Comparison of Mechanistic Models for the Sulfation Reaction in a

All three models give almost identical results for the highest particle sizes using the product ... These models revealed marked effects of the type o...
0 downloads 0 Views 197KB Size
Download PDF
2190

Ind. Eng. Chem. Res. 1996, 35, 2190-2197

Comparison of Mechanistic Models for the Sulfation Reaction in a Broad Range of Particle Sizes of Sorbents Juan Ada´ nez,* Pilar Gaya´ n, and Francisco Garcı´a-Labiano Instituto de Carboquı´mica, P.O. Box 589, 50080 Zaragoza, Spain

Three different structural models, a random pore model, a changing grain size model, and a distributed pore size model (DPSM), have been used to predict the sulfation conversion (Xs) vs time (t) curves of seven limestones with different pore structure in seven narrow particle size cuts between 100 and 2000 µm. None of the models were found able to predict the experimental results Xs-t with all the particle sizes considered. The results were improved by changing the relation of the effective diffusivity in the pores with conversion and, specifically, the change of tortuosity with porosity. Thus, the results of this study showed that the values of the exponent b in the equation to calculate the tortuosity had to be determined for each particle size, except for the DPSM model with limestones with widespread pore size distribution. All three models give almost identical results for the highest particle sizes using the product layer diffusion coefficient Ds and the exponent b of the tortuosity equation as characteristic of each limestone. Introduction In recent years there has been considerable interest in the use of limestones as the fluidizing medium in fluid bed coal combustors in order to reduce sulfur dioxide emissions. In atmospheric combustors, limestone decomposes instantaneously to yield its calcined form and then reacts with SO2 leading to the in situ desulfuration.

CaO + SO2 + 1/2O2 f SO4Ca

(1)

The above reaction is a noncatalytic gas-solid reaction, involving a porous structure which evolves temporally. The calcination reaction leads to formation of well-developed pore structure, with high values of porosity and internal surface area. The sulfation of the lime is a very complex reaction characterized by pore closure and formation of inaccessible pore space, both being a consequence of the fact that CaSO4 (the solid product) occupies more space than CaO. Pore closure is unable to achieve stoichiometric utilization of CaO. Most theoretical research in the area of gas-solid reaction with solid product has been directed toward development of mathematical models describing pore structure evolution. These are usually classified as pore models and grain models. The grain models assume that the solid structure consists of a matrix of very small grains, usually spherical in shape. The first model was done by Szekely and Evans (1970), who applied the grain model theory by assuming that the shrinking grains of solid reactant are surrounded by a dense layer of solid product of uniform thickness. Most early grain models assumed that the overall grain size remained constant in the course of the reaction, but Georgakis et al. (1979) showed that density differences between the solid product and the reactant may readily be incorporated in the grain model by means of an equation relating the overall grain size to the local conversion. There are some models that consider the distribution of grain size, such as Bartlett et al. (1973) and Szekely and Propster (1975). These models revealed marked effects of the type of the grain size distributions on the conversion vs time behavior. S0888-5885(95)00389-7 CCC: $12.00

The main lack of these models is the hypothesis of nonoverlapping grains; thus Lindner and Simonsson (1981) allowed the grains in the structure to overlap. The notion of overlapping was further pursued by Sotirchos and Yu (1988), who derived analytical expressions for the structural properties of porous media whose solid phase is represented by a population of randomly overlapping grains of uniform or distributed size. Recently, Efthimiadis and Sotirchos (1993) used an overlapping distribution grain model showing the important effect of grain size in solid behavior. In the second category, the pore models go back to the original model developed by Petersen (1957) of an idealized network of randomly intersecting cylindrical pores. A more refined random pore model for a distributed pore size system was presented by Bhatia and Perlmutter (1981a,b). Christman and Edgar (1983) presented a distributed pore size model considering infinitely long pores relative to the dimensions of the porous medium, and Simons and Garman (1986) assumed a treelike structure with each pore connected to at least one larger pore located closer to the external surface of the porous solid. The effects of pore overlapping were considered by Sotirchos and Yu (1985) and further using the percolation theory of Yu and Sotirchos (1987) and Reyes and Jensen (1987), whose solid structures can be represented by a network of cylindrical pores. To use gas-solid reaction models in modeling of sulfur retention in fluidized bed boilers, it is necessary to know the sulfation kinetics in the broad range of particle sizes employed. Previous works on sulfation reaction have been limited to representing conversion vs time data in a narrow range of particle sizes and a few limestones. In this work, different models have been tested over a wide type of limestones and using a broad range of particle sizes (100-2000 µm). The models have been compared with respect to their ability to represent the data for the sulfation reaction over the extended times and over a broad range of particle sizes by using only structural parameters of the limestone. The models selected for this work were the changing grain size model (CGSM) by Georgakis et al. (1979) and the random pore model (RPM) by Bhatia and Perlmutter (1981a,b) as corresponding to the two big groups of © 1996 American Chemical Society

