A NONLINEAR MODEL OF SULPHATION OF
CALCIUM CARBONATE STONES:
NUMERICAL SIMULATIONS AND PRELIMINARY
LABORATORY ASSESSMENTS
C. GIAVARINI† , M. INCITTI† , M. L. SANTARELLI† , R. NATALINI , V. FURUHOLT]
Abstract. Sandstones, limestones, and marbles are particularly vulnerable
to the effects of sulphur dioxide, since they can react to form gypsum (black
crust formation). In this paper we consider a differential model which gives
a quantitative description of these phenomena, by presenting a numerical and
experimental validation. This approach is relevant in order to assess and to
prevent further damages on the surface of historical monuments, by taking into
account the local geometry, the dynamic of the polluted air, and the different
exposures. It improves the current tools used to study the evolution of chemical
damage in stones, as for instance the Lipfert formula, based on statistical
models of atmospheric corrosion. The previous models are valuable for civil
uses, but not sufficiently precise for artistic and historical hand-works, since
they yield only an averaged description of the damage. On the contrary, this
approach gives a quantitative law for the progression of the front of sulphation
inside of the stone, with a good agreement with the experimental data.
Keywords.
Calcium carbonate stone, sulphation, black crusts, reaction
diffusion systems, thickness of gypsum.
1. Introduction
Although in the last decade the concentration of sulphur dioxide has been greatly
reduced in urban areas, sulphation is still important for monuments decay.
The effects of air pollution on limestone were already known in the last century;
however, only in the last decades they have attracted the interest of the scientists.
Various papers deal with the reaction mechanisms of calcium carbonate stone degradation [2, 7]. Among the reactions that favor the degradation of calcium carbonate,
the sulphation reaction is the most important and produces a gypsum layer on the
stone surface. The reaction in the urban atmosphere can be represented by the
following steps [2]:
1
(1.1)
SO2 + O2 → SO3 ,
2
(1.2)
and
(1.3)
SO3 + H2 O → H2 SO4 ,
H2 SO4 + CaCO3 + H2 O → CaSO4 · 2H2 O + CO2 .
† CisteC, Centro Interdipartimentale di Scienza e Tecnica per la Conservazione del Patrimonio
Storico-Architettonico dell’Università ”La Sapienza” di Roma, via Eudossiana 19, Rome, Italy.
E-mail: [email protected]
Istituto per le Applicazioni del Calcolo “M. Picone”, Consiglio Nazionale delle Ricerche, Viale
del Policlinico 137, I–00161 Rome, Italy. E-mail: [email protected]
] Present address: Centre for Ships and Ocean Structures, Norwegian University of Science and
Technology, Otto Nielsens vei 10, N-7491 Trondheim, Norway, E-mail: [email protected].
1
2
C. GIAVARINI, M. INCITTI, M. L. SANTARELLI, R. NATALINI, V. FURUHOLT
where step 1.1 is strongly affected by the presence of the atmospheric oxygen and
by various catalysts usually carried by solid pollutants (carbonaceous particulate,
fine powders, etc.). A number of experimental works emphasized the presence of
calcium sulphite (CaSO3 ) on the sulphation of the laboratory samples [6, 12, 8].
This is probably due to the high SO2 concentration (lack of oxygen) and to the
absence of catalytic effects.
The main types of degradation on carbonate stone surface (marble, limestone,
sandstone, etc.) have been classified by some authors in term of white, grey and
black areas [7, 6].
White areas can be found in monument surfaces exposed to rainfall, which causes
the washing away of the dry deposition and therefore the erosion of the stone material. The grey areas, mainly constituted of gypsum, are found in monument
portions with little humidity; the gray color depends on the dry deposition of carbon particles, without substantial alteration of the stone morphology. These areas
represent the initial phase of surface attack, where the pollutant particles do not
remain in contact with the stone for the time needed by a complete reaction. The
black areas, also called black crusts, are compact and heterogeneous incrustations
present in protected portions not in contact with the rainfall. In these zones, there
is an accumulation of atmospheric particles and polluting gases. The black crust is
composed mainly of gypsum, present as needle-shaped crystals that grow perpendicularly to the stone surface, [14].
