Dissolution of Finely Ground Limestone Particles in Acidic Solutions

Batch dissolution of fine limestone particles in HCl solutions was studied in a ... Employing a step-wise titration method under semi-slow reaction re...
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Ind. Eng. Chem. Res. 2001, 40, 5378-5385

Dissolution of Finely Ground Limestone Particles in Acidic Solutions Francesco Pepe* Dipartimento di Ingegneria Chimica, Mineraria e delle Tecnologie Ambientali (DICMA), Universita` di Bologna, Viale Risorgimento 2, 40136 Bologna, Italy

Batch dissolution of fine limestone particles in HCl solutions was studied in a baffled tank reactor stirred with a Rushton turbine using a pH-drift technique. The effects of both acid strength and stirring rate were analyzed by varying the initial pH in the range of 3.7-5 and the impeller Reynolds number in the range of 104-105. The main experimental result is that, after a very fast step that leads to a quick rise of the solution pH, the dissolution rate is strongly affected by CO2 gas-liquid mass transfer, the intensity of which depends on the rate of stirring. A diffusional model based on film theory was used to analyze the data. The model confirmed that, for the operating conditions considered, a very short time is necessary for the first step of the dissolution process to take place. Furthermore, by taking into account the two limiting cases of the absence of CO2 gas-liquid transport and of infinitely fast transport, it was possible to identify lower and upper bounds for the experimental Ca2+ concentration. Introduction Coupling between mass transfer and chemical reaction during the dissolution of fine solid particles plays an important role in a number of industrial applications. In particular, the dissolution of finely ground limestone particles often is the limiting step in the wet limestonegypsum flue gas desulfurization (FGD) process.1 Limestone dissolution in acid solutions has mainly been studied in connection with the neutralization of acid waters2-4 and wet limestone FGD.5,6 The dissolution rate can be controlled by both the rate of transport of reactants and products between the limestone surface and the bulk solution and the rate of heterogeneous reaction at the solid surface. Experimental results indicate that, in systems with low CO2 partial pressures, the dissolution rate is controlled by diffusional phenomena for pH values lower than 5.6,7 Plummer and coworkers8,9 found that, for relatively high pH, surface reactions control the kinetics of limestone dissolution. This result was confirmed by Terjesen et al.,10 who showed that Cu2+ and Sc3+ can inhibit limestone dissolution. Other species that, under conditions of relatively high pH, might inhibit the dissolution process are aluminum-fluoride complexes11 and sulfite ion.7,12 On the other hand, Lancia et al.13-15 studied limestone dissolution in aqueous solutions with pH ) 2-4 using different acids and confirmed that, under these conditions, the dissolution process is controlled by mass transfer in the liquid-solid boundary layer. They proposed a model based on film theory and were able to evaluate the concentration profiles of the different species in the liquid film, outlining the presence of two reaction planes in the film. Rochelle and co-workers studied the role that CO2 plays in the limestone dissolution rate under conditions typical of FGD applications. Chan and Rochelle5 indicated that a high CO2 partial pressure can enhance the dissolution rate via the CO2 hydrolysis reaction. Toprac * Author to whom correspondence should be addressed. Tel.: +39-051-2093152. Fax: +39-051-581200. E-mail: [email protected].

and Rochelle6 suggested an evaluation of the dependence of the mass transfer coefficient on the stirring rate by means of the equation16

Sh ) 2 + 0.12Ret1/2 Sc1/3

(1)

in which Sh is the Sherwood number, Ret is the turbulent Reynolds number (Ret ) d4/31/3/ν, where d is the particle diameter,  is the average power input per unit mass of fluid, and ν is the kinematic viscosity), and Sc is the Schmidt number. However, they confirmed the relevant role played by CO2 and suggested that a correction factor be introduced into eq 1 that depends on the CO2 partial pressure to take this effect into account. Their approach was later followed by Ukawa et al.17 and Brogren and Karlsson.12 With regard to the evaluation of the solid-liquid mass transfer coefficient, Kulov et al.18 pointed out that a 0.75 exponent for Ret applies only to rather large particles (roughly, d > 100 µm). In papers specifically dealing with the dissolution of small particles, Asai et al.19 and Armenante and Kirwan20 proposed values of 0.58 and 0.62, respectively, as the exponent for Ret. More recently, Vishnevetskaya et al.,21 on the basis of the Kolomogorov theory of isotropic turbulence, proposed that the following equation be used for solid-liquid mass transfer involving very small particles