Ind. Eng. Chem. Res., Vol. 35, No. 7, 1996 2191

mechanistical models. These models are of easy application and they need only a few physical parameters, which can be determined by experimental measures. Among grain models, the changing grain size model by Georgakis et al. (1979) introduced a modification in the radius of the grain core during the process not considered by other models. The distributed size grain models need measurements of the particle structure to calculate this distribution, which were difficult to determine. The random pore model by Bhatia and Perlmutter (1981a,b) is one of the most well-known pore models and includes a structural parameter of the initial pore distribution. Finally, another pore model was selected for comparison because of its ability to consider the progressive pore blockage. The distributed pore size model (DPSM) by Kocaefe (1986), which is based on that proposed by Christman and Edgar (1983), was chosen because it uses a distribution of the pore radii, and so, it takes into account the change in the radius of the pores with reaction time. Moreover, it is less mathematically complex than other similar types of models such as those proposed by Yu and Sotirchos (1987) or Reyes and Jensen (1987). Thus, a comparison of the three models (CGSM, RPM, DPSM) to predict conversion vs time sulfation curves in a broad range of particle sizes has been made to find out which model better fits the results obtained with seven different limestones.

concentration in the pores of the particle and the concentration at the reaction interface; (3) an equation for the rate movement of reaction interface which is determined from the chemical reaction rate taking place at the reaction interface. The mass balance for diffusion and reaction of the gas over a differential volume element in a spherical particle, assuming the pseudo steady state, is common for the three mechanistic models.

Gas-Solid Reaction Modeling

The effective diffusivity depends on the type of gas diffusion occurring in the pores: molecular, Knudsen, or a combination of both. Because of the variation in the size of the pores during the sulfation reaction, the gas diffusivity was calculated as a combination of the molecular and Knudsen diffusion:

The sulfation process is a specific case of a gas-solid reaction where structural changes take place in the solid as the reaction proceeds. The gaseous reactants must diffuse in order to react with the solid reactant, CaO. Four resistances to the overall chemical reaction rate have been considered: diffusion through a boundary layer surrounding the particle; diffusion through the porous medium into the reacting spherical particle; diffusion through the solid product layer developed on the walls of the reacting pores; surface chemical reaction at the solid interface reactant-product. In the usual sulfation conditions corresponding to fluidized bed coal combustion, the more important resistances are those corresponding to gas diffusion in the pores and in the product layer. Moreover, some common assumptions in the three models tested have been adopted: (1) The solid reactant is a porous spherical particle the outer radius of which remains constant during reaction. (2) The solid is an isothermal particle. (3) Changes in the solid structure occur slowly enough so that the pseudo-steady-state hypothesis for the gas concentration profile in the particle is valid. (4) The chemical reaction between the gas and the solid reactant is first order with respect to the SO2 and irreversible. The analyses for the overall reaction rate in the grain and pore models used are similar since they consider the same mass transfer resistance. The calculation of the overall reaction rate in terms of the solid conversion as a function of time requires the solution of the following equations with appropriate boundary conditions: (1) a differential mass balance equation for the gas diffusion and reaction within the particle which yields the gas concentration profile through the particle; (2) a differential mass balance equation for the gas in the product layer which gives a relation between the

δC 1 δ δC )0 DeR2 - (r)s ) 2 δR δR δt R

(

)

(2)

The boundary conditions required for the solution of this equation are the following:

De

δC ) kg(Co - C) at R ) Ro and t > 0 (3) δR δC ) 0 at R ) 0 and t > 0 δR

(4)

The mass transfer coefficient (kg) was determined by using the Ranz and Marshall (1952) equation to calculate the Sherwood number. The effective diffusivity is calculated as a function of the particle porosity and the tortuosity factor of the particle by

De ) Dgp/τ

Dg ) [DSO2-1 + DK-1]-1

(5)

(6)

The tortuosity was calculated using the equation of Wakao and Smith (1962) and later modified by EliasKohav et al. (1991) by introducing a parameter, b, to define different porous structures of the sorbents.