The black crusts are rarely continuous and impermeable and do not constitute a
protective layer for the limestone surface. In fact, the greater solubility of calcium
sulphate with respect to calcium carbonate is responsible for water infiltration and
recrystallization phenomena; sulphation and corrosion processes can continue due to
the presence of cracks on the crust surface. Therefore, the deterioration proceeds
under the crust, which appears incoherent and not adherent to the surfaces [7].
The chemical composition and physical characteristics of the crust are different as
regards to the original stone surface and increase the rate of the stone materials
degradation for various reasons:
• volume increase: gypsum has a greater volume than calcium carbonate; its
formation generates remarkable stresses inside the stone structure, causing
decohesion and separation of large fragments.
• Different thermal expansion: this effect is more important for the black
layers, because the dark color causes larger absorption of sun radiations.
• Permeability reduction: it favors the water retention in the stone with all
the related damages.
As pointed out, the characteristic dark color of the black crusts is due to particles
of various origin, such as: carbon particles from vehicles, domestic and industrial
emissions, allumosilicates spherical particles from the combustion, etc. These particles have a fundamental role in the sulphation reaction because of their catalytic
effect [2].
2. Aim of the work
The described phenomena are very complex, therefore, for a quantitative evolution and prediction it is necessary to have a simplified approach to the problem.
The first author to define a quantitative function of damage was Lipfert (1988)
[11] , who introduced an empiric relation that correlates the mass loss of stone
materials with the deposition rate, and pH of rainfall. The mass loss is attributed to
the combined effects of acid rain and deposition of atmospheric pollutants. Lipfert’s
equation is still used to evaluate the mass loss from limestone; however, it has the
limit to be an empiricist equation which correlated the mass loss only to the washing
A NONLINEAR MODEL OF SULPHATION
3
away by the rainfall, without considering the characteristic and chemistry of the
material. More recent studies (Skoulikidis [17, 16]) and Gauri et al. [6, 12, 8]) have
tried to define the kinetics of the reaction of SO 2 with limestone materials.
This kinetics have been proposed on the basis of the reaction mechanism determined though XRD analysis in laboratory samples to justify and represent the
observed phenomena; however, they are not suitable to supply a provisional model.
On the contrary the aim of this paper is to apply a recently proposed mathematical model [3], which describes the diffusion and the chemical action of sulphur
dioxide on calcium carbonate stones; the purpose is to create a scientific instrument
to forecast the progress with time of the sulphation reaction on carbonate materials. The availability of a satisfactory quantitative model is of the great importance
for planning the monument restoration before the aesthetic and structural damage
becomes irreparable. In this work, a series of experimental tests were put into execution to verify the mathematical hypothesis and the rate of gypsum formation on
the stone surface.
The sulphation, carried out in the laboratory with a concentrated SO 2 flow, was
assumed to be governed by both the following global reactions:
(2.1)
CaCO3 + SO2 + nH2 O → CaSO3 · nH2 O + CO2 ,
(2.2)
1
CaCO3 + SO2 + 2H2 O + O2 → CaSO4 · 2H2 O + CO2 ,
2
with 0, 5 ≤ n ≤ 1. Considering the experimental conditions and the model characteristics, both reactions are suitable; in fact, the model is only affected by the
stechiometry of the main reagents (SO2 and CaCO3 ) and products (CaSO4 and
CaSO3 ). The plan of the paper is as follows. In Section 3, a presentation of the
model is given, with some information about the qualitative behavior of the solutions and some numerical simulations. Section 4 is devoted to the description of
the experimental part of our investigation. Finally, in Section 5, numerical and
experimental results are compared.
