Rate Equation Theory for the Carbonation Reaction of CaO with CO2

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Rate Equation Theory for the Carbonation Reaction of CaO with CO2 Zhenshan Li,* Hongming Sun, and Ningsheng Cai Key Laboratory for Thermal Science and Power Engineering of Ministry of Education Beijing Municipal Key Laboratory for CO2 Utilization and Reduction Department of Thermal Engineering, Tsinghua University, Beijing 100084, China ABSTRACT: Rate equation theory for CaO carbonation reaction was developed to replace the assumption of critical product layer thickness with a complete calculation of the rates of CaCO3 nucleation and growth. The model includes rate expressions for the island size distribution function with time evolution, and the surface reaction, surface diffusion, grain boundary and lattice diffusion, and Ostwald ripening were considered in this theory to describe CaCO3 product islands growing on the CaO surface. The macroscopic behavior of a gas−solid reaction can be calculated with the island size distribution. The carbonation reaction of CaO with CO2 including both early islands growth stage and later product layer diffusion stage was calculated with the developed theory and was validated with experimental data, and the relative importance of some elementary steps was discussed. This rate equation theory provides a link between the macroscopic behavior of a gas−solid reaction with the microscopic mechanisms from the molecular level.

1. INTRODUCTION The calcium looping concept by use of the carbonation/ calcination cycles of CaO/CaCO3 is a promising technology for the capture of CO2 for both postcombustion 1−5 and precombustion.6−10 Recent reviews have extensively covered the large volume of literature published to date.11,12 The application of calcium looping requires a detailed understanding of the carbonation reaction of CaO with CO2. Previous studies have shown that the reaction of CaO with CO2 in the initial stage is fast and is controlled by chemical kinetics.13−15 After the fast initial stage and a transitional stage, a slower stage controlled by diffusion in the pores or in the solid product layer takes place, resulting in slower reaction rates.13−15 Another observed phenomenon is that the temperature has a significant effect on CaO conversion; that is, increasing the temperature will increase the carbonation conversion.16,17 In order to understand or predict above carbonation behaviors of CaO with CO2, different gas−solid reaction models have been used, and they are summarized in Figure 1. The simplest form of gas−solid reaction model is the apparent model,18,19 where a simple rate expression (dXCaO/dt = k(1−(XCaO/Xu))m(CCO2 − CCO2,e)n) including the maximum carbonation conversion Xu, the practical CaO conversion XCaO, the kinetic constant k, and the average CO2 concentration (CCO2 − CCO2,e). The second type of model is shrinking core model,20 as shown in Figure 1a, where the particle is assumed as a nonporous sphere and a layer of solid product is outside of the solid reactant. The third type of gas−solid reaction model is the grain model,21,22 as shown in Figure 1b. A fourth group of researchers also applied the pore models, as shown in Figure 1c. Bhatia and Perlmutter13 first applied the random pore model (RPM) to the carbonation reaction. Recently, Grasa et al.23 have described the cyclic carbonation reaction with the RPM model in order to represent CaO conversion with time as function of the cycle number. A concept of discrete pore size was also used by Sun et al.24 to describe the carbonation reaction. Dennis et al.25,26 also used the pore model to describe © 2012 American Chemical Society

the carbonation behavior of synthetic sorbent. The nucleation and growth model,27 as shown in Figure 1d, describes the kinetic behavior of some gas−solid reactions that present a sigmoid curve when solid conversion is plotted vs time. The models shown in Figure 1 belong to the macroscopic level of model, and they consider the following:28 (1) external mass transfer, (2) internal mass transfer or CO2 diffusion inside the pore, (3) diffusion through the product layer, and (4) chemical reaction. For a gas−solid reaction such as the carbonation of CaO with CO2, the formation and growth of the solid CaCO3 product is the most important step. First, the formation and growth of solid product will occur at the interface between solid reactant and solid product, making gas more difficult to contact with the reactant surface, the gas or ion have to diffuse through the solid product layer. Second, the formation and growth of solid product will result in plugging of the porous structure by filling the pores, the diffusion of gas through the pore structure become more difficult. It is obvious that the product layer has a critical effect on the diffusion in pore, surface reaction, and product layer diffusion. About the reaction mechanism of carbonation in product layer diffusion stage, Bathia et al.22 have proposed the O2− ion diffuse outwardly and CO32− ion diffuse inwardly for the carbonation reaction in the product layer controlled stage. Mess et al.14 observed the grain boundary diffusion. These works from the literature mainly focus on the slower product layer diffusion stage, while the initial faster reaction stage is more important from both fundamental science and practical application point of view. The thickness of the carbonate layer formed on the free surfaces of CaO is a critical parameter to mark the end of the fast reaction period; it was reported that the average value for the critical CaCO3 product layer thickness was about 49 nm.29 The theory of a critical product layer thickness is now used almost in all carbonation models (as shown in Figure 1) to Received: April 11, 2012 Revised: June 5, 2012 Published: June 7, 2012 4607