τ ) 1/pb

(7)

The changes in porosity inside the particle with sulfation conversion were calculated using the Hartman and Coughlin (1976) expression, as a function of the initial porosity, o, and the expansion ratio, Z:

p ) o - (Z - 1)(1 - o)Xs

(8)

where Z is defined as

Z ) 1 + Freal(VMSO4Ca - VMCaO)/MCaO

(9)

being

Freal ) FsxCaO/(1 - o) The rate of reaction per unit of particle volume is proportional to the chemical reaction rate constant and to the gas concentration at the reaction interface. Due to the diffusional resistance through the solid product layer, the gas concentration in the pores is not the same as that existing at the reaction interface. The mass

2192

Ind. Eng. Chem. Res., Vol. 35, No. 7, 1996

balance equation for the diffusion of the gas through the product layer is characteristic of each model and it will be analyzed below. The variation of the sulfation conversion with time at each position inside the particle can be calculated with this balance and that mentioned above (eq 2). The mean conversion in the whole particle is calculated by integration of local conversions using the following equation:

∫0R 4πR2Xs(R,t) dR o

X h s(t) )

4/3πRo3

(10)

The main equations defining the three mechanistical models used in this work are shown below. (1) Changing Grain Size Model (CGSM). The changing grain size model by Georgakis et al. (1979) assumes that the particle consist of a number of spherical grains of uniform initial radius, ro. As the reaction proceeds the grain size grows, r1, while the size of the unreacted core shrinks, r2. In order to find the relation between the gas concentration in the pores and the concentration in the reaction interphase, a mass balance in spherical coordinates was done, which analytical solution allows us to express the local reaction rate in terms of the gas concentration in the pores by this equation:

(r)s )

CSoks(r2/ro)2 r2/ro 1+ (1 - r2/r1) β1

(11)

(12)

(2) Random Pore Model (RPM). The random pore model by Bathia and Perlmutter (1981a,b) considers the reaction surface to be the result of the random overlapping of a set of cylindrical surfaces of size distribution f(r). The total length of the system LE0, and the surface area SE0, are related to the structural parameter ψ, by means of the expression

ψ)

4πLE0(1 - o) SE02

(13)

A mass balance in the pore must be done to relate the concentration in the pores to that of the interfacial, assuming a lineal gradient in the product layer. With this balance and the preceding equations, Bhatia and Perlmutter found an expression for the calculation of the local reaction rate:

dXs ) dt

CSoks(1 - Xs)x1 - ψ ln(1 - Xs) βZ (1 - o)Fmolar 1 + (x1 - ψ ln(1 - Xs) - 1) ψ (14)

[

]

where

β ) 2ks(1 - o)/(DsSo)

(r)s )

(r2/ro2)kηk

∑k (r)sk ) 2oCks∑k 1+

(r2/ro)k βk

(15)

(16)

ln(r2/r1)k

Once the new pore radius at each region is known, the local conversion is calculated as the sum of the local conversions at each pore interval considered:

Xs(R,t) )

o

∑k ηkXs(R,t)k ) 1 -  ∑k ηk((r2/ro)k2 - 1) o

To calculate the local conversion at each time and position inside the grain the following equation is used:

Xs(R,t) ) 1 - (r2/ro)3

This model does not consider the progressive pore plugging, because the structural parameter (ψ) has a constant value. (3) Distributed Pore Size Model (DPSM). The distributed pore model by Kocaefe (1986) allows for a pore size distribution rather than using a single pore size. The more general model uses a population balance for a changing pore size distribution, as suggested by Christman and Edgar (1983). The distribution is divided into a number of regions within randomly distributed through the particle. The pores initially have a radius of ro,k in the k region, which is decreasing during the reaction, r1,k, taking different values with time. The concentration in the product layer is calculated as a function of the concentration in the pores by means of a mass balance in cylindrical coordinates. Thus, the local reaction rate can be expressed as the sum of the reaction rates at each region by