3. The Hydrodynamic Model
In this work a hydrodynamic model is considered, which has been originally
introduced in [3] to describe the transformation vs. time of the calcium carbonate
(Calcite) stones under the chemical aggression of sulphur dioxide.
The model is based on basic physical relations, on the balance laws of the chemical reactions and on the Fick’s law, neglecting the permeability of the medium.
The reaction path of SO2 with calcite is assumed to be governed by the simplified
reactions (2.1) and (2.2). This means that for one molecule of calcium carbonate
and one of sulphur dioxide, the reaction produces one molecule of gypsum and carbon dioxide by addition water and oxygen. All heat effects have been neglected,
assuming that the air contains enough water to give rise to the reaction. Moreover,
it was assumed that changes in concentration of oxygen, water, and carbon dioxide
do not affect the reaction. Let Ω be the domain occupied by the specimen of calcite
under consideration and set ρs for the total concentration of SO2 , c for the density
of calcite and γ for the density of gypsum.
Following previous works, [3, 4, 13], it was assumed that ρ s and c satisfy the
following balance laws

k
 ∂t ρs + ∇ · (ρs Vs ) = − mc ρs c,
(3.1)

∂t c = − mks ρs c,
4
C. GIAVARINI, M. INCITTI, M. L. SANTARELLI, R. NATALINI, V. FURUHOLT
where Vs is the sulphur dioxide “fluid” velocity, k is the reaction rate, m c , ms are
the masses of single molecules of calcite and SO2 .
Notice that we assume the conservation of the total density; then the density of
gypsum can be recorded as a function of calcite, by the formula
mc
mc
(3.2)
c+
γ = c0 +
γ0 ,
mγ
mγ
where c0 and γ0 are the given initial densities, respectively for calcite and gypsum,
and setting mγ for the mass of one molecule of gypsum.
Next, we introduce the porosity of the calcite specimen ϕ, which cannot be
assumed constant, since the transformation of calcite in gypsum alters the volume
of void (occupied by air and sulphur dioxide). Therefore, it is reasonable to regard
it as a function of the amount of gypsum or, equivalently, as a function of the
amount of calcium carbonate, that is ϕ = ϕ(c). As shown in [1], the porosity of
the specimen during the reaction can be expressed as a linear combination of the
porosity of the pure calcite specimen, ϕ0 , and the porosity of the final sulphate
m
product, ϕγ̃ , namely for c = 0 and γ̃ = mγc c0 + γ0 :
c
(3.3)
ϕ(c) = ϕγ̃ + (ϕ0 − ϕγ̃ )
c0
where s is the porous concentration of SO2 , which is defined as the concentration
taken with respect to the volume of the pores. The porous concentration is related
to the total concentration by
(3.4)
ρs = ϕ(c)s.
At the same time, the seepage velocity vs is related to the fluid velocity Vs by the
classical Dupuit-Forchheimer relation
(3.5)
vs = ϕ(c)Vs .
To close system (3.1), an expression for the seepage velocity v s is necessary. Following previous studies [3], it is assumed that all the contributions given by the
pressure gradient to the seepage velocity, usually driven by the Darcy law, is to
be neglected. Therefore, we shall consider only the component of v s given by the
Fick’s law [13] (see also [15]):
(3.6)
svs = −D(c)∇s,
where D(c) = dϕ(c), and d is the (scalar) effective molecular diffusive coefficient.
This yields a closed model for the new unknown (s, c):

k
 ∂t (ϕ(c)s) = − mc ϕ(c)sc + d∇ · (ϕ(c)∇s),
(3.7)

∂t c = − mks ϕ(c)sc.
Restricting the attention to the case of a mono-axial symmetry and making the
simple change of variables
r
k
kt
mc
(3.8)
y=
x, τ =
, s̃ =
s,
dmc
mc
ms
the following space dimensional system in its adimensional form can be written:

 ∂t ϕ(c)s − ∂x (ϕ(c)∂x s) = −ϕ(c)sc,
(3.9)

∂t c = −ϕ(c)sc,
with x ∈ [0, 1], t ≥ 0. The initial conditions are taken as
(3.10)
s(x, 0) = 0, c(x, 0) = c0 ,
A NONLINEAR MODEL OF SULPHATION
5
where c0 is a positive constant; the boundary conditions are
∂ϕ(c)s
(1, t) = 0.