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Figure 1. Gas−solid reaction models.

explain the “maximum” conversion. Because of the simplified description of the reaction process, above models are relatively simple and easy to use. However, these models are often not sufficient for assessing the mechanism of solid product formation and growth involved in the initial stage, and many phenomena are not well explained by these models. Therefore, it is necessary to study the nucleation, islands formation, and growth of solid product during the carbonation reaction. In previous work by our group,30−33 by using atomic force microscopy (AFM) and a single crystal sample, it was found that the solid product shows three-dimensional island shaped morphology on the reactant surface.30 High density groups of islands with smaller size are formed at lower temperatures, while low density, larger sizes of islands are formed at high temperatures.31 Surface diffusion is responsible for the morphology changing.32 The islands growth mode, product nucleation and growth, grain boundary and lattice diffusion were discussed.33 However, how to apply the mechanism we got using AFM and a single crystal reactant to the gas−solid reaction model for solid particle reactants is not clear in the previous work, we will investigate this problem in this study. The purposes of this study were as follows: (1) to develop the rate equation theory considering surface reaction and diffusion, grain boundary and lattice diffusion, and Ostwald ripening, to analyze the evolution of island size distribution with time, and to replace the assumption of critical product layer thickness with a complete calculation of the rates of CaCO3 nucleation and growth, (2) to study the numeric algorithm for the rate equation and to validate the model with experimental data, and (3) to discuss the effects of temperature, CO2 concentration, and grain boundary diffusion on the carbonation reaction.

Figure 2. Principle of nucleation and growth processes of CaCO3 product during the carbonation reaction of CaO with CO2.

by the inclusion of the step on the surface, and it proceeds via incorporation into the surface, rather than mono- or bidentate adsorption above the surface. Besson et al.41 presented a detailed study of CO2 adsorption on CaO by means of density functional theory, and found that calcite formation should imply a nucleation process occurring on a localized scale. From these discussions, the CaCO3 formation mechanism via dry gas−solid carbonation between the adsorbed CO2 and O2− ion onto CaO surfaces was well studied by many researchers. Beside of the surface reaction discussed, the nucleation and growth of CaCO3 products on CaO surface, as shown in Figure 2, are critical steps and are controlled both by the thermodynamics and the kinetics.33 The product island growth involves the diffusion of product molecules or ions along the surface and the boundary between the solid reactant and product. Surface diffusion plays an important role in product island nucleation and growth steps because the nucleation and growth in the initial reaction stage are competing processes, as shown in Figure 2, and the outcome of this competition is determined by both the chemical reaction rate and surface diffusion. At the same time, the larger islands can grow at the expense of smaller ones (Ostwald ripening) due to the further reduction of the surface energy,33 as shown in Figure 2. During the product layer diffusion controlled stage, both grain boundary and lattice diffusion control the growth, as shown in Figure 2. The boundary between islands is the most important diffusing path. Less tightly bound ions at the nonlattice grain boundary and dislocation are expected to be

2. RATE EQUATION THEORY For the carbonation reaction of CaO with CO2, the first step for the product formation and growth is the surface reaction, as shown in Figure 2. About the interaction mechanism between adsorbed CO2 and CaO, some published papers34−37 already reported this issue. Pacchioni et al.34 have found that a chemically bound carbonate is formed by coordination to an oxygen ion at the CaO surface. Freund and Roberts35 claimed the O2− ion at the surface is stabilized by the Madelung potential of the ionic crystal of CaO, thus leading to a higher basicity and reactivity of CaO. Metastable induced electron spectroscopy (MIES) in combination with ultraviolet photoelectron spectroscopy (UPS) has been applied to the study of the interaction of CO2 with CaO,36,37 and it was found that the chemisorption occurs; it takes place in form of carbonate (CO3) complexes whereby the CO2 adsorbates interact with surface oxygen anions. For CaO results of first-principles, calculations are available.38,39 Allen et al.40 found that carbonation is indeed a favorable process, particularly enhanced 4608

