Article Cite This: Energy Fuels 2017, 31, 12466-12476
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Dynamic Coupling of Pore Structure Evolution with Carbonation Kinetics of CaO-Based Sorbents: Experiments and Modeling Mingchun Li,* Hao Yang, Lanting Song, and Yusheng Wu School of Material Science and Engineering, Shenyang University of Technology, Shenyang 110870, Liaoning, PR China ABSTRACT: The microstructure and pore structure evolution of CaO-based sorbents in three particle size ranges of 0.075−0.1, 0.15−0.18, and 3 mm were investigated using field emission scanning electron microscopy with energy dispersive spectroscopy and nitrogen adsorption−desorption techniques. A clear heterogeneous distribution of elemental carbon across the sorbent particles was found. A particle carbonation reaction model considering the structural evolution effects and the heterogeneously distributed reaction profile were established and verified. It was found that the pore structure of different particle size sorbents all exhibited a transition from the bimodal distribution to the unimodal distribution, which has a great influence on the dynamic reactive characteristics during carbonation. Within the defined particle size range, the carbonation reaction regimes all transform from interface reaction control into mass-transfer control, and the obtained critical product layer thickness that marks the transformation of control regime are 22, 46, and 74 nm, respectively, at 923 K. The maximum relative errors between experimental data and the simulation results calculated separately under interface reaction control (X < 50%) and mass-transfer control (X > 55%) with the effect of pore structure evolution are 12.7% and 12.1% over the defined particle size range.
1. INTRODUCTION CaO-based sorbents as high-temperature solid sorbents used for CO2 capture from the burning of fossil fuels have attracted a great deal of research attention, because of their low cost, high CO2 uptake capacity, and good resistance to high temperature.1−3 In addition, the calcium-looping technology that makes use of the reversible reaction between CaO and CO2 can be combined with clean combustion technology,4 coal gasification hydrogen production technology,5 integrated gasification combined cycle,6 and other advanced energy utilization technologies. These techniques can result in the prevention of global warming, leading to a cleaner atmosphere. The carbonation reaction of CaO with CO2 is composed of two successive steps, namely, a rapid step controlled by chemical reaction followed by an abrupt transition to the slow diffusion-controlled step.7 Results of the past studies suggest that most of the achievable uptake of CO2 occurs in the period of fast reaction, and the capture efficiency appears to be limited by the volume available inside the pores smaller than a certain diameter and also by the maximum thickness of the product layer beyond which the carbonation rate becomes very slow.8,9 For example, Sun et al.10 reported that the pore volume shrinks for smaller pores narrower than 300 nm can be used for marking the turning point from the fast stage to the slow stage during carbonation over a particle size range of 38−45 μm. Alvarez and Abanades11 estimated a critical thickness of product layer around 50 nm for 0.4−0.6 mm particles to assess the turning point of these two stages under the assumption that an even distribution of carbonate exists in the carbonated particle. Form these studies, it can be found that the pore structure evolution and the modification in product layer thickness during carbonation, which affect significantly the kinetics of carbonation, should be the keys to understanding the transition mechanism of the fast and slow reaction stages. The structural characteristics of CaO-based sorbent before and after carbonation have been studied by many researchers,12−14 © 2017 American Chemical Society
but less attention has been paid to the microstructure and pore structure evolution during carbonation over a wide particle size range; no consideration about the dynamic coupling among the structural evolution, product layer thickness and carbonation kinetics were reported. Therefore, it is important to demonstrate the detailed structural evolution during carbonation including pore size distributions and grain size variation and to establish a suitable model describing the dynamic coupling of the pore structure evolution with the critical product layer thickness and the carbonation reaction kinetics. Generally, the carbonation behavior with structural change in a CaO-based sorbent can be modeled according to the unreacted shrinking core model (USC),15,16 random pore model (RPM),17,18 and grain model (GM).19,20 Because of the large porosity of initial CaO-based sorbent particles, carbon dioxide diffuses within the pores and reacts simultaneously on the surface of CaO grains, which leads to a continuous decrease in both porosity and active CaO surface during carbonation. Hence, a desirable characteristic for the mathematical model describing the carbonation behavior is that it should adequately represent the simultaneous diffusion and reaction over the sorbent particles. In addition, the radial distribution of the reaction rate and the effect of varied pore structure associated with carbonate reaction also should be described. The unreacted shrinking core model hypothesizes the reaction to occur at a sharp interface that moves progressively toward the particle center. Zevenhoven et al.21 pointed out that the use of a USC-type model was somewhat self-contradicting when pore diffusion inside the particle is not a limiting mechanism, because a sharp interface at radial position is not present in this case. The USC model is therefore inappropriate for the Received: July 27, 2017 Revised: October 25, 2017 Published: October 25, 2017 12466
DOI: 10.1021/acs.energyfuels.7b02196 Energy Fuels 2017, 31, 12466−12476
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Energy & Fuels carbonation behavior because the carbonation reaction and pore diffusion take place simultaneously over the sorbent particles. The RPM model considers the pore structure as a network of randomly interconnected pores and correlates the carbonation behavior with the internal pore structure based on this geometry. The RPM model is also not fully suitable in this case, because the radial diffusion resistances through the pore network of the sorbent particles used to be ignored in its application.18 The GM model assumes that the sorbent is composed of small grains. The CO2 diffuses through the pores between grains, and each grain is converted according to the unreacted shrinking core model, which is capable of describing the simultaneous diffusion and reaction and illustrating the radial conversion profiles as well as the structure change.22−24 For this study, it is preferable to use a GM model to deal with the coupling between pore structure evolution and carbonation kinetics. The objectives of this research are to investigate the structural evolution and the underlying mechanisms controlling the transformation in reaction regimes of CaO-based sorbent in three particle size ranges of 0.075−0.1, 0.15−0.18, and 3 mm during carbonation and obtain a kinetic model in terms of pore structure evolution for a better understanding of the dynamic carbonation behavior in accordance with the GM model. The dynamic coupling among the pore structure evolution, mass transport, and carbonation reaction regime could be taken into account. The validation of the particle reaction model and the pore structure evolution effect are presented. In addition, the product layer thickness is evaluated and the influences of particle size on the critical product layer thickness and the dynamic reaction features are discussed.
