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Application of an Extended Shrinking Film Model to limestone dissolution Vincenzo Russo, Tapio Salmi, Claudio Alberto Carletti, Dmitry Yu. Murzin, Tapio Westerlund, Riccardo Tesser, and Henrik Grénman Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.7b01654 • Publication Date (Web): 27 Jun 2017 Downloaded from http://pubs.acs.org on June 30, 2017

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Industrial & Engineering Chemistry Research

Application of an Extended Shrinking Film Model to limestone dissolution Vincenzo Russo1-2, Tapio Salmi1, Claudio Carletti1, Dmitry Yu. Murzin1, Tapio Westerlund1, Riccardo Tesser2, Henrik Grénman1,3,* 1

Åbo Akademi University, Laboratory of Industrial Chemistry and Reaction Engineering, Johan Gadolin Process Chemistry Centre, FI-20500 Turku/Åbo, Finland 2

3

Università di Napoli ‘Federico II’, Chemical Sciences Department, IT-80126 Naples, Italy

Åbo Akademi University, Molecular Process and Materials Technology, FI-20500 Turku/Åbo, Finland * [email protected]

Abstract

The reactions of soluble and reactive solids with components in the liquid phase are of high relevance in the field of chemical engineering. A mathematical model was recently developed applying an extended film theory, where the reactive solid material dissolves in the liquid phase and diffuses through a dynamic liquid film surrounding the particle. In the present work, this Extended Shrinking Film Model (E.S.F.M.) was applied to a very challenging reaction, the limestone dissolution in acid environment. The model was applied to experimental data collected under a wide range of operation conditions, i.e. varying temperature, particle size, stirring rate and type of limestone. A very good fit of the model to experimental data was obtained and the chemical and physical phenomena were clearly identified, significantly contributing to understanding of the reaction kinetics. The work clearly demonstrates that the data interpretation can be considerably enhanced by rigorously taking into account the physical phenomena and that the E.S.F.M. can be used in planning larger reactors, due to its flexibility in predicting the reaction kinetics at different conditions.

Keywords: solid-liquid reactions, limestone dissolution, film theory, extended shrinking film model, mathematical modelling, numerical solution

1

ACS Paragon Plus Environment


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1. Introduction

Limestone dissolution is a key step in sulfur dioxide removal technologies. Even though the process is of uttermost importance for the chemical industry, dissolution kinetics is still not well understood. For example, the reported activation energies vary within a very broad range, form under 10 kJ/mol to over 60 kJ/mol, as described by Carletti et al.1 Moreover, different kinetic models are typically employed depending on the pH regime, which indicates that the overall mechanism and model are not fully described. One of the possible reasons for these discrepancies is that experimental data have been modelled and interpreted with rather simplistic models, which do not take into account the physical phenomena involved in the process.2-14

Several approaches to modeling limestone dissolution are presented in the literature, and the well-known shrinking sphere model with 2/3-order kinetics with respect to the solid phase has often been applied.

1-4

Close to first-order kinetics with respect to the hydronium ions has also

been considered for low pH values,6,12-14 as well as first-order kinetics with respect to CaCO3.15,16 The dissolution rate has been reported to depend on the agitation,14-18 which has led to the conclusion that mass transfer has controlled the observed reaction rate. It has also been reported that the effect of stirring is more prominent with larger particle sizes 12 and that even in the case of first-order kinetics, diffusion towards the solid surface has in fact been experimentally observed.19 Different assumptions can be found in the literature regarding the role of mass transfer and chemical surface reaction derived from the experimentally recorded values of apparent activation energies. Lund et al. reported a high (62.8kJmol−1) apparent activation energy for calcite dissolution under acidic conditions and they suggested that the dissolution can be described by mass transfer and reaction rate resistances in series, where the rate determining step could be the surface reaction. 20 Plummer et al. reported a low (8.4 kJ/mol) value for the activation energy at low pH (pH 5.5. Several apparent activation energy values for calcite and two for dolomite found in literature with some major remarks, were presented by Carletti et al.1 The range of apparent activation energies reported in the literature is broad; however, the experimental conditions have not always been comparable.

The challenges in modeling limestone dissolution lay partly in the dynamic equilibrium, which changes during the course of the experiment with pH. Typically, commercial WFGD scrubbers operate within the pH range up to 6,5 therefore, a model capable of describing the dissolution rate of limestone over a broad range is highly encouraged. The aim in the current work was to model the dissolution kinetics of both limestone samples in the industrially relevant pH range 2.4–6 with two different particle sizes without the need for dividing the studied pH regime into sections.

