conclusion is not so clear. Again we restrict to S D = ΨD (C D ) . If ΨD is such that S D → S D max for CD → ∞ , then due to (7)
we would have the unnatural consequence of Ψ( x , t , C H (x , t )) → 0 , i.e. in particular no co-sorption. If on the other hand Π is linear according to (28) of part I, Π C Hf = K HD C Hf , and also ΨD, ΨD (CD ) = K D C D , then
( )
ρ Ψ( x , t , C H (x , t )) = ΨHb ρ
ΨH
k
KD C H (x , t ) K HD
(9)
which may be interpreted as a representation of co-sorption. Dependence on carrier concentration: enhanced or reduced mobility The dependence of the effective isotherm for fixed CH(x,t) on the carrier concentration CD is shown by d Ψ( x , t , C H ) = dC D
(
) (
Π C Hf ρ ρ ΨHb C C C − ΨHf − − ⋅ Ψ Ψ ′ ′ Hf ρ ΨH Hf Hf ρ ΨH Hb H 1 + CD ( x , t )Π ′ C Hf
(
)
(
)
(10) where CHf is given by G (CD ( x , t ), CH ). The factor after the square brackets has to be interpreted as 0 for CHf = 0. Define A( CH ) by A( C H ) =
ϕ ΨHf
ϕ ΨH
ϕ ΨHf ′ ( CHf ) − ΨHb
ϕ ΨH
ΨHb ′ (CHb ) ,
(11)
4
)
where CHb = CH − CHf . Thus the behaviour is determined by the expression A: If A ≥ 0, then Ψ is monotone decreasing in CD, i.e. an increase of carrier decreases the overall sorption. This is the case, if the sorption of free HOC CHf to soil dominates the sorption of carrierbound HOC CHb to soil and this is exactly quantified by the requirement A ≥ 0. It is this case, which corresponds to an increase in mobility due to the binding of HOC to the carrier. Such a situation is given if there is no sorption of carrier-bound HOC, a case usually considered only in the literature (see section Special Cases in part I). If A ≤ 0, then Ψ is monotone increasing in CD, i.e. an increase of carrier concentration increases the overall sorption. This is the case if the sorption of CHb to soil dominates the sorption of CHf to soil, exactly quantified by the requirement A ≤ 0. It is this case which corresponds to a decrease in mobility due to the binding of HOC to the carrier. Estimation of mobility changes for linear isotherms For the special case, where all isotherms are linear, considered in eq. (32)-(34) part I, the dependence of the effective isotherm on the carrier concentration can be made even more explicit: We define the following quantities α = α (x , t ) = K HD CD ( x , t )
β=
ρΨHb KHb , ρΨHf KHf
(12a)
(12b)
where α is the mass of carrier bound HOC related to the mass of carrier, and β the ratio of sorption capacity to soil of carrier-bound HOC and of free HOC, both related to the total volume of an REV. The effective isotherm (eq. 33, part I) may be written in the following form: ρ K Hf C H , Ψ(x , t , C H ) = f (α , β ) ΨHf ρ ΨH
(13)
where the function f is defined by f(
α, β ) =
1 + αβ 1+ α
.
(14)
Thus the function f characterizes the deviation of the effective isotherm from the isotherm describing the sorption of free HOC to soil: The mobility is enhanced, if f (α , β ) < 1, which is exactly the case for β < 1. The mobility is reduced, if f (α , β ) > 1, which is true for β > 1. Thus the size of β gives a direct estimation of the principal influence of the carrier on the mobility independent of the carrier concentration, or more exactly of α. The general criterion based on (11) can be recovered, as here, independent of CH :
5
ρ A( C H ) = ΨHf
ρ ΨH
K Hf (1 − β )
(15)
Also, the limit cases (6), (8) and the special case (34) of part I can be expressed in terms of f. They correspond to f (0, β ) = 1, f (∞, β ) = β , f (α ,0) =
1 1+ α
(16)
respectively. Figure 1 displays breakthrough-curves for different values of β, figure 2 the graph of f. A similar analysis is possible, if the sorption of HOC to the carrier is still linear (according to (28) in part I)), but the isotherms ΨHf and ΨHb are of nonlinear Freundlich type:
( )
p ΨHf C Hf = K Hf C Hf
(
(17)
)
q ΨHb C Hb = K Hb C Hb
with the parameters K Hf , KHb , p, q > 0. We define α and β as in (12), where the interpretation of β has to be modified and β is only dimensionless for p=q. The generalization of identity (13) then reads ρ p K Hf C H , Ψ(x , t , C H ) = g(α , β , C H ) ΨHf ρ ΨH
(18)
where the function g is defined by
g(α , β , C H ) =
q
α q− p + β CH , p 1 + α (1 + α ) 1
(19)
The picture is more complicated, as the factor g describing the deviation of the effective isotherm from the isotherm for the free HOC depends not only on p and q, but also on the total concentration CH. For the sake of simplicity we restrict ourselves to the situation p=q, where g is independent of CH and simplifies to
g(α , β ) =
1 + βα p
(1 + α ) p
.
