Ind. Eng. Chem. Res. 2007, 46, 1161-1166
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Dry Regenerable Metal Oxide Sorbents for SO2 Removal from Flue Gases. 2. Modeling of the Sulfation Reaction Involving Copper Oxide Sorbents Vasudeo S. Gavaskar and Javad Abbasian* Illinois Institute of Technology, 10 W 33rd Street, Chicago, Illinois 60616
The expanding grain model is used to describe the sulfation reaction rate involving sol-gel-derived copperbased sorbents and sulfur dioxide. The intrinsic reaction rate constant was estimated from the experimental data presented in Part 1 to be in the range of 0.014-0.0047 cm/min for the sorbents with copper content ranging from 11.2% (Cu-1) to 26.5% (Cu-5). The grain radius and the expansion factors were calculated from the physical and chemical properties of sol-gel-derived copper-based sorbents, such as the BrunauerEmmett-Teller (BET) surface areas, particle porosities, and true densities, as well as chemical composition. The grain radius of the sorbents ranged from 8.89 to 9.14 nm, while the expansion factor ranged from 1.5 to 2.1. The model, which contained two adjustable parameters [i.e., the product layer diffusion (Dg) and the tortuosity parameter (R)], was shown to be capable of describing the experimental data of the sulfation reaction with excellent accuracy. On the basis of the model results, the optimum copper content of this type of the sol-gel-derived sorbent was determined to be 13.5%. The expanding grain model was also incorporated in a fluid-bed reactor model to describe the experimental data of the sorbent’s performance in a batch fluidizedbed reactor with good accuracy. Introduction
Table 1. Physical and Chemical Properties of the Sorbent Formulations
study1
Part 1 of this described the formulation, development, and experimental evaluation of sol-gel-derived dry regenerable alumina supported copper oxide sorbents for flue gas desulfurization (FGD) application. A systematic study was performed in a thermogravimetric analyzer (TGA) and a lab-scale fluidizedbed reactor to obtain a relatively complete set of experimental data describing the physical and chemical characteristics (i.e., reactivity, physical strength, long-term durability, etc.) of these sorbents. It was observed that the overall sulfation reaction rate is controlled by the transport of the gaseous SO2 through the sorbent particle, which is dependent on the physical and chemical characteristics of the sorbent. In Part 1 of this paper, it was shown that the intrinsic sulfation reaction rate is first order with respect to the SO2 concentration, with an activation energy of 8.2 kcal/mol. The results of the characterization of the different sorbent formulations revealed that the increasing copper content apparently had no significant effect on the poresize distributions of the sorbent formulations. However, the physical characteristics indicated a distinct relationship between the surface areas and particle porosities of the sorbent formulations with the copper content (see Table 1). This paper focuses on the modeling of the experimental results presented in Part 1 of this study by associating the sulfation characteristics of the copper oxide based sorbents with their physical and chemical properties, such as Brunauer-EmmettTeller (BET) surface areas, copper contents, porosities, etc., using the expanding grain model.2 Design Equations for the Expanding Grain Model The expanding grain model, also known as the particle-pellet model, has been used to describe the sulfation reaction of calcined limestone.2-7 In the grain model, the porous solid is described as an assemblage of a large number of small * To whom correspondence should be addressed. Phone: (312) 5673047. Fax: (312) 567-8874. E-mail:
[email protected].
initial reaction BET rate expansion grain Cu Al surface factor, radius constant content content area particle (nm) (cm/s) Zv sorbent (%) (%) (m2/g) porosity Cu-1 Cu-2 Cu-3 Cu-4 Cu-5
11.2 14.1 17.4 21.3 26.5
44.3 43.2 41.2 37.4 34.2
151 150 141 137 134
0.4824 0.4814 0.4689 0.4671 0.4662
1.5 1.6 1.7 1.9 2.1
8.89 9.01 9.07 9.12 9.14
0.014 0.011 0.0079 0.0057 0.0047
nonporous grains. Surrounding these grains are macropores through which the gaseous reactant (i.e., SO2) diffuses to reach the surface of the grains. A schematic diagram of such a porous particle along with the assemblage of nonporous grains is given in Figure 1a. The reaction occurs at the surface of the nonporous grains according to the unreacted shrinking core model. As the reaction proceeds, the grains expand to a size corresponding to the differences in the molar volumes of the solid reactant and product. This expansion causes a decrease in the local porosity of the sorbent, thus decreasing the diffusivity of the gaseous reactant through the porous sorbent matrix. A product layer is formed in the outer regions of the grains, which leads to additional diffusional resistance to the reaction. Therefore, the overall rate of reaction is governed by three resistances: the diffusion of SO2 through the pores of the sorbent, the diffusion of SO2 through an expanding shell of the sulfated product layer formed on the grains, and the chemical reaction at the reaction interfaces of the grains. Detailed description of various versions of the grain models as well as the governing equations have been extensively reported in the literature.2,3,6,8 In this study, the reacting sorbent particles were assumed to be spherical with a diameter of 270 µm with uniform CuO composition. The nonporous grains constituting the sorbent particle were also assumed to be spheres of uniform radius. No overlapping of the grains was considered (i.e., the grains expand uniformly to the size corresponding to their maximum possible conversion). The governing differential equations describing the
10.1021/ie061363w CCC: $37.00 © 2007 American Chemical Society Published on Web 01/12/2007
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Figure 1. Schematic diagrams of (a) a porous sorbent particle and the assemblage of the nonporous grains and (b) a partially reacted grain.