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islands of size s. The fourth and fifth terms account for the single molecule escape from islands of size s − 1 and s. A single molecule is involved in the nucleation and surface growth of islands. A molecule balance is necessary to account for single molecule density changes due to surface reaction, molecule capture, and escape from islands, and the rate equation for monomer density is

more mobile than lattice ions. In order to describe the gas− solid reaction from microscopic point of view, above elementary steps are included in the rate equation theory, we will discuss this in this study. To express above elemental steps described in Figure 2, the rate equation theory was developed to describe the product nucleation and growth for the carbonation reaction of CaO with CO2. The rate equation theory is widely used in the field of surface physics,42,43 but, to the best of our knowledge, this is the first time application of this theory in the field of gas−solid reaction. In the concept of rate equation theory, the dynamic variables are the average densities of islands composed of s CaCO3 molecules, Ns. The density for single CaCO3 molecule is N1. The CaO surface occupied by CaCO3 is θ = ∑s =∞2NsSs, and Ss is the interface between CaO substrate and CaCO3 island of size of s. Therefore, the unoccupied CaO surface area is θs = 1−θ. Ds is the surface diffusion coefficient. The elemental steps considered in the rate equation were shown in Figure 3

dN1 = Fθs − DsN1(2σ1N1 + dt

∞

∞

∑ σsNs) + (2η2N2 + ∑ ηsNs) s=2

s=3

(2)

where the first term on the right-hand side results from the flux of CaCO3 product formed on the CaO surface due to the chemical reaction. The second term accounts for the loss of single molecule to island formation. The factor of 2 in the second term is present because the formation of island with two molecules results in the loss of two molecules. The third term is due to the single molecule escape from islands. In these formulas, F is the chemical reaction rate, ks is the rate at which an island of size s captures product molecule produced directly onto itself and this process is used to describe the grain boundary and lattice diffusion steps, ηs is the mean rate at which a molecule escapes from an island of size s after detachment, and the capture number σs is a measure of the efficiency with which an island of size s captures molecule. The chemical reaction rate can be expressed as F = k(CCO2 − CCO2,e)Nmolecular

(3)

3

where k (m /s) is the chemical reaction rate coefficient; Nmolecular is the number of molecules per mole, 6.0225 × 1023; CCO2(mol/m3) is CO2 concentration in gas phase; and CCO2,e (mol/m3) is equilibrium concentration of CO2 and can be calculated as follows:44

Figure 3. Elemental steps considered in the rate equation (grain boundary and lattice diffusion are not shown in this figure).

CCO2,e =

and include (1) surface reaction (F) and the formation of solid product molecule; (2) surface diffusion of single molecule (Ds); (3) single molecule captured by islands (σs); (4) single molecule escape from islands (ηs); (5) grain boundary and lattice diffusion, and then reaction with gas on islands (ks). An island may grow to a size either because of capture of a molecule, grain boundary, and lattice diffusion on smaller islands or because of the escaping of a molecule from larger islands. Likewise, an island may shrink because of either its molecule escape to a smaller size island or the capture of a molecule to form a larger island or the grain boundary and lattice diffusion to form a larger island. The change in island size distribution (for the density of islands of size s > 2) due to capture or escape of a molecule can be mathematically expressed as

(2 ≤ s ≤ ∞)

(4)

where R is constant, 8.314, and T (K) is reaction temperature. The diffusion coefficient can be written as ⎛ −E ⎞ Di = D0i exp⎜ i ⎟ ⎝ RT ⎠

(5)

where i represents surface (S), grain boundary (GB), and lattice (L) diffusion; Ei (kJ/mol) is the activation energy for diffusion. The activation energy for surface diffusion is typically smaller than those of the other diffusion mechanisms; therefore, the initial product island’s growth is dominated by surface diffusion. During the product layer diffusion controlled stage, both GB and lattice diffusion control the growth. The effective product layer diffusion coefficient can be expressed as45

dNs = F(ks − 1Ns − 1 − ksNs) + Dsσs − 1Ns − 1N1 − DsσsNsN1 dt − ηsNs + ηs + 1Ns + 1