Figure 1. Schematic diagram of the experimental apparatus. and exited from the end of the tube. The gas flow rate, which was controlled by a flow meter, was held constant at 0.6 L/min. When the desired carbonation degree was achieved, the sample holder was removed and cooled to room temperature. The variation of sample mass after carbonation was weighed by a delicate electronic balance. The morphologies of the original and carbonated particles were obtained using a field emission scanning electron microscope equipped with an energy dispersive spectroscopy instrument. To expose the inner parts of the carbonated samples to the electron beam, small subsamples were mildly crushed by pressing them between two pieces of glass. The pore size distribution and the specific surface area of the calcined and carbonate samples at different carbonation stages were determined by the nitrogen adsorption−desorption method. A series of runs at different reaction stages yield the conversion−time response and the pore structure evolution characteristics.
3. RESULTS AND DISCUSSION 3.1. Particle Reaction Model. The field emission scanning electron microscopy (FESEM) analysis could be used to demonstrate the appropriate reaction model followed by the studied samples in this work. The pore size distributions of the original limestone samples with different particle sizes before calcination are presented in Figure 2a, and the corresponding FESEM images of the three samples are also shown by Figure 2b−d. As Figure 2a shows that the original limestone samples with different particle sizes all have a small number of mesopores in the range of 2−50 nm, and the majority of pores are in the range of 2−3 nm. All of the samples show a similar III adsorption isotherm with H4-type hysteresis loops, suggesting the presence of a few slit macropores. The mesopores in samples 0.15−0.18 mm are more than that in the other two samples, which may be due to the presence of more microcracks caused by crushing. As can be seen from Figure 2b,c, the surface of the original 0.075−0.1 and 0.15− 0.18 mm samples is a kind of crushing morphology with some particulate debris, clear cleavage plane, and microcracks. As to the coarse 3 mm limestone particles, a relatively complete structure with sparite, larger slit pores, and microcracks can be observed in Figure 2d. It can be concluded that the original limestone samples before calcination show a dense structure and contain only a few slit pores and microcracks. The comparisons of the surface microstructures for CaObased sorbents with 0.075−0.1, 0.15−0.18, and 3 mm diameters during carbonation, including the initial, middle, and latter carbonation stages, are presented in Figure 3. As Figure 3a,e,i shows, the surface of 0.075−0.1 mm sorbents before carbonation is a kind of porous structure accumulated by sphere-shaped CaO grains with a large number of mesopores between the grains, which changes to a porous framework with a larger pore size range formed by fused bonding grains as the
2. EXPERIMENTAL METHODS Screened natural limestone from Liaoning was the only natural limestone used in this investigation, yielding 56.7% calcined material. The chemical components of the original limestone as measured by Xray fluorescence analysis are shown in Table 1. The particle size of the parent-sorbents was in the ranges of 0.075−0.1, 0.15−0.18, and 3 mm.
