Mass Transfer between a Fixed Bed of Limestone ... - ACS Publications

The model, based on the film theory, takes into account the effect of the electric potential of diffusion on the dissolution rate. In order to validat...
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Ind. Eng. Chem. Res. 1997, 36, 3859-3865

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Mass Transfer between a Fixed Bed of Limestone Particles and Acid Solutions† Amedeo Lancia,*,‡ Dino Musmarra,§ and Francesco Pepe| Dipartimento di Ingegneria Chimica, Universita` di Napoli “Federico II”, P.le Tecchio 80, 80125 Napoli, Italy, Istituto di Ricerche sulla Combustione, CNR, P.le Tecchio 80, 80125 Napoli, Italy, and Facolta` di Scienze Ambientali, Seconda Universita` di Napoli, Via Arena 22, 81100 Caserta, Italy

Limestone dissolution in acid solutions was experimentally studied by means of a fixed-bed reactor. A diffusive model was proposed to describe experimental results. The model, based on the film theory, takes into account the effect of the electric potential of diffusion on the dissolution rate. In order to validate the model, experiments were carried out to study the effect of both particle size and liquid-solid relative velocity on the dissolution rate. Furthermore, the limestone dissolution rate in aqueous solutions of SO2(aq), HCl, and H2SO4 was measured at different acid concentrations. The results showed that the diffusive model, associated with the correlation proposed by Chu et al. [Chu, J. C.; Kalil, J.; Wetteroth, W. A. Chem. Eng. Prog. 1953, 49 (3), 141] for the evaluation of the liquid-solid mass-transfer coefficient, is capable of describing the dependence of the dissolution rate on both the size of the limestone particles and the liquidsolid relative velocity. In addition, it was found that, in agreement with the physical meaning of the film theory, the thickness of the liquid film is not dependent on the kind of acid present in the solution nor on its concentration but only on the fluid dynamic conditions of the liquidsolid system. Introduction The wet limestone-gypsum flue gas desulfurization (FGD) process is widely used to treat exhaust gas from power plants. In this process the SO2 contained in the flue gas is absorbed by means of a limestone (CaCO3) suspension, and the sludge produced is oxidized to gypsum, which can be used in the construction industry. One of the most important steps in the wet limestonegypsum process is CaCO3 dissolution, which provides the dissolved alkalinity necessary for SO2 absorption (Rochelle and King, 1977). It is important to study in depth CaCO3 dissolution because, due to the relatively low reactivity of CaCO3, the dissolution step may limit the whole process. Moreover, since commercial gypsum has to be CaCO3 free, complete utilization of this reagent is essential, and this has to be reflected in the design of the process equipment. Limestone dissolution in acid solutions has been mainly studied in connection with neutralization of acid waters (Santoro et al., 1973; Barton and Vatanatham, 1976; Volpicelli et al., 1981) and wet limestone FGD (Chan and Rochelle, 1982; Toprac and Rochelle, 1982). The dissolution rate can be controlled both by the rate of transport of reactants and products between the limestone surface and the bulk solution and by the rate of heterogeneous reaction at the solid surface. Experimental results indicate that in systems with low CO2 partial pressure the dissolution rate is controlled by diffusional phenomena for pH lower than 5 (Gage and Rochelle, 1992). Plummer and Wigley (1976) and Plummer et al. (1979) found that, for relatively high pH, surface reactions control the kinetics of limestone dissolution. This result was confirmed by Terjesen and co* Corresponding author. Telephone: [39](81)768-2243. Fax: [39](81)239-1800. E-mail: [email protected]. † A reduced version of this work was presented at Fluid Particle Interaction IV, Davos, Switzerland, May 12-17, 1996. ‡ Universita ` di Napoli “Federico II”. § CNR. | Seconda Universita ` di Napoli. S0888-5885(96)00670-7 CCC: $14.00