Sh ) 2 + 0.55Ret1/2 Sc1/3

(2)

Despite the relevant work on limestone dissolution, many important issues still remain to be addressed. Among them, the possible role played by particle morphology on dissolution under conditions of high limestone utilization was pointed out by Brogren and Karlsson,12 who suggested the existence of an “aging” effect for the dissolving particles. Another point to be addressed is the inhibiting effect of sulfite. Such an effect, first observed by Chan and Rochelle,5 was studied in detail by Gage and Rochelle.7 On the other hand, in a recent paper, Kiil et al.22 successfully modeled a wet FGD pilot plant without considering the inhibiting effect

10.1021/ie001119j CCC: $20.00 © 2001 American Chemical Society Published on Web 10/12/2001

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Figure 1. Sketch of the experimental apparatus and the Rushton turbines used.

of sulfite. Similarly, Lancia et al.15 did not detect this effect, probably because of the low pH at which their experiments were carried out. The main objective of the present work is to study the dissolution kinetics of small limestone particles (d e 100 µm) in aqueous HCl solutions, focusing attention on the role of CO2 gasliquid mass transfer. A baffled laboratory-scale stirred tank without pH control was used, and experimental data were obtained for variations in both the HCl concentration and the intensity of stirring. On the basis of the experimental results, the main reactions involved in the dissolution process are discussed using a diffusive model based on film theory, and the suggestion is made that a relevant role is played by CO2 gas-liquid mass transfer. Experimental Apparatus and Technique Limestone dissolution in HCl solutions was studied by carrying out batch experiments in a stirred tank using a pH-drift technique. The experimental apparatus, a sketch of which is presented in Figure 1, was an open cylindrical tank made of Plexiglas with a flat bottom. The tank diameter was T ) 190 mm, and it was filled with 5 L of suspension, giving a liquid height H that was roughly equal to T. Four equally spaced vertical baffles, each having a width of T/10, were glued to the tank wall. The suspension was stirred by means of a Rushton turbine driven by a 70-W variable speed motor. Two different “standard” Rushton turbines having diameters D ) 76 and 115 mm (see Figure 1) were used alternately, and both were placed at a clearance from the bottom of the tank of C ) T/3. In the experiments carried out with the smaller turbine, the stirrer speed n was varied in the range of 3.5-12.8 s-1, whereas in the experiments carried out with the larger turbine, n was varied in the range of 1.5-8.4 s-1. For both turbines, the lower speed was fixed with the