(17)

Solution Procedure. An orthogonal collocation method was used to solve the differential equations including the different mass balances above shown for the different models. The desired precision (0.1% in sulfation conversion) was achieved with eight collocation points on spherical symmetry in dimensionless radial coordinates. Applying the orthogonal collocation method to the general mass balance equation (2) in a dimensionless way, a system of linear equations (one for each collocation point) is obtained, for whose solution the local conversion at each time is needed. These local conversions are calculated depending on the model used (eq 11, 14, or 16) by using the Runge-Kutta integration method to solve the differential equations, one for each collocation point. Once the new conversions at each point are known, the new gas concentrations can be calculated by solving again the orthogonal system. The mean conversion of the particle was known by integrating the local conversions obtained at each collocation point (eq 10). Results and Discussion The above models were used to interpret the behavior during sulfation of different limestones. The experimental curves of sulfation conversion were obtained in a previous work (Ada´nez et al., 1994) in a thermobalance at 850 °C, 2500 ppm SO2, and 10% CO2. Seven limestones (five with unimodal pore size distribution and two with a widespread pore size distribution) in seven narrow-cut particle sizes between 100 and 2000 µm, with mean particle diameters of 158, 353, 632, 894, 1118, 1414, and 1788, were used. The maximum

Ind. Eng. Chem. Res., Vol. 35, No. 7, 1996 2193

Figure 1. Pore size distributions of the limestones Blanca, Sa´stago, and Alborge. Table 1. Physical Parameters and Calculations for Applying the Models to the Seven Limestones limestone

porosity (%)

dpore (µm)

So (m2/g)

r1 (µm)

LE0 × 10-8 (m/g)

Blanca Morata Muel Arin˜o Bocafoz Sa´stago Alborge

52. 47. 50. 46. 46. 66. 59.

0.055 0.053 0.052 0.052 0.059 0.435 0.255

14.4 15.2 14.5 15.5 9.8 4.6 7.5

0.055 0.063 0.064 0.063 0.098 0.161 0.126

3.35 2.8 2.0 0.86 2.4 1.8 3.9

sulfation conversions were always reached by the finest particles and by the limestones with a widespread pore size distribution. As example, Figure 1 shows the pore size distributions of the calcined limestones Sa´stago and Alborge (wide pore size distribution) and Blanca (unimodal pore size distribution). The rest of the calcined limestones have pore size distributions similar to Blanca limestone. The parameters required for comparison of the three mechanistical models are obtained from the experimental conditions (bulk gas concentration, particle radius, etc.) and from physical measurements. The initial porosity, specific surface area, and pore size distribution of the limestones were measured by Hg porosimetry and N2 adsorption (BET) techniques. These physical parameters are shown in Table 1. The average grain size, necessary for the CGSM model, and the characteristic length for the RPM model are also shown. The average grain size was calculated from the specific surface area by assuming spherical grain shape. For the DPM model, five pore size intervals were used with limestones with unimodal pore size distribution and eight intervals with the widespread pore size distribution limestones. Values ranging from 66 × 10-3 to 15 × 10-6 m/s for the kinetic constant and from 6 × 10-13 to 3 × 10-10 m2/s for the diffusivity in the product layer have been found in the literature (Borgwardt et al., 1987). Due to the wide ranges, initially the values for these variables (chemical reaction constant and product layer diffusivity) were determined, assuming the tortuosity exponent in eq 7 equals 1 for all limestones. To determine these variables, the Nelder and Mead method (1964) was employed by curve fitting to the experimental data of Xs-time. The fitting confirmed that the chemical reaction rate constant for the range of experimental conditions studied did not affect the predictions of the models. This confirms the fact that the reaction is mainly controlled by diffusion, as found by Bhatia and Perlmutter (1981b),