(20)
Again, g( 0, β ) = 1 and for α > 0 there is an enhancement of mobility, if g(α , β ) < 1 and a reduction, if g(α , β ) > 1. These cases are characterized by 6
g(α , β ) < 1 ⇔ β < h(α )
(21)
g(α , β ) > 1 ⇔ β > h(α ) where the function h is defined by h(α ) =
(1 + α ) p − 1
(22)
αp
i. e. the characterizing threshold for β is now α- (and p-) dependent. More precisely, for p < 1 we always have h(α ) < 1 and h(α ) = 1 only for α = ∞ , i. e. an abundance of carrier. This means that compared to the totally linear case (14), the range of β-values, for which the mobility is enhanced, decreases. It vanishes in the limit α → 0, i. e. for vanishing carrier concentration, as h(α ) → 0 for α → 0 . For p > 1 the picture is reversed: Then we have always h(α ) > 1 and h(a ) = 1 only for α = ∞ . Compared to the totally linear case (14), the range for β values, for which the mobility is enhanced, increases and for small carrier concentrations there is only enhancement, as h(α ) → ∞ for α → 0 . In figure 3 sample graphs of g, l ≡ 1, and h as the intersection curve are shown. Shape of the effective isotherm We now turn to the shape of the effective isotherm. Like its constituents, the effective isotherm is monotone increasing in the dissolved concentration CH: We have d ρ Ψ (x , t , C H ) = ⋅ ΨHf Ψ ′ C Hf + ρ ΨH Hf dC H
( )
+ ρ ΨHb
ρ ΨH 1
( )) CD (x, t )Π′(CHf )]⋅
(
ΨHb ′ CD (x , t )Π C Hf
(
1 + C D ( x , t )Π ′ C Hf
(23a)
)
where again CHf is given by C Hf = G(CD ( x , t ), C H )
(23b)
( )
and CD (x , t ) is fixed. The term CD Π ′ C Hf CD = CHf = 0. Therefore for fixed (x,t) d Ψ(x , t , C H ) ≥ 0 . dC H
has to be interpreted as 0 for
(23c)
More precisely, the slopes of both contributing isotherms ΨHf, ΨHb are weighted by
7
(
)
γ x , t , C Hf =
1
( )
1 + C D (x , t )Π ′ C Hf
,
(23d)
i.e. 0 ≤ γ ≤ 1, and 1 − γ , respectively. This shows that not only the values of the single constituents are diminished because of CHf ≤ CH , CHb ≤ CH , but also their slopes. E.g.,
in the case of no sorption of C Hb (ρ ΨHb ρ ΨH = 0) the effective isotherm lies below ΨHf and has a smaller slope. The overall picture becomes more complicated with respect to the curvature. Even in the case of linear Π, where the first term of Ψ has the same curvature as ΨHf and analogously the second the same as ΨHb, their interplay can lead to an effective isotherm changing from concave to convex shape and vice versa, also if ΨHf, ΨHb have no change in curvature. In van Duijn and Knabner [1992b] this is worked out for the combination of Freundlich isotherms. Another possible reason for the change in curvature is a nonlinear Π. A general discussion can be based on the second derivative of Ψ with respect to CH, computed from (23a), but for reasons of briefness, we restrict ourselves to the following example:
( )
r Π C Hf = K HD C Hf with 0 < r < 1,
i.e.
there
( )
is
a
strong
formation
(24a) of
carrier-bound
HOC.
If
p = K Hf C Hf with p > 0 and ρ Hb ΨHb = 0 , i.e. there is only sorption to the soil
ΨHf C Hf of the free HOC, then:
For p ≥ 1, Ψ is strictly convex,
(24b)
for r
0 and ϕ Hf ΨHf = 0 , i.e. there is only sorption to the
soil of the carrier-bound HOC, then: For q ≥ 1 r , Ψ is strictly convex,
(24e)
for 1 < q < 1 r , Ψ changes from convex to concave,
(24f)
for q ≤ 1, Ψ is strictly concave.
(24g)
Strict concavity is a sufficient condition for the existence of traveling waves (see van Duijn and Knabner, 1992b). We see, that in (24b, c, d) the usual condition p < 1 is sharpened to p < r , showing a stronger dispersion of fronts in this case of enhanced
8
mobility. Statements (24e, f, g) can be interpreted in a similar way. We will discuss the existence of traveling waves in detail in a subsequent paper. Effect of variable carrier concentration If the carrier concentration CD is constant, the model, i. e. (12)-(13) of part I has the form of the multiple site adsorption model, such that no new features with respect to the shape of concentration profile or breakthrough curves will appear for example in the case of an inflow experiment. If the concentration CD is space-, but not timedependent, then again equations (12)-(13) of part I have the form of the adsorption model, now for a heterogeneous porous medium, where the heterogeneity is defined by the carrier profile (if not the porous medium itself is heterogeneous, too). This situation occurs if CD is a limit profile produced by a constant source, e.g. situated in the surface horizons of a soil. If the concentration CD depends on time, this situation can be considered as the adsorption model with additional distributed sources or sinks, as ∂ ∂ d Ψ(x , t , C H ) = Ψ(x , t , C H )∂ t C H + Ψ(x , t , C H ) ∂C H ∂t dt
(25)
where the first term is the storage term due to the equilibrium adsorption, whereas the second one can be interpreted as an additional source or sink term of the form −
∂ ∂ Ψ(x , t , C H ) = − Ψ(C D , C H )∂ t CD (x , t ) ∂t ∂C D
=
(
A(C H )Π C Hf
)
( )
1 + C D (x , t )Π ′ C Hf
(26) ∂ t CD (x , t )
with A from (11). I.e., whether there is a source or a sink, depends on the sign of A(C H )∂ t C D . We first consider situations where ∂ t CD ≥ 0, e.g. flow regimes with inflowing carrier CD. This is the situation found commonly in forest soils with DOC leached from the forest floor material and entering the mineral surface horizons of soils. There is a source exactly for A≥0, which has been identified above as the case where the sorption of free HOC CHf dominates the sorption of carrier-bound HOC CHb. There is a sink in the reverse case A ≤ 0 . For ∂ t CD ≤ 0, e.g. flow regimes with leaching of initially present carrier, a situation we have to face in agricultural soils, the picture is reversed: a sink for A ≥ 0 and a source for A ≤ 0. In the case of an additional source we expect the possibility of local accumulation of the HOC, in the case of an additional sink of local depletion. In particular, the first case is of paramount importance for risk assessment studies.