∂x
More general models and a complete mathematical and numerical justification
of the present model can be found in [1, 3, 9], respectively.
(3.11)
s(0, t) = sb,
3.1. Numerical behavior of the calcite density and the SO2 concentration. Following [3], a finite elements scheme has been used to solve numerically
the model, looking for s as a continuous piecewise linear function, and c as a piecewise constant function. In order to confirm the above mentioned hypothesis and
to assess the time dependence of the described chemical phenomena a large number of numerical tests were carried out in laboratory. In Figure 1 (top), numerical
solutions for c, are shown on a domain of length L = 1 and taking k = 10 5 , for
three different values of time; other values of the parameter k will only influence
the curve by the shifting (3.8) in the graph, but without changing the curve shape.
The initial calcite density was set at c0 = 1.0. Each curve must be considered
as a profile of the density of calcite in the one-dimensional specimen, exposed to
an SO2 -contaminated atmosphere. Comparing the three graphs, the shape of the
boundary layer appear to be unchanged with the time. Figure 1 (bottom) shows the
numerical approximations of s, corresponding to the calcite profiles, with the initial
concentration of SO2 set to ŝ = 0.1. The concentration decreases linearly with x,
with a “rounds off” to s = 0. Comparing the curves of s and c, we notice that the
calcite boundary layer coincides with the value of x, when the SO 2 concentration
comes to zero.
3.2. Asymptotic behavior of the solutions. A numerical analysis of the asymptotic behavior for higher time values of the solutions is given elsewhere [3].
The main result was the following: for t → ∞ and in the half-line x > 0, it holds
x
x
lim s(x, t) − S √ = 0, lim c(x, t) − C √ = 0,
t→∞
t→∞
t
t
where the limit functions (S, C) are explicitly given by

Rξ 1 2 2
 sb − ξ02c0 0 e 4 (ξ0 −η ) dη, C = 0, for ξ ∈ (0, ξ0 );
(3.12)
(S(ξ), C(ξ)) =

S = 0,
C = c0 , for ξ > ξ0 .
Figure 2 shows how the shapes of the calcite density and SO 2 concentration profiles
change as t → ∞. Notice that, due to the self-similar structure of the asymptotic
profiles, the solutions are plotted in the scaled coordinate √xt . By increasing the
time t, the calcite density profiles become gradually steeper, while the SO 2 concentration profile appears to split into two distinct parts. The part closest to the
boundary shows a nearly linear decrease in concentration, while in the second part
the concentration is zero. The asymptotic limit functions (3.12) are displayed as
dotted lines. Figure 3 shows the gypsum front and the scheme of the asymptotic
profile, plotted in the standard (x, t)-plane. Figure 4 shows a plot of the position√of
the front of gypsum, obtained from the numerical simulations, as a function of t,
together with the corresponding function ζ(t) for the time asymptotic limit. The
numerical results were found using 300 finite elements, a time step ∆t = 1/3000
and a reaction coefficient k = 105 , and confirm the validity of the time asymptotic
expansion.
The analytical position of the gypsum front ζ(t) in the asymptotic limit is given
by
√
(3.13)
ζ(t) = ξ0 t ,
6
C. GIAVARINI, M. INCITTI, M. L. SANTARELLI, R. NATALINI, V. FURUHOLT
where ξ0 is a constant depending on ŝ and c0 , that can be found inverting the
following strictly increasing function [3]:
Z ξ0
2
2
1
2ŝ
e 4 (ξ0 −η ) dη =
(3.14)
F (ξ0 ) = ξ0
.
c0
0
Notice that the constant ξ0 depends on the physical parameters and on the initialboundary conditions defined above, through the following relation, now expressed
in the physical (unscaled) variables:
2mc sbϕγ̃ √
(3.15)
ξ0 = G
d,
ms c 0
where the function G has the following approximated expression:
q
√
G(θ) =
9 + 6θ − 3 + o(θ 5/2 ).