Christman and Edgar (1983), and Hartman and Coughlin (1976). For this reason a constant value of 66 × 10-3 m/s, as proposed Hartman and Coughlin (1976), was considered. The values of the variables’ product layer diffusivity coefficient, Ds, and exponent of the tortuosity, b, were found for each limestone. Comparison with Experimental Data. A model has to show its validity by accurately predicting experimental results. If these predictions are not good, it becomes necessary to modify the model in order to make it suitable to the system. The three models analyzed, CGSM, RPM, and DPSM, had been checked in the literature using few limestones and very few particles sizes (two or three). In this work, their validity has been tested with seven limestones and seven particle sizes. In this case, important limitations of the models have arisen. If the models had worked well, each limestone would have been characterized by a solid diffusivity Ds that depends on the limestone composition. Using this variable, the models should accurately predict the conversion-time curves of each limestone in all particle size ranges. When this was done, none of the models were found able to predict the experimental results Xs-t with all the particle sizes considered. This is the reason we modified the models. We approached the problem looking into the change of effective diffusivity in the pores with conversion and, specifically, the change of tortuosity with porosity. Assuming that eq 8 is correct, the analysis was based on the variation of the tortuosity with particle porosity, that is, eq 7. Therefore, we searched for the values of the exponent b that fitted the experimental results for all the particle sizes used with each limestone. The Nelder and Mead method was used to find the values of Ds and b that fit the experimental Xs-t best. The three mechanistical models mentioned above were used for comparison purposes. A pair of values of Ds and b for each limestone, and valid in all the range of particle sizes, was obtained in this way with each model. Table 2 shows the Ds and b values obtained for each limestone and model considered. As a general finding, a greater product layer diffusion coefficient, Ds, and a smaller exponent in the tortuosity equation, b, were obtained with those sorbents showing wider and higher pore size system with all the models tested. The greatest values of b were found for the limestones with a unimodal porous system, leading to a small value for the effective pore diffusion coefficient. For this reason, together with the smaller diffusion coefficient in the product layer obtained in the fitting, the low values of maximum sulfation conversion attainable for this type of limestones were obtained. Limestones with a widespread pore size distribution showed low values of b and, thus, low tortuosity of the pore system, with the effective diffusivity also high. Moreover, this type of limestone shows high product layer diffusivity and thus can achieve higher maximum sulfation conversions than limestone characterized by a unimodal porous system. For this reason, limestone Blanca (unimodal pore size distribution) and Sa´stago (widespread pore size distribution) have been used as representatives of both types of behavior. Figures 2 and 3 present comparisons of the experimental conversion vs time plots with the predictions given by the three models for the limestones Blanca and Sa´stago, respectively. As examples, some plots of the experimental and predicted curves of four particle size

2194

Ind. Eng. Chem. Res., Vol. 35, No. 7, 1996

Table 2. Parameters Obtained by Fitting of Xs-t Curves with CGSM, RPM, and DPSM with the Seven Limestones Considered model CGSM RPM DPSM

Ds × 1013 b Ds × 1013 b Ds × 1013 b

Blanca

Muel

Arin˜o

Morata

Bocafoz

Sa´stago

Alborge

0.91 1.8 0.84 1.8 0.34 1.5

2.3 1.5 2. 1.63 3.2 2.2

1.5 1.4 1.7 1.48 1.5 1.6

1.5 0.6 1.4 0.56 3.4 1.04

2.4 1.5 1.6 1.48 1.6 1.6

25 0.7 8.1 0.6 24 0.6

13 0.6 4.4 0.4 18 0.97

Figure 2. Comparison between experimental conversions and those predicted by CGSM, RPM, and DPSM with Blanca limestone: s, experimental; ‚‚‚, CGSM; - -, RPM; - ‚-, DPSM.

cuts with mean values of 158, 353, 632, and 1788 µm of the seven used in all this work are shown. In general, similar trends were obtained for the CGSM and RPM models, giving similar Xs-t curves for slightly different Ds values. The three models overestimate the conversion at long reaction times for the smallest particle sizes. For the Blanca limestone, good and similar predictions of the experimental results were found with the three models for the highest particle sizes (above 900 µm). However, in a general way, a bad fitting was found with this type limestone when the particle size decreased. For the Sastago limestone, the three models fit reasonably well for the largest particle sizes, but for the smallest sizes the only acceptable predictions were reached with the distributed pore model. This model gives much better predictions of the experimental conversion than the other models for the fine particles of widespread limestones, because it allows the small pores to become filled with reaction product while larger pores remain open for reaction. The results indicate that the use of an average grain (CGSM model) or pore size (RPM model) is an oversimplification for the limestones with a wide pore size distribution, which suffer structural changes during the reaction. The best predictions for limestones with wide pore size distributions were achieved by using the distributed pore size model.