Table 1. X-ray Fluorescence Analyses for the Elemental Compound of Limestone component
wt %
component
wt %
component
wt %
CaO CO2 SiO2 MgO Al2O3
54.24 43.30 0.718 0.625 0.6
Fe2O3 K2O Na2O TiO2 SO3
0.242 0.106 0.0940 0.0682 0.0517
SrO MnO P2O5 others
0.0252 0.0229 0.0039 0.0031
The dual fixed bed reactor with a calciner and a carbonator operated under atmospheric pressure was employed to test the structural evolution and its coupling with the kinetics of carbonation. The schematic diagram of the experimental apparatus is shown in Figure 1. Two quartz tube furnaces were used to perform calcination and carbonation, which can be operated at temperatures up to 1200 °C. The diameter of the quartz tube is 60 mm, and the length is 1000 mm. The constant temperature zone in the reactor was greater than 200 mm. The porcelain boat containing about 0.25−1 g samples was first calcined at 900 °C in pure N2, which provided sufficient samples to allow later N2 adsorption−desorption analysis. The calciner was heated from room temperature to the preset calcination temperature at a heating rate of 15 K/min. Once calcination was completed, the sorbents were subsequently circulated into the carbonator operating at the carbonation temperature in a synthesized mixture of 20% vol. CO2 and 80% vol. N2. The reacting gases flowed axially over the samples 12467
DOI: 10.1021/acs.energyfuels.7b02196 Energy Fuels 2017, 31, 12466−12476
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Figure 2. Pore size distributions and FESEM images of the original limestone samples with different particle sizes: (a) pore size distributions and nitrogen adsorption−desorption isotherms; (b) 0.075−0.1 mm; (c) 0.15−0.18 mm; (d) 3 mm.
Figure 3. FESEM images of CaO-based sorbents with different particle sizes before and after carbonation: (a) 0.075−0.1 mm, before carbonation; (b) 0.075−0.1 mm, carbonated partially about 20%; (c) 0.075−0.1 mm, carbonated partially about 40%; (d) 0.075−0.1 mm, carbonated partially about 75%; (e) 0.15−0.18 mm, before carbonation; (f) 0.15−0.18 mm, carbonated partially about 20%; (g) 0.15−0.18 mm, carbonated partially about 40%; (h) 0.15−0.18 mm, carbonated partially about 72%; (i) 3 mm, before carbonation; (j) 3 mm, carbonated partially about 20%; (k) 3 mm, carbonated partially about 45%; (l) 3 mm, carbonated about 71%.
0.075−0.1 mm in different stages are displayed in Figure 3b−d. It can be seen that with increasing of carbonation time, the collective recrystallization of CaCO3 product occurs and leads
particle size increases to 0.15−0.18 mm and 3 mm; the larger the particle size, the more obvious the sintering and fusion of grains. The surface morphologies of the carbonated samples 12468
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Figure 4. Field emission scanning electron microscopy with energy dispersive spectroscopy (FESEM-EDS) spectrum of the surface and fracture cross section of CaO-based sorbent 0.15−0.18 mm after carbonation: (a) FESEM image; (b) EDS spectrum; (c) elemental mapping image of calcium; (d) elemental mapping image of carbon.
Figure 5. Cross-sectional FESEM images of 0.15−0.18 mm sorbents carbonated partially about 60%: (a) cross section near the surface; (b) cross section away from the surface.
to an obvious coarsing and densification of the initial granular structure. The surface of sorbents 0.15−0.18 mm and 3 mm in the early stage of carbonation still has a porous matrix structure, which is composed of a large number of small crystals of nascent CaCO3, as can be seen from Figure 3f,j. With extended carbonation time, the large size CaCO3 crystals was gradually rich (see Figure 3g,k), and a compact granular texture with very few unclosed macropores and cracks was formed in the subsequent carbonation process, as shown in Figure 3h,l. It can be concluded from Figure 3 that the smaller the particle size of CaO-based sorbent, the smaller the CaCO3 grains on the surface of carbonate product. The morphology and compositions of CaO-based sorbent 0.15−0.18 mm after carbonation, including the surface and the fracture cross section, are presented in Figure 4a−d. Figure 4a shows the fracture morphology of CaO-based sorbent after carbonation. It can be seen from Figure 4a that although the surface has been dense, there are still a certain number of isolated pores and internal channels inside the carbonated sorbents. The heterogeneous structure means that the
carbonation conversion ratio over the carbonated sorbents is heterogeneous because of the closure of the surface pores and the increasing diffusion resistance of CO2 from the surface to interior. This can also be confirmed from Figure 4b. Figure 4b presents the EDS spectrum of the carbonated sorbents at different positions. It can be seen from Figure 4b that the surface of the sorbents is mainly calcium carbonate, while the interior contains a certain amount of calcium oxide. The corresponding elemental mapping images of calcium and carbon are shown by panels c and d of Figure 4, respectively. It can be observed that the distribution of elemental calcium is almost homogeneous, while that of elemental carbon is heterogeneous across the sorbent particles. Therefore, the assumption of no radial conversion profiles, which was commonly used in describing the carbonation behavior, is not suitable for the carbonation of natural calcium-based sorbents under atmospheric pressure. Panels a and b of Figure 5 show the cross-sectional FESEM images near the surface and away from the surface of the carbonated samples, respectively. It can be concluded from 12469
