A Free Boundary Problem for CaCO
3
Neutralization of AcidWaters
Lorenzo Fusi, Angiolo Farina, Mario Primicerio,Universit`a degli Studi di FirenzeDipartimento di Matematica “Ulisse Dini”Viale Morgagni 67/A, I50134 Firenze, Italy
Abstract
In this paper we propose a mathematical model for the reaction kinetics of CaCO
3
in an acidsolution. In particular we study the system in planar geometry showing that the problem is a freeboundary problem with an intrinsic multiscale structure. We rescale the problem and show howthe equations simplify according to the speciﬁc scale considered. We present also a result thatallows to calculate the diﬀusivity coeﬃcient from experimental measurements of pH.
1 Introduction
Environmental pollution due to acid mine drainage is probably the most critical problem in miningindustry. This phenomenon occurs naturally as part of the weathering process of rocks but is enhancedby mining activities, especially within rocks containing an abundance of sulﬁde minerals (see [10]).Though many options are available for remediation purposes, the main being the ones based onchemical reactions neutralizing the acid water. A typical approach to the neutralization process is thesocalled lime neutralization, which basically consists in the addition of lime to the acid water in orderto raise the pH (so that dissolved heavy metals can precipitate).Limestone is a cheap neutralizing agent (mainly formed by calcium carbonate, CaCO
3
) that isparticularly useful when available in crushed or pulverized form (because of the large available reactionsurface). The neutralization process can be described as the one in which excess of CaCO
3
(w.r.t.the stoichiometric amount) results in an increase of the pH to an equilibrium value, providing also agranular network capable of retaining precipitates.In this paper we propose a mathematical model which describes the evolution of a the processdescribed above. First, we derive a mathematical model for the acid water CaCO
3
chemical reactionoccurring at the interface between the solid and the liquid acid solution. The model is developed in ageneral 3D setting (section 2), assuming a chemical reaction of the ﬁrst order. Once the main physicalquantities are deﬁned (such the concentration of ions H
+
or, equivalently, the solution
pH
) we write amass balance equation between CaCO
3
and the acid solution as well as a convection diﬀusion equationfor H
+
concentration. Actually, the mathematical problems turns out to be a free boundary problem1
since, due to the chemical reaction, the surface separating the liquid phase from the solid evolves intime.Then we focus on a onedimensional setting (section 3), considering a slab ﬁlled by an acid solutionand neglecting convection. The left part of the slab is a CaCO
3
layer whose surface is exposed tothe acid solution. The surface location (a priori unknown) evolves in time. We do not focus on thewell posedeness of the mathematical problem (since an abundant literature is already existent) but weanalyze the qualitative behavior of the solution. The problem is multiscale in both time and spaceand the nondimensional formulation of the problem puts in evidence the existence of three time scaleshaving precise physical meaning. Depending on which time scale we are interested in, we may simplifythe problem so that pieces of information concerning the spacetime behavior of the solution can beeasily obtained. In particular, according to the value of
θ
, which is the ratio between the diﬀusivecharacteristic time and the reaction characteristic time, deﬁned by formula (3.7), two scenarios arepossible. The latters are analyzed in detail in section 4.Focussing on the longest time scale (section 5) we are able to derive a formula for estimating
D
∗
, i.e.the H
+
ions diﬀusivity in the acid solution. In particular, measuring the rate at which the acid solutionneutralizes (i.e. the pH growth rate) we may infer the value of
D
∗
. Of course, such a experimentalmeasurements have to be operated when the asymptotic regime is fully developed. This occurs on atime scale, which we explicitely estimate.
2 Derivation of the model
We assume that the system is constituted by a given domain Ω
⊂
R
3
, such that
1
Ω = Ω
l
(
t
∗
) + Ω
s
(
t
∗
)
,
where Ω
l
(
t
) is the volume occupied by acid water and Ω
s
(
t
∗
) is the volume occupied by the solidreacting material (for instance CaCO
3
). The surface separating the liquid and solid regions is denotedwith Σ(
t
∗
), and it is supposed to be evolving with time, but its evolution it is a priori unknown. Σ isthus a free boundary. In particular, we assume that the surface Σ is represented by
S
(
x
∗
,t
∗
) = 0 (with
S
suﬃciently smooth), so that its normal (pointing toward the liquid region Ω
l
) is given by
n
Σ
=
∇
∗
S
∇
∗
S