NUMERICAL APPROXIMATION In this section we indicate how existing codes for the numerical simulation of adsorption models can be modified to deal with the model developed here, i.e. (13) 9
of part I, with the definition (12), (11) (or (10)) of part I, supplemented with appropriate initial and boundary conditions. The modification is only concerned with the more evolved evaluation of isotherms, rate functions and possibly their derivatives. Therefore the following applies independent of the spatial dimensions of the problem considered and the type of spatial discretization used (finite differences, finite elements or finite volumes). For a more specific treatment of a finite-element approximation we refer to Knabner [1992], where the algorithmic background of the SOTRA family of codes has been described. The computations of the following section have been performed by a modification of SOTRA-1D along the following lines, called CARRY. Consider as an example the following adsorption model: ∂ t (θC ) + ρ1∂ t ϕ 1(x , t , C ) + ρ 2∂ t S 2 + ρ 3∂ t S 3 − (27a) − div (θ D∇C − qC ) = 0 ∂ t S 2 = k2 (ϕ 2 (x , t , C ) − S 2 )
(27b)
∂ t S3 = k3 (ϕ 3 (x , t , C ) − S3 )
(27c)
The transformed model (13) of part I is easily to be recognized for the specific rate functions of the specific form (2), (3), i.e. ϕ1, ϕ2, ϕ3 are composed of the explicitly
given isotherms ΨHf , ΨHb , ϕ Hf , ϕ Hb and the only implicitly defined G (CD , C H ) . Assume a spatial (finite element) discretization of the underlying domain with nodes P1, ...., Pn, and a time stepping t 0 : = 0; t m+1 = t m + ∆t , where ∆t m+1 = ∆t m possibly is determined by a step size control. Assume that at time level t = t
(
)
(
m
approximations
)
C m = C m (Pi ) , S l m = S l m (Pi ) , l = 2, 3, at the nodes are known, which is true for m=0 by evaluations of the initial data. Finite differencing in time, combined e.g. with the implicit Euler method and a (finite element) discretization in space, with an incorporation of the boundary conditions, leads to the following set of equations for m+1 the values C , which is nonlinear if one of the isotherms is nonlinear (see Knabner, 1992, for details):
( M m+1 + ∆t K m+1)C m+1 = M m C m + bm+1 + (28a) 3
+∑ Ml l =1
(
)i
m m +1 , C m+ 1 S l − ϕ l Pi , t (∆tkl ) + 1 1
1
Here, M m are the mass matrices for t = t m , depending on the possibly timedependent water content θ, M l the mass matrices depending on the bulk densities ρl , K m is the stiffness matrix for t = t m originating from the discretization of the linear m+1 part of the equation dealing with advective and dispersive transport. Finally b 10
contains the contribution from a flux boundary condition and possibly an additional m+1 source term, and k1:= ∞ , i.e. 1 (∆tk1) = 0 . After C has been computed by the (approximate) resolution of (28a), S l m+1is given by the evaluation of
(S1m+1)i = ϕ1 Pi , t m+1, (C m+1)i (S lm+1)i =
1 1
(∆tkl )
(28b)
(
)i +
ϕ l Pi , t m +1, C m +1 +1
(28c) 1 +
(∆tkl )
1
(∆tkl )
S lm ) , ( i +1
l = 2,3, i = 1,...., n
Usually node oriented quadrature rules are applied in the evaluation of the mass matrices, which leads to mass lumping, i. e. a diagonal form of M m and M l , l=1,2,3. We will assume mass lumping such that the set of nonlinear equations (28a) has the following structure: A m +1C m +1 = b
m +1
(
+ g im +1 Pi , t m +1, (C )i
)
m +1
(29)
i. e. the nonlinearity gm+1 is such that its i-th component only depends on the i-th m+1 component of the unknown C . There are various iteration schemes with which (29) can be resolved approximately. We will discuss fixed point (Picard's) iteration and Newton's method: Given the ν-th iteration , for ν=0 e. g. C m+1,0 = C m , the (ν + 1) st iteration is given by C m+1, ν+1 = C m+1,ν + ∆ C m+1, ν+1
(30)
where the increment ∆C m+1,ν+1 is given for fixed point iteration by Am+1 ∆ C m+1 = − d ν+1
(31)
the defect d ν+1 being defined by d ν +1 = Am +1C m +1,ν − b
(
m +1
(
)
− g m +1 C m +1,ν ,
(32)
)
m +1 for short. g m +1(C ) = gim +1 Pi , t m +1, (C )i
For Newton's method, the increment is computed by
11
(A − Dg (C ))∆C m +1
m +1
m +1,ν +1
= − d ν +1 .
(33)
Here Dg m+1(C ) is the Jacobimatrix of g m+1 at C , i. e. a diagonal matrix with the i-th
(
)
∂ m +1 g Pi , t m +1, (C )i . ∂C i To perform one iteration step, one has to evaluate the isotherm ϕ l Pi , t m +1, C m +1,ν for all nodal values of the last iteration for the computation of i the defect, and in case of Newton's method also to evaluate its derivative ∂ ϕ l Pi , t m +1, C m +1,ν for all nodal values of the last derivative to set up the i ∂C m+1 Jacobian Dg . diagonal entry
(
)
(
)
For our specific model the evaluation of the isotherm , say ϕ1 , which corresponds to the effective isotherm Ψ of (12) of part I , requires the computation of , . The approximation of C P , t m +1 due to eq. (23) of part G C D Pi , t m +1 , C m +1ν D i i I is given exactly by the algorithm described above, but now with explicitly given isotherms and their derivatives. Therefore this procedure should be combined in the same time stepping scheme, where first the computation of the carrier concentration and then of the contaminant concentration is performed, say at the time level t m+1 .
(
)(
(
)
(
)
)
Therefore we consider CD Pi , t m +1 as to be known and return to the computation of G. The problem of evaluation of C Hf = G(CD , C H ) for given values
(
)
(CD , CH ) is
equivalent to the resolution of C H = C Hf + CD Π C Hf (see (10) of part I). This equation can be resolved iteratively by Newton's method. Usually Π is either a concave or a convex function and thus Newton's method is guaranteed to converge from below or above, if it is started from below the solution, e. g. with CHf = 0, or from above the solution, e. g. with CHf = C H , respectively. For Newton's method the evaluation of the derivatives of the isotherms requires in addition
(
)(
d G CD Pi , t m +1 , C m +1,ν dC
(
)(
)i = 1+ C
(
1
) (
m +1 Π ′ C Hf D Pi , t
)
(34)
)
, , as it is to be seen for the effective isotherm where C Hf = G CD Pi , t m +1 , C m +1ν i Ψ in (23). This is an explicit formula, as the vector of values CHf appearing in (34) has already been computed for the corresponding evaluation of ϕl, discussed above. The structure of the resulting overall algorithm is summarized in Table 1. This sketch has to be supplemented by possible repetitions of the time step for smaller ∆t, if one of the iterations diverges or the solution does not fulfill certain qualitative criteria (e.g. is not positive). But note that more involved iteration schemes are available, for which convergence is guaranteed independent of ∆t, which is not the case for (31) and (32) .