DOI: 10.1021/acs.energyfuels.7b02196 Energy Fuels 2017, 31, 12466−12476
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Energy & Fuels Figure 5 that most of the mesopores have been filled up completely in addition to a few residual large pores, and closer to the sample surface there are fewer remaining pores. The results show that the full carbonation within the CaO-based sorbent can not be completed because of the closure of the surface pores and the increasing difficulty of CO2 diffusion as the carbonation reaction proceeds. According to the change of the pore structure characteristics before and after carbonation of CaO-based sorbents with different particle size, it can be assumed that the carbonation reaction takes place simultaneously over the particles, and the degree of conversion has a certain gradient distribution along the particle radial direction because of the difference of diffusion distance and the resistance of CO2 diffusion. The closer to the particle surface, the higher the carbonation conversion rate. That is, the assumption of even distribution of carbonation as well as the shrinking core model adopted in many studies would not appear to be fully adequate. As a result, the carbonation behavior of the CaO-based sorbent in this work is assumed to follow a modified grain model with a porous structure and variable effective diffusion coefficient. 3.2. Dynamic Coupling Kinetic Model. Some of the classical assumptions are removed in the present model. The internal concentration gradient and the radial distributions for both the product layer thickness and the conversion ratio within the particles can be considered. The diffusion inside the pores of sorbents and the diffusion through the solid product layer as well as their performance in carbonation are also included in the proposed model. The structural changes in the reactant particles are taken into account by the inclusion of a variable diffusion coefficient according to the actual measurement of pore structure characteristics. On the basis of the above FESEM and EDS analyses, the fresh sorbent particles produced from natural limestone can be assumed to be composed of nonporous, uniform spherical CaO grains, which grow during the reaction process because of the solid product formation around the grain inner core, so the particle void-fraction is reduced gradually and the diffusion is increasingly hindered as the reaction proceeds. Transport and carbonation reactions occur simultaneously over the spherical samples with constant radius R0. The CO2 concentration and the radius of unreacted core rc in each grain are all varied with time and the radial coordinate within the particles. The physical model of the CaO-based sorbents is depicted schematically in Figure 6. Take a thin spherical shell along the radial direction R as the representative elementary volume, which consists of a number of small grains of initial radius r0. Considering the CO2 diffusion and carbonate reaction take place simultaneously in
the porous particle, the mass balance for reaction gas CO2 on the representative elementary volume scale is given by De[∂ 2CCO2(R , t )/∂R2 + (2/R )∂CCO2(R , t )/∂R ] − RA(R , t ) = ∂CCO2(R , t )/∂t
(1)
The initial and boundary conditions for the problem are CCO2(R , 0) = 0,
rc(R , 0) = r0
(2)
−De(R , t )[∂CCO2(R , t )/∂R ]|R = R 0 = hd[CCO2(R , t ) − C0], [∂CCO2(R , t )/∂R ]|R = 0 = 0
(3)
The initial radius of CaO grain r0 can be deduced from the initial specific surface area S0 as r0 = 3(1 − ε0)/(S0ρ)
(4)
The grain actual radius that varies with the carbonation reaction can be calculated as ⎡ ⎤1/3 MCaCO3ρCaO 3 3 3 ⎢ r0(R , t ) = (r0 − rc (R , t )) + rc (R , t )⎥ ⎢⎣ MCaOρCaCO ⎦⎥ 3 (5)
Based on the pseudosteady assumption, the overall reaction rate RA(R,t) at reaction time t in mass-transfer controlled regime per unit volume can be obtained as RA(R , t ) = [3(1 − ε0)/r03(R , t )] [(r0(R , t )rc(R , t )/(r0(R , t ) − rc(R , t ))]De,g CCO2(R , t ) (6)
According to the stoichiometric relationship of carbonate reaction, the molar amount of the reaction gas consumed in unit volume per unit time RA(R,t) equals that of the consumed calcium oxide, that is r0(R , t )rc(R , t ) De,g CCO2(R , t ) r0 (R , t ) [r0(R , t ) − rc(R , t )]
3(1 − ε0) 3
=−
4πrc 2(R , t )ρCaO drc MCaO dt 4πr0 3(R , t )/3 (1 − ε0)
(7)
Separating the variables in eq 7 and integrating ⎛ ⎞2 ⎤ r0 2(R , t )ρCaO ⎡ r 2(R , t )ρCaO ⎢1 − ⎜ rc(R , t ) ⎟ ⎥ − 0 2MCaO ⎣⎢ 3MCaO ⎝ r0(R , t ) ⎠ ⎥⎦ ⎡ ⎛ ⎞3 ⎤ ⎢1 − ⎜ rc(R , t ) ⎟ ⎥ − D e,g ⎢⎣ ⎝ r0(R , t ) ⎠ ⎥⎦
∫0
t
CCO2(R , t )dt = 0 (8)
The carbonation fractional conversion X can be expressed as ⎡ r (R , t ) ⎤ 3 X (R , t ) = 1 − ⎢ c ⎥ ⎣ r0 ⎦
(9)
The distributions of concentration of reaction gas CCO2(R, t) and the carbonation fractional conversion X(R, t) can be calculated from eqs 1, 2, 3, 6, 8, and 9. When the overall reaction rate is controlled by the interface chemical reaction, RA(R, t) can be given as
Figure 6. Schematic diagram of the porous sorbent and the CaO grain during carbonation: (a) CaO-based sorbent; (b) cross section; (c) CaO grain during carbonation. 12470
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Figure 7. Pore size distribution evolution of CaO-based sorbent with different particle sizes during carbonation at 650 °C: (a) 0.075−0.1 mm; (b) 0.15−0.18 mm; (c) 3 mm.