Equation (3.15) shows the dependence of the sulphation crust growth on the SO 2
concentration.
4. Experimental evaluation of the model
To evaluate the model, a laboratory reaction chamber was designed and constructed (Figure 5). The reaction chamber was constituted by the two glass compartments 3 and 6, between which a Teflon sample holder 4 was placed; the stone
sample 5 was inserted in the Teflon padder. The SO 2 gas flow was measured in the
flow-meter 2, before reaching the reactor. The humidifier 7 was necessary for the
reaction progress (humid deposition of SO3 and acid attack). The reaction chamber 6 was vacuumed by a vacuum pump to establish a pressure gradient between
3 and 6, and to force the gas through the sample. The rector was planned to have
a mono-dimensional symmetry of the gas flow through the sample, as required by
the mathematical model. In this way, only a face of the sample was exposed to
the artificial atmosphere. The operative conditions were: high relative humidity
(100%), room temperature and pressure of 1.8 bar. To define the right conditions
for testing the chamber efficiency a pure SO2 gas flow was used. The pure SO2 gas
facilitated the reaction on the surface of the sample and increased the reaction rate.
In this way a large number of samples were produced in relatively short time. The
selected stone was a Carrara marble (Statuario). This stone was used for its lower
porosity and permeability with respect to other calcium carbonate stones; hence the
first approximation required for the hypothesis of the model was satisfied. Table 1
Material
Porosity (%)
Carrara Marble
0,58
Verona red stone
8
Pietra di Lecce
38
Table 1. Porosity measures for different calcium carbonate stones.
shows porosity values of different calcium carbonate stones. About sixty samples
were exposed at different time intervals (from 1 hour to 430 hours). The sulphation
layer of the reacted samples was measured by using the Scanning Electron Microscope (SEM, mod. Cambridge Stereoscan 360). The chemical composition of the
sulphation layer, in terms of CaSO4 · 2H2 O, CaSO3 · nH2 O, was determined by the
Thermal Gravimetric Analysis (TGA, mod. 2950 TA Instruments). The Energy
Dispersive Spectrometer (EDS, mod. LEO 1450 VP INCA 300) was finally used to
map the elements (as calcium and sulphur) on the surface.
A NONLINEAR MODEL OF SULPHATION
7
5. Analysis of the Results
The experimental work was necessary to verify that the calcite front and therefore
the
√ thickness of crust deposed on the sample surface, was a linear function of the
t, as predicted by equation (3.13).
In order to verify this function, thickness measures were done using the scanning electron microscopy (SEM), (Figure 6). Table 2 shows the gypsum thickness
Time (Hr) Thickness (µ)
0
0
24
214
48
357
72
450
96
506
120
620
144
700
168
788
Table 2. Gypsum thickness measured by SEM at different reaction times.
values. The thickness of the gypsum increased √
with the time of reaction. These
values were plotted in Figure 7 as a function of t; therefore, this graph based on
experimental results, is comparable with that Figure 7 based on the model values.
Comparing the evolution of the numerical simulations with the experimental measures it’s possible to define the rightness of the mathematical model hypothesis.
Experimental measures are in good agreement with the numerical simulations and,
therefore, confirm and validate the proposed model. Figure 8 shows the boundary
layer between gypsum and unreacted calcite; as predicted by the model, it appears
to be thin and well defined front.