condition that complete suspension of the solid particles was reached, whereas the upper speed was dictated by the power available (see the Appendix). Before each experiment, 5 L of HCl solution with a concentration varying between 0.01 and 0.2 mol/m3 (initial pH ) 3.7-5) was placed in the reactor, and stirring was started with the required value of n. After a few minutes, a small amount of solution was sampled to measure the initial Ca2+ and Cl- concentrations, and the temperature was read by means of a thermometer. In this regard, it is important to observe that, despite the great care with which the tank was washed after each experiment, in most cases, a small Ca2+ concentration (in the range of 0.01-0.03 mol/m3) was detected when the solution just placed into the reactor was analyzed. However, because the initial concentrations were much lower than the concentrations observed in the course of the experiments (see Figures 3-5 below) in all cases, their values are neglected in the discussion that follows. After the initial samples had been analyzed, the experiment began with the addition to the solution of a mass Mo of limestone (Mo ) 25 or 50 g, corresponding to mass fractions of X ) 0.5 or 1%, respectively). The dissolution process was followed by the withdrawal, at fixed time intervals, of small liquid samples (ca. 20 mL) from the reactor by means of syringes fitted with cellulose acetate filters in order to avoid sampling of CaCO3 particles (Bernard et al.23). Minisart filters (Sartorius of Go¨ttingen, Germany) were used, with a diameter of 27 mm and a pore diameter of 0.45 µm. Their use required approximately 20-40 s for each sample. The Ca2+ concentration was measured by EDTA titration using muresside as the indicator. Similarly, neutralization titration with NaOH and methyl red as the indicator was employed to measure the HCl concentration. Temperature readings were always in the range of 20 ( 1 °C. The working solutions were prepared using distilled water; furthermore, all chemicals used were reagent-grade, and premeasured vials were used to prepare the HCl, NaOH, and EDTA solutions. The pulverized limestone used in the experiments was provided by ENEL, an Italian power producer. It had a purity greater than 98%, and its particle size distribution (PSD) was evaluated by means of a laser particle sizer (Analyzette 22 by Fritsch of Idar-Oberstein, Germany). The granulometric analysis indicated that the PSD for the limestone used is rather wide, with particle diameters ranging from 1 to 100 µm, and a Sauter (surface-volume) mean diameter of d23 ) 40.3 µm. To have an analytical description of the PSD obtained by the particle sizer, the log-γ density function was used.6,22 According to this model, if λ, the argument of the γ function, is assumed to be an integer, it is possible to obtain the following analytical expression for the cumulative mass distribution F

F(Y) )

[

e-YYλ-1 λ - 1 (λ - 1)(λ - 2) + 1+ + ... + Y (λ - 1)! Y2 (λ - 1)(λ - 2)‚‚‚ × 2 × 1 Yλ-1

]

(3)

Here, Y is given by the expression

Y)

( )

d100 3 ln β d

(4)

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rived only from CaCO3 dissolution was imposed by writing that the total carbonate concentration was equal to the Ca2+ concentration

cC(IV) ) cCO2(aq) + cHCO3- + cCO32- ) cCa2+

(9.i)

Alternatively, the gas-liquid equilibrium for CO2 was described in terms of Henry’s law

CO2(g) ) CO2(aq) HCO2 ) 2.54 × 103 m3 Pa/mol (9.ii) Figure 2. Comparison between experimental and analytical cumulative mass distributions.

where d100 is the maximum particle size (d100 ) 100 µm) and β is a parameter whose value depends on the spread of the distribution.22 A regression analysis indicated that the best fit between eqs 3 and 4 and the experimental results is reached when λ ) 3 and β ) 0.72. In Figure 2, both the experimental and the analytical cumulative mass distributions are reported, indicating that the limestone used is characterized by a PSD that, despite being wider than the one used by Gage and Rochelle,7 is relatively narrow. Modeling of the Dissolution Process Bulk Composition. The solution in the tank contains Cl- ions deriving from HCl dissociation, together with calcium and carbonate ions deriving from CaCO3 dissolution. Furthermore, the carbonate concentration (and, in turn, the limestone dissolution rate) is affected by CO2 absorption/desorption, which can take place through the gas-liquid interface. However, because the attention here is focused on solid-liquid mass transfer rather than on gas-liquid mass transfer, only the upper and lower limits of the carbonate concentration in the liquid bulk were estimated. Such limits were alternately found by assuming either that no mass transfer took place, and therefore that carbonate was present in the solution only as a result of CaCO3 dissolution, or that gas-liquid equilibrium existed for CO2. Therefore, the following species were assumed to be present in the liquid bulk: Ca2+, H+, OH-, CO2(aq), HCO3-, CO32- and Cl-, where CO2(aq) is to be considered as a “pseudospecies” lumping CO2(aq) and H2CO3 together. Among the species considered, the following reversible reactions were taken into account