Current approach

Limestone dissolution in acidic environment is described by the following reaction scheme,

1. CaCO3(S) ⇄ Ca2+(L) + CO32-(L) 2. CO32-(L) + H+ ⇄ HCO33. HCO3- + H+ ⇄ CO2 + H2O 4. HCO3- + H+ ⇄ H2CO3

Step 1 is the limestone dissolution, which can be considered rapid in a well-stirred system. Step 2 is an acid catalyzed reaction. Further rapid proton equilibria occur according to reaction 3 or 4, depending on the pH.12 At very low pH, reaction 4 dominates, while at mildly acidic and neutral pH, reaction 3 is prominent. These rapid protonation reactions are here assumed to be in quasiequilibrium and do, thus, not influence directly the dissolution kinetics.1 3


However, reaction 2


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can be considered as the rate determining step of the overall dissolution, as apparent first order kinetics in the proton concentration has been observed for the reaction kinetics6,13,14 at low pH, also in the present work. Reaction 2 can also be considered irreversible in the studied pH range, as the reverse reaction is thermodynamically favored only at high pH12. The reaction system can be described by the following steps: (i) solid dissolution, (ii) diffusion and acid reaction in the film phase, (iii) reaction in the bulk phase. Very recently, a generic model utilizing an extension of the well-known film theory21-24 was developed by Salmi et al.25 for describing reactions between dissolving solids and liquids. Limestone dissolution in acidic environment is a challenging prime candidate for testing the newly developed model. Recently, Carletti et al. published kinetic data on limestone dissolution in the presence of HCl1. Experimental data were collected in a wide range of operation conditions, i.e. varying temperature, particle size, stirring rate and limestone type. The collected data were rather successfully modelled in a wide pH range by assuming a Langmuir-type kinetic rate law, not considering however the physical phenomena encountered in the film surrounding the particles. The goal of present work was to apply the film theory to the limestone dissolution, in order to evaluate its capability to describe the broad range of experimental data when applying identical kinetic parameter values for different particle sizes and even for limestones of different origin. Moreover, the kinetics were quantified and the influence of film diffusion in different experimental conditions was demonstrated.

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2. Materials and Methods 2.1 Extended shrinking film model (E.S.F.M.) The film theory is very much used in the description of gas-liquid systems in cases that the size of the gas bubble remains constant. Moreover, in the dissolution of solids, the liquid film thickness changes during the course of the reaction: as the particle diminishes, the liquid film becomes thinner. The developed Extended Shrinking Film Model (E.S.F.M.) is based on the fact that a solid reactant, dispersed in a liquid phase, first dissolves and then reacts with a liquid component. The reaction occurs both in the stagnant liquid film surrounding the particle and in the bulk of the liquid phase. As the reaction proceeds, the solid particles diminish in size and consequently the liquid film thickness decreases. These concepts have been formulated by writing opportune mass balance equations for: (i) the liquid bulk phase; (ii) the liquid film; (iii) the solid phase. The dimensionless mass balances equations are reported in Eqs. 1-3. Liquid bulk:

Liquid film:

dy 'i τ R  n j  = r ' − A2 0  dθ c0 δ  n0 j 

s s +1

∂yi ∂x

, A2 = x =1

Di aτ R0

∂yi A1  ∂ 2 yi s (δ / R0 ) ∂yi  τ Dτ  2 +  + r , A1 = i 2 = 2  ∂θ (δ / R0 )  ∂x R / R0 + (δ / R0 ) x ∂x  c0 R0

(1)

(2)

s

Solid phase:

dn' j / n0 j dθ

=

D j a0τ

δ

 n  s +1 ∂y j A4  j   n  ∂x  0j 

, A4 = x =0

c0VL n0 j

(3)

The following boundary conditions are needed to solve the PDE. •

x=0: yi=yi* (saturation of the solid components at the surface)

•

x=0: ∂yi/∂x = 0 (liquid-phase component at the surface)

•

x=1: yi=yi’ (bulk-phase conditions valid at the end of the film)

By considering that the mass balances have been written in a non-dimensionless form, the initial value of each variable is 1. The reaction rate is included in both the liquid bulk and the liquid film mass balance equations, where r is the reaction rate in the liquid film and r’ in the liquid bulk.