(2.1)We also introduce Γ
l
= Ω
l
∩
∂
Ω and Γ
s
= Ω
s
∩
∂
Ω.We denote by
c
∗
(
x
∗
,t
∗
) the molar concentration ([
c
∗
] =
mol/vol
) of the H
+
ions. The dynamics of
c
∗
is governed by the following equation
∂c
∗
∂t
∗
+
q
∗
·∇
∗
c
∗
=
D
∗
∆
∗
c
∗
, x
∗
∈
Ω
l
(
t
∗
)
, t
∗
0
,
(2.2)where:
1
Throughout this paper the superscript “
∗
” means that the quantity has physical dimension.
2
•
q
∗
(
x
∗
,t
) is the convective ﬂux ([
q
∗
] =
length/time
). We assume mechanical incompressibilityof the liquid phase and hence
∇
∗
·
q
∗
= 0;
•
D
∗
is the diﬀusivity coeﬃcient assumed to be constant ([
D
∗
] =
length
2
/time
).The boundary Γ
l
is impermeable to water and to ions so that
∇
∗
c
∗
·
n
l
= 0
,
and
q
∗
·
n
l
= 0
, x
∗
∈
Γ
l
, t
∗
0
,
(2.3)where
n
l
is the outward unit normal to Γ
l
.The chemical reaction occurs on Σ and it is described by the socalled “velocity” of the reaction
v
∗
([
v
∗
] =
mol/time
·
length
2
), which represents the rate, per unit surface, at which the H
+
ions areneutralized on Σ. Hence, assuming a reaction of the ﬁrst order we have
Σ
v
∗
dσ
∗
=
−
k
∗
Σ
(
c
∗
−
c
∗
o
)
+
dσ
∗
,
(2.4)where:
•
k
∗
is the chemical reaction rate constant ([
k
∗
] =
length/time
).
•
c
∗
o
is the “neutral concentration”, i.e. the concentration at which the solution is neutralized([
c
∗
o
] =
mol/vol
).
•
(
·
)
+
denotes the positive part.Of course, by standard mass conservation arguments,
Σ
v
∗
dσ
∗
equals also the rate at which the solidmaterial (namely CaCO
3
) is consumed. Therefore we can also write
2
Σ
v
∗
dσ
∗
=
ddt
∗
Ω
s
(
t
∗
)
ρ
∗
d
3
x
∗
,
where
ρ
∗
is the molar density (assumed to be uniform and constant) of the solid reacting material([
ρ
∗
] =
mol/vol
). Hence introducing
3
w
∗
, the normal velocity of the surface Σ, we have
Σ
v
∗
dσ
∗
=
ddt
∗
Ω
s
(
t
∗
)
ρ
∗
d
3
x
∗
=
ρ
∗
Σ
w
∗
dσ
∗
.
(2.5)Obviously
w
∗
=
−
1
∇
∗
S

∂S ∂t
∗
,
(2.6)
2
The stoichiometric coeﬃcient of the reaction is taken to be 1. This means that one mole of H
+
is reacting with onemole of CaCO
3
. Of course, depending on the particular acid solution considered, the coeﬃcient may have a diﬀerentvalue. For instance when considering HCl we have 2 moles of H
+
reacting with one mole of CaCO
3
, and hence thecoeﬃcient is 2.
3
w
∗
>
0 means a growing solid phase.
3
and hence, (2.4), (2.6) and (2.5) yield
ρ
∗
∇
∗
S

∂S ∂t
∗
=
k
∗
(
c
∗

Σ
−
c
∗
o
)
+
,
(2.7)which is the evolution equation of Σ.Next, we have to write the boundary condition for
c
∗
on Σ. The latter, essentially, is a “sink”surface, so that the classical RankineHugoniot condition [4] gives[
c
∗
(
q
∗
·
n
Σ
−
w
∗
)
−
D
∗
∇
∗
c
∗
·
n
Σ
]
Σ
=
v
∗
.
(2.8)Finally we consider the following initial data
c
∗
(
x
∗
,
0) =
c
∗
in
(
x
∗
)
, S
(
x
∗
,t
∗
) =
S
∗
o
(
x
∗
)
.
(2.9)The free boundary problem, considering
q
∗
known, is thus given by (2.2), (2.3), (2.7), (2.8), and (2.9).
3 One dimensional model
In this section we write the problem in a onedimensional setting. The domain Ω is constituted by aslab [0
,L
∗
]
.
The region [0
,s
∗
(
t
∗
)] is Ω
s
, i.e. the part occupied by the solid and Ω
l
= [
s
∗
(
t
∗
)
,L
∗
]. Theinterface Σ separating the acid water from the solid is
S
(
x
∗
,t
∗
) =
x
∗
−
s
∗
(
t
∗
) = 0, yielding
w
∗
= ˙
s
∗
.We consider a static situation so that
q
∗
= 0
.
Following (2.2) we can write
c
∗
t
∗
=
D
∗
c
∗
x
∗
x
∗
, while (2.3)gives
c
∗
x
∗
(
L
∗
,t
∗
) = 0. From (2.7) we obtain
ρ
∗
˙
s
∗
=
−
k
∗
(
c
∗
(
s
∗
,t
∗
)
−
c
∗
o
)
+
.
Concerning (2.8), we ﬁrst observe that, exploiting (2.5), it can be rewritten as
−
D
∗
∇
∗
c
∗