transport of SO2 through the porous particle and the rate of reaction of the individual grains along with the respective boundary conditions are given below (assuming pseudo-steady state)
(
)
[
]
ri2 ∂CR DgCR 1 ∂ 2 D R 3(1 )k )0 e 2 ∂R 3 D + kr (1 - r /r′ ) ∂R R rg g i i g (1) with boundary conditions
∂CR ) 0 at R ) 0 ∂R
(1a)
CR ) Cb at R ) R0
(1b)
(
)
DgCR dri k , for ri > 0 )dt Fmt Dg + kri(1 - ri/r′g)
(2)
ri ) rg at t ) 0
(2a)
dri ) 0 at ri ) 0 dt
(2b)
The solid dependent reaction rate constant (k) is determined using the initial sulfation reaction rate data and the initial grain radius, which can be calculated from the measurable physical properties of the sorbents
( )( ) 1 dX Cg dt rg )
1 (S MW tf0 g CuO)
3 SgFreactant
(3)
(4)
The radii of the reacting interface, ri, is obtained by solving
() ( ( ))
XL ) 1 XO )
3 RO3
∫0R
O
ri rg
3
R2 1 -
ri rg
(5)
3
dR
(6)
Structural Changes in the Solid during Reaction. In the expanding grain model, because of the differences in the molar volume of the product (CuSO4) and the reactant (CuO), as depicted in Figure 1b, the radius of the grain, rg′, is expected to increase with increasing conversion, which can be determined by material balance using the expansion factor, Zv, which is the ratio of the molar volume of the product to the reactant.
r′g ) rg[1 + (Zv - 1)XL]1/3 Zv )
with the following initial condition:
k)
eqs 1 and 2 and is used to calculate the local and the overall conversions.
FreactantMWproduct FproductMWreactant(1 - s)
(7) (8)
The value of the porosity of the product layer (s) has been estimated in the literature5 to be between 0.05 and 0.2. In this study, s was assumed to be 0.1 for the first sulfation of the sol-gel-derived CuO/Al2O3 sorbents. Values of Zv < 1 indicate that the particles shrink during reaction, and with Zv > 1, swelling occurs. The expansion factors for these sorbent formulations were shown1 to range from 1.5 to 2.1, indicating that the grains expand during the sulfation. It can be seen that, if 0 is greater than (Zv - 1)/Zv, then the complete conversion of the grain, and thus of the particle, is possible; otherwise, pore closure will take place, and the conversion of the solid particle will level off below the complete conversion. The initial effective diffusivity, De0, of the gaseous reactant through the particle is calculated using the following equation:
De0 )
(
0 1 1 + τ Df Dk
)
-1
(9)
The Knudsen coefficient, Dk, was calculated using the dusty gas model,2 while the free gas (molecular) diffusion coefficient, Df, was calculated using the equation developed by Fuller et al.9 Diffusion of the gaseous reactant through the porous solid reactant is, therefore, characterized by Df, Dk, and (/τ). Unfortunately, little or no experimental data is available on tortuosities for noncatalytic gas-solid reaction systems. Therefore, the effective diffusivity of the SO2 through the solid particle and its dependence on the particle conversion is modeled using empirical equations with adjustable parameters that are determined from the experimental data. A comprehensive list of empirical models for effective diffusivity can be found elsewhere.10-12 In this study, an exponential function [i.e., exp(RXO)] was used to describe the tortuosity parameter R. A similar functional form has also been used by Shaaban13 and Karnik,14 while modeling the sulfation of calcined limestone. Therefore, the dependence of effective diffusivity on the sorbent conversion is determined from
De ) exp(-RXO) De0 0
(10)
Equations 1 and 2 were approximated by finite difference formulation and solved as a boundary value problem. The detailed method of the solution of these differential equations can be found elsewhere.5 Preliminary calculations showed that a net of 150 grid points on the particle radius and a time step of 0.05 min provided a numerical solution of sufficient accuracy. The model was solved numerically to determine the adjustable parameters Dg and R by regression analysis, involving the minimization of the mean sum of squares of errors/ residuals between the calculated and measured overall CuO conversion.