It must be noted that the majority of the mechanistic models proposed in the literature have been validated for narrow ranges of particle sizes. Bhatia and Perlmutter (1981b) applied their model to the data of Hartman and Coughlin (1976), obtained with one limestone and only three particle sizes (565, 900, and 1120 µm). Even so, some discrepancies between predicted values and experimental with the finest particle sizes were found. Georgakis et al. (1979) applied their grain model to the same data of Hartman and Coughlin (1976) and to the data of Borgwardt (1970) with one limestone and three particle sizes (8, 25, and 130 µm). Christman and Edgar (1983) validated their model with three limestones and only one particle size (600 µm) using as adjustable parameters the Sherwood number and the limestone tortuosity. There are more authors that attempt to validate the models using very few particle sizes and two adjustable parameters (generally Ds and tortuosity). Thus, Simons and Garman (1986) used particles of 0.5 and 1 µm; Reyes and Jensen (1987) used three sizes (1200, 1500, and 1800 µm); Sotirchos and Zarkanitis (1993) two sizes of 96 and 322 µm; and Dam-Johansen et al. (1991) used particles of 2000 µm. Korbee et al. (1991) and Lin et al. (1992) applied their models to four limestones and four particle sizes (249, 917, 1295, and 1540 µm), determining the values of b of their own experimental data. They found that the accessibility to a porous

Ind. Eng. Chem. Res., Vol. 35, No. 7, 1996 2195

Figure 3. Comparison between experimental conversions and those predicted by CGSM, RPM, and DPSM with Sa´stago limestone: s, experimental; ‚‚‚, CGSM; - -, RPM; - ‚-, DPSM.

Figure 4. Values of parameter b vs particle diameter for Blanca limestone.

Figure 5. Values of parameter b vs particle diameter for Sa´stago limestone.

system (the tortuosity) is more important than pore size in the values of effective diffusivity. In general, models claim good fitting to the experimental results. However, the partial validity of these models has been shown in this work when a wider particle size range is considered. The sulfation behavior of limestones with an unimodal pore size distribution, especially with the finest particles, cannot be predicted only with a unique pair of values of Ds and b for each limestone. This was possible only with limestones showing a widespread pore size distribution. In order to see the relation between b and the particle size of the sorbent, the values of b were determined by fitting for each particle size considered. The same Ds value for each limestone and the DPSM were used for that. Figures 4 and 5 show plots of b vs particle size for limestones Blanca (unimodal pore size distribution) and Sa´stago (widespread pore size distribution), respectively. Similar values of the b parameter for the different particle sizes were found for the limestones with a widespread pore size distribution, which was expected

due to the good fitting previously found (Figure 3). In this case, a unique value of b would be valid to reasonably predict the experimental sulfation curves in all the range of particle sizes. Oppositely, in those limestones characterized by a unimodal and small pore size distribution, a clear variation of the parameter b with the particle size was found. In this case, two zones can be distinguished. For particle sizes below 900 µm, a decrease of b with increasing particle size was noted; meanwhile for the rest of particle sizes, similar values of b were found. Therefore, the experimental data could be satisfactorily predicted with a unique value of b for particle sizes above 900 µm. However, with this value of b corresponding to higher particle sizes, it is not possible to predict the sulfation curves for the smallest particles, as previously noted. The necessity of using a b value for each particle size invalidates the advantages of the mechanistical gas-solid reaction models, because it would be necessary to determine the Xs-t curves for each particle size of limestone.

2196

Ind. Eng. Chem. Res., Vol. 35, No. 7, 1996

Figure 6. Experimental sulfation conversions for Blanca limestone and predicted by the DPSM using the b value corresponding to each particle size: s, experimental; - ‚-, DPSM.