CO2(aq) + H2O ) H+ + HCO3K ) 4.16 × 10-4 mol/m3

(5)

HCO3- ) H+ + CO32- K ) 4.13 × 10-8 mol/m3

(6)

+

-

H2O ) H + OH

-9

K ) 6.83 × 10

2

mol /m

6

1 d 2 (r NI) ) -σI r2 dr

(10)

(7)

whereas HCl was considered to be completely dissociated. In reactions 5-7, K is the thermodynamic equilibrium constant at 20 °C.23 Furthermore, because ionic species are present, the electroneutrality equation was imposed by writing

∑IzIcI ) 0

where HCO2 is the Henry constant for CO2.23 Overall, the equilibrium equations associated with reactions 4-7, together with eq 8 and either eq 9.i or 9.ii, allow for the composition of the liquid phase to be evaluated once cCa2+ and cCl- are known. In modeling the experimental results, the Cl- concentration was assumed to be constant, whereas eq 24 (see below) was used to describe the variation with time of the Ca2+ concentration. Diffusional Model for Limestone Dissolution. Limestone dissolution requires the removal of Ca2+ and CO32- ions, which originated from the dissolution reaction, from the solid surface. The reaction of CO32- with H+, produced by the dissociation of HCl, allows the concentration of Ca2+ ions to increase and, therefore, enhances the dissolution rate. As a consequence, limestone dissolution takes place via mass transfer of the species originally present at the solid surface and of those produced by chemical reactions in the liquid phase. A diffusive model derived from the one proposed by Lancia et al.15 was set up to describe this process. Such model assumes that the limiting step for limestone dissolution is the diffusion of ions and molecules in the mass transfer boundary layer and that thermodynamic equilibrium exists among the species involved. Using film theory to model liquid phase mass transfer, the transport equations have to describe the diffusion and chemical reactions that simultaneously take place in a stagnant film adhering to the liquid-solid interface, and the material balances for the seven species indicated above. The thickness of the liquid film, δ, is a function of both the physical characteristics of the solid-liquid system (particle size, liquid viscosity and density, etc.) and the stirring intensity. Because, for very small particles, δ is of the same order of magnitude as the particle size, the mass transfer equations have to take into account the fact that the film curvature is not negligible (Brogren and Karlsson12). Therefore, assuming particles of spherical shape, the material balance in the film for any species I can be written as

(8)

where cI is the concentration of species I and zI is its electric charge. The condition that carbonate was de-

where NI is the molar flux of species I, σI is its rate of consumption (number of moles per unit time and volume), and r is the radial coordinate in the system having its origin at the center of the particle. It is important to observe that, in transport processes involving charged species, the molar flux of species I has to be expressed, not by Fick’s law, but by the expression26

NI ) -DI

dcI DI dΦ -F h zc dr RT h I I dr

(11)

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In this equation, DI is the diffusivity of species I,27 F h is the Faraday constant, R is the gas constant, T h is the temperature, and dΦ/dr is the gradient of the electric potential. It has to be observed that, to minimize the differences between the results of film theory and those of the penetration or surface renewal theories, the ratio between the square roots of the diffusivities should be substituted for the ratio between the diffusivities.28,29 This substitution leads to the following equation for each species

D′I ) DCa2+

xDI

xDCa

(12)

2+

where DI is the “true” diffusivity of species I and D′I is its “normalized” value and therefore the value used to evaluate the molar fluxes. As seen in eq 12, Ca2+ was arbitrarily chosen as the reference species, and the normalization was carried out with respect to the Ca2+ diffusivity. Because no net charge transport takes place between the liquid bulk and the solid-liquid interface, the following equation should be used for the gradient of electric potential

dΦ )dr F h RT h

∑

(13)