⎛ 19680 ⎞ 1.826 × 1012 ⎟ exp⎜ − ⎝ RT T ⎠

ks = fDGB + (1 − f )DL

(6)

where f is the volume fraction of the grain boundary. The boundary between islands is the most important diffusing path, less tightly bound ions at the nonlattice grain boundary and dislocation are expected to be more mobile than lattice ions; therefore, only grain boundary diffusion is considered in this study (i.e., ks = f DGB). The volume fraction of the grain boundary is dependent on the interface between CaO and CaCO3, island size and number, island morphology, and other factors; in this study, following equation is used to estimate f:

(1)

s is the number of molecules, and Ns is the number of islands of size s. The first term accounts for the island increasing or decreasing due to the grain boundary diffusion and lattice diffusion along an island of size s − 1 and s, respectively. The second and third terms on the right-hand side of this equation are the rate at which diffusing molecules are added to an island of size s − 1 and s multiplied by the total density of islands of that size. This process increases or decreases the number of 4609

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1 1 θ

4 − 9d f

+ 10

−d f V CaCO 3

(7)

VCaO,0

The geometric approximation was used to estimate the capture number,46 σs = 2.0 + (rs/r1)1/2.7

=

(8)

=

Molecule escape from islands is important for the nucleation and growth, and determines the islands size distribution. The reason for molecule escape from islands is the Ostwald ripening or coarsening, which had been originally described by physical chemist W. Ostwald.47 Ostwald ripening has been observed in many systems where particles with various sizes are dispersed in a matrix; however, as a result of the large surface area present, the mixture of particles is not in thermodynamic equilibrium. The total energy of the system can be decreased via an increase in the size scale of particles and thus a decrease in total interfacial area.48 Molecule escape from smaller islands and growth on larger islands originates from the concentration gradients around the islands. The effective molecule concentration inside a smaller island is larger, forcing them to flow through the concentration gradients both from the smaller islands to surface and from the surface to the larger islands, and the average size of the islands will increase. The density of a molecule, N1e,s (number of molecule with size of s), at the surface of a spherical island with diameter ds (containing s molecule) is48 ⎛ γΩ 1 ⎞ N1e,s = Ne exp⎜ ⎟ ⎝ RT ds ⎠

i≥1

π 6

M VCaO,0VCaCO 3 M VCaO M VCaO,0VCaCO 3

∑ Nvi i i≥1

(13)

VM CaO

vi = qi − 1 × vi − 1

(14)

where vi−1 and vi are the volume for nodes i − 1 and i, q is the geometric spacing factor. Decreasing the geometric spacing factor will increase the accuracy, but it will result in the large computational requirements. In this study, the geometric spacing factor is 2, and there are 10 size nodes per order of magnitude in the particle diameter space; in total, 40 size nodes are required to cover the island size range. Based on this node spacing, the rate equation in eq 1 can be changed as

(9)

⎛ ⎞ dNk v1 v1 kk − 1Nk − 1 − kkNk ⎟ = F⎜ vk + 1 − vk dt ⎝ vk − vk − 1 ⎠ v1 v1 Dsσk − 1Nk − 1N1 − DsσkNkN1 + vk − vk − 1 vk + 1 − vk v1 v1 ηk Nk + η Nk + 1 (2 ≤ k ≤ 40) − vk − vk − 1 vk + 1 − vk k + 1

∑ Ndi i3 i≥1

M VCaCO3VCaO

3. NUMERIC ALGORITHM For the carbonation reaction of CaO with CO2, the size of CaCO3 product islands range from ∼0.49 nm (one CaCO3 molecule, i.e., v1 = 0.061 nm3) to about 3 μm. On the volume scale this size range corresponds to ∼6.15 × 10−29 m3 to ∼1.4 × 10−17 m3. That is to say, about 1012 CaCO3 molecules will be included in the largest island, and the number for the rate equation (eq 1) will be 1012. It is difficult to solve this huge number of differential equations. In order to solve this difficulty and cover the 12 orders of magnitude for the volume range, the total volume range for the solid product is divided into nodes; it is assumed that islands exist only at these nodes, this assumption simplifies the computation. The nodes are spaced linearly with equal spacing,50

(10)

∑ Nvi i =

M VCaO

where and are the molar volume of CaCO3 and 3 CaO, 36.9 cm /mol for CaCO3 and 16.9 cm3/mol for CaO, respectively. In this study, it is assumed VCaO,0 = 1 × hCaO,0. It should be noted that all variables are specified on the basis of the unit surface area; that is, all variables should be divided by 1 m2.