The higher values of b found for the particles below 900 µm could be due to several physical factors, which effect could be incorporated into b during this type of fitting. The existence of different tortuosity in the smallest particles by variations in the porosity, or the different rearrangement of the solid during sulfation as a function of the particle size, could lead to the obtained results. Moreover, the equation used to predict the porosity change with conversion could only be fulfilled until a certain radius of the particle. In any case, further physical measurements would be necessary to probe these assumptions. Figure 6 shows the experimental and predicted Xs vs time curves for the smallest particles sizes of limestone Blanca, when using the distributed pore model and the b values corresponding to each particle size. A good fit was found for all sizes and similar predictions were found for the other limestones similar to this one. Analyses of the three models made have shown how the DPSM is valid for fitting the Xs-time sulfation curves when working with limestones with widespread pore size distribution in a broad range of particle sizes. However, it is necessary to do individual fitting for each particle size when working with limestones with unimodal pore size distribution and sizes lower than 900 µm. Conclusions During model evaluation it was found that CGSM and RPM are not adequate to predict the sulfation conversion vs time plots of limestones (unimodal or widespread pore size distribution) in a broad particle size range (100-2000 µm). The DPSM has been found to be appropriate to predict the conversion-time curves of limestones with wide pore size distribution in a broad interval of particle diameters, by using the product layer diffusion coefficient and the tortuosity of the limestone as fitting parameters. However, the DPSM is not valid to predict the Xstime curves of the limestones with uniform pore size distribution but only with these parameters in the range of particle sizes studied. In this case, it would be necessary to use a Ds for each limestone and a specific b for each particle size considered.

Nomenclature b ) tortuosity exponent in eq 7 C ) SO2 local concentration (mol/m3) Co ) SO2 concentration around the particle (mol/m3) dpore ) mean pore diameter (µm) De ) effective diffusivity (m2/s) Dg ) SO2 diffusion coefficient (m2/s) DK ) Knudsen diffusion coefficient (m2/s) DSO2 ) Molecular diffusion coefficient (m2/s) Ds ) product layer diffusion coefficient (m2/s) kg ) mass transfer coefficient (m/s) ks ) kinetic constant (m3/m2s) LE0 ) characteristic length of porous system (m/m3 of particle) ro ) initial pore or grain radius (m) r1 ) pore or grain radius (m) r2 ) reaction surface radius of pore or grain (m) rgrain ) mean grain radius (mm) R ) radial coordinate (m) Ro ) particle radius (m) (r)s ) reaction rate (mol/m3 s) SE0 ) characteristic surface of porous system (m2/m3 of particle) So ) initial specific surface BET (m2/m3) t ) time (s) VMCaO ) molar volume of CaO (cm3/mol) VMSO4Ca ) molar volume of SO4Ca (cm3/mol) xCaO ) CaO mass fraction Xs ) sulfation conversion Xs ) global sulfation conversion in the particle Z ) expansion ratio defined in eq 9 Greek Symbols β ) parameter defined in eq 15 β1 ) parameter defined as (Ds/ksro) βk ) parameter defined as (Ds/ksro,k) o ) initial particle porosity p ) particle porosity ηk ) fraction of pore volume in region k τ ) tortuosity ψ ) structural parameter defined in eq 13 Fs ) solid density (kg/m3) Freal ) real solid density (kg/m3) Fmolar ) molar solid density (kg/m3) Subcripts k ) relating to a region of pore radii

Acknowledgment The authors thank the DGICYT (AMB92-0888-C0201) for funding the project of which this study is part. Also, P.G. thanks the Spanish Ministry of Education for a FPI grant.

Literature Cited Ada´nez, J.; Garcı´a-Labiano, F.; de Diego L. F.; Abanades, J. C. Methods for characterization of sorbents used in fluidized bed boilers. Fuel 1994, 73 (3), 355.