carbonation conversion proceeds, the first peaks of small-scale mesopores about 3 nm in different particle size samples all increase at the beginning of carbonation, while the subsequent peaks of large-scale mesopores about 48 nm all decrease sharply with an increasing tendency in the most probable pore size at the same time, within a pore size range from 2 to 200 nm. However, when the carbonation fractional conversion exceeds about 25%, both the number of small-scale mesopores and the number of large-scale mesopores decrease steadily, then the pore size distribution tends to be unimodal distribution at the latter stage of carbonation. A probability density function that can simulate both the bimodal and unimodal distributions was used to construct the pore volume percent of interval i (di − Δd/2, di + Δd/2) based on the actual pore structure characteristics for carbonated sorbents, as presented in previous work.25
RA(R , t ) = [(1 − ε0)/(4πr0 3(R , t )/3)]4πrc 2(R , t )kcCCO2(R , t ) (10)
Similar to the derivation of eq 8, the following equation can be easily obtained: r0(R , t )ρCaO ⎡ r (R , t ) ⎤ ⎢1 − c ⎥− kcMCaO ⎣ r0(R , t ) ⎦
∫0
t
CCO2(R , t )dt = 0 (11)
Then, for the interface reaction controlled regime, the concentration of reaction gas CCO2(R, t) and the carbonation conversion ratio X(R, t) can be calculated from eqs 1−3 and 9−11. The product layer thickness rlayer(R, t) can be calculated as a function of the unreacted core: rlayer(R , t ) =
⎡ (ln d − ln μ )2 ⎤ i 1 ⎥ exp⎢ − φ(di) = 2 ⎢⎣ diσ1 2π 2σ1 ⎦⎥
1/3 ⎡⎡ ⎧ ⎛ rc(R , t ) ⎞3⎤ MCaCO3ρCaO ⎛ rc(R , t ) ⎞3⎤ ⎪ ⎢ ⎥ r0⎨ ⎢1 − ⎜ +⎜ ⎟⎥ ⎟ ⎪⎢⎢ ⎝ r0 ⎠ ⎥⎦ MCaOρCaCO3 ⎝ r0 ⎠ ⎥⎦ ⎩⎣⎣
⎫ r (R , t ) ⎪ ⎬ − c ⎪ r0 ⎭
x
+ (12)
⎡ (ln d − ln μ )2 ⎤ 1−x i 2 ⎥ exp⎢ − ⎢⎣ ⎥⎦ diσ2 2π 2σ2 2
(13)
where x is the probability of the first peak for small-scale mesopores. The rate of mass transfer within the pore volume is governed by both the Knudsen and molecular diffusion mechanisms according to the scale of pore size. Considering the effect of variation in pore size during carbonation, the effective diffusion coefficient of CO2 in porous particles resulting from the two contributions mentioned above is given by
3.3. Pore Structure Evolution and Effective Diffusion Coefficient. The effective diffusion coefficient of gaseous reactant CO2 through the pores between grains needs to be modified with the pore structure evolution during carbonation. The pore size distributions of CaO-based sorbents with three particle sizes at different carbonated stages, including the initial, middle, and latter reaction stages, were inferred from N2 adsorption measurements as shown in Figure 7. As the 12471
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Figure 8. Comparison of the experimental data with the calculated results under different carbonation reaction regimes at 650 °C: (a) 0.075−0.1 mm; (b) 0.15−0.18 mm; (c) 3 mm.