workers (Terjesen et al., 1961; Nestaas and Terjesen, 1969), who, working with pH > 5, showed that the process is controlled by surface phenomena and, in particular, that such cations as Cu2+ and Sc3+ may inhibit limestone dissolution. Other species which in conditions of relatively high pH may inhibit the dissolution process are aluminum-fluoride complexes (Mori et al., 1981) and sulfite ion SO32- (Gage and Rochelle, 1992). On the other hand, Lancia et al. (1991, 1994) studied limestone dissolution in sulfurous solutions with pH ) 2-4 and confirmed that in these conditions the dissolution process is controlled by mass transfer in the liquid-solid boundary layer. They proposed a model based on the 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. In the present work limestone dissolution in aqueous solutions of SO2, HCl, and H2SO4 is studied. A diffusive model based on the film theory, derived from the one previously presented by Lancia et al. (1991, 1994) and modified to take into account the specific situations encountered here, is used to describe the experimental results, particularly as far as the dependence of the dissolution rate on the liquid-solid relative velocity and on the particle size is concerned. The model proposed requires the knowledge of the liquid-solid masstransfer coefficient, which can be calculated by means of empirical correlations. A literature survey is presented, and it is shown that, if the mass-transfer coefficient is evaluated by means of the correlation proposed by Chu et al. (1953), the diffusive model is capable of describing the experimental results. Diffusive Model A diffusive model derived from the one proposed by Lancia et al. (1991, 1994) was set up to describe limestone dissolution in aqueous solutions of SO2, HCl, and H2SO4. This model, on the basis of the work of Volpicelli et al. (1981), assumes that the limiting step © 1997 American Chemical Society

3860 Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997

for limestone dissolution is the diffusion of ions and molecules in the mass-transfer boundary layer and that thermodynamic equilibrium exists between the species involved. Limestone dissolution requires the removal of Ca2+ and CO32- ions from the solid surface. The reaction of CO32- with H+, which comes from acid dissociation, allows the increase of the concentration of Ca2+ ions. As a consequence, limestone dissolution takes place via mass transfer of both the species originally present at the solid surface and those produced by chemical reactions in the liquid phase. In particular, for the experiments of limestone dissolution in SO2 solutions the following reactions were taken into account:

NI ) -DI

dcI DI dΦ - F zIcI dx RT dx

In this equation DI, cI, and zI are respectively the diffusivity, the concentration, and the electric charge of the I species, F is the Faraday constant, R is the gas constant, T is the absolute temperature, and dΦ/dx is the gradient of electric potential. Since there is no net applied potential, the following equation, proposed by Vinograd and McBain (1941), can be used for the diffusional potential gradient:

dcI

∑I zIDI dx

dΦ 2-

H2CO3 ) H + HCO3 +

HCO3- ) H+ + CO32-

3

mol/m (1)

K ) 4.57 × 10-8 mol/m3 (2) K ) 1.00 × 10

H2O ) H + OH +

K ) 4.25 × 10

-4

-8

SO2(aq) + H2O ) H+ + HSO3HSO3- ) H+ + SO32-

2

mol /m

6

(3)

K ) 13.9 mol/m3 (4)

K ) 6.50 × 10-5 mol/m3 (5)

where K is the thermodynamic equilibrium constant at 25 °C. Values of K for reactions (1)-(3) were evaluated from data reported by Brewer (1982), while for reactions (4) and (5) those reported by Goldberg and Parker (1985) were used. For experiments carried out in HCl solutions, HCl was considered as completely dissociated, and therefore only reactions (1)-(3) were taken into account. Eventually, for experiments carried out in H2SO4 solutions, only the first dissociation was considered complete, while for the second the following equilibrium reaction was taken into account:

HSO4- ) H+ + SO42-

K ) 12.0 mol/m3

(6)

where K was evaluated from data reported by Brewer (1982). Using the film theory to model liquid phase mass transfer, the transport equations have to describe diffusion and chemical reactions which simultaneously take place in a stagnant film of thickness δ adhering to the liquid-solid interface. The species involved in the dissolution process are Ca2+, H+, OH-, H2CO3, HCO3CO32-, SO2(aq), HSO3-, SO32-, Cl-, HSO4-, and SO42-, and for each of these the material balance can be written as:

dNI ) rI dx

(7)

where NI is the molar flux of the I species, rI is the rate of generation of such species per volume unit, and x is the normal coordinate in a system with its origin on the solid surface. It is worth noting that, in transport processes involving charged species, the molar flux of the I species has to be expressed by means of the following expression (Onsager and Fuoss, 1932), rather than by using the Fick’s law:

(8)

)dx

DI

(9)