12
SIMULATION RESULTS The following section represents the application of the model to virtual and real world data sets. For all simulations we used the previously introduced simulation algorithm CARRY, which is available as a computer program running on various operating systems like UNIX, OS/2 and DOS. Virtual Data Sets We now turn to the results of numerical simulation runs with nonexperimental (virtual) data sets. We start with the representation of some general features of the model exemplified by simulations with virtual data sets. Figure 4a and figure 4b show the principal influence of increasing carrier concentration on the breakthrough of a virtual contaminant through a porous medium. The applied physicochemical conditions for the numerical simulations are summarized in Table 2. The porous medium was assumed to be saturated and homogeneous with an unimodal pore size distribution, i.e. neither preferential water flow nor multiple sorption sites were considered. Therefore, only equilibrium sorption sites are provided by the porous medium, i.e. ρΨHf = ρ . The formation of the carrier bound contaminant was assumed to be instantaneous and linear. Both carrier and contaminant were fed continuously at a constant rate to the porous medium. The simulation were individually run for four different increasing carrier concentrations, i.e. 0, 0.5, 1.0 and 5.0*10-3 kg m-3, respectively. The increasing carrier concentration leads to a significant increase of mobility of the substances. Figure 4a shows this mobility increase plotted for the total concentration of the contaminant, figure 4b represents the plot for the free fraction of the contaminant. The mobility increase is due to the fact, that the substances apparent water solubility is increased by the formation of the carrierbound contaminant. As one may expect and can be seen by (10), (11) of part I and its consequence, e.g. (7), the free concentration of the contaminant is reduced for increasing carrier concentration. This situation is met within experimental situations described e.g. by Dunnivant et al. [1992], which reported increased mobility for hexachlorobiphenyl and cadmium in the presence of DOC. We will discuss their findings with respect to numerical simulation in the following section on experimental data sets. As for this example the model is just the standard adsorption model with linear equilibrium adsorption, numerical simulation could also be performed by evaluating numerically the closed form series solution, which is available (see for example van Genuchten and Alves [1982]). Experimental Data Sets To illustrate the usefulness of the proposed model in describing real world data sets we will employ experimental data reported by Dunnivant et al. [1992] on hexachlorobiphenyl (HCB) and cadmium (Cd) transport and our own data on the transport of polycyclic aromatic hydrocarbons (PAH). In both cases, the data are resulting from laboratory column experiments. For easier comprehension, we briefly summarize the conditions of the experiments. For a detailed description of the experiments, the reader may consult Dunnivant et al. [1992] and Totsche et al. [1995]. The soil column apparatus used in the experiments of Dunnivant et al. [1992] was a glass chromatography column with 0.01m inner diameter. These columns were uniformly packed to a final bulk density of approximately 1.6*10-3 kg m-3 with aquifer material originating from the water saturated zone of a sandy, siliceous, 13
thermic, psammentic Hapludult. The DOC percolation solution was prepared by collecting DOC bearing water from a stream channel downstream a peat deposit. After collection it was filtered and finally diluted with deionized water. In the case of Cd transport studies, the columns were washed with a DOC solution to saturate the bulk soil material with respect to organic carbon content. Soil column dispersivities ( λ ) were calculated by model fitting the observed breakthrough of a step input of KBr tracer at a constant flow rate (8.3*10-9 m3 s-1) to the classical advection dispersion equation described by Parker and van Genuchten [1984]. They were found to range between λ=0.047 m and λ=0.075 m (column Peclet numbers between 38 and 60), indicating a advection dominated flow regime. The transport experiments were conducted by applying a continuous step input of Cd or HCB in combination with DOC at a constant flow rate (8.3*10-9 m3 s-1). The soil column apparatus used in the experiments with PAH was a specifically constructed soil column system for the conduction of transport experiments with hydrophobic organic substances. The soil containment was made of stainless steel with 0.10 m inner diameter. They were uniformly packed with surface soil material originating from a shallow, frigid, sandy, siliceous, spodic Udipsamment. The final bulk density ranged between 1.43*10-3 and 1.45*10-3 kg m-3. The DOC percolation solution was prepared by leaching forest floor material prior to use within the column experiments. The forest floor material was collected at the same site where the soil material was sampled. Soil column dispersivities were calculated according to the same procedure as above with the Br-tracer replaced by a Cl-tracer and ranged between 0.015 m and 0.026 m (column Peclet numbers between 2.7 and 4.8) indicating a slightly advection dominated flow regime. The transport experiments were performed by applying a continuous step input of PAH in combination with DOC at a constant flow rate (2*10-8 m3 s-1). The percolation solution was sucked out the column by applying a suction pressure at the column outflow, which was keyed to atmospheric pressure. Thus a constant gradient was achieved which guaranteed for steady state unsaturated flow conditions in the porous medium. Figure 5a shows the measured versus the simulated breakthrough of HCB for different carrier concentrations. Dunnivant et al. [1992] found best fit of simulated versus measured values for a one site kinetic model with linear adsorption to the bulk soil material and a linear partition type interaction for PCB between carrier DOC and water. Adsorption of the carrier was not considered as the aqueous material was equilibrated with DOC prior to the experiments. Therefore, neither sorption of carrier DOC nor sorption of carrier-bound PCB has to be considered within the simulations. In terms of our model, that means that the fraction of bulk soil material providing nonequilibrium sorption sites for PCB covers the total bulk soil material, i. e. ρϕHf = ρ = 1. 655 * 103 kg m −3 . The kinetic rate parameter was found to be equal to 2.06*10-5 s-1, the linear Freundlich partition coefficient was equal to 0.245 m3kg-1 and the partition coefficient for PCB between carrier DOC and liquid phase was found to be equal to 123 m3 kg-1. With this parameter set we found good agreement between observed and simulated breakthrough of PCB. The observed increase in mobility as a consequence of increasing carrier DOC concentration could also be simulated with good agreement. In the case of Cd, Dunnivant et al. [1992] found best fit for a two-site nonequilibrium model with linear adsorption to the bulk soil material and a linear partition type interaction for Cd between carrier DOC and water. Also no sorption of carrier DOC was considered as the column was saturated with DOC prior to the experiments. 14