Ind. Eng. Chem. Res., Vol. 35, No. 7, 1996 2197 Bartlett, R. W.; Krishnan, N. G.; Van Hecke, M. C. Micrograin models of reacting porous solids with approximations to logarithmic solid conversion. Chem. Eng. Sci. 1973, 28, 2179. Bhatia, S. K.; Perlmutter, D. The effect of pore structure on fluidsolid reactions: Application to the SO2-lime reaction. AIChE J. 1981a, 27 (2), 226. Bhatia, S. K.; Perlmutter, D. A random pore model for fluid-solid reactions. II Diffusion and transport effects. AIChE J. 1981b, 27, 247. Borgwardt, R. H. Kinetics of the reaction of SO2 with calcined limestone. Environ Sci. Technol. 1970, 4, 59. Borgwardt, R. H.; Bruce, K. R.; Blake, J. An investigation of product-layer diffusivity for CaO sulfation. Ind. Eng. Chem. Res. 1987, 26, 1993. Christman, P. G.; Edgar, T. Distributed pore-size model for sulfation of limestone. AIChE J. 1983, 29, 388. Dam-Johansen, K.; Ostergard, K. High-temperature reaction between sulphur dioxide and limestone- II An improved experimental basis for a mathematical model. Chem. Eng. Sci. 1991, 46, 839. Efthimiadis, E. A.; Sotirchos, S. V. A partially overlapping grain model for gas-solid reactions. Chem. Eng. Sci. 1993, 48 (7), 1201. Elias-Kohav, T.; Sheintuch, M.; Avnir, D. Steady-state diffusion and reactions in catalytic fractal porous media. Chem. Eng. Sci. 1991, 46 (11), 2787. Georgakis, C.; Chang, C. W.; Szekely J. A changing grain size model for gas-solid reactions. Chem. Eng. Sci. 1979, 34, 1072. Hartman, M.; Coughlin, R. W. Reaction of sulfur dioxide with limestone and the grain model. AIChE J. 1976, 22, 490. Kocaefe, D. Ph.D. Thesis. An experimental and theoretical study on the kinetics of basic oxide. Chem. Eng. Dept., Univ. New Brunswick, Fredericton, Canada´, 1986. Korbee, R.; Wolff, E. H.; van den Bleek, C. M. A general approach to FBC sulfur retention modeling. Proc. 11th Int. Conf. on FBC 1991, 907. Lin, W.; Svoboda, K.; van den Bleek, C. M. Sulfur capture and its interactions with NOx emissions in fluidized bed combustion of coal: general analytical model based on particle gas-solid reactions. Chem. Eng. Sci. 1992, 47 (9-11), 2425. Lindner, B.; Simonsson, D. Comparison of structural models for gas-solid reactions in porous solids undergoing structural changes. Chem. Eng. Sci. 1981, 36 (9), 1519.

Nelder, J. A.; Mead, R. A simplex method for function minimization. Comput. J. 1964, 7, 308. Petersen, E. E. Reaction of porous solids. AIChE J. 1957, 3, 443. Ranz, W. E.; Marshall, W. R., Jr. Evaporation from drops. Chem. Eng. Prog. 1952, 48 (4), 173. Reyes, S.; Jensen, K. Percolation concepts in modelling gas-solid reactions. Application to sulfation of calcined limestone. Chem. Eng. Sci. 1987, 42 (3), 565. Simons, G. A.; Garman, A. R. Small pore closure and the deactivation of the limestone sulfation reaction. AIChE J. 1986, 32 (9), 1491. Sotirchos, S. V.; Yu, H. C. Mathematical modelling of gas-solid reactions with solid product. Chem. Eng. Sci. 1985, 40 (11), 2039. Sotirchos, S. V.; Yu, H. C. Overlapping grain models for gas-solid reactions with solid product. Ind. Eng. Chem. Res. 1988, 27, 836. Sotirchos, S. V.; Zarkanitis, S. A distributed pore size and length model for porous media reacting with disminishing porosity. Chem. Eng. Sci. 1993, 48 (8), 1487. Szekely, J.; Evans, J. W. A structural model for gas-solid reactions with a moving boundary. Chem. Eng. Sci. 1970, 25, 1091. Szekely, J.; Propster, M. A structural model for gas-solid reactions with a moving boundary. VI The effect of grain size distribution on the conversion of porous solids. Chem. Eng. Sci. 1975, 30, 1049. Wakao, N.; Smith, J. M. Diffusion in catalyst pellets. Chem. Eng. Sci. 1962, 17, 825. Yu, H. C.; Sotirchos, S. V. A generalized pore model for gas-solid reactions with pore closure behaviour. AIChE J. 1987, 33 (3), 382.

Received for review June 21, 1995 Revised manuscript received April 15, 1996 Accepted April 15, 1996X IE950389C

X Abstract published in Advance ACS Abstracts, June 1, 1996.