Σ
·
n
Σ
= (
ρ
∗
+
c
∗

Σ
)
w
∗
,provided
q
∗
= 0. We thus get[
ρ
∗
+
c
∗
(
s
∗
,t
∗
)]˙
s
∗
=
−
D
∗
c
x
∗
(
s
∗
,t
∗
)
.
Therefore, in the 1D case, the free boundary problem becomes
c
∗
t
∗
−
D
∗
c
∗
x
∗
x
∗
= 0
, s
∗
< x
∗
< L
∗
, t
∗
>
0
c
∗
(
x
∗
,
0) =
c
∗
in
(
x
∗
)
, s
∗
o
< x
∗
< L
∗
c
∗
x
∗
(
L
∗
,t
∗
) = 0
, t
∗
>
0
,ρ
∗
˙
s
∗
=
−
k
∗
(
c
∗
(
s
∗
,t
∗
)
−
c
∗
o
)
+
, t
∗
>
0[
ρ
∗
+
c
∗
(
s
∗
,t
∗
)] ˙
s
∗
=
−
D
∗
c
∗
x
∗
(
s
∗
,t
∗
)
, t
∗
>
0
,s
∗
(0) =
s
∗
o
,
0
< s
∗
o
<
1
.
(3.1)Before studying problem (3.1) we rescale the variables to obtain a non dimensional formulation. Forthis purpose we deﬁne
c
∗
A
>
0 as the molar concentration corresponding to a strong acid solution (e.g.
c
∗
A
= 10
−
1
mol/lt
), and introduce the characteristic times
t
∗
D
=
L
∗
2
D
∗
, t
∗
R
=
L
∗
k
∗
, t
∗
E
=
L
∗
ρ
∗
k
∗
c
∗
A
,
(3.2)4
representing the diﬀusive characteristic time, the reaction characteristic time and the erosion time scale,respectively. We also introduce the non dimensional parameters
λ
=
ρ
∗
c
∗
A
=
t
∗
E
t
∗
R
, δ
=
c
∗
o
c
∗
A
.
(3.3)When considering CaCO
3
in an acid solution typical values are
ρ
∗
= 2
.
7
×
10
−
2
mol/ℓt, c
∗
o
∼
10
−
7
mol/ℓt
(3.4)which yields
λ
≈
2
.
7
·
10
3
and
δ
∼
10
−
6
. Introducing then the following non dimensional quantities
c
=
c
∗
c
∗
A
, c
in
=
c
∗
in
c
∗
A
, t
=
t
∗
t
∗
D
, x
=
x
∗
L
∗
, s
=
s
∗
L
∗
, s
o
=
s
∗
o
L
∗
,
(3.5)the nondimensional version of (3.1) becomes
c
t
−
c
xx
= 0
, s < x <
1
, t >
0
c
(
x,
0) =
c
in
(
x
)
, s
o
< x <
1
c
x
(1
,t
) = 0
, t >
0
,λ
˙
s
=
−
θ
[
c
(
s,t
)
−
δ
]
, t >
0[
λ
+
c
(
s,t
)] ˙
s
=
−
c
x
(
s,t
)
, t >
0
,s
(0) =
s
o
,
0
< s
o
<
1
,
(3.6)where
θ
=
t
∗
D
t
∗
R
=
L
∗
k
∗
D
∗
.
(3.7)
3.1 Integral formulation and stationary solution
We integrate equation (3.6)
1
over
D
t
= (
s,
1)
×
(0
,t
), by means of Green formulas, and we get
s
(
t
)
−
s
o
= 1
λ
1
s
(
t
)
c
(
x,t
)
dx
−
1
s
o
c
in
(
x
)
dx
.
(3.8)We can prove the following
Theorem 1
If the asymptotic solution of system (3.6) exists, it is given by
c
(
x,t
)
≡
δ
and
θ
Proof.
Let us assume for a moment that a classical solution of (3.6) exists for all times. Then thestationary concentration is clearly
c
=
δ
. Moreover suppose that the function
s
(
t
) is diﬀerentiable.From the integral formulation we see thatlim
t
→∞
s
(
t
)
=
−∞
.
5