[
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9Dg(1 - bed)z Cg ) exp Cin RO3rg3Cg,avg
∫0R
O
(
R2kri2CR
)]
( )
ri Dg + kri 1 r′g
dR
(15)
Results and Discussions Model Results of the Sulfation Reaction of the Sorbents. The physical and chemical properties of the sorbents prepared in this study are given in Table 1, indicating that the initial grain radii of these sorbents increase with increasing copper content. The model equations were initially solved to match the experimental results obtained in the TGA with the five different sorbent formulations listed in Table 1. Comparison of the experimental sulfation results of the different sorbent formulations, obtained at 450 °C with a SO2 concentration of 2500 ppmv, with those generated by the model are shown in Figure 2, indicating excellent fit to the experimental results. The product layer diffusivity, Dg, is plotted as a function of the copper content of the sorbents in Figure 3, indicating that Dg decreases from ∼7.0 × 10-12 to 2.1 × 10-12 cm2/min as the copper content (measured) increases from 11.2% (Cu-1) to 26.5% (Cu-5). This expected decrease of Dg with the increasing copper content results in a lower overall reaction rate, as evident from Figure 2. This relationship between the product layer diffusivities and the copper content is consistent with the observations reported in the literature for zinc titanate based sorbents.16 Similar findings have also been reported for Ca-
Design Equations for a Fluidized-Bed Reactor Model The expanding grain model was incorporated into a uniform fluidization model2,15 to predict the performance of the copperoxide based sorbents in a laboratory-scale batch fluidized-bed reactor. A differential mass balance on the reacting gas assuming pseudo-steady state behavior gives
Ug
〈 〉
∂Cg dX )0 + (1 - bed)Fs ∂z dt z
(11)
Figure 2. Model fit of the experimental sulfation data of various sorbents in TGA at 450 °C and 2500 ppmv SO2.
The quantity in 〈 〉 is taken at its average value and can be rewritten as2,3
〈dXdt 〉 ) 〈dXdt 〉
Cg(z) Cg,avgCg,avg
(12)
( ( ))
(13)
z
Cg,avg )
〈 〉
9Dg dX ) 3 3 dt RO rg Fmt
Cin - Cexit Cin ln Cexit
∫0R
O
(
R2kri2CR
)
( )
ri Dg + kri 1 r′g
dR
(14) Figure 3. Dependence of the product layer diffusivity (Dg) on chemical composition.
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Figure 4. Dependence of tortuosity parameter (R) on the expansion factor (Zv).
Figure 6. Model fit of the experimental sulfation data of Cu-2 at 450 °C.
Figure 7. Effect of temperature on the product layer diffusivity of Cu-2. Figure 5. Local conversion and local porosity profiles of Cu-1 and Cu-5 (after 60 min).
based sorbents.17,18 The dependence of the tortuosity parameter, R, on the expansion factors of the various sorbents is shown in Figure 4. The increasing expansion of the grain radius results in a reduction of the local porosities, making the diffusion path increasingly tortuous, as is suggested by the higher values of R (Figure 4). This inference can be further supported by comparing the local porosity and conversion profiles for Cu-1 and Cu-5 sorbent particles after 60 min of sulfation (shown in Figure 5). In the case of Cu-1 (11.2% Cu), it is clearly evident that, although the grains at the outer surface have reached complete conversions, the dimensionless local porosities (/0) reduce only to 0.5, suggesting that complete CuO conversion is possible. This is in sharp contrast to the case of Cu-5 (26.5% Cu), where the local porosity at the outer surface of the sorbent particle dropped to zero, indicating the pore closure, resulting in the leveling off of the CuO conversion well-below 100%. Figure 6 shows a comparison of the experimental data with those predicted by the model using the values of Dg and R obtained for Cu-2 (shown in Figures 3 and 4), indicating that the model is capable of accurately describing the experimental data. The dependence of Dg on the reaction temperature (350-450 °C) is shown in Figure 7, indicating an apparent activation energy of ∼41 kcal/mol, which is similar with those previously reported in the literature for the sulfation of calcined limestone.5,19-21 A comparison of the predicted and experimental results at different temperatures is shown in Figure 8, indicating that the grain model provides satisfactory fits to the experimental data. The calculated values of product layer diffusivity (Dg) and tortuosity parameter (R) shown in Figures 3 and 4 were used to
Figure 8. Model fit of the experimental sulfation data of Cu-2 at 2500 ppmv SO2 concentration.