The volume of all CaCO3 islands, VCaCO3(m3), VCaCO3 =

M VCaCO 3

VM CaCO3

where Ne is the density in equilibrium with island of infinite diameter, γ is the free energy of the solid product (here, 0.6J/ m2 for CaCO349), Ω is the molar volume of the solid product (36.9 × 10−6m3/mol for CaCO3), ds (m) is the diameter of island with size of s. The escape number ηs can be calculated using following equation:43

ηs = DN s 1e, s

VCaCO3

XCaO =

(11)

(15)

where vi (m3) and di are the island volume and diameter with size of s. The interface between CaO substrate and all CaCO3 islands, θ (m2),

The initial conditions are as follows: t=0

Nk = 0

(1 ≤ k ≤ 40)

(16)

(12)

There are 40 ordinary differential equations with initial conditions. The “ode15s.m” stiff-variable integrated in the time domain method in MATLAB was used to obtain the value of Nk.

where si (m2) is the interface between CaO substrate and CaCO3 island with size of s. The conversion of CaO is

4. RESULTS AND DISCUSSIONS The output parameter of the rate equation is island density (Ns), and the macroscopic behavior such as CaO conversion

θ=

∑ Nsi i = i≥1

π 4

∑ Ndi i2 i≥1

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6), and Ne (eq 9). k, Ds, and Ne determine the island size distribution in the initial fast reaction stage, where the surface of CaO can contact directly with CO2. ks determines the reaction behavior of product layer diffusion stage. Therefore, k, Ds, and Ne can be obtained by fitting the calculation with the experimental data in the initial fast stage, and ks can be obtained through the fitting of the experimental data in product layer diffusion stage. A summary of the values of the parameters used in the calculation is given in Table 1. It can be seen from Table

can be calculated from the island density. Thermogravimetric analysis (TGA) tests can be used to get CaO conversion and validate the model. The calculated results of rate equation for CaO carbonation reaction at different temperature and CO2 concentration were validated by the experimental data from the TGA test, as shown in Figure 4 and 5. For the TGA (Q500)

Table 1. Values of Parameters Used in Calculation param. 515 590 640 690 720 720 743

°C °C °C °C °C °C °C

CO2 vol %

CCO2 mol/m3

CCO2,e mol/m3

14 14 14 14 14 25 14

2.165 1.977 1.869 1.772 1.718 3.068 1.679

0.004 0.032 0.105 0.304 0.546 0.546 0.837

k m3/s 1.33 1.36 1.33 1.34 1.33 1.33 6.28

× × × × × × ×

10−5 10−5 10−5 10−5 10−5 10−5 10−7

Ne no./mol 2.84 2.20 1.56 1.04 5.60 1.44 3.60

× × × × × × ×

1011 1011 1011 1011 1010 1011 109

1 that the chemical reaction rate coefficient k is almost constant in the temperature range 515∼720 °C; Bhatia and Perlmutter13 also obtained the similar conclusion from their TGA data. Ne decreases with the increasing of temperature, while it increases with the increasing of the CO2 concentration, as shown in Table 1. The value k at 743 °C is smaller than that at lower temperatures; this may be because the decomposition rate of CaCO3 is enhanced when the temperature is increased to 743 °C. For all conditions, the values of surface diffusion and grain boundary diffusion can be estimated with following equations:

Figure 4. Effect of temperature on the carbonation of CaO with CO2 (CaO: ∼20 μm. CO2: 14 vol %. N2: 86 vol %. hCaO,0 = 2.52 × 10−8 m. Dot: experimental data. Line: calculated results).

⎛ −110 × 103 ⎞ Ds = 5.0 × 10−3 exp⎜ ⎟ RT ⎝ ⎠ ks = fDGB =

2FDs10−8 1 −d f + θ 104 − 9df V CaCO3

(17)

df = 3 (18)

From Table 1 and eqs 17 and 18, it can be seen that the unknown parameters in the developed rate equation theory are k, Ds, and Ne. Compared with the other models such as shrink core, grain, or pore models where the unknown parameters are k, Deff (product layer diffusion coefficient), and hcritical (critical product layer thickness), the number of unknown variables in rate equation is same, but the assumption of the critical product layer thickness is replaced. The rate equation can assess the effect of real physical and chemical steps such as solid state diffusions (surface, grain boundary, and lattice diffusion) and solid product nucleation and growth on the kinetic behavior. The experimental and calculated results of CaO carbonation are shown in Figure 4 and 5, where the CaO conversions at different conditions are plotted versus time. It can be seen from Figure 4 and 5 that the calculated results agreed with experimental data well for the CaO carbonation reaction. From Figure 4 and 5, it can be seen that the reaction of CaO with CO2 in the initial stage is fast, and after this stage, a stage with slower reaction rate takes place, and there is a critical CaO conversion that is the indicator of the end of initial fast reaction stage and the start of the slower reaction stage. The critical CaO conversion correspond the critical product layer thickness, as proposed by Abanades et al.29 It can be seen from Figure 4 that the temperature has a significant effect on the critical CaO