∑I zI2RTcI

F

from which it is obtained that dΦ/dx is not zero since the diffusivities are not all equal (see Table 1). In order to resolve the system obtained from the balance equations (7), it is useful to observe that reactions (1)-(6) are proton-exchange reactions and therefore very fast; thus, in considering them as instantaneous, it is possible to apply the “total material balance” approach introduced by Olander (1960). In this way, instead of writing three differential equations for H2CO3, HCO3-, and CO32-, a single transport equation for carbonate can be used, in conjunction with the two algebraic equations describing the equilibria of reactions (1) and (2) (see Appendix). Similarly, it is possible to use just one transport equation for sulfite, in conjunction with the equilibrium equations relative to reactions (4) and (5), and one transport equation for sulfate, in conjunction with the equilibrium equation relative to reaction (6). Finally, as far as H+ and OHare concerned, the two transport equations can be replaced by the algebraic equation relative to the equilibrium of reaction (3) and by the equation stating that there is no charge generation in the film. Overall the model relative to limestone dissolution in sulfurous solutions is described by the following three differential equations for transport of sulfite, calcium and carbonate, strongly coupled due to the presence of the term dΦ/dx:

d (N + NHSO3- + NSO32-) ) 0 dx SO2(aq)

(10.i)

d (N ))0 dx Ca2+

(11)

d (N + NHCO3- + NCO32-) ) 0 dx H2CO3

(12)

associated with the five algebraic equations relative to equilibria of the reactions (1)-(5) and to the electroneutrality equation:

∑I zIcI ) 0

(13)

For limestone dissolution in HCl solutions eq 10.i is substituted by the following equation for transport of chloride ion:

d (N ) ) 0 dx Cl-

(10.ii)

associated with eqs 11 and 12, to the algebraic equations

Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997 3861 Table 1. Diffusivities in Water at 25 °C D × 103 (mm2/s)

species

D × 103 (mm2/s)

Ca2+

0.79a

H+ OHH2CO3 HCO3CO32-

9.30a 5.27a 2.0a 1.20a 0.70a

SO2(aq) HSO3SO32ClHSO4SO42-

1.76b 1.33a 0.77a 0.87 1.33 1.06

species

a From Rochelle et al. (1983). b From Pasiuk-Bronikowska and Rudzinski (1991).

relative to the equilibria of reactions (1)-(3) and to eq 13. Eventually, for dissolution in H2SO4 solutions, the following equation for transport of sulfate is used:

d (N + NSO4-) ) 0 dx HSO4associated with eqs 11 and 12, to the algebraic equations relative to the equilibria of reactions (1)-(3) and (6) and to eq 13. The boundary conditions for the system of eqs 10-12 at x ) 0 (liquid-solid interface), depending on the acid used, are one among the following:

NSO2(aq) + NHSO3- + NSO32- ) 0

(14.i)

NCl- ) 0

(14.ii)

NHSO4- + NSO42- ) 0

(14.iii)

Figure 1. Sketch of the experimental apparatus.

and with the equilibrium equation of the dissolution reaction:

rate, once the liquid bulk composition is known and a value for δ has been assigned. On the contrary, values of δ can be obtained by integrating the model equations once the flux of calcium has been experimentally determined. With the aim of validating the model of eqs 1-19, the experiments of limestone dissolution were carried out in a fixed-bed reactor varying the composition of the liquid phase, the liquid-solid relative velocity, and the particle size and measuring the outlet Ca2+ concentration.

CaCO3 ) Ca2+ + CO32-

Experimental Apparatus and Technique

associated with the following equation:

NCa2+ ) NH2CO3 + NHCO3- + NCO32-

(15)

K ) 3.32 × 10-3 mol2/m6 (16)

Such conditions ensure the following: the absence of transport of sulfite, chloride, and sulfate through the liquid-solid interface (eqs 14.i-iii); the stoichiometric restriction that the rate of calcium ion generation is equal to that of carbonate (eq 15); and the equilibrium of the CaCO3 dissolution reaction (16), for which the equilibrium constant is reported by Gage and Rochelle (1992). On the other hand, the boundary conditions between film and liquid bulk (x ) δ), depending on the acid used, are among the following:

cSO2(aq)|x)δ + cHSO3-|x)δ + cSO32-|x)δ ) cS(IV)|b

(17.i)

cCl-|x)δ ) cCl|b

(17.ii)

cHSO4-|x)δ + cSO42-|x)δ ) cS(VI)|b

(17.iii)

associated with the following equations:

cCa2+|x)δ ) cCa|b

(18)

cH2CO3|x)δ + cHCO3-|x)δ + cCO32-|x)δ ) cC(IV)|b (19) where cI|x)δ is the concentration of the I species at x ) δ, and cS(IV)|b, cCl|b, cS(VI)|b, cCa|b, and cC(IV)|b are the total liquid bulk concentrations of sulfite, chloride, sulfate, calcium, and carbonate, respectively. Equations 1-19 allow the evaluation of the flux of Ca2+ ion or, in other words, the limestone dissolution