predict the performance of the sorbent in the fluidized-bed unit and the “optimum” sorbent composition corresponding to maximum sulfur loading (at an exit SO2 concentration of 100 ppmv) in the fluidized-bed reactor. Figures 9-12 provide comparisons of the predicted and the experimental results at various operating condition obtained in the fluidized bed, confirming that the grain model coupled with the uniform fluidization model provides an excellent fit to the experimental data. The estimated values of bed were in the range of 0.550.57 for the five sorbent formulations, which is consistent with the values of bed porosities reported in the literature, for
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Figure 9. Model fit of the fluidized-bed breakthrough curves with various sorbents at 450 °C with 2500 ppmv SO2.
Figure 12. Model fit of the fluidized-bed breakthrough curves Cu-2 (temperature ) 450 °C, SO2 concentration ) 2500 ppmv).
Figure 10. Model fit of the fluidized-bed breakthrough curves Cu-2 (inlet SO2 conc. ) 2500 ppmv, space velocity ) 4000 h-1).
Figure 13. Optimum copper content as predicted by the expanding grain model. Table 2. Estimated Properties of the Optimum Sorbent copper content, % aluminum content, % (by difference) surface area, m2/g particle porosity original grain radius (rg × 103), microns product layer diffusion, cm2/min tortuosity parameter (R) expansion factor (Zv)
13.5 44.3 152 0.484 8.96 6.9 × 10-10 2.9 1.53
Conclusions
Figure 11. Reactor model fit of the fluidized-bed breakthrough curves Cu-2 (temperature ) 450 °C, space velocity ) 4000 h-1).
fluidization of coarse materials.3 The dependence of the sulfur loading in the fluid bed on the copper content of the sorbents is shown in Figure 13, clearly showing that the sulfur loading of these sol-gel-derived sorbents passes through a maximum. As shown in Figure 8, the maximum sulfur loading corresponds to the copper content of ∼13.5%. The estimated values of the physical and chemical properties of the optimum sorbent are given in Table 2.
A modified expanding grain model was used in this study to describe the kinetics of the reaction between a regenerable CuO based sorbent and SO2. The model incorporated the physical and chemical properties of the sorbent in its basic equations, which were solved to model the experimental conversion-time data. The adjustable parameters of the model parameters [i.e., product layer diffusivity (Dg) and tortuosity parameter (R)] were related to the physical/chemical characteristics of the CuO based sorbents. The model provides an excellent fit to the experimental TGA data. The expanding grain model, coupled with a uniform fluidized-bed model, is shown to predict the performance of the sorbents in a fluidized bed with excellent accuracy. The optimum copper content is estimated to be 13.5%. Nomenclature Cg ) concentration of SO2 in the bed, mol/cc CR ) The gaseous reactant within the porous particle, mol/cc
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De ) effective diffusivity through the particle, cm2/min Df ) molecular diffusion, cm2/min Dg ) effective diffusivity through the product layer on the grain, cm2/min Dk ) Knudsen diffusion, cm2/min MWj ) molecular weight of species j, g/mol k ) reaction rate constant, cm/min R ) radial coordinate of the reacting particle, cm rg ) The initial grain radius, cm ri ) The radius of the unreacted part of the grain, cm rg′ ) The radius of the expanded grain, cm Sg ) BET surface area of the sorbent, cm2/g XL ) local conversion XO ) overall conversion of the solid reactant (CuO) Ug ) superficial gas velocity at the operating conditions, cm/ min Zv ) expansion factor, dimensionless z ) axial coordinate of the sorbent bed, cm ) local particle porosity, dimensionless bed ) porosity of the fluidized bed, dimemsionless s ) porosity of the product layer owing to the cracks and fissures, dimensionless Fj ) density of the species j, g/cm3 Fmt ) molar density of CuO in the sorbent particle, mol/cc Fs ) The molar density of the solid reactant (CuO), mol/cc τ ) The tortuosity factor, dimensionless Literature Cited (1) Gavaskar, V. S.; Abbasian J. Dry Regenerable Metal Oxide Sorbents for SO2 Removal from Flue Gases. 1. Development and Evaluation of Copper Oxide Sorbents. Ind. Eng. Chem. Res. 2006, 45 (17), 5859. (2) Szekely, J.; Evans, J.W.; Sohn, H. Y. Gas-Solid Reactions; Academic Press: New York, 1976; Chapter 2. (3) Fenouil, L. A.; Lynn, S. Design of Entrained-Flow and Moving-, Packed-, and Fluidized-Bed Sorption Systems: Grain-Model Kinetics for Hot Coal-Gas Desulfurization with Limestone. Ind. Eng. Chem. Res. 1996, 35, 1024. (4) Bardakci, T. Diffusional Study of the Reaction of Sulfur Dioxide with Reactive Porous Matrices. Thermochim. Acta 1984, 76, 287. (5) Hartman, M.; Trnka, O. Influence of Temperature on the Reactivity of Limestone Particles with Sulfur Dioxide. Chem. Eng. Sci. 1980, 35, 1189.