A visualization of the E.S.F.M. is depicted in Figure 1.

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Solid particle

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Film

Bulk i

Film

Solid particle R

r

Bulk

j

R+δ

δ

t Figure 1 – A schematic representation of the E.S.F.M., where j stands for the concentration of the dissolved solid phase compound (CO32-) and i is the component coming from the liquid bulk phase (H+) and δ is the film thickness.

The particle size coordinate (x) is dimensionless (with x=0 for the solid surface and x=1 for the film-bulk interface) and the film thickness was considered to be dependent on both physical parameters related to mass transfer (i.e. stirring rate, viscosity, diffusivity) and the conversion of the solid (Eq. 4). However, the film thickness was considered component non-specific, as the sizes and the diffusivities of the ionic compounds could be considered to be in the same range. 1 /( s +1)

δ  n j  = R0  n0 j 

2 /( 3( s +1)) 1/ 3 1/ 6 2 1/ 3   1 +  ε   υ   R0   n j     υ 3   D   2   n    i    0j   

−1

(4)

A detailed derivation of Eq. 4 was reported by Salmi et al.25, valid for stirred tanks. Eq. 4 can be re-written by introducing a dimensionless adjustable parameter, dependent on the physical properties of the system and on the stirring rate (Eq. 5), 1 /( s +1)

δ  n j  = R0  n0 j 

−1

2 /( 3( s +1)) 1/ 3 1/ 6 2 1/ 3   1 + α  n j   , α = A =  ε   υ   R0   3    3   n   υ   Di   2  0j    

6


(5)

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Finally, the relative contributions of the film and bulk reactions can be calculated from the momentaneous contribution of each reaction occurring in each phase (film or bulk), and can be calculated from Eqs. 6-7, VF

λF = V

∫ rdV

F

0

F

∫ rdV 0

F

+ r 'VL

, λ B = VF

r 'VL

∫ rdV

F

(6)

+ r 'VL

0

Further details of the E.S.F.M. can be found in the article of Salmi et al.25. The coupled ordinary differential equations (ODEs) and partial differential equations (PDEs) system was solved numerically by using gPROMS Model Builder v.4 software26-28. The PDEs were solved by choosing a second order centered finite difference formula for the spatial derivatives (the number of discretization points was 40).

2.2 Limestone dissolution experiments All the experimental data used for the parameter estimation were obtained by Carletti et al.1. The main details of the experiments are described here. Two different natural limestone samples originating from Parainen, Finland and Wolica, Poland were used in the experiments. Limestone in the soil of Parainen is 1900 - 2000 million years old being very white, crystalline and marble. The other limestone sample is “only” 150 million years old. It is softer and more porous than Parainen limestone, better suited for grinding; however, the specific surface area of both samples was below 0.6 m3/g showing very low porosity. Both samples were about 99 w-% pure CaCO3, with magnesium being the major impurity (~0.3 w-%) and the density of the samples was 2720 kg/m3 and 2703 kg/m3 for Parainen and Wolica, respectively. The qualities of the reactive stone make it excellent for flue gas cleaning. Each sample was prepared by grinding and sieving in two size fractions (74–125µm and 212–250µm), named “small” and “large”. The limestone sample (1g) was placed in contact with 1.5L of pre-heated hydrochloric acid (HCl) solution in a 2L glass reactor working at constant temperature, under intensive agitation and equipped with a gas bubbling system for an inert gas. The pH of the solution was constantly measured on-line until reaching a stable pH (initial pH≅2-3, final pH≅7), corresponding to almost total HCl and c.a. 3040% solid conversion. Different experiments were conducted in a wide range of operation conditions i.e. temperature, stirring rate and particle dimensions. The main experimental conditions, used in the parameter estimation analysis, are listed in Table 1. 7



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Table 1 – Experimental conditions used in the limestone dissolution experiments. Carletti et al.1. For all experiments, cS,0 = 6.66mol/m3. aTests 17,18,19 were conducted respectively at 1800, 1200 and 800rpm. Test

Material

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17a 18a 19a

Parainen Parainen Parainen Parainen Parainen Parainen Parainen Parainen Wolica Wolica Wolica Wolica Wolica Wolica Wolica Wolica Wolica Wolica Wolica

Size [µm] 74-125 74-125 74-125 74-125 212-250 212-250 212-250 212-250 74-125 74-125 74-125 74-125 212-250 212-250 212-250 212-250 212-251 212-252 212-253