2 IzI DIcI

from which one obtains that dΦ/dr is not zero because the diffusivities are not equal among the different species. To solve the system obtained from the balance equations (eq 10), reactions 5-7 were assumed to be at equilibrium throughout the film. This assumption allowed just one transport equation to be used for the total carbonate, associated with the two algebraic equations related to the equilibria of reactions 5 and 6, instead of three differential equations for CO2(aq), HCO3-, and CO32- (see Brogren and Karlsson12 and Lancia et al.13-15). However, it is important to note that the validity of the assumption was questioned by Chan and Rochelle5 and Toprac and Rochelle.6 For H+ and OH-, the two transport equations can be replaced by the algebraic equation related to the equilibrium of reaction 7 and by the equation stating that no charge is generated in the film

1 d 2 (r r2 dr

∑IzINI) ) 0

(15)

d 2 [r (NCO2(aq) + NHCO3- + NCO32-)] ) 0 dr

(16)

d 2 (r NCl-) ) 0 dr

(17)

d 2 (r dr

∑IzINI) ) 0

(18)

associated with the three algebraic equations related to the equilibria of reactions 5-7. The boundary conditions for the system of eqs 15-18 at r ) d/2 (liquid-solid interface) are

NCa2+ ) NCO2(aq) + NHCO3- + NCO32-

(19.i)

NCl-) 0

(19.ii)

∑IzINI ) 0

(19.iii)

associated with the equilibrium equation for the dissolution reaction

CaCO3 ) Ca2+ + CO32- K ) 3.52 × 10-3 mol2/m6 (19.iv)

dcI

∑IzIDI dr

d 2 (r NCa2+) ) 0 dr

(14)

Furthermore, following the suggestion of Glasscock and Rochelle,30 Fick’s law (rather than eq 11) was used to express the molar fluxes, and at the same time the electroneutrality condition (eq 8) was neglected. Indeed, as pointed out by Brogren and Karlsson,31 who studied this problem in the context of limestone dissolution, the results are only slightly affected if this technique is used; yet, it is possible to decouple the transport eq 11 and, therefore, to integrate the equations more easily. On the whole, the model for limestone dissolution is described by the following four differential equations for the transports of calcium, carbonate, chloride, and electric charge

These conditions impose the stoichiometric restriction that the rate of calcium ion generation is equal to that of carbonate (eq 19.i); the absence of transport of chloride and net electric charge through the liquid-solid interface (eqs 19.ii and iii); and the equilibrium of the CaCO3 dissolution reaction (eq 19.iv), for which the equilibrium constant at 20 °C was taken from Gage and Rochelle.7 On the other hand, the boundary conditions between the film and liquid bulk (r ) d/2 + δ) are

cCa2+|r)d/2+δ ) cCa2+|b

(20.i)

cCO2(aq)|r)d/2+δ + cHCO3-|r)d/2+δ + cCO32-|r)d/2+δ ) cC(IV)|b cCl-|r)d/2+δ ) cCl-|b

(20.ii) (20.iii)

∑IzIcI|r)d/2+δ ) ∑IzIcI|b

(20.iv)

where the subscript b refers to the liquid bulk. Integration of eq 15 leads to the following expression for the dissolution rate

Ndiss ) NCa2+|r)d/2 ) DCa2+ 2 1+ (c | - cCa2+|r)d/2+δ) (21) δ Sh Ca2+ r)d/2

(

)

where Sh ) d/δ (Brogren and Karlsson12) and the interfacial Ca2+ concentration can be calculated, together with all other interfacial concentrations, by integrating the system of eqs 15-18. Batch Dissolution in a Stirred Vessel. To apply the diffusive model described above to the dissolution process in the stirred reactor, the limestone particles fed to the reactor at the beginning of each experiment (t ) 0) were divided in k fractions, the ith of which was composed of all particles having initial diameters be-

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Table 1. Operating Conditions Used for the Experiments Reported in Figures 3-5 number

Mo (kg)

n (s-1)

D (mm)

Rei

 (W/kg)