The experimental apparatus used to measure the limestone dissolution rate is sketched in Figure 1. It consists of a tubular fixed-bed reactor made of acrylic (i.d. 16 mm, length 60 mm) containing the limestone particles, a peristaltic pump for liquid phase feeding, and the feed and discharge lines. In order to operate with a small reactive surface and with a long enough bed to avoid channeling, the reactor was loaded with 1.6 g of limestone particle mixed with 16 g of glass beads used as inert. The diameter of the glass beads, which constituted about 90% of the packing, was about 1 mm, and the void fraction of the packing  was 0.41. The limestone used was a technical-grade solid (>98% CaCO3); it was dry ground, and after grinding, in order to remove the adhering dust, it was washed in distilled water until the filtrate was clear. The washed limestone was sieved, and the following particle sizes were selected for the experiments: 200-425, 400-600, 600-850, 1120-1400, and 2000-3000 µm. The inlet liquid stream was obtained by acidifying distilled water with one of the following: SO2, HCl, or H2SO4 (reagent-grade concentrated solutions). The diluted solution was heated at 25 ( 1 °C and was placed into the reservoir tank, from where it was fed to the reactor by means of the peristaltic pump. Iodometric titration was employed to measure the total sulfite concentration of the stream fed to the reactor, while neutralization titration with KOH and methyl red as indicator was employed to measure the HCl or H2SO4 concentrations. The calcium concentration in the exit stream was measured, in samples taken from the outlet

3862 Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997

Figure 2. Outlet Ca2+ concentration extrapolated at t ) 0 versus H+ concentration for limestone dissolution in SO2 solutions (a), HCl solutions (b), and H2SO4 solutions (c): (0) experimental results; (s) model calculations.

liquid stream at different times, by EDTA titration using muresside as the indicator. Two different series of limestone dissolution experiments were carried out to explore the capability of the diffusive model described above. In the first series the nature and the concentration of the acid present in the solution was varied using limestone particles with a fixed particle size (namely, 1120-1400 µm) and keeping the liquid flow rate L constant at approximately 1.30 × 10-5 m3/s. In the experiments of this series the concentrations of sulfite, sulfate, and chloride were varied, giving a pH in the range of 1.8-4.3. In the second series of experiments the influence of particle size and of liquid-solid relative velocity was studied. In these experiments SO2 was used as the acidifying agent, while the particle size was varied between 200-425 and 2000-3000 µm and L between 1.75 × 10-5 and 5.70 × 10-5 m3/s, corresponding to superficial velocities ν ranging from 0.087 to 0.284 m/s. Results The experimental runs led to the evaluation of the outlet Ca2+ concentration from the reactor as a function of time. Since the system works under unsteady-state conditions, due to nonuniform limestone consumption along the reactor, the Ca2+ concentrations in the outlet reactor are a function of time. Due to nonuniform limestone consumption along the reactor, the system works under unsteady-state conditions. For this reason the Ca2+ concentrations in the outlet liquid stream from the reactor were extrapolated at t ) 0, following the procedure indicated by Lancia et al. (1991, 1994). In this way it was possible to refer the outlet Ca2+ concentration to a condition in which the total limestone surface could be evaluated by assuming that the limestone particles were equally sized spheres with diameter equal to the average of the particle sizes used. The experimental results for the first series of experiments (L ) 1.30 × 10-5 m3/s, particle size 1120-1400 µm) are reported in Figures 2a-c as concentration of dissolved calcium extrapolated at t ) 0 versus concentration of H+ ion for each of the three acid solutions

Figure 3. Outlet Ca2+ concentration extrapolated at t ) 0 versus sulfite concentration for limestone dissolution in SO2 solutions. (a) Particle size: 200-425 µm. (b) Particle size: 2000-3000 µm. (]) L ) 3.93 × 10-5 m3/s; ([) L ) 1.75 × 10-5 m3/s; (3) L ) 5.70 × 10-5 m3/s; (1) L ) 1.81 × 10-5 m3/s; (s) model calculations.