Figure 5b shows the measured and simulated Cd breakthrough for different carrier concentrations for the linear case. In the case of HCB better agreement between observed and simulated data could be found when allowing for nonlinear adsorption in the case of HCB adsorption to bulk soil material. As there where no data available on nonlinear parameters we applied the following method: Starting with the fitted parameters supplied by the above mentioned HCB breakthrough experiment, we calculated nonlinear Freundlich type partition coefficients by computation of equation (35) for different values of C0 and p with given values of Klin:
Knlin ⋅ C0 pnlin = Klin ⋅ C0 , i. e. (35)
Knlin = K lin ⋅
C0 C0 pnlin
Here, Knlin denotes the Freundlich type partition coefficient for the nonlinear case, Klin the equilibrium partition coefficient estimated by parameter identification, pnlin the Freundlich type adsorption exponent and C0 the feed concentration. By doing so, we were able to provide plausible sets of parameters for the simulations which kept the nonlinear isotherm keyed to the linear isotherm derived from the experiments. Figure 6 gives the graph of the goodness as a function of the exponent. In order to get a measure for a"better" or a "worse" fit of measured versus simulated data we compute the least square difference of measured versus simulated data. Note that the lower the value of the goodness, the better the agreement between measured and simulated data. We found best agreement for pϕHf -values deviating significantly from 1, indicating that the adsorption process is highly nonlinear (goodness linear case with : 0.209, goodness nonlinear case (pnlin= 0.74): 0.079, see figure 7). Figure 6 shows the measured and simulated breakthrough curve of PCB. The dotted line indicates the linear situation ( pϕHf =1), whereas the nonlinear situation ( pϕHf = 0.74) is indicated by the solid line. The intersection at approximately 1000 pore volumes between measured and simulated curve observed for the linear case was not observed in the nonlinear case. Note that the solid curve in figure 6 is the simulated breakthrough curve in the nonlinear case and not a fit to the experimental data. We now turn to the discussion of the simulation results for PAH through soil columns. As the results are qualitatively the same for all PAH-species used within our experiments, we will restrict our discussion on the experimental findings on pyrene. Note that the column breakthrough studies were run under unsaturated flow conditions and that the bulk soil material was not saturated with DOC prior to the experiments. Therefore we have to consider sorption of both the carrier DOC and the carrier bound contaminant to the bulk soil material. Table 4 lists the parameters used for the simulation of pyrene through the soil columns. In order to understand the experimental findings with respect to DOC mobility, we briefly indicate the findings on our DOC transport experiments without presenting the data. The breakthrough of DOC exhibited a significant retardation with tailing to very long times. Analysis of the breakthrough curve with curve-fitting procedures resulted in strongly kinetically controlled adsorption of DOC to BSM with high K-values of 15
approximately 0.5 m3 kg-1 and rate parameters kD of about 1.6*10-4 s-1. As a toxic was added to the columns, we could exclude loss of DOC due to microbial decay. Moreover, mass balance analysis with respect to DOC resulted in a mass deficiency smaller than 5 % of total DOC, indicating that no other sink-term is acting on DOC. We suppose that the breakthrough of the DOC is not the breakthrough of only one species, but the superposition of at least two breakthrough curves, a fast breakthrough attributable to hydrophilic fractions of the DOC and a second, much more slower breakthrough, attributable to the hydrophobic fraction of the DOC. To run the simulation we assumed that there is no distinction in the sorption of free DOC and DOC-bound pyrene, meaning that the sorption process of the carrier bound pyrene is determined by the physicochemical properties of DOC. Therefore, the sorption of the carrier-bound pyrene is described by the same sorption parameters as the sorption of DOC (i. e. KϕHb = KϕD and k Hb = k D ) and there is no distinction between the fractions of bulk soil material providing sorption sites for the free DOC and the carrier bound pyrene, i.e. ρϕHb = ρϕD . Moreover, the sorption of free pyrene is mainly controlled by the amount of soil organic matter, which is approximately 0.35 % of the total mass. For KΨHf we used values reported by Karcher et al. [1988] for pyrene equilibrium KOC values. Figure 8a shows the initial breakthrough of pyrene with solid squares indicating the situation with no DOC present in the mobile phase, the solid circles indicating the situation with DOC present in the liquid phase and the solid lines indicate the simulation results. In both cases we observe quite good agreement between measured and simulated data. Therefore, the assumptions on the adsorption chemistry seem to be in good accordance with the "real world" situation. Furthermore, the significant decrease of the mobility of pyrene in the presence of DOC is supported by the β-value for the given situation which is considerably larger than one (β = 1.69). Let us now elucidate the specific property of CARRY to distinguish between free and carrier bound fraction of pyrene. Figure 8b gives the graph of the measured versus simulated breakthrough of pyrene, by which the dotted line indicates the breakthrough of the free substance and the solid line the breakthrough of total pyrene, i. e. the sum of free and carrier bound pyrene. There is no significant difference in the breakthrough curves between the free and the total pyrene at times smaller than approximately 250 porevolumes. After that, the curves for total and for free pyrene diverge significantly and the graph of the measured values is much better represented by breakthrough of total pyrene. This simulation result is in good agreement with the observation that there was no significant breakthrough of the hydrophobic part of DOC before approximately 300 porevolumes were exchanged. According to the high affinity of pyrene to the hydrophobic fractions of DOC rather than to the hydrophilic part, total pyrene breakthrough is mainly due to the breakthrough of free pyrene. After 300 porevolumes are exchanged, the hydrophobic fractions of the DOC break through the column and, of course, also the pyrene bound to these fractions.
EXTENSIONS In this section we briefly indicate some extensions which will be the focus of subsequent studies. If we also want to consider degradation processes, we have to add further sink terms in the right hand side of equations (3),(4) of part I, say
16
hHf(CHf) and hHb(CHb), respectively. In the transformed model the consequence is a sink term in the right hand side of (13a) of part I in the form
(
)
(
)
h(x , t , C H ) = hHf G(CD (x , t ), C H ) + hHb C H − G (CD (x , t ), C H )
(36)
If we want to include the pore size exclusion effect discussed in the section The Model of part I, we can do so by using different water contents θ1, θ2 in equations (3),(4) of part I, with θ 2 ≤ θ1 . The reduced water content θ2 is a consequence of the reduced pore space, which is accessible to the HOC-DOM particles. If we modify the definition (7) of part I to CH = CHf + θ 2 θ CHb ,
(37)
1
i.e. we use the pore space on which θ1 is based as a reference, then equation (10) of part I is modified to CH = CHf + θ 2
θ1
CD Π( CHf ) ,
(38)
The resolution for CHf for given CH will again be denoted by G, but is now also (x,t) θ dependent for (x,t) dependent 2 θ , i.e. 1
G = G ( x , t , CD , CH ).