Figure 5. Effect of CO2 concentration on the carbonation of CaO with CO2 (CaO: ∼20 μm. Temperature: 720 °C. hCaO,0 = 2.52 × 10−8 m. Dot: experimental data. Line: calculated results).

experiment, about 8−10 mg analytical pure CaO with average particle size of ∼20 μm was loaded in the TGA instrument and pure N2 was fed into it for 10 min at a flow rate of 100 mL/ min. The temperature was increased at a rate of 20 °C/min to the carbonation temperature and was maintained for 10 min. When the temperature was stable, a gas mixture of 14 vol % CO2 and 86 vol % N2 at a flow rate of 100 mL/min was introduced into the TGA instrument to induce the carbonation reaction. The adjustable parameters contained in rate equation are chemical reaction rate coefficient k (eq 3), surface diffusion coefficient Ds (eq 5), grain boundary diffusion coefficient ks (eq 4611

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Figure 6. Island size distribution at different temperatures (conditions are same as those in Figure 4; here, island size is expressed with number, the real island diameter can be calculated with eq 14).

phenomena, as shown in Figure 4, can be explained as follows. At a low temperature such as 515 °C, the initial carbonation reaction is fast, because some fresh CaO surface is available for direct contact between CaO and CO2. The product molecules or ions diffuse slowly at lower temperature compared higher temperature, resulting in a high density of small islands, as shown in Figure 6a. With the reaction proceeding, the product islands gradually cover the CaO surface. When the surface is fully covered by the product, the reaction enters into the product layer diffusion stage, and the CO2 or ions have to diffuse through the CaCO3 product layer, and this stage is controlled by both GB diffusion and lattice diffusion; island size will increase while island density will decrease, as shown in Figure 6a. As the temperature is increased to 690 °C, surface diffusion rate was enhanced, and ions or atoms on the surface diffuse faster than at a low temperature (515 °C). More important, the island density is reduced with the formation of large islands at higher temperatures, as shown in Figure 6b. When the temperature is increased to 743 °C, both surface diffusion and Ostwald ripening are fast, while the chemical reaction rate becomes slower due to the thermodynamic limitation. In this case, larger islands are formed due to fast surface diffusion, but the single molecule number (N1) is not enough due to the slow reaction rate; therefore, the island density is small, as shown in Figure 6c.

conversion (i.e., critical product layer thickness), the critical CaO conversion increases with the increasing in the temperature. CO2 concentration has no big effect on the critical CaO conversion, as shown in Figure 5. An interesting phenomenon, as shown in Figure 4, is that there is no critical CaO conversion before 20 min when the temperature is increased to 743 °C and the carbonation rate become very slow. The effect of temperature on CaO carbonation reaction is complex, and the possible mechanism will be discussed with the developed rate equation theory. In the concept of rate equation theory, the nucleation and subsequent growth of the solid product are the necessary intermediate steps during the formation of the new product phase. Product molecules or ions can diffuse on the surface and will come into contact to create the possibility of nucleating a new island or finding and joining an existing island, or product molecules or ions escape from small islands and join larger ones. Surface diffusion is believed to be responsible for the transport of ions and atoms that are necessary for the reaction, nucleation, and growth of the solid product. Island growth will compete with island nucleation; surface diffusion is important. The increase in temperature enhances surface diffusion and causes the formation of larger islands, making fresh CaO surface available to the CO2. Based on this mechanism for product layer formation and growth, the experimental 4612

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Figure 7. Surface coverage and average island volume (conditions are same as those in Figure 4).