considered. The figures show that, for a given particle size and liquid-solid relative velocity, the dissolution rate increases almost linearly with H+ concentration. Moreover, a comparison between the figures allows the identification of the buffer effects due to the presence in the solutions of the SO2(aq)-HSO3--SO32- system and of the HSO4--SO42- system. Since HCl is a stronger acid than HSO4- and SO2(aq)‚H2O, which, in turn, are stronger than HSO3-, in the range of pH considered here the fractions of undissociated HSO4-, SO2(aq)‚H2O, and HSO3- give a considerable contribution to limestone dissolution. As a result of this, for given H+ concentration, the limestone dissolution rate in SO2 solution is higher than that in H2SO4 solution, which, in turn, is higher than that in HCl solution. The dependence of the limestone dissolution rate on the liquid-solid relative velocity and on the particle size was studied by carrying out a number of experiments in which both these parameters were varied, and SO2 was used as the acidifying agent. For some experiments carried out with the smallest and the largest particle sizes, the values of cCa2+|out at t ) 0 versus total sulfite concentration are reported in Figures 3a,b. In particular, Figure 3a refers to experimental runs carried out with particles in the range of 200-425 µm and liquid flow rates of 1.75 × 10-5 and 3.93 × 10-5 m3/s, while Figure 3b refers to particles in the range of 2000-3000 µm and liquid flow rates of 1.81 × 10-5 and 5.70 × 10-5 m3/s. Parts a and b of Figure 3 show that, for all the flow rates considered, there is an almost linear dependence of cCa2+|out on the total sulfite concentration, confirming the results of Figures 2a-c. Furthermore, it appears that, as expected, cCa2+|out increases when either the particle size or the liquid-solid relative velocity decreases. In Figure 4 the values of cCa2+|out extrapolated at t ) 0 are reported as a function of L for the five particle sizes used. The figure confirms the results previously reported and, namely, that cCa2+|out decreases when either the particle size or the liquidsolid relative velocity increases. Discussion As indicated by Lancia et al. (1994), the knowledge of outlet Ca2+ concentration allows the integration of

Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997 3863

by Hobson and Thodos (1949), Gaffney and Drew (1950), Chu et al. (1953), and Calderbank and Moo-Young (1961) refer to situations similar to those described in the present work, i.e., mass transfer between a fixed bed of solid particles and a fluid phase, and have therefore been used to compare the model predictions with the experimental results reported in Figures 2-4. According to Hobson and Thodos (1949), who studied mass transfer between water and fixed beds of porous spheres soaked with isobutyl alcohol or methyl ethyl ketone, the following equation expresses the dependence of the Sherwood number (Sh ) dp/δ, where dp is the particle diameter) on the Reynolds number (Re ) Fvdp/ µ, where F and µ respectively are the density and the viscosity of the liquid phase) and the Schmidt number (Sc ) µ/FD): Figure 4. Outlet Ca2+ concentration extrapolated at t ) 0 versus liquid flow rate for limestone dissolution in SO2 solutions. (]) Particle size: 200-425 µm. (O) Particle size: 400-600 µm. (4) Particle size: 600-850 µm. (0) Particle size: 1120-1400 µm. (3) Particle size: 2000-3000 µm. (s) Model calculations.

the model equations. Indeed, even though fluxes and concentrations change along the bed length l, they can be calculated by integrating the following equation:

NCa2+|l dS ) LdcCa2+|b

(20)

log(Sh) ) 0.78630 + 0.0825 log(Re) + 0.0817[log(Re)]2 + 1/3 log(Sc) (24) with 3 < Re < 4000. On the other hand, Gaffney and Drew (1950), studying the dissolution of fixed beds of succinic or salicylic acid pellets in organic solvents, reported a diagram from which the following equation is obtained, valid for 0.7 < Re/(1 - ) < 5000:

(1Re- )

where

Sh ) 2.04 dS )

S dl l*

(21)

In these equations NCa2+|l is the Ca2+ flux at length l along the reactor, S is the limestone surface in contact with the liquid, l* is the reactor length, and S/l* gives the limestone surface per unit length of the reactor, which is assumed constant. Integration of eq 20 with the boundary condition:

cCa2+ ) 0

at l ) 0

(22)

dl ) LcCa2+|out

(23)

gives

S l*

∫0l*NCa

2+

where cCa |out is the outlet concentration from the reactor. The value of NCa2+ along the bed, which depends on the film thickness, can be evaluated by integrating the model equations (1)-(19). According to the physical meaning of δ, the integration has to be carried out giving δ a value and imposing such value as constant along the bed. Once eqs 1-19 are integrated at l ) 0, where the bulk concentrations are known, Ca2+ concentration at l ) l + ∆l can be calculated by means of eq 20, while the remaining bulk concentrations at any l + ∆l can be calculated using the equilibrium equations of reactions (1)-(6), the electroneutrality equation (13), and four balance equations for carbonate, sulfite, sulfate, and chloride. In order to compare model and experimental results, it is necessary to have a value for δ, which depends on both the liquid-solid relative velocity and the particle size. In literature many correlations have been proposed aimed at evaluating the liquid-solid masstransfer coefficient, and thus δ, for transport processes between single particles or beds of particles and surrounding fluids. In particular, the correlations proposed 2+