⎡ M ⎛ 8R gT dε De = ⎢(DCO2ε /τ ) ∑ ⎜⎜ i φ(di) ⎢ πMCO2 i = 1 ⎝ 3τ ⎣ ⎡ /⎢DCO2ε /τ + ⎢ ⎣
M
⎛dε
∑ ⎜⎜ i=1
i
⎝ 3τ
φ(di)
⎞⎤ ⎟⎥ ⎟⎥ ⎠⎦
8R gT ⎞⎤ ⎟⎥ πMCO2 ⎟⎠⎥⎦
the effect of pore structure evolution are much lower than the experimental results at the initial stage, while in good agreement with the experimental results at the middle and latter stages (X>55%); the maximum relative errors for 0.075− 0.1, 0.15−0.18, and 3 mm particles are 12.1%, 10.7%, and 8.1% respectively. The results calculated by the mass-transfer controlled regime without the pore blockage effect (the effective diffusion coefficient of CO2 in porous particle is considered as a constant), which are presented by dashed lines in Figure 8, can not fit the experimental data well, especially at the stage of abrupt transition from fast reaction to slow reaction, and the maximum relative error between experimental data and these calculated curves reaches 27.3% as X > 55%. This shows that the pore evolution effects, as a relevant mechanism to prevent carbonation, are important in marking the transition between the fast and slow reaction regimes. Hence, it is necessary to discuss and identify the effects of pore structure evolution for a gas−solid reaction system with solid product layer during the research of kinetics. Figure 9 shows the comparison of the experimental data and the calculated results including the influence of pore structure evolution for the three sorbent particles at 700 °C. It can also be found that the results calculated by the interface reaction controlled regime are consistent with the experimental data at the early carbonation stage, and the transformation points of mass-transfer controlled regime rise slightly compared to those in Figure 8. As the carbonation fractional conversion exceeds 60%, the maximum relative error between the experimental data and the results calculated by the mass-transfer controlled regime with pore structure evolution effect is less than 13.2% over the defined particle size range. This shows that the established model considering the pore structure evolution can better simulate the carbonation of CaO-based sorbents within the range of carbonation temperature 650−700 °C.
(14)
Because the average pore radius in the solid product layer of CaO grain is less than 10 nm, the Knudsen mechanism is predominant and the effective diffusion coefficient De,g depends on the molecular velocity and the pore radius. Assuming a cylindrical pore structure for the CaO grain, the average pore radius rav changed with the carbonation degree is given by22 ⎧ 6M CaO [1 − X(R , t ) + X(R , t )MCaCO3ρCaO rav = 2/⎨ r ( R , t )ρCaO ⎩ 0 ⎪
⎪
⎫ /(ρCaCO MCaO)]2/3 ⎬ 3 ⎭ ⎪
⎪
(15)
Then the effective diffusion coefficient De,g can be calculated as De,g = 2ravεg[8R gT /(πMCO2)]1/2 /(3τ )
(16)
3.4. Carbonation Rates and Product Layer Thickness. Comparisons were made between the experimental data and the calculated results obtained by the established mathematical model under different carbonation reaction regimes, as shown in Figure 8. It can be seen in Figure 8 that the experimental results at the early carbonation stage (X < 50%) of sorbents over the defined particle size range are all in good agreement with the results calculated by the interface reaction controlled regime, the maximum relative error being less than 12.7%. In contrast, the simulation results under mass-transfer control with 12472
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aggravates gradually with extended calcination time, and a pore structure with larger pore size range was formed (see Figure 3), which can accommodate more product to fill the pore network available in the calcined sample. In addition, combined with Figure 8, the carbonation conversion ratio corresponding to critical product layer thickness are about 66%, 65%, and 63%, respectively, over the defined particle size range.
4. CONCLUSIONS Systematic theoretical models that take the effects of porous structure evolution, mass transport, and carbonation reaction regime into account were established and verified for predicting the carbonation reaction rate and critical product layer thickness over a defined particle size range. A clear heterogeneous distribution of elemental carbon across the sorbent particles was found by FESEM-EDS, and a modified grain model assuming the carbonation reaction takes place simultaneously over the particles with radial conversion distributions was established and solved. The evolution characteristics of pore structure and its coupling with reaction regimes were discussed. At the middle and latter carbonation stages, the differences between the experimental data and simulation results with the effect of pore structure evolution decrease gradually with the increase in particle size. The critical product layer thickness that marks the onset of the slow carbonation rate are diverse for different particle size sorbents. As the particle size increases from 0.075−0.1 to 0.15−0.18 and 3 mm, the critical product layer thickness calculated numerically are 22, 46, and 74 nm, respectively. The developed model was capable of predicting the experimental trends of this kind of gas−solid reaction with solid product layer and varied pore structure.
Figure 9. Comparison of the experimental data with the calculated results at 700 °C.