on island size distribution and can be calculated with Ns. Figure 7 shows the comparison of θ with average island volume at different temperatures. From the nucleation theory, if CO2 concentration in gas phase is far larger than the equilibrium concentration, the nucleation rate is fast (we refer this as “growth controlled”); while CO2 concentration in gas phase approaches the equilibrium concentration, the nucleation is slow while the growth is fast (we refer this as “nucleation controlled”). At 515 °C, the carbonation reaction is in the “growth controlled” stage because the difference of (CCO2 − CCO2,e) is larger; while at 743 °C the carbonation reaction is in the “nucleation controlled” stage because the difference of (CCO2 − CCO2,e) is small. The rate of θs reduction is fast at relatively low temperatures such as 515 and 690 °C in the “growth controlled” stage because of large island density (as shown in Figure 6) and small island size (as shown in Figure 7). At 743 °C, the nucleation rate is slow because the carbonation reaction is slow and most of CaCO3 produced from the reaction involve the growth of the existing nucleus; this results in the small island density and larger island size (as shown in Figure 6d). Therefore, the rate of θ increasing is slow at 743 °C, as shown in Figure 7a. Figures 6 and 7 can be used from the microscopic point to explain the mechanism responsible for the macroscopic behavior occurring in Figure 4. During the product layer diffusion controlled stage, both GB and lattice diffusion control the growth. Grain boundaries are known to be transition regions between two neighboring grains, and diffusion along grain boundaries often controls the evolution of the microstructure and properties of materials at elevated temperatures. Grain boundaries of CaCO3 were already observed by Mess et al.14 in the carbonation reaction of CaO with CO2. For the carbonation reaction of CaO with CO2, grain boundary diffusion is particularly important because the movement of atoms or ions take place mostly at the grain surface. The treatment of GB diffusion can be found in the paper published by Fisher.51 The effective product layer diffusion coefficient can be related to the lattice diffusion and grain boundary diffusion coefficients, as shown on eq 6. It should be noted that GB changes with the increase of reaction time, and therefore, f also changes. When the grain has a small diameter, the boundary between grains becomes a dominating factor, having a grain boundary diffusion coefficient as the effective diffusion coefficient. With the increase of reaction time, grains become larger and the grain boundary decreases,

The nucleation of solid products is a key step for a gas solid reaction. From a thermodynamic point of view, the critical radius of nuclei, r*, is mainly dependent on the free energy change of the reaction. The solid product islands with radii smaller than r* are not stable and will shrink by losing molecules. Only islands larger than r* are stable, and they will grow larger while lowering the overall system energy. At the same time, the reaction of CaO with CO2 is exothermic, thus, (∂r*/∂T) > 0; that is to say, r* is smaller at lower temperatures than at higher temperatures under otherwise identical conditions. This can be seen from Figure 6. At 515 °C, the starting point for island size distribution is about 13 (d13 = 7.8 nm), indicating that islands with radii smaller than 7.8 nm are not stable, while at 690 °C, this starting size was increased to 16 (d16 = 15.7 nm). This clearly demonstrates that the critical radius of nuclei calculated with rate equation increased with the increasing of temperature, as shown in Figure 6a and b. From Figure 6b, it can be seen that the average island diameter after 20 min at 690 °C is d21 = 49.7 nm, and this agree well with the reported critical product layer thichness of 49 nm,29 indicating that the rate equation can predict a reasonable product layer thickness. Figure 6c shows the island size distribution of 743 °C; because the CO2 equilibrium concentration increases with temperature, the island nucleation and growth are controlled by the chemical reaction rate. The size and density of islands increase gradually with the reaction time, as shown in Figure 6c. Figure 6d shows the comparison of island size distribution at different temperatures; it can be seen that, with the increasing of temperature, the island density decreases while the islands size increase. Of the parameters that enter into the rate equation (eq 2) for the heterogeneous reaction, the reactive surface area, that is, the unoccupied CaO surface area θs, is a critical variable. This is because the CaO surface area is not constant with time, covered gradually and modified by CaCO3 product islands. That is to say, reduction in the directly available surface area of solid reactant for reaction is caused by the formation of product islands. Therefore, θs changes with time, and the flux of CaCO3 product formed on the CaO surface due to the chemical reaction (Fθs in eq 2) will decrease with time. The directly available surface area for the carbonation reaction, is, therefore, expected to be significantly smaller than the total. This is most likely one of the main reasons for the initial fast decrease in CaO conversion rate during the carbonation of CaO as observed in Figure 4. The reduction in reactant surface depends 4613

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leading to the decrease of f. In this case, the importance of lattice diffusion through the product layer crystals increases with time relative to the diffusion through the grain boundaries. It was found by Mess et al.14 that diffusivity along the grain boundary is significantly higher than diffusivity in the inner part of the grain (lattice diffusion), most likely due to the unsaturated and more defective nature of the grain surface; therefore, only grain boundary diffusion is considered in this study. The effect of grain boundary diffusion rate on the carbonation of CaO at 515 °C was calculated with the rate equation, and the results are shown in Figure 8. It was observed