0.529

Sc1/3

(25)

The correlation proposed by Chu et al. (1953), who studied gas- and liquid-solid mass transfer for solid particles of regular shapes in fixed and turbulent beds, is the following, valid for 20 < Re < 6000:

Sh ) 1.77Re0.56Sc1/3(1 - )0.44

(26)

Finally, according to Calderbank and Moo-Young (1961), who studied a variety of mass-transfer phenomena using the theory of local isotropic turbulence, the following correlation best describes liquid-solid mass transfer in a fixed bed of granular particles when the fluid undergoes turbulent flow:

Sh ) 0.318Re2/3Sc1/3

(1 - )

1/3

(27)

Ca2+

With the aim of identifying the correlation which best describes the experimental results, the measured values of cCa2+|out were compared with the values predicted by the diffusive model using the δ given by each one in eqs 24-27. The results of this comparison are reported in Figures 5a-d, where E, the deviation between model and experimental results in the evaluation of cCa2+|out, is plotted as a function of the experimental value of cCa2+|out itself. In particular, Figure 5a refers to the correlation proposed by Hobson and Thodos (1949) (eq 24), Figure 5b refers to eq 25 (Gaffney and Drew, 1950), Figure 5c refers to eq 26 (Chu et al., 1953), and Figure 5d refers to eq 27 (Calderbank and Moo-Young, 1961). The comparison of the four figures indicates that eq 25 yields overestimated values of cCa2+|out; eq 27, on the contrary, leads to the underestimation of cCa2+|out, while eqs 24 and 26 do not yield systematic deviations between predicted and measured values of cCa2+|out. Overall, it appears that the correlation proposed by Chu et al. (1953) leads to the best agreement between model and experimental results, with an average error of 8.7%, and for this reason eq 26 has been used to predict the

3864 Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997

SO2 solution. A diffusive model, based on the film theory, has been used to describe the dissolution process, and a good agreement has been found between model and experimental results. It has been shown that, if the liquid-solid mass-transfer coefficient is evaluated by means of the correlation proposed by Chu et al. (1953), the model is capable of correctly describing the dependence of the dissolution rate on both the composition of the liquid phase and on the fluid dynamic conditions of the liquid-solid system considered. List of Symbols

Figure 5. Deviation between model and experimental results versus experimental results. Liquid film thickness evaluated by means of different correlations: (a) Hobson and Thodos (1949), eq 24; (b) Gaffney and Drew (1950), eq 25; (c) Chu et al. (1953), eq 26; (d) Calderbank and Moo-Young (1961), eq 27. (]) Particle size: 200-425 µm. (O) Particle size: 400-600 µm. (4) Particle size: 600-850 µm. (0) Particle size: 1120-1400 µm. (3) Particle size: 2000-3000 µm.

values of δ relative to the fluid dynamic conditions used in the experiments. The model results obtained using eq 26 to evaluate δ were reported in Figures 2-4. In such figures the model predictions for cCa2+|out are reported as continuous lines. Again, the comparison between model and experimental results shows that the diffusive model proposed is capable of describing the limestone dissolution process in acid solutions, once eq 26 has been used for the dependence of the liquid film thickness on the particle size and on the liquid-solid relative velocity. In particular, it is interesting to observe that the same value of δ allows the description of the results relative to the three different acids used in the whole range of pH. The fact that the value of δ depends neither on the kind of acid nor on its concentration is a good test of the diffusional model, since it incorporates the assumption that the film thickness depends on the fluid dynamic conditions of the liquid-solid system considered but is independent of the liquid phase composition. However, a larger deviation between model and experimental results is found for the dependence of cCa2+|out on L for the smallest particles considered (upper curve of Figure 4). The reason for this deviation lies in the fact that eq 26 evaluates δ considering for each class of particles the arithmetic average between the largest and the smallest diameters in the interval. Therefore, for the particles in the interval of 200-425 µm the relative deviation between the average diameter and the true diameters is the largest, and this yields larger errors in the estimates of δ and in the predicted values of the limestone dissolution rate. Conclusions Limestone dissolution plays a relevant role in the wet limestone-gypsum FGD process, since it provides the dissolved alkalinity necessary for SO2 absorption. In this work the dissolution of limestone particles in aqueous solutions of SO2, H2SO4, and HCl has been experimentally studied by varying the acid concentration, the particle size, and the liquid-solid relative velocity. The buffer effect of the undissociated species has been put in evidence, showing that for a given pH the dissolution rate in HCl solution is lower than that in H2SO4 solution, which, in turn, is lower than that in