1.83 × 104 6.74 × 104 1.80 × 104 6.07 × 104 1.01 × 105

0.107 5.36 0.34 2.59 11.9 0.107 0.202 0.327 0.508 0.768 0.973 1.5 2.45 5.36

1 2 3 4 5

0.025 0.025 0.025 0.025 0.025

cHCl ) 0.01 mol/m3 76 3.48 76 12.8 115 1.50 115 5.05 115 8.40

6 7 8 9 10 11 12 13 14

0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050

cHCl ) 0.1 mol/m3 76 3.48 76 4.30 76 5.05 76 5.85 76 6.72 76 7.27 76 8.40 76 9.88 76 12.8

1.83 × 104 2.26 × 104 2.65 × 104 3.07 × 104 3.53 × 104 3.82 × 104 4.41 × 104 5.19 × 104 6.74 × 104

15 16 17 18

0.025 0.025 0.025 0.025

cHCl ) 0.2 mol/m3 115 1.50 115 3.48 115 5.05 115 8.40

1.80 × 104 4.19 × 104 6.07 × 104 1.01 × 105

Figure 3. Ca2+ concentration as a function of time for experiments 1-5 (see Table 1). b, exp 1; 9, exp 2; [, exp 3; 2, exp 4; 1, exp 5; - - -, model results.

0.34 0.85 2.59 11.9

tween do,i-1 and do,i (obviously, do,o ) 0, and do,k ) dmax ) 100 µm). Making the hypothesis that all particles belonging to the ith fraction had an initial diameter of do,i, the number of such particles was calculated as

Zi )

6Mo[F(do,i) - F(do,i-1)] Fsπ(do,i)3

(22)

Furthermore, the following differential equation was used to describe the dissolution of these particles

d(di) 2M h )N dt Fs diss

(23)

where M h is the limestone molecular weight and, according to eq 21, Ndiss depends on di, the Ca2+ concentrations in the liquid bulk and at the solid-liquid interface at time t, and the liquid film thickness. δ was evaluated from Sh (δ ) d/Sh, see above), and the correlation proposed by Vishnevetskaya et al.21 (eq 2) was chosen to describe the dependence of Sh on d and the intensity of stirring. Eventually, the variation with time of the Ca2+ concentration in the liquid bulk was described by means of the equation

(24)

Figure 4. Ca2+ concentration as a function of time for experiments 6-14 (see Table 1). (a) b, exp 6; 9, exp 8; [, exp 10; 2, exp 12; 1, exp 14; - - -, model results. (b) b, exp 7; 9, exp 9; [, exp 11; 2, exp 13; - - -, model results.

Experiments were carried out with variations in both the HCl concentration in the liquid bulk and the stirring intensity. In Table 1, the operating conditions used in each experiment (namely, the amount of limestone used, acid concentration, stirrer size, and stirrer speed) are reported, together with the impeller Reynolds number, Rei, and the average power input per unit mass, . The corresponding experimental results are reported in Figures 3-5. The figures show that dissolution is quite fast. Indeed, the first sample, taken in most cases less than 1 min after limestone had been added to the acid solution, already shows a relatively high Ca2+ concentration (usually, 0.2-0.3 mol/m3). Furthermore, this concentra-

tion is practically independent of the intensity of stirring. However, even after the Ca2+ concentration reaches this relatively high value, it continues to grow, albeit much more slowly, reaching an apparent plateau after a few hours (cCa2+ up to 0.8 mol/m3). Quite interestingly, this second part of the dissolution process shows an appreciable dependence on the stirring rate. In particular, it appears that, for a given kind of impeller, the higher the stirring rate, the higher the dissolution rate (Figures 4 and 5). On the other hand, the comparison between the results obtained with different impellers is less straightforward: Figure 3 shows that the results of experiments 1 and 3 almost coincide, despite the different values of the stirring speed and of the total mechanical power provided to the suspension. Similarly, the dissolution rate of experiment 4 is slightly higher than that of experiment 2, despite the latter being

dcCa2+|b dt

)