Figure 9. Effect of initial CaO thickness on the carbonation reaction of CaO with CO2 at 515 °C (CaO: ∼20 μm. CO2: 14 vol %. N2: 86 vol %. Dot: experimental data. Line: calculated results).

area changing of solid reactant on the gas−solid reaction can be reflected in the developed rate equation theory. It should be noted that the initial CaO is assumed to be a “thin layer” (see Figure 3) and the pore structure characteristic was not considered in current rate equation theory. Most of practical solid reactants are porous, and the pore size distribution inside the solid has some effects on the gas−solid reaction; the importance of the pore volume for the carbonation reaction has been highlighted by some authors, such as Pacciani et al.52 In order to describe accurately the effect of pore size distribution on the gas diffusion and reaction, a particle scale model such as the grain model or pore model will be used, and how to integrate the rate equation into the particle scale model will be studied in the future.

Figure 8. Effect of grain boundary diffusion on the carbonation reaction of CaO with CO2 at 515 °C (CaO: ∼20 μm. CO2: 14 vol %. N2: 86 vol %. Dot: experimental data. Line: calculated results).

that grain boundary diffusion becomes dominant in the product layer diffusion stage, but if the grain boundary diffusion rate is enhanced to 25 times of its original value, grain boundary diffusion also happens in the initial fast reaction stage. However, in this case, this is no a sudden transition from the initial fast reaction stage to the slower product layer diffusion stage, as shown in Figure 8. During the carbonation reaction, CaO is substrate as a solid reactant, the solid product CaCO3 grow on the substrate surface. It can be seen from eq 13 that the thickness of CaO is an important variable for the CaO conversion. The thickness of CaO can be calculated as follows: hCaO,0 = 1/(ρCaO SCaO,0) = 3 × 10−7 /SCaO,0

5. CONCLUSIONS A rate equation theory was developed to provide information on the variations in island size distribution with time evolution. The elemental steps of surface reaction, surface diffusion, and grain boundary and lattice diffusion were included in these rate equations. The macroscopic solid conversion can be calculated by use of the island size distribution information. Through the comparison of calculated results with the experimental information, the developed rate equation theory was demonstrated to explain successfully some macroscopic behaviors, such as the temperature effects. With the increase of reaction temperature, CaCO3 product island sizes increase while the island density decreases, and this lead to the increase of solid conversion in the fast reaction stage (i.e., critical product layer thickness). When the carbonation reaction approaches equilibrium, the rates of nucleation and growth are fast, while the surface reaction is the limited step and results in the macroscopic slow CaO conversion rate. Contrasted with the conventional gas−solid reaction models, such as shrink model, grain model, and pore model, etc., the rate equation theory describing the time dependent change of island densities in term of the processes, which occur on the surface, is a microscopic approach and opens the way to model the kinetics of nucleation and growth processes and build the link between the macroscopic behaviors of a gas−solid reaction with the microscopic mechanism from the molecular level. The

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where ρCaO = 3.31 × 10 (g/m ) is CaO density, SCaO,0 (m /g) is CaO specific surface area, and hCaO,0 (m) is the thickness of CaO substrate. In this study, the specific surface area of CaO is about 11.9 m2/g. Therefore, hCaO,0 is about 2.52 × 10−8 m. In order to investigate the effect of surface area of CaO on the carbonation, different SCaO,0 values were used to get the hCaO,0, and then, the CaO conversion was calculated with the developed rate equation, the results are shown in Figure 9. In this calculation, the SCaO,0 values were increased to 11.9 m2/g, 17.7 m2/g, and 28.0 m2/g from the original 4.8 m2/g. From Figure 9, it can be seen clearly that the CaO conversion increases with the decreasing of CaO substrate thickness (i.e., the increasing of CaO surface area), and the results from the rate equation are reasonable. That is to say, the effect of surface 6

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developed rate equation can be used for not only carbonation reaction of CaO, but also other gas−solid reactions.

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AUTHOR INFORMATION

Corresponding Author

*Telephone: 86-10-62789955. Fax: 86-10-62770209. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research was supported by the National Natural Science Funds of China (No. 51061130535) and by the National Basic Research Program of China (No. 2011CB707301).

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