Aγ ) Debye-Hu¨ckel constant, m3/2 mol-1/2 a ) activity, mol/m3 B ) Debye-Hu¨ckel parameter c ) concentration, mol/m3 D ) diffusivity, m2/s dp ) particle diameter, m E ) deviation between model and experimental results, dimensionless F ) Faraday constant, s‚A/mol FI ) ionic strength, mol/m3 K ) thermodynamic equilibrium constant L ) liquid flow rate, m3/s K ) thermodynamic equilibrium constant l ) reactor axial coordinate, m l* ) reactor length, m N ) molar flux, mol/m2‚s r ) reaction rate, mol/m3‚s R ) gas constant, J/mol‚K Re ) Reynolds number, dimensionless S ) section area, m2 Sc ) Schmidt number, dimensionless Sh ) Sherwood number, dimensionless T ) temperature, K v ) superficial velocity, m/s x ) normal coordinate in the film, m z ) axial coordinate along the reactor, m Greek Symbols R ) stoichiometric coefficient, dimensionless γ ) activity coefficient, dimensionless δ ) film thickness, m  ) bed void fraction, dimensionless µ ) viscosity, kg/m‚s F ) density, kg/m3 Φ ) electric potential, J/s‚A Subscripts b ) liquid bulk I ) I species M ) cations out ) outlet from the reactor x ) anions

Appendix The chemical reactions taken into account can be written in the following general form:

∑I RII ) 0

(A1)

where RI is the stoichiometric coefficient of the I species and is assumed positive for the reactants and negative for the products. The equilibrium condition for reaction (A1) is

K)

∏I RI-R

I

where aI is the activity of the I species.

(A2)

Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997 3865 Table 2. Debye-Hu 1 ckel Parameters for Eqs A4 and A5 from Abdulsattar et al. (1977) species

B

species

B

Ca2+ H+ OHHCO3CO32-

-0.035 0.087 -0.012 -0.039 -0.087

HSO3SO32ClHSO4SO42-

-0.013 -0.087 0.156 -0.013 -0.09

The activity aI is related to the molar concentration by

aI ) cIγI

(A3)

where γI is the activity coefficient. Values of the activity coefficients for anions (M) and cations (x) can be calculated using the extended version of the Debye-Hu¨ckel theory proposed by Bromley and co-workers (Abdulsattar et al., 1977). According to those authors, it is

log(γM) ) -

log(γx) ) -

AγzM(FI)1/2 1 + (FI)1/2

Aγzx(FI)1/2 1 + (FI)1/2

∑x cx + ∑x Bxcx

(A4)

cM + ∑BMcM ∑ M M

(A5)

+ BM

+ Bx

In these equations Aγ is the Debye-Hu¨ckel constant, the value of which is given by Colin et al. (1980), and FI is the ionic strength, which can be evaluated by means of the following equation:

FI )

1

zI2cI ∑ 2 I

(A6)

The values of the Debye-Hu¨ckel parameters B are reported in Table 2, taken from Abdulsattar et al. (1977). Literature Cited Abdulsattar, A. H.; Shridar, S.; Bromley, L. A. Thermodynamics of the sulfur dioxide-seawater system. AIChE J. 1977, 23, 62. Barton, T.; Vatanatham, T. Kinetics of limestone neutralization of acid waters. Environ. Sci. Technol. 1976, 10, 262. Brewer, L. Thermodynamic values for desulfurization processes. In Flue Gas Desulfurization; Hudson, J. L., Rochelle, G. T., Eds.; ACS Symposium Series 188; American Chemical Society: Washington, DC, 1982. Calderbank, P. H.; Moo-Young, M. B. The continuous phase heat and mass-transfer properties of dispersions. Chem. Eng. Sci. 1961, 16, 39. Chan, P. K.; Rochelle, G. T. Limestone dissolution. In Flue Gas Desulfurization; Hudson, J. L., Rochelle, G. T., Eds.; ACS Symposium Series 188; American Chemical Society: Washington, DC, 1982. Chu, J. C.; Kalil, J.; Wetteroth, W. A. Mass transfer in a fluidized bed. Chem. Eng. Prog. 1953, 49 (3), 141.