π

k

∑[Zi(di)2Ndiss]

Vi)1

Results and Discussion

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Figure 5. Ca2+ concentration as a function of time for experiments 15-18 (see Table 1). b, exp 15; 9, exp 16; [, exp 17; 2, exp 18; - - -, model results.

characterized by an almost double power input to the suspension. The most likely interpretation of the results shown is that, in the first phase, limestone dissolves very quickly via the reaction

CaCO3 + H+ ) Ca2+ + HCO3K ) 8.53 × 104 mol/m3 (25) Afterward, once the H+ ions have been consumed and the solution pH has increased, the prevailing mechanism for CaCO3 dissolution becomes its reaction with CO2(aq)

CaCO3 + CO2(aq) + H2O ) Ca2+ + 2HCO3K ) 35.5 mol2/m6 (26) The fact that, in the second phase, the dissolution rate is quite sensitive to stirring, suggests that a relevant role is played by CO2 transport through the gas-liquid interface. Indeed, an increase in the stirring intensity leads to greater turbulence at the gas-liquid interface and to greater air entrainment in the proximity of the impeller, and both of these factors enhance the gasliquid mass transfer. It is useful to observe that a beneficial effect of stirring on limestone dissolution in the presence of CO2 has already been observed by Toprac and Rochelle,6 who attributed the effect to the finite rate of CO2(aq) hydrolysis (reaction 5) in the solidliquid film. To have a clearer picture of the overall results, a modeling effort was attempted using the approach outlined in the previous section. In Figure 6a,b, the model results obtained using eq 9.ii are presented as curves of cCa2+ and pH in the liquid bulk vs time. The figures refer to cHCl ) 0.01 and 0.2 mol/m3 and  ) 0.1 and 5.0 W/kg (corresponding to n ) 3.4 and 12.6 s-1, respectively, for the smaller stirrer). The results presented confirm that, in the absence of pH control, dissolution is very rapid and goes to completion within a few tenths of a second. Because of the high rate of the solid-liquid mass transfer, no comparison could be attempted between the model and the experimental results. It is interesting, however, to observe that, at least under the experimental conditions considered, the role played by the intensity of stirring is relatively weak. Indeed, Figure 6a,b shows that the time required to complete the first phase of the dissolution process is roughly halved when the power input to the liquid is

Figure 6. (a) Modeling results for Ca2+ concentration and pH as a function of time with cHCl ) 0.01 mol/m3. s,  ) 0.1 W/kg; - - -,  ) 5 W/kg. (b) Modeling results for Ca2+ concentration and pH as a function of time with cHCl ) 0.2 mol/m3. s,  ) 0.1 W/kg; - -,  ) 5 W/kg.

increased by a factor of 50. To obtain an approximate estimate of the effect of CO2 absorption, the two limits described by eqs 9.i and 9.ii, namely, the absence of gas-liquid transport and infinitely fast transport, were alternately incorporated in the model. The model results are reported, together with the experimental values, in Figures 3-5. It is important to observe that, because of the absence of a specific model for CO2 gas-liquid mass transfer, the model only leads to semiquantitavive results. However, these results indicate that CO2 concentration, and therefore CO2 gas-liquid transport, plays a relevant role in the dissolution process, with the equilibrium Ca2+ concentrations varying by almost 1 order of magnitude between the two cases. Furthermore, the comparison between the model and the experimental results indicates that the two hypotheses regarding CO2 gas-liquid transport set upper and lower limits for the dissolution rate. Indeed, the experimental Ca2+ concentration always lies between these two limits, and the higher the intensity of stirring the closer the Ca2+ concentration to the upper limit. These results are in good agreement with the interpretation proposed above, according to which, under conditions of relatively high pH, the dissolution rate is limited by the rate CO2 absorption from the gas phase. Conclusions The dissolution of fine limestone particles in HCl solutions was studied using a baffled tank reactor in the absence of pH control. Two different Rushton turbines were used, driven by a variable-speed motor. Experiments were carried out with variations in both the HCl concentration in the solution (0.01-0.2 mol/ m3) and the stirring rate (Rei ) 104-105). Despite the limits of the experimental technique adopted, the main experimental result obtained was the