The chemical characterization by TGA revealed the presence of the CaSO 4 ·
2H2 O and of CaSO3 · nH2 O, whose concentration are given by Table 3. The presence of CaSO3 · nH2 O is due to the high concentration of SO2 that reacted directly
with CaCO3 without undergoing the oxidation to SO3 . This reaction is probably
not important in urban atmosphere because of the presence of catalysts (such as
O3 or those contained in the dispersed particulate) which favor the oxidation. As
previously mentioned, the presence of CaSO3 · nH2 O was already identified by others authors [12, 8, 17] in similar experimental conditions. However, the formation
Time (Hr) %Gypsum %CaSO3 · nH2 O
0
0
0
24
11.33
88,67
48
11
90,18
72
14.07
85,6
96
16.24
84,49
120
11.97
90,4
144
13.5
86,11
192
16.21
83,98
Table 3. CaSO4 · 2H2 O and CaSO3 · nH2 O concentrations measured at different reaction times.
8
C. GIAVARINI, M. INCITTI, M. L. SANTARELLI, R. NATALINI, V. FURUHOLT
of such compounds is not relevant in order to give a full assessment of our macroscopic model; in fact the model relays only on SO2 concentration and on the crust
thickness.
6. Conclusion
A macroscopic hydrodynamic model for the evolution of the gypsum fronts in
calcium carbonate stones has been considered.
A specific reaction chamber was employed to perform the tests and obtain a onedimensional symmetry of the gas flow. A humidified, 100% SO 2 flow was used. The
preliminary results are in a good agreement with those of the numerical simulations.
It was
√ confirmed that the thickness of the sulphation layer grows as a linear function
of t.
Other experiments will be realized in the next future, to obtain condition similar
to the external atmosphere (diluted SO2 flow and presence of catalysts). In this
way it will be possible to better quantify the real damage phenomena on the stone
materials.
Acknowledgments
This work was partially supported by Soprintendenza ai Beni Culturali Regione
Autonoma Valle d’Aosta, within the project ”Analisi di processi di corrosione di
materiali ferrosi”, 2002-2005. The research activity reported in this paper has been
also partially conducted within the European Union RTN HYKE project: HPRNCT-2002-00282.
References
[1] G. Alı̀, V. Furuholt, R. Natalini, I. Torcicollo, Sulphite chemical aggression of calcite specimens with high permeability: a mathematical model, Working paper 2003.
[2] G. G. Amoroso, V. Fassina, Stone decay and conservation - Atmospheric Pollution, Cleaning,
Consolidation and Protection. Amsterdam: Elsevier Science Publishers, 1983.
[3] D. Aregba-Driollet, F. Diele, R. Natalini, A mathematical model for the SO 2 aggression
to calcium carbonate stones: numerical approximation and asymptotic analysis, Quad IAC
2003; 6.
[4] G.I. Barenblatt,V.M.Entonov, V.M. Ryzhik, Theory of fluid flows through natural rocks.
Dordrecht: Kluver Academic Publ., 1990.
[5] R. H. Borgwardt, Kinetics of the reaction of the SO2 with calcined limestone, Environ.
Science & Tecnol. 1970; 4: 59-63.
[6] K. L. Gauri and G. C. Holdren, Pollutant effects on stone decay, Environmental Science and
Technology. 1981; 5: 386-390.
[7] V. Fassina, Environmental Pollution in Relation to Stone Decay, Durability of Building
Materials. 1988; 5: 317-358.
[8] K. Gauri, N. P. Kulshreshtha, A. R. Punuru, A. N. Chowdhury, Rate of decay of marble in
laboratory and outdoor exposure, J. Materials Civil. Eng. 1989; 2: 73-85.
[9] F.R. Guarguaglini, R. Natalini, Global existence of smooth solutions to a nonlinear model
for sulphation phenomena of calcareous stones Quad IAC 2003; 9.
[10] B. G. D. Hoke and D. L. Turcotte , Weathering and damage, J.Geophys. Res. 2002;
107(B10).
[11] W. T. Lipfert, Atmospheric damage to calcareous stone: comparison and reconciliation of
recent experimental findings, Atmos. Environ. 1989; 23: 415-429.