Small Small Small Small Large Large Large Large Small Small Small Small Large Large Large Large Large Large Large

T [K]

cH+,0 [mol/m3]

cH+,0/cS,0 [mol/mol]

293.15 303.15 313.15 323.15 293.15 303.15 313.15 323.15 293.15 303.15 313.15 323.15 293.15 303.15 313.15 323.15 323.15 323.15 323.15

3.43 3.55 3.72 2.89 3.53 3.89 3.83 3.81 3.27 3.56 3.91 2.85 3.41 3.46 3.96 4.00 3.84 3.81 3.83

0.515 0.534 0.559 0.434 0.529 0.584 0.575 0.571 0.491 0.534 0.588 0.428 0.512 0.519 0.594 0.601 0.601 0.601 0.601

Most of the experiments were carried out at 1600 rpm. Carletti et al. proposed that with this agitation, the system could be considered to operate in the kinetic regime making the liquid-solid film mass transfer limitations negligible1. This suggestion was confirmed both by applying existing correlations and with dedicated experiments varying the impeller rate on the “small” Wolica sample. Only a minor improvement was observed changing the stirring speed from 1600 to 1800 rpm.

8


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3. Modelling results and Discussion The extended film theory was applied to the limestone dissolution data. First, a reaction rate expression needed to be defined. The rate for the limestone dissolution process was based on the second order irreversible kinetics previously shown to be consistent with the reaction mechanism1. As the reaction occurs in both bulk and film phases, it is necessary to define the relative reaction rate expressions (Eq. 7),

r = kcH + cS , r' = kcH + ' cS '

(7)

where CS refers to the CO32- concentration. The subscript S was chosen to reflect that this species comes from the solid reactant. In order to solve the mass balance equations, some important physical properties were defined. In particular, the diffusivities of H+ and CO32- were estimated by using the Nernst-Haskell correlation29, Eq. 8.

0 D AB =

RT [(1 / z + ) + (1 / z − )] = ai T F 2 [(1 / λ0+ ) + (1 / λ0− )]

(8)

As both the valence (z+ and z-) and the limit conductance (λ+0 and λ-0) are constant for every species, Eq. 8 is solely dependent on temperature. The remaining terms were merged in the ai coefficients, reported in Table 2, together with the other necessary physical properties. The results obtained by applying this correlation were compared with some literature findings1. As an example, the calculated value for D0H+(25°C) was 9.31·10-9m2/s, while 9.30·10-9m2/s was reported in the literature1.

9



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Table 2 – Physical properties used in modelling.

s cS,* MWS ρS τ aH+ aS

Value

Unit

2 0.14 100.09 2930 1 3.12·10-11 4.78·10-12

mol/m3 g/mol kg/m3 s m2/s/K m2/s/K

λ+0 (H+) z+ (H+) λ-0 (CO32-) z- (CO32-)

Value

Unit

349.8 1 44.5 2

(A/m2) (V/m) (g-equiv/m3) (A/m2) (V/m) (g-equiv/m3) -

The characteristic arbitrary time, needed to make our calculation time dimensionless, was chosen arbitrarily to be τ=1s. Based on the SEM images in our previous publication1 a shape factor greater than 2 could be plausible. However, the nitrogen adsorption measurements show low values (below 0.6 m2/g) for the specific surface area, which is why internal mass transfer is not considered. Moreover, the film surrounding the particles “evens out” the surface roughness making the area where the reactions happen rather spherical. This is the basis for assigning a value of s=2 for the shape factor.

Modelling of the experimental data was made by performing parameter estimation for all experimental data simultaneously, comprising different particle size and limestone type. It is important to underline that in this way, it is possible to estimate the kinetic parameters covering a wide range of experimental conditions, enhancing the diapason of validity and the physical meaning of the obtained parameters. This approach is very unconventional in the field of limestone dissolution kinetics, even if common in homogeneous reaction kinetics, helping in assessing the process kinetics.

The particle radius was considered to be the average of the range reported in Table 1. We tested the sensitivity of our modeling to our particle size ranges and did not observe any significant change in the results when the higher or lower boundary value was used. Therefore, only the average values were used.