(6) Hartman, M.; Coughlin, R. W. Reaction of Sulfur Dioxide with Limestone and the Grain Model. AIChE J. 1976, 22, 490. (7) Hartman, M.; Coughlin, R.W. Reaction of Sulfurdioxide with Limestone and the Influence of Pore Structure. Ind. Eng. Chem. Process Des. DeV. 1974, 13, 248. (8) Ramachandran, P.A.; Doraiswamy, L.K. Modeling of Noncatalytic Gas-Solid Reaction. AIChE J. 1982, 28, 881. (9) Fuller, E. N.; Schettler, P. D.; Giddins, J. C. A New Method for Prediction of Binary Gas-Phase Diffusion Coefficients. Ind. Eng. Chem. 1966, 58 (5), 19. (10) Calvelo, A.; Cunningham, R. E. Kinetics of Gas-Solid Reactions. J. Catal. 1970, 17, 1. (11) Fan, L. S.; Miyanami, K.; Fan, L. T. Transients in Isothernal FluidSolid Reaction SystemssModeling of the Sigmoidal-Conversion-Time Behavior of a Gas-Solid Reaction. Chem. Eng. J. 1977, 13, 13. (12) Wen, C. Y. Non-Catalytic Heterogenous Solid-Fluid Reaction Models. Ind. Eng. Chem. 1968, 60, 34. (13) Shaaban, A. F. Determination of the Kinetic Parameters of the Reaction between SO2 and CaO Using Thermogravimetric Technique. Thermochim. Acta 1991, 180, 9. (14) Karnik, A. P. Effects of External Factors on the Measurement of Gas-Solid Reaction Rates. Master’s Thesis, Illinois Institute of Technology, Chicago, IL, 2004. (15) Kunni, D; Levenspiel, O. Fluidization Engineering; ButterworthHeinemann: Boston, MA, 1977. (16) Lew, S.; Sarofim, A. F.; Flytzani-Stephanopoulous, M. Modeling of the Sulfidation of Zinc-Titanium Oxide Sorbents with Hydrogen Sulfide. AIChE J. 1992, 38 (8), 1161. (17) Borgwardt, R. H.; Bruce, K. R.; Blake, J. An investigation of Product-Layer Diffusivity for CaO Sulfation. Ind. Eng. Chem. Res. 1987, 26, 1993. (18) Mahuli, S. K.; Agnihotri, R.; Chauk, S.; Fan, L. S. Combined Calcination, Sintering and Sulfation Model for CaCO3-SO2 Reaction. AIChE J. 1999, 45 (2), 367. (19) Bhatia, S. K.; Perlmutter, D. D. A. Random Pore Model for FluidSolid Reactions. II. Diffusion and Transport Effects. AIChE J. 1981, 27, 247. (20) Christman, P. G.; Edgar, T. F. Distributed Pore-Size Model for Sulfation of Limestone. AIChE J. 1983, 29, 388. (21) Marsh, D.W.; Ulrichson, D. L. Rate and Diffusional Study of the Reaction of Calcium Oxide with Sulfur Dioxide. Chem. Eng. Sci. 1985, 40 (3), 423.
ReceiVed for reView October 23, 2006 ReVised manuscript receiVed November 26, 2006 Accepted November 29, 2006 IE061363W