[12] N. P. Kulshreshtha, A. R. Punuru, K. L. Gauri, Kinetics of reaction of SO 2 with marble, J.
Mater. Civ. Eng. 1989; 1: 60-72.
[13] D. A. Nield and A. Bejan, Convection in Porous Media, Springer, 1992.
[14] C. Sabbioni, G. Gobbi, O. Favoni, Black crusts on ancient mortars, Atmospheric Environment 1998; 32: 215-223.
[15] S. Majid Hassanizadeha and A. Leijnsea, , A non-linear theory of high-concentration-gradient
dispersion in porous media, Advances in Water Resources 1995; 18: 203-215.
A NONLINEAR MODEL OF SULPHATION
9
[16] T. Skoulikidis and D. Charalambous, Mechanism of sulphation by atmospheric SO 2 of the
limestone and marble of ancient monuments and statues, II. Hypothesis concerning the rate
determining step in the process of sulphation, and its experimental confirmation, Brit. Corr.
J. 1981; 16: 70-76.
[17] T. Skoulikidis and E. Papakonstantinou-Ziotis, Mechanism of sulphation by atmospheric
SO2 of the limestone and marble of ancient monuments and statues, I. Observations in situ
(Acropolis) and laboratory measuremants, Brit. Corr. J. 1981; 16: 63-69.
[18] R. A. Smith, The Beginnings of a Chemical Climatology . London: Longman Green 1872.
C. GIAVARINI, M. INCITTI, M. L. SANTARELLI, R. NATALINI, V. FURUHOLT
10
0.1
0.09
Sulphite concentration
0.08
0.07
0.06
0.05
0.04
0.03
0.02
T1=0.1
T2=0.5
T3=1.0
0.01
0
0
0.1
0.2
0.3
0.4
x
0.5
0.6
0.7
0.8
0.9
1
1
0.9
0.8
Calcite density
0.7
0.6
0.5
0.4
0.3
0.2
T1=0.1
T2=0.5
T3=1.0
0.1
0
0
0.1
0.2
0.3
0.4
0.5
x
0.6
0.7
0.8
0.9
1
Figure 1. Numerical simulations of calcite density (top) and SO 2
concentration (bottom) profiles for a non-dimensional problem
with a dimensionless reaction constant k = 105 and dimensionless domain length L = 1. The reaction rate for the three graphs
are T1 = 0.1, T2 = 0.5 and T3 = 1.0 as indicated. In all cases
the results were obtained using 300 finite elements and a time step
length ∆t = 1/3000.
A NONLINEAR MODEL OF SULPHATION
11
0.1
0.09
0.08
SO2 concentration
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1/2
x/t
1
0.9
0.8
Calcite density
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
x/t1/2
Figure 2. Calcite density (top) and SO2 concentration√(bottom)
profiles for a selection of values of t in the range 10 2 ≤ t ≤ 105 .
The dotted lines represent the corresponding time asymptotic limit
functions given by equation (3.11).
1
C. GIAVARINI, M. INCITTI, M. L. SANTARELLI, R. NATALINI, V. FURUHOLT
Figure 3. Scheme of the asymptotic limit system solution in the
(x, t)-plane
0.45
0.4
0.35
0.3
Crust Thickness
12
0.25
0.2
0.15
0.1
0.05
0
Simulation
Fast reaction limit
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1/2
t
√
Figure 4. Position of the calcite front as a function of t. The
simulated results were obtained using 300 finite elements, ∆t =
1/3000 and k = 105 .
A NONLINEAR MODEL OF SULPHATION
Figure 5. Scheme of the reaction chamber.
Figure 6. Measure of the gypsum thickness of a sample exposed
to SO2 for 144 hours.
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C. GIAVARINI, M. INCITTI, M. L. SANTARELLI, R. NATALINI, V. FURUHOLT
Figure 7. Crust thickness measured in function of time reaction.
Figure 8. SEM-EDS map of the chemical elements on the sample
surface exposed for 144 hours (Calcium: right Sulphur: left).