Fits obtained for the experiments performed with the Parainen and Wolica samples are depicted in Figures 2-3. The obtained parameter values, together with the related statistics, are displayed in Table 3. 10


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1.0

0.6

0.4

Parainen "large" T = 293.15K T = 303.15K T = 313.15K T = 323.15K

0.8

cH+/cH+,0 [-]

cH+/cH+,0 [-]

1.0

Parainen "small" T = 293.15K T = 303.15K T = 313.15K T = 323.15K

0.8

0.2

0.6

0.4

0.2

0.0

0.0

0

50 100 150 200 250 300

0

250

t [s]

500

750

1000

t [s]

Figure 2 – Dimensionless proton concentrations as a function of time for experiments performed at different temperatures with “small” and “large” Parainen limestone. 1.0

1.0

Wolica "small" T = 293.15K T = 303.15K T = 313.15K T = 323.15K

0.6

0.4

Wolica "large" T = 293.15K T = 303.15K T = 313.15K T = 323.15K

0.8

cH+/cH+,0 [-]

0.8

cH+/cH+,0 [-]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.2

0.6

0.4

0.2

0.0

0.0

0

50 100 150 200 250 300

0

250

t [s]

500

750

1000

t [s]

Figure 3 – Dimensionless proton concentrations as a function of time for experiments performed at different temperatures with “small” and “large” Wolica limestone.

11



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Table 3 – Parameters obtained in modelling for “large” and “small” Parainen and Wolica limestones (tests 1-16 of Table 1). C.I.: confidence interval. Parameter Ea [J/mol] k (293.15K) [m3/(mol·s)] α [-]

Value

95%C.I.

4.03·104 2.48·105 50.36

0.23·104 0.12·105 0.75

As revealed by Figures 2 and 3, a good fit of the model to all experimental data was obtained. Moreover, the largest error in the parameter values was about 5%. The results for different temperatures and CaCO3 particle sizes and for the different limestone were properly described applying the same kinetic parameters, which has not been achieved with previous theories. One very notable observation is that the results with both Parainen and Wolica samples could be very well described by applying the same values for the activation energy and pre-exponential factor, which according to conventional practice and literature findings would be considered impossible, or at least improbable. Even if Parainen and Wolica limestone samples are different in nature (metamorphic and sedimentary, respectively) and are of different geological ages (Proteozoic and Jurassic, respectively), they display the same chemical activity when the shrinking film is taken into account in the modelling. This physically reasonable, as the samples are chemically very similar exhibiting high CaCO3 content and similar particle sizes, even if porosity and density are somewhat different.

The activation energy estimated by using the extended film theory model (40 kJ/mol) is higher than the one obtained with the previously used Langmuir approach1 (16-21 kJ/mol for Wolica and 16.5-17.9kJ/mol for Parainen limestone). This can be attributed to the fact that the extended film theory model takes into account the physical phenomena surrounding the particles involved in the process, increasing the accuracy of the obtained parameter values. Low activation energies obtained for chemical reactions can be an indication of either simplification in the reaction mechanism or the presence of mass transfer limitations. The values obtained in this work can be considered physically reasonable both numbers and the relative errors, which are less than 5%. The fact that the value 40 kJ/mol obtained for the chemical kinetics is in the higher end of the observations reported in literature is also logical, as most of the models reported do not differentiate between mass transfer and chemical kinetics, as described in the introduction part.1

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Another interesting aspect can be noticed when modelling the experiments performed at different stirring rates. The mixing parameter (α) was obtained by parameter estimation for experiments performed at different stirring rates, with the “small” Wolica limestone at 50°C (experiments 1619 in Table 1). The parameter estimation was conducted by using the implemented Maximum Likelihood algorithms available in gPROMS ModelBuilder v.4. The objective function is given by Eq. 9.

f obj =

 NE NVi NM ij  (z ijk − z ijk )2   N 1 ln(2π ) + min ω ∑∑ ∑ ln(σ ijk2 ) +  2 2 σ ijk2    i =1 j =1 k =1 

(9)

The fit of the model to the experimental data along with the obtained parameter values is displayed in Figure 4.

4.0

60

3.5

55

3.0

50

α [-]

cH+ [mol/m3]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


2.5 2.0

40

800 rpm 1200 rpm 1600 rpm 1800 rpm

1.5 1.0 0

A

45

5

35 30

10 15 20 25 30

800

B

t [s]

1200

1600

2000

S.R. [rpm]

Figure 4 – A. Proton concentration as a function of time for experiments performed at different stirring rates with the “small” Wolica limestone at 50°C. B. Dependence of the mixing parameter (α) as a function of the stirring rate.