Colin, E.; Clarke, W.; Glew, D. N. Evaluation of Debye-Hu¨ckel limiting slopes for water between 0° and 150 °C. J. Chem. Soc., Faraday Trans. 1980, 76, 1911. Gaffney, B. J.; Drew, T. B. Mass transfer from packing to organic solvents in single phase flow through a column. Ind. Eng. Chem. 1950, 42, 1120. Gage, C. L.; Rochelle, G. T. Limestone dissolution in flue gas scrubbing. Effect of sulfite. J. Air Waste Manage. Assoc. 1992, 42, 926. Goldberg, R. N.; Parker, V. B. Thermodynamic of solution of SO2(g) in water and of sulfur dioxide solutions. J. Res. Natl. Bur. Stand. 1985, 90, 341. Hobson, M.; Thodos, G. Mass transfer in the flow of liquids through granular solids. Chem. Eng. Progr. 1949, 45 (8), 517. Lancia, A.; Musmarra, D.; Pepe, F.; Volpicelli, G. Concentration profiles in the diffusional film in the calcium carbonate dissolution process. Chem. Eng. Sci. 1991, 46, 2507. Lancia, A.; Musmarra, D.; Pepe, F.; Volpicelli, G. Characteristic times for limestone particle dissolution in the production of gypsum from the wet flue gas desulphurisation process. Environ. Sci. Technol. 1991, 28, 1031. Mori, T.; Matsuda, S.; Nakajima, F.; Nishimura, T.; Arikawa, Y. Effect of Al3+ and F- in the desulfurization reaction in the limestone slurry scrubbing process. Ind. Eng. Chem. Process Des. Dev. 1981, 20, 144. Nestaas, I.; Terjesen, S. G. The inhibiting effect of scandium ion upon the dissolution of calcium carbonate. Acta Chem. Scand. 1969, 23, 2519. Olander, D. R. Simultaneous mass transfer and equilibrium chemical reaction. AIChE J. 1960, 6, 233. Onsager, R.; Fuoss R. M. Irreversible processes in electrolytes. Diffusion, conductance and viscous flow in arbitrary mixtures of strong electrolytes. J. Phys. Chem. 1932, 36, 2689. Pasiuk-Bronikowska, W.; Rudzinski, K. J. Absorption of SO2 into aqueous systems. Chem. Eng. Sci. 1991, 46, 2281. Plummer, L. N.; Wigley, T. M. L. The dissolution of calcite in CO2saturated solutions at 25 °C and 1 atm total pressure. Geochim. Cosmochim. Acta 1976, 40, 191. Plummer, L. N.; Parkhurst, D. L.; Wigley, T. M. L. Critical review of the kinetics of calcite dissolution and precipitation. In Chemical Modeling in Aqueous Systems; Jenne, E. A., Ed.; ACS Symposium Series 33; American Chemical Society: Washington, DC, 1979. Rochelle, G. T.; King, C. J. The effect of additives on mass transfer in CaCO3 or CaO slurry scrubbing of SO2 from waste gases. Ind. Eng. Chem. Fundam. 1977, 16, 67. Rochelle, G. T.; Chan, P. K. R.; Toprac, A. T. Limestone dissolution in flue gas desulfurization process. EPA Report EPA-6/7-83043, 1983. Santoro, L.; Ciambelli, P.; Volpicelli, G. Attacco di calcare con soluzioni solforose. Chim. Ind. (Milan) 1973, 55, 577. Terjesen, S. G.; Erga, O.; Thorsen, G.; Ve, A. Phase boundary processes as rate determining steps in reactions between solids and liquids. Chem. Eng. Sci. 1961, 14, 277. Toprac, A. J.; Rochelle, G. T. Limestone dissolution in stack gas desulfurization. Environ. Prog. 1982, 1, 52. Vinograd, J. R.; McBain, J. W. Diffusion of electrolytes and the ions in their mixtures. J. Am. Chem. Soc. 1941, 63, 2008. Volpicelli, G.; Caprio, V.; Santoro, L.; Ciambelli, P. Neutralization of acid waters containing aluminum ions by means of limestone. Chem. Eng. J. 1981, 21, 29.

Received for review October 21, 1996 Revised manuscript received April 21, 1997 Accepted May 1, 1997X IE9606707 X Abstract published in Advance ACS Abstracts, June 15, 1997.