Because the controlling step of carbonation reaction rate changes with the continuous thickening of the solid product layer, the critical thickness of the product layer is an important parameter to mark the transition in reaction regime. Figure 10 portrays the variations of the average product layer thickness with reaction time over the defined particle size range, which were calculated by the established model considering the effect of pore structure evolution. It can be seen in Figure 10 that a faster growth rate of the average product layer thickness was followed by a slow growth stage. The carbonate layer thickness corresponding to the slope change point of the calculated curves can be used as the critical thickness to assess whether the carbonation reaction rate is controlled by mass transfer. For particle sizes of 0.075−0.1, 0.15−0.18, and 3 mm at 650 °C, the critical thickness of product layer obtained from Figure 10 are 22, 46, and 74 nm, respectively. This shows that the critical product layer thickness is dependent on the particle size. As the particle size of the parent carbonates increases, the sintering
Figure 10. Average product layer thickness for different particle sizes during carbonation at 650 °C: (a) 0.075−0.1 mm; (b) 0.15−0.18 mm; (c) 3 mm. 12473
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APPENDIX
r0(R , t )rc(R , t ) De,g CCO2(R , t ) r0 3(R , t ) [r0(R , t ) − rc(R , t )]
3(1 − ε0)
A.1. Derivation for Equation 1
Take a thin spherical shell [R, R + ΔR] along the radial direction R as the representative elementary volume (ΔR→0). The CO2 diffusion and carbonate reaction take place simultaneously in the porous particle. The mass change induced by diffusion on the representative elementary volume scale Md can be expressed by
=−
De,g
r0(R , t )MCaO
drc
(A9)
∫0
t
CCO2(R , t )dt = 0 (A10)
A.3. Interface Reaction Control, Derivation for Equations 10 and 11
For a single CaO grain, the reaction rate mc in interface reaction controlled regime can be given as mc = 4πrc 2(R , t )kcCCO2(R , t )
(A4)
(A11)
Then, the overall reaction rate RA(R, t) at reaction time t in interface reaction controlled regime per unit volume can be obtained as
Then, the mass balance (Mt = Md − Mr) for reaction gas CO2 in the representative elementary volume can be given by
RA(R , t ) = mc[(1 − ε0)/(4πr0 3(R , t )/3)]
De[∂ 2CCO2(R , t )/∂R2 + (2/R )∂CCO2(R , t )/∂R ]
= [(1 − ε0)/(4πr0 3(R , t )/3)]4πrc 2(R , t )kcCCO2(R , t )
(A5)
(A12)
Similar to the derivation of eq A8, the following equation can be obtained according to the stoichiometric relationship of carbonate reaction
A.2. Mass-Transfer Control, Derivation for Equations 6 and 8
For a single CaO grain, the diffusion rate (md) of CO2 through the product layer in the mass-transfer controlled regime can be expressed as md = 4πr2 De,g dCCO2/dr, which can be considered as a constant on the basis of pseudo-steady-state assumption. Separating the variables in the expression of md and integrating in the intervals of [r0(R, t), rc(R, t)] and [ CCO2(R, t), 0] gives
(1 − ε0) 3
4πr0 (R , t )/3
4πrc 2(R , t )kcCCO2(R , t )
4πrc 2(R , t )ρCaO drc =− MCaO dt 4πr0 3(R , t )/3 (1 − ε0)
md = 4πDe,g [(r0(R , t )rc(R , t )/(r0(R , t ) − rc(R , t ))]CCO2(R , t )
(A13)
Separating the variables in eq A13 and integrating in the intervals of [0, t] and [r0(R, t), rc(R, t)] gives
(A6)
Then, the overall reaction rate RA(R, t) at reaction time t in mass-transfer controlled regime per unit volume can be obtained as follows
kc
∫0
t
CCO2(R , t )dt = −
rc(R , t )
∫r (R ,t) 0
ρCaO MCaO
drc
(A14)
Furthermore
3
r0(R , t )ρCaO ⎡ r (R , t ) ⎤ ⎢1 − c ⎥− kcMCaO ⎣ r0(R , t ) ⎦
RA(R , t ) = md [(1 − ε0)/(4πr0 (R , t )/3)] 3
= [3(1 − ε0)/r0 (R , t )][(r0(R , t )rc(R , t ) /(r0(R , t ) − rc(R , t ))]De,g CCO2(R , t )
rc(R , t )[r0(R , t ) − rc(R , t )]ρCaO
⎡ ⎛ ⎞3 ⎤ ⎢1 − ⎜ rc(R , t ) ⎟ ⎥ − D e,g ⎢⎣ ⎝ r0(R , t ) ⎠ ⎥⎦
(A2)
The mass change in the representative elementary volume per unit time can be expressed by
− RA(R , t ) = ∂CCO2(R , t )/∂t
rc(R , t )
∫r (R ,t)
⎛ ⎞2 ⎤ r0 2(R , t )ρCaO ⎡ r 2(R , t )ρCaO ⎢1 − ⎜ rc(R , t ) ⎟ ⎥ − 0 2MCaO ⎣⎢ 3MCaO ⎝ r0(R , t ) ⎠ ⎥⎦
(A3)
M t = 4πR2ΔR ∂CCO2(R , t )/∂t
CCO2(R , t )dt
Furthermore
The mass consumed by carbonate reaction in the representative elementary volume can be given by M r = 4πR ΔRRA(R , t )
t
0
Md = 4πR2ΔRDe[∂ 2CCO2(R , t )/∂R2
2
∫0
=−
(A1)
The first term on the right side of eq A1 can be expanded by the Taylor series expansion, then
+ (2/R )∂CCO2(R , t )/∂R ]