(39)
The definition of the effective isotherm (12) of part I and rate functions (13) of part I has to be modified correspondingly. Then in the transformed model equation (13a) of part I has to be substituted by ∂ t (θ 1C H ) + ρ ΨH ∂ t Ψ(x, t , C H ) + ρ ϕHf ∂ t S Hf +
(
)
+ ρ ϕHb∂ t S Hb − div θ 1 D∇C H − qC H = θ − div θ 1 D∇ 2 θ CD Π(G (x , t , C H , C D )) 1
(40)
θ − div q1 − 2 θ CD Π(G(x , t , C H , C D )) 1 From the two new terms on the right hand side, the first is only present for space dependent θ 2 θ1 . Together they can be interpreted as an additional transport term due to convective transport acting on the carrier-bound HOC mass concentration CHb induced by the 'flow field' θ 1 D∇ θ 2 θ + q 1 − θ 2 θ 1 1
(41)
For space independent θ 2 θ1 this is the underlying water flow field reduced in magnitude by the factor (1 − θ 2 θ 1) . This explains the experimentally observed fact of a larger travel speed for the case under consideration compared to inert transport 17
(see e.g. Enfield et al. [1989]) without using an unphysical fitting of individual fluxes to experimental data (see the discussion of Enfield et al. [1989] in the section on Special Cases in part I). The incorporation of pore space reduction due to the sorption of carrier to the soil, as it has also been proposed by Corapcioglu and Jiang [1993] and Jiang and Corapcioglu [1993] leads to the following modifications: In the equations (3) and (4) θ − ρ ϕD S D + ρ ΨD ΨD (C D ) of part I θ has to be substituted by , where ρ D [M L-3] ρD is the density of the carrier DOC. As this may be interpreted as a new, diminished and (x,t) dependent water content, all the subsequent analysis remains valid. Formally we are back to our original model for ρ D = ∞. The equation (23a of part I) for the transport of the carrier has to be modified as follows:
(
)
ρ ϕD S D + ρ ΨD ΨD (CD ) CD + ∂ t θ − ρD + ρ ΨD∂ t ΨD (CD ) + ρ ϕD ∂ t S D −
(42)
ρ ϕD S D + ρ ΨD ΨD (CD ) ⋅ D∇CD − qCD = 0 − div θ − ρD For the sake of simplicity we only consider equilibrium reactions of the carrier, i. e. ρϕD S D = 0. Then the volume effect results in a reduced sorption isotherm
(1− CD
ρ D )ΨD (C D ) instead of ΨD (CD ) . As CD ρ D , the volumetric ratio of carrier and REV, in general will be rather small, we do not expect qualitative changes in the modified isotherm, which are possible in principle, e.g. (1− CD ρ D )ΨD (C D ) may
decrease for large values of CD. In particular, θ − ρ ΨD ρ D ΨD (CD ) , the remaining pore space for the fluid phase, has to be positive, such that the modification in the dispersive flux brings in an additional nonlinearity, but does not change the mathematical nature of the equation. One of our main assumptions was the description of the HOC-soil (and HOC-carrier) reactions as adsorption processes without competition. Here we were guided by the limits of experimental information, not of the modelling approach. Instead of the we could also think of a 'competitive' isotherm isotherm ΨHf C Hf
(
(
)
)
ΨHf C Hf , C Hb , CD , which then would appear in the effective isotherm Ψ, with CHf, CHb substituted by (11a,b) of part I. CONCLUSIONS For vanishing carrier or carrier in abundance the model has the desired limiting behaviour, in particular the contaminant behaves as being inert in the later case, if there is no co-sorption. Whether there is enhanced or reduced mobility in general, can be seen directly from a function derived from the isotherms (11). For more specific situations, this simplifies to the question of size for a characteristic 18
parameter ((12), (21)). The shape of the resulting effective isotherm may be quite involved, even if the constituents are simple. This may have impact on displacement profiles to be observed. Time dependent carrier concentration can lead to local accumulation or depletion of HOC. The incorporation of pore-size exclusion of the large carrier-bound HOC particles is possible and explains quantitatively a travel speed higher than the interstitial flux. Existing simulation codes for the usual adsorption model can be modified with minor effort to deal with the model discussed here. We were able to validate the model by reproducing satisfactorily experimental data for breakthrough experiments, both from the literature and from our own experiments. It turns out that the possibility of nonlinear isotherms and rate functions leads to an improved representation of the experimental data. ACKNOWLEDGEMENTS We like to thank Dr. Phil Jardine, Oak Ridge National Labs, who was so kind to provide us with the breakthrough data of cadmium and HCB and G. Leykam, University of Bayreuth, who performed the numerical simulations with our code CARRY. We acknowledge financial support by BMBF (Verbundprojekte Anwendungsorientierte Mathematik) and by EU Environment and Climate Programme under contract EV5V-CT94-0536.
19
REFERENCES Corapcioglu, M. Y. and S. Jiang, Colloid-facilitated groundwater transport, Wat. Res. Res., 29, 2215-2226, 1993 van Duijn, C. J. and P. Knabner, Travelling waves in the transport of reactive solutes through porous media: Adsorption and binary ion exchange, Part I, Transport in Porous Media, 8, 167-194, 1992a van Duijn, C. J. and P. Knabner, Travelling waves in the transport of reactive solutes through porous media: Adsorption and binary ion exchange, Part II, Transport in Porous Media, 8, 199-226, 1992b Dunnivant, F. M., Jardine, P. M., Taylor, D. L. and J. F. McCarthy, Cotransport of cadmium and hexachlorobiphenyl by dissolved organic carbon through columns containing aquifer material, Environ. Sci. Technol., 26, 360-368, 1992 Enfield, C. G., Bengtsson, G. and R. Lindqvist, Influence of macromolecules on chemical transport, Environ. Sci. Technol., 23, 1278-1286, 1989 van Genuchten, M. Th. and W. J. Alves, Analytical Solutions of the One-Dimensional Convective-Dispersive Solute Transport Equation, U. S. Dept. of Agric., Techn. Bull., No 1661, 151p., 1982 Jiang, S. and M. Y. Corapcioglu, A hybrid equilibrium model in porous media in the presence of colloids, Colloids and Surfaces A, 73, 275-286, 1993 Karcher, W., S. Ellison and M. Ewald, Spectral atlas of polycyclic compounds, Vol. 2D, reidel publishing company, Dordrecht, The Netherlands, 1988 Knabner, P., Finite element approximation of solute transport in porous media with general adsorption processes, in Xiao, S. T. (ed.), Flow and transport in porous media, World Scientific Publishing, Singapore, 1992