A very good agreement between the experimental and the calculated values was obtained. As can be seen in Figure 4 B, by increasing the impeller stirring rate, there is an obvious increase of the stirring parameter (α) until a plateau is reached. This behavior displays a shift from a mass transfer limited regime to kinetically limited regime. This result is in agreement with the 13



observations of Carletti et al.1, who noticed that when working at stirring rates greater than 1600 rpm the system was operated in the kinetic regime. This result demonstrates the flexibility of the E.S.F.M. as it is able to describe systems ranging from mass transfer limited to kinetic regimes. For better visualization of the fitting, an overall parity plot is shown in Figure 5. All the experimental data are well predicted within an error margin of ±10%. 1.0

Calculated data

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0.8

0.6

0.4

0.2

± 10% ±±20% 0.0 0.0

0.2

0.4

0.6

0.8

1.0

Experimental data Figure 5 – Parity plot for all performed experiments.

As described previously, the reaction occurs simultaneously in both the bulk and the film phases. A very interesting theoretical aspect is the identification of conditions when the dominating reaction front shifts from the film to the bulk and, additionally variations with the stirring rate. In order to study this, the kinetic parameter values were fixed to the ones previously obtained, and a contribution analysis was performed. The results are plotted in Figure 6.

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1.0

800 rpm 1200 rpm

1.0

0.8

0.8

0.6

0.6

λF [-]

λF [-]

0.4

0.2

0.0

0.0

100

125

150

1600 rpm 1800 rpm

0.4

0.2

175

50

t [s]

75

100

125

t [s]

Figure 6 – Contribution analysis for experiments performed at different stirring rates with the “small” Wolica limestone at 50°C.

As can be noticed, the shift of the predominant reaction regime from the film to the bulk decreased when the stirring rate was increased. This is even more evident in Figure 7 where the time at the inflection point is plotted against the stirring rate. 140 130 120

tf [s]

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110 100 90 80 70 800

1000

1200

1400

1600

1800

S.R. [rpm] Figure 7 – Time at the inflection point (tf) for the contribution analysis. The time of the inflection point decreases with increasing stirring rate. At low stirring rates, the overall kinetics was dominated by the film mass transfer limitation, for a long period during the 15



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Page 16 of 23

experiment. At higher stirring rates, the overall kinetics was determined more by the chemical reaction. Considering that the reaction occurs both in the film and the bulk phases, even small changes in the stirring rate can lead to large differences in the ratio between the two. This can be of uttermost importance in designing and scaling-up reactors for solid-liquid reactions, as it is important to be able to identify how intense mixing needs to and is worthwhile applying to the system in order to gain the desired results in an economically benign way.

4. Conclusions The recently developed Extended Shrinking Film Model (E.S.F.M.) was applied to experimental data on solid-liquid reaction kinetics, namely to limestone dissolution in HCl. An overall parameter estimation was performed with all collected data, covering a wide range of experimental conditions. A very good fit of the model to the experimental data was obtained. Applying a second order rate law for the chemical kinetics, the model was able to correctly take into account all the involved physical phenomena in experiments performed with different limestone samples, granulometry and stirring rates employing the same rate laws and kinetic parameters independent of the sample. This unifying approach emphasizes ability of the applied methodology to reveal the kinetics of limestone dissolution and can be extended to various solidliquid reactions where film diffusion limitations contribute significantly to the kinetics. The model can be considered thus a generic one capable of covering a wide range of operative conditions and showing high flexibility. Limestone dissolution kinetics was thoroughly modelled allowing to conclude that low apparent activation energies often observed in the literature are mainly due to film diffusion limitations and not chemical kinetics.

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Notation a

surface area-to-volume ratio, m2/m3

ai

diffusivity coefficient, m2/(sK)

AN

dimensionless quantity with N=1-4, -

c

concentration, mol/m3

D

diffusion coefficient, m2/s

D0AB

ion diffusivity, m2/s

Ea

activation energy, J/mol

F

Faraday, 96.5 C/g-equiv

fobj

Objective function

k

reaction rate constant, m3/(mols)

MW

molecular weight, g/mol

n

amount of substance, mol

N

total number of measurements taken during all the experiments, -

NE

number of experiments performed, -

NMij

number of measurements of the jth variable in the ith experiment, -

NVi

number of variables measured in the ith experiment, -

R

ideal gas constant, J/(Kmol)