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indication that the dissolution process proceeds through two distinct steps. In the first step, which is very fast, the solution pH quickly rises. Afterward, dissolution continues more slowly, limited by both CO2 absorption from the atmosphere and its hydrolysis to HCO3- in the liquid bulk. The rate of the second phase of the dissolution process strongly depends on the stirring rate, probably because of the effect of this variable on the gas-liquid mass transfer coefficient for CO2. A modeling effort was undertaken using film theory and taking into account the fact that, in the case of small particles, the film thickness and particle diameter are comparable and, therefore, the film curvature cannot be neglected. The model showed that, for the operating conditions considered, a few tenths of a second are necessary for dissolution to take place and for the pH to rise to a value greater than 7. Eventually, by incorporating into the model the two limiting cases of absence of CO2 gas-liquid transport and of infinitely fast transport, it was possible to identify lower and upper bounds for the equilibrium Ca2+ concentration that were in agreement with the experimental results. Acknowledgment The author thanks Mr. Leopoldo Vigliotta and Miss Michela Girotti for the help given in carrying out the experimental work. Furthermore, useful discussions with Prof. Franco Magelli and Dr. Giusi Montante of the DICMA, University of Bologna, are gratefully acknowledged. Appendix. Particle Suspension and Mechanical Power Provided to The Suspension The stirring intensity influences both the particle suspension and the energy provided to the suspension, which, in turn, affects the solid-liquid mass transfer coefficient. For the first problem, the correlation proposed by Ibrahim and Nienow32 was used to determine the minimum stirrer speed at which all of the limestone particles would be suspended, njs. According to this correlation, it is

(

)

Fs - Fl Fl

njs ) S g

0.45

d0.2X0.13ν0.1D-0.85

(A1)

where Fs is the density of the limestone used (Fs ) 2430 kg/m3), g is the coefficient of gravitational acceleration, and S is a dimensionless constant that is a function of the stirrer type and of the D/T and C/T ratios. Using the values of S ) 6.05 and 3.19 for the smaller and larger impellers, respectively,32 and taking for a conservative estimate d ) 100 µm and X ) 0.01, values of 3.75 and 1.41 s-1 were obtained for njs. On the other hand, the mechanical power provided to the suspension, P, was evaluated using the data of Ibrahim and Nienow,33 who reported the power number Po (Po ) P/Fln3D5) as a function of the impeller Reynolds number Rei (Rei ) nD2/ν). For the conditions considered here, Rei varied between 1.74 × 104 and 9.75 × 104, and according to Ibrahim and Nienow,33 for the experimental apparatus used, it is Po ) 5; therefore, P varied between 0.31 and 54 W, and  varied between 6.2 × 10-2 and 10.9 W/kg. List of Symbols C ) stirrer clearance from the bottom, m cI ) concentration of species I, mol/m3

D ) stirrer diameter, m DI ) diffusivity of species I, m2/s d ) particle diameter, m F ) cumulative mass fraction H ) liquid height in the tank, m K ) thermodynamic equilibrium constant, mol/m3 or mol2/ m6 M ) mass of limestone, kg NI ) molar flux of species I, mol/(m2 s) n ) stirrer speed, s-1 Po ) power number r ) radial coordinate, m Re ) Reynolds number Sc ) Schmidt number Sh ) Sherwood number T ) tank diameter, m t ) time, s X ) mass fraction of solid Zi ) number of particles having given initial size zI ) electric charge of species I Greek Symbols δ ) film thickness, m  ) power per unit mass, W/kg ν ) kinematic viscosity, m2/s F ) density, kg/m3 Φ ) electric potential, J/(s A)

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