(A8)
Separating the variables in eq A8 and integrating in the intervals of [0, t] and [r0(R, t), rc(R, t)] gives
Md = 4π (R + ΔR )2 [De∂CCO2(R , t )/∂R ]R +ΔR − 4πR2[De∂CCO2(R , t )/∂R ]R
4πrc 2(R , t )ρCaO drc MCaO dt 4πr0 3(R , t )/3 (1 − ε0)
∫0
t
CCO2(R , t )dt = 0 (A15)
(A7)
A.4. Derivation for Equations 14 and 16
According to the stoichiometric relationship of carbonate reaction, the molar amount of the reaction gas consumed in unit volume per unit time RA(R, t) equals that of the consumed calcium oxide, that is
The rate of mass transfer within the pore volume is governed by both the Knudsen and molecular diffusion mechanisms according to the scale of pore size. Then, the effective diffusion coefficient De can be calculated from the Knudsen diffusion 12474
DOI: 10.1021/acs.energyfuels.7b02196 Energy Fuels 2017, 31, 12466−12476
Article
Energy & Fuels
De = effective diffusion coefficient of CO2 within pores, variable with X (m2/s) De,g = effective diffusion coefficient in product layer of grain, variable with X (m2/s) di = average pore size of interval i (m) hd = interfacial mass -transfer coefficient (m/s) kc = reaction rate constant (m/s) MCO2 = molecular mass of CO2 (g/mol) MCaO = molecular mass of CaO (g/mol) MCaCO3 = molecular mass of CaCO3 (g/mol) R = coordinate variable in radial direction (m) R0 = initial radius of CaO-based sorbent particle (m) RA(R, t) = overall reaction rate, variable with R and t (mol/ m3·s) Rg = gas constant (J/K·mol) r0 = initial radius of CaO grain (m) r0(R, t) = actual radius of CaO grain (m) rc(R, t) = unreacted core radius of grain, variable with R and t (m) rlayer(R, t) = product layer thickness, variable with R and t (m) rav = average pore radius in product layer of CaO grain (m) S0 = initial specific surface area of CaO-based sorbent (m2/g) T = carbonation temperature (K) t = reaction time (s) x = probability of first peak for small-scale mesopores X(R, t) = carbonation fractional conversion,variable with R and t
coefficient DK and the effective molecular diffusion coefficient of CO2 DM
D K DM D K + DM
De =
(A16)
DK can be calculated from the following equation when the pore size distribution is uniform26 DK = (εd pore/3τ ) 8R gT /(πMCO2)
(A17)
where dpore is pore size. In this work, the actual pore size distributions are obtained by experimental measurements. Then, DK can be expressed as M
DK =
∑ i=1
εdi φ(di) 8R gT /(πMCO2) 3τ
(A18)
DM is calculated by DM = DCO2ε /τ
(A19)
Then, the effective diffusion coefficient De can be expressed as ⎡ M ⎛ 8R gT dε De = ⎢(DCO2ε /τ ) ∑ ⎜⎜ i φ(di) ⎢ πMCO2 i = 1 ⎝ 3τ ⎣ ⎡ /⎢DCO2ε /τ + ⎢ ⎣
M
⎛dε
∑ ⎜⎜ i=1
i
⎝ 3τ
φ(di)
⎞⎤ ⎟⎥ ⎟⎥ ⎠⎦
8R gT ⎞⎤ ⎟⎥ πMCO2 ⎟⎠⎥⎦
Greek Symbols
(A20)
Because the Knudsen mechanism is predominant in the solid product layer of the CaO grain, the effective diffusion coefficient De,g mainly depends on the average pore radius rav. According to eq A17 and the expression of rav, the effective diffusion coefficient De,g can be calculated as
■
De,g = 2ravεg[8R gT /(πMCO2)]1/2 /(3τ )
(A21)
■
AUTHOR INFORMATION
Corresponding Author
*Tel: +86 13604187687. Fax: +86 24 25496301. E-mail:
[email protected]. No. 111, Shenliao West Road, Economic & Technological Development Zone, Shenyang, 110870, PR China.
ε = initial porosity of CaO-based sorbent εg = porosity of the product layer of CaO grain τ = tortuosity σ1, σ2 = distribution parameters μ1, μ2 = location parameters φ(di) = pore volume percent of interval i ρ = density of CaO-based sorbent (kg/m3) ρCaO = density of CaO grain (kg/m3) ρCaCO3 = density of CaCO3 product (kg/m3)
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ORCID
Mingchun Li: 0000-0001-5515-8903 Notes
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS This work was financially supported by the Natural Science Foundation of Liaoning Province (2015020229), the National Natural Science Foundation of China (51004071), Liaoning BaiQianWan Talents Program, and the Scientific Research Fund of Liaoning Province Education Department (LT2014004).
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NOMENCLATURE CCO2(R, t) = concentration of CO2 within porous particle, variable with R and t (kmol/m3) C0 = concentration of CO2 in main stream (kmol/m3) DCO2 = molecular diffusion coefficient of CO2 (m2/s) 12475
DOI: 10.1021/acs.energyfuels.7b02196 Energy Fuels 2017, 31, 12466−12476
Article
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DOI: 10.1021/acs.energyfuels.7b02196 Energy Fuels 2017, 31, 12466−12476