R0

particle radius at t=0, m

r, r’

reaction rate, mol/(m3s)

S.R.

stirring rate, rpm

s

shape factor, -

T

temperature, K

t

time, s

V

volume, m3

x

dimensionless film coordinate, -

y

dimensionless concentration in the film phase, -

y’

dimensionless concentration in the bulk phase, -

z+,z-

valence, -

zijk

kth predicted value of variable j in experiment i

zijk

kth measured value of variable j in experiment i

17



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Greek letters α

dimensionless number related to dissipated energy (ε), -

δ

film thickness, m

ε

dissipated energy, W/kg

θ

dimensionless time, -

λ

contribution term, -

λ0+, λ0-

limiting zero concentration ionic conductances, (A/m2) (V/m) (g-equiv/m3)

υ

kinematic viscosity, m2/s

ρ

density, kg/m3

σ2ijk

variance of the kth measurement of variable j in esperiment i, -

τ

characteristic reaction time, s

ω

set of model parameters to be estimated

Subscripts and superscripts

B

bulk phase

i

liquid-phase and general component index

f

inflection point

F

film phase

j

solid-phase component index

L

liquid phase

S

solid

0

initial quantity

*

saturated state

ACKNOWLEDGEMENT This work is a part of activities at the Johan Gadolin Process Chemistry Centre (PCC), a centre of excellence in scientific research financed by Åbo Akademi University. Financial support from Academy of Finland (Tapio Salmi), from Johan Gadolin Scholarship program (Vincenzo Russo) and from Graduate School in Chemical Engineering (GSCE) (Claudio Carletti) is gratefully acknowledged.

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References (1) Carletti, D.; Grénman, H.; De Blasio, C.; Mäkilä, E.; Salonen, J.; Murzin, D., Salmi, T.; Westerlund, T. Revisiting the Dissolution Kinetics of Limestone – Experimental Analysis and Modelling. J. Chem. Technol. Biot. 2016, 91, 1517-1531. (2) Carletti C.; Bjondahl, F.; De Blasio, C.; Ahlbeck, J.; Järvinen, L.; Westerlund, T. Modeling Limestone Reactivity and Sizing the Dissolution Tank in Wet Fue Gas Desulfurization Scrubbers. Environ. Prog. Sustain. 2013, 32, 663–672. (3) De Blasio, C.; Carletti, C.; Westerlund, T.; Järvinen, M. On Modeling the Dissolution of Sedimentary Rocks in Acidic Environments. An Overview of Selected Mathematical Methods with Presentation of a Case Study. J. Math. Chem. 2013, 51, 2120–2143. (4) Shih, S. M; Lin, J. P.; Shiau, G. Y. Dissolution Rates of Limestones of Different Sources. J. Hazard. Mater. 2000, B79, 159–171.

(5) Toprac, A.; Rochelle, G. T. Limestone Dissolution in Stack Gas Desulfurization. Environ. Prog. 1982, 1, 52–58.

(6) Barton, P.; Vatanatham, T. Kinetics of Limestone Neutralization of Acid Waters. Environ. Sci. Technol. 1976, 10, 262–266.

(7) Gibert, O.; de Pablo, J.; Cortina, J. L.; Ayora, C. Evaluation of Municipal Compost/Limestone/Iron Mixtures as Flling Material for Permeable Reactive Barriers for In-situ Acid Mine Drainage Treatment. J. Chem. Technol. Biot. 2010, 85, 1208–1214. (8) Busenberg, E.; Plummer, L. N, A Comparative Study of the Dissolution and Crystal Growth Kinetics of Calcite and Aragonite. In Studies in Diaganesis; F. A. Mumpton Ed.: NewYork, 1986, 139–168. (9) Morse, J. W. Dissolution Kinetics of Calcium Carbonate in Sea Water. III: A New Method for Study of Carbonate Reaction Kinetics. Am. J. Sci. 1974, 274, 97–107. (10) Morse, J. W.; Arvidson, R. S.; The Dissolution Kinetics of Major Sedimentary Carbonate Minerals. Earth-Sci. Rev. 2002, 58, 51–84. (11) Coto, B.; Martos, B.; Peña, L.; Rodríguez, R.; Pastor, G. Effects in the Solubility of CaCO3: Experimental Study and Model Description. Fluid. Phase. Equilib. 2012, 324, 1–7. (12) Plummer, L. N.; Busenberg, E. The Solubilities of Calcite, Aragonite and Vaterite in CO2H2O Solutions Between 0 and 90°C, and an Evaluation of the Aqueous Model for the System CaCO3-CO2-H2O. Geochim. Cosmochim. Ac. 1982, 46, 1011–1040. 19