20
Knabner, P., K. U. Totsche and I. Kögel-Knabner, The modeling of reactive solute transport with sorption to mobile and immobile sorbents - part I: experimental evidence and model development, Wat. Res. Res., in press, 1995. Kreft, A. and A. Zuber, On the physical meaning of the dispersion equation and its solutions for different initial and boundary conditions, Chem Eng. Sci., 33, 1471-1480, 1978 Lafrance, P., Banton, O., Campbell, P. G. C., and J.-P. Villeneuve, Modeling solute transport in soils in the presence of dissolved humic substances, The Science of the Total Environment, 86, 207-221,1989 Parker, J. C. and M. T. van Genuchten , Deterministic transport parameters from laboratory and field tracer experiments, Virginia Agricultural Experimental Research Station, USDA-ARS, Bulletin 84-3, 1984 Totsche, K. U., I. Kögel-Knabner and J. Danzer: DOC-enhanced retention of polycyclic aromatic hydrocarbons in soils: implications from miscible displacement experiments, in prep. van Genuchten, M. T. and W. J. Alves, Analytical solutions of the one-dimensional convectivedispersive solute transport equation , Tech. Bull. 1661, 151pp, U.S Dept. of Agric., Washington, DC, 1982
21
APPENDIX A: Notation: ∂ or ∂ t : ∂t div () ⋅ ,∇ :
Differentiation with respect to time Operator for divergence and gradient in space
α:
[M M-1]:
Ratio of mass of HOC-DOC-particle to HOC-particle
β:
[1]:
Ratio of sorption capacity to soil of carrie-bound substance and free substance, both related to the total volume of an REV
θ
[L3 L-3]:
θ1
[L3 L-3]:
Volumetric water content with respect to unit volume of porous medium Volumetric water content with respect to unit volume of porous
θ2
[L3 L-3]:
medium accessible to all solutes Volumetric water content with respect to unit volume of porous
χHf
[M M-1]
medium accessible to larger particles (e.g. carrier-bound substance) Mass concentration of free HOC momentary sorbed to the fraction
χHb
[M M-1]
of BSM providing nonequilibrium sites Mass concentration of carrier-bound HOC momentary sorbed to the fraction of BSM providing nonequilibrium sites
λ Π
[L]: [M M-1]:
ρ ρD
[M L-3]: [M L-3]:
Soil column dispersivities Mass of HOC bound to carrier DOC relative to the total mass of dissolved carrier Density of BSM related to unit volume of porous medium Bulk density of the fraction of the BSM providing both equilibrium
ρΨΗ
[M L-3]:
adn nonequilibrium sorption sites for carrier DOC Sum of fractions of bulk density providing equilibrium sorption sites
ρΨHb
[M L-3]:
for both free and carrier-bound HOC Bulk density of the fraction of the BSM providing equilibrium
ρΨHf
[M L-3]:
sorption sites for carrier-bound HOC Bulk density of the fraction of the BSM providing equilibrium
[M L-3]:
sorption sites for free HOC related to unit volume of porous medium Bulk density of the fraction of the BSM providing nonequilibrium
[M L-3]:
sorption sites for free HOC related to unit volume of porous medium Bulk density of the fraction of the BSM providing nonequilibrium
ρϕHf
ρϕHb
sorption sites for carrier-bound HOC
22
Ψ
[M M-1]:
ΨD
[M M-1]:
Mass concentration related to fraction of mass of BSM sorbed to equilibrium sites Total mass concentration of DOC composed of free and carrier
ΨHb
[M M-1]:
DOC sorbed to equilibrium sorption sites. Mass of carrier-bound HOC sorbed at equilibrium to the fraction of
ΨHf
[M M-1]:
BSM providing equilibrium sorption sites Mass concentration of free HOC sorbed at equilibrium to the
ϕHf
[M M-1]:
fraction of the BSM providing equilibrium sorption sites mass concentration of free HOC sorbed at equilibrium to the
ϕHb
[M M-1]:
fraction of the BSM providing equilibrium sorption sites mass concentration of free HOC sorbed at equilibrium to the
C C0
[M L-3]: [M L-3]:
fraction of the BSM providing equilibrium sorption sites Mass concentration of solute related to the water filled pore volume Feed concentration: Mass concentration of solute related to the
CD CH CHb CHf
[M L-3]: [M L-3]: [M L-3]: [M L-3]:
water filled pore volume Total mass concentration of DOC dissolved in liquid phase Total mass concentration of HOC Mass concentration of carrier-bound HOC dissolved in liquid phase Mass concentration of free HOC dissolved in liquid phase
D fHf fHb
[L2 T-1]: [M M-1T-1]: [M M-1 T-1]:
Matrix of sum of molecular diffusion and mechanical dispersion Effective rate function for free HOC sorption to nonequlibrium sites Effective rate function for carrier-bound HOC sorption to
gHf gHb
[M M-1T-1]: [M M-1 T-1]:
nonequlibrium sites Reaction rate of the nonequilibrium sorption of free HOC Reaction rate of the nonequilibrium sorption of carrier-bound HOC
G
[M L-3]:
hHf hHb kD kHf kHb
[M L-3 T-1]: [M L-3 T-1]: [T-1] [T-1] [T-1]
Mass concentration of free HOC in liquid phase in the presence of carrier DOC Additional sink term for free substance (e.g. degradation) Additional sink term for carrier-bound substance (e.g. degradation) Rate constant for nonequilibrium sorption of carrier DOC Rate constant for nonequilibrium sorption of free substance Rate constant for nonequilibrium sorption of carrier-bound
KϕHb
[L3 M-1]:
substance Linear Freundlich type partition coefficient of carrier-bound
[L3 M-1]:
substance between liquid phase and and fraction of BSM providing nonequilibrium sites Linear Freundlich type partition coefficient of carrier DOC between
KϕD
liquid phase and fraction of BSM providing nonequilibrium sites
23
Klin
[L3 M-1]:
Linear Freundlich type partition coefficient of HOC between liquid
Knlin
[L3 M-1]:
phase and BSM Nonlinear Freundlich type partition coefficient of HOC between
KHD KHf KHb
[L3 M-1]: [L3 M-1]: [L3 M-1]:
liquid phase and BSM Partition coefficient of HOC between carrier DOC and liquid phase Partition coefficient of free HOC between liquid phase and BSM Partition coefficient of carrier-bound HOC between liquid phase and
KD
[L3 M-1]:
BSM Partition coefficient of carrier DOC between liquid phase and BSM
p pΨHf
[1]: [1]:
Freundlich type exponent for free substance Freundlich type exponent for sorption of free substance to fraction
pϕHf
[1]:
of BSM providing equilibrium sorption sites Freundlich type exponent for sorption of free substance to fraction
pnlin
[1]:
of BSM providing nonequilibrium sorption sites Freundlich type exponent
P q q
[M L-3 T-1]: [1]: [L T-1]:
Volumetric consumption or production rate Freundlich type exponent for carrier-bound substance Specific discharge
r SD
[1]: [M M-1]:
Freundlich type exponent for formation of carrier-bound substance Mass concentration of both free and carrier-bound DOC momentary
SHb
[M M-1]:
sorbed to the fraction of BSM providing nonequilibrium sorption sites Mass concentration of carrier-bound HOC momentary sorbed to the
SHf
[M M-1]:
fraction of BSM providing nonequilibrium sites Mass concentration of free HOC momentary sorbed to the fraction
[T]: [L]:
of BSM providing nonequilibrium sites Variable of time Variable of space
t x
24
Table 1: Structure of the numerical algorithm t=0; CD, CH given by initial data Time stepping loop: till final time compute ∆t, t=t+∆t set up equation for CD, SD according to (28) Nonlinear equations loop: till convergence: One step of the iterative method (e.g. (31) or (33)) Set up equations for CH, SHf, SHb according to (28) Nonlinear equation loop: till convergence: For all nodal values of last iterative of CH: Scalar nonlinear equations loop: till convergence: One step of Newton's method Compute isotherms (and derivatives) for all nodal values of last iterative of CH one step of the iterative method (e.g. (31) or (33))
25
TABLE 2. Parameter values for virtual data sets Parameter
Value
L ρ θ λ q CH CD KΨHf KHD
0.1 1.5*103 0.5 0.5*10-3 0.83*10-4 1*10-3 (0, 0.5, 1.0, 5.0)*10-3 5*10-3 5*10-3
[m] [kg m-3] [m3m-31] [m] [m s-1] [kg m-3] [kg m-3] [m3 kg-1] [m3 kg-1]
26
TABLE 3. Parameter Values of Breakthrough-Experiments
Parameter L [m] d [m] ρ [kg m-3] θ [m3 m-3] OC [kg kg-1] λ [m] q [m s-1] CH [kg m-3] CD [kg m-3]
Value Dunnivant et al., 1992 Cd HCB 0.0798-0.0805 0.01
0.01 1.66*103 0.397-0.409
0.00058
0.00039 0.047-0.075 1.061*10-4
0.25*10-3 (0,5.2,20.4,58.1)*10-3
(0.4-0.6) * 10-9 (0,5.2,20.4)*10-3
PAH Pyrene 0.073 0.1 1.44*103 0.23-0.25 0.003-0.005 0.015-0.026 2.9*10-6 54.72*10-6 35*10-3
27
TABLE 4: Parameter Values for Pyrene-Simulations Value Parameter ρΨHf [kg m-3] ρΨHb [kg m-3] ρϕD [kg m-3] KΨHf [m3 kg-1] KϕHb [m3 kg-1] KϕD [m3 kg-1] KHD [m3 kg-1] kHb [s-1] kD
[s-1]
no DOC 0.510*103 83.176 -
DOC 0.510*103 1.4349*103 1.4349*103 83.176 0.5 0.5 41.687 1.6*10-4
-
1.6*10-4
28
List of figures Fig. 1: Breakthrough of a virtual substance as influenced by β. Fig. 2: Plot for f Fig. 3: Plot for g, nonlinear case Fig. 4a: Breakthrough of a virtual substance as influenced by increasing carrier concentrations, Fig. 4b: Breakthrough of the free dissolved fraction of a virtual substance as influenced by increasing carrier concentrations. Fig. 5a: Measured and simulated breakthrough of HCB at different carrier concentrations Fig. 5b: Measured and simulated breakthrough of Cd at different carrier concentrations Fig. 6: Differences for linear and nonlinear simulation vor HCB, Fig. 7: Dependence of the goodness on varying pϕHf. Fig. 8a: Measured and simulated breakthrough of pyrene Fig. 8b: Measured versus simulated breakthrough of pyrene: Total pyrene and free dissolved pyrene
29
Figure Captions Fig. 1: Breakthrough of a virtual substance as influenced by varying β. pv: porevolume; C/C0: reduced concentration;
_____
no carrier present;
0.2; ¬ β = 0.6; ¢ β = 1.0; u β = 1.5; n β = 1.8.
u
no sorption,
+β=
Fig. 2: Plot of function f , characterizing the deviation of the effective isotherm from the isotherm describing sorption of free substance in the linear case. Dotted surface: l(α,β) = 1; Area above this surface: collection of parameters of α, β with reduced mobility, area below this surface: increased mobility. Fig. 3: Plot of function g, characterizing the deviation of the effective isotherm from the isotherm describing sorption of free substance in the nonlinear case. Freundlich, (p = q = 0.5); Dotted area: l(α,β) = 1; Solid line h, intersection curve of surfaces g and l: threshold value for β for increased or reduced mobility. Fig. 4: Breakthrough of total concentration (a) and free dissolved fraction (b) of a virtual substance as influenced by increasing carrier concentrations. The carrier and the carrier-bound substance are assumed to be nonreactive.
A no carrier present, þ
Ccarrier = 0.5 *10-3 kg m-3; l Ccarrier = 1.0 *10-3 kg m-3 ;u Ccarrier = 5.0 *10-3 kg m-3. Fig. 5: Measured and simulated breakthrough of HCB (a) and Cd (b) at different carrier
____
l no DOC; A CDOC = 5.2 *10-3 kg m-3; Û CDOC = 20.4 *10-3 kg m-3. In (b) additionally: u CDOC = 58 *10-3 concentrations;
simulated breakthrough. In (a)
kg m-3 Fig. 6: Linear versus nonlinear simulated breakthrough of HCB in comparison to measured data.
........
simulation with linear isotherm;
_____
Simulation with nonlinear
isotherm (pϕHf = 0.74). Fig. 7: Dependence of the goodness of simulated versus measured breakthrough of HCB on varying pϕHf. Fig. 8: (a) Measured and simulated breakthrough of pyrene with and without DOC in the liquid phase;
........
simulated breakthrough; ¡ 35 *10-3 kg m-3 ; ¨ no DOC present.
30
(b) Simulated breakthrough of free and total concentration of pyrene in comparison to measured values of total concentration: pyrene;
........
______
simulated total concentration of
simulated breakthrough of free fraction of pyrene,
l measured total
concentration of pyrene.
31
32
f l 1.1 1 0.9 0
0.05
0.1
α
0.15
0.2
0
0.5
1
1.5
2
β
33
l h g
2.5 2 1.5 1 0.5
0.5
α
1 2
1.5
1.5 1
2 2.5
0.5
β
34
35
36
37
38
39