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(13) Ahlbeck, J.; Engman, T.; Faltén, S., Vihma, M. Measuring the Reactivity of Limestone for Wet-flue Gas Desulfurization. Chem. Eng. Sci. 1995, 50,1081–1089. (14) Plummer, L. N.; Wigley, T. M. L.; Parkhurst D. L. The Kinetics of Calcite Dissolution in CO2 Water Systems at 5 and 60°C and 0.0 to 1.0atm CO2. Am. J. Sci. 1978, 278, 179–216. (15) Sjöberg, E.; Rickard, D, Calcite Dissolution Kinetics: Surface Speciation and the Origin of the Variable pH Dependence. Chem. Geol. 1984, 42, 119–136. (16) Fusi, L.; Monti, A; Primicerio, M. Determining Calcium Carbonate Neutralization Kinetics from Experimental Laboratory Data. J Math. Chem. 2012,50,2492–2511. (17) Sjöberg, E; Rickard D. The Influence of Experimental Design on the Rate of Calcite Dissolution. Geochim. Cosmochim. Acta. 1983, 47,2281–2285. (18) Lund K.; Fogler S.; McCune, C. Acidization I. The Dissolution of Dolomite in Hydrocloric Acid. Chem. Eng. Sci. 1973, 28, 691–700. (19) Lund K.; Fogler, H.; McCune, C.; Ault, J. Azidization - II. The Dissolution of Calcite in Hydroclhoric Acid. Chem. Eng. Sci. 1975, 30, 825–835. (20) Sjöberg E. L.; Rickard, D. T. Temperature Dependence of Calcite Dissolution Kinetics Between 1 and 62°C at pH 2.7 to 8.4 in Aqueous Solution. Geochim. Cosmochim. Acta. 1984, 48,485–493.

(21) Higbie, R. The Rate of Absorption of a Pure Gas Into Still Liquid During Short Periods of Exposure. Trans. Am. Inst. Chem. Eng. 1935, 35, 365. (22) Danckwerts, P. V. Gas-liquid reactions; McGraw Hill: New York, 1970. (23) Levenspiel, O. Chemical Reaction Engineering (3rd ed.); Wiley: New York, 1990. (24) Trambouze, P.; van Landeghem, H.; Wauquier, J.P. Chemical Reactors: design, engineering, operation; Editors Technip: Paris, 1988.

(25) Salmi, T.; Russo, V.; Carletti, C.; Kilpiö, T.; Tesser, R.; Murzin, D.; Westerlund, T.; Grénman, H. Application of Film Theory on the Reactions of Solid Particles with Liquids: Shrinking Particles with Changing Liquid Films. Chem. Eng. Sci. 2017, 160, 161-170. (26) Russo, V.; Kilpiö, T.; Di Serio, M.; Tesser, R.; Santacesaria, E.; Murzin, D. Y.; Salmi., T. Dynamic Non-isothermal Trickle Bed Reactor with Both Internal Diffusion and Heat Conduction: Arabinose Hydrogenation as a Case Study. Chem. Eng. Res. Des. 2015, 102, 171185.

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gPROMS

Model

Builder,

Process

System

Enterprise

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Page,

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List of figure captions Figure 1 – A schematic representation of the E.S.F.M., where j stands for the concentration of the dissolved solid phase compound (CO32-) and i is the component coming from the liquid bulk phase (H+) and δ is the film thickness.

Figure 2 – Dimensionless proton concentrations as a function of time for experiments performed at different temperatures with “small” and “large” Parainen limestone.

Figure 3 – Dimensionless proton concentrations as a function of time for experiments performed at different temperatures with “small” and “large” Wolica limestone.

Figure 4 – A. Proton concentration as a function of time for experiments performed at different stirring rates with the “small” Wolica limestone at 50°C. B. Dependence of the mixing parameter (α) as a function of the stirring rate.

Figure 5 – Parity plot for all performed experiments. Figure 6 – Contribution analysis for experiments performed at different stirring rates with the “small” Wolica limestone at 50°C.

Figure 7 – Time at the inflection point (tf) for the contribution analysis.

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