Rate-equation-based Grain Model for Carbonation of CaO with CO2

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Rate-equation-based Grain Model for Carbonation of CaO with CO2 Hui Wang, Zhenshan Li, Fan Xiaoxu, and Ningsheng Cai Energy Fuels, Just Accepted Manuscript • DOI: 10.1021/acs.energyfuels.7b02751 • Publication Date (Web): 27 Nov 2017 Downloaded from http://pubs.acs.org on December 6, 2017

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Rate-equation-based Grain Model for Carbonation of CaO with CO2 Hui Wang1, Zhenshan Li1*, Xiaoxu Fan2, Ningsheng Cai1 1 Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Thermal Engineering, Tsinghua University, Beijing 100084, China 2 Shandong Academy of Sciences, Jinan 250014, China

Abstract: A rate-equation-based grain model was developed to describe and predict the complex behaviors of the carbonation reaction of CaO with CO2 in calcium looping. In the model, the assumption of a uniform CaCO3 film at the grain scale was replaced with the product island morphology, and the rate equation theory was used to calculate the growth of product islands; this modified grain model was integrated into a particle scale model in which gas diffusion inside a CaO particle and the pore plugging effect were considered. The macroscopic kinetics of the carbonation reaction—including the initial fast stage and the later product layer diffusion stage—could be predicted successfully using the developed theory and was validated by comparison with experimental data. The effects of structural parameters on the carbonation kinetics were discussed. Further, a nanometer-scale grain design criterion for CaO sorbent was proposed to optimize particle structure and achieve maximum CaO conversion in the initial fast stage; this concept was validated and supported by the developed rate-equation-based grain model. This developed model provides a link between microscopic mechanisms at the grain level

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and the gas diffusion behavior inside a sorbent particle, and it can be used to describe the macroscopic kinetics of a gas–solid reaction.

Key words: carbonation, grain model, rate equation, calcium looping

1. Introduction CaO-CO2 carbonation is a typical gas–solid reaction that is of great importance in the field of calcium looping; examples of its applications in this field include sorption-enhanced hydrogen production

1-3

, CO2 separation from flue gas

4-5

or syngas 6, and coal gasification 7. CaO-CO2

carbonation involves the simultaneous occurrence of several physical/chemical steps 8: (1) external diffusion of CO2 from the gas phase to the CaO particle surface; (2) internal diffusion of CO2 into CaO particles; (3) surface reaction of CO2 with CaO, which results in the formation and growth of the solid product CaCO3; (4) product layer diffusion of O2- and CO32- ions through the solid product layer

9-10

; and (5) change in the pore structure owing to the formation of the solid

product CaCO3. These physical/chemical steps are complicated and coupled to control and influence the macroscopic behaviors of the CaO-CO2 carbonation reaction through parameters such as temperature, gas concentration, particle size, porosity, pore size distribution, and grain size. To gain a deeper understanding of the apparent behavior of CaO-CO2 carbonation, mathematical models need to be developed; these models can be further used to improve CaO sorbent performance. Various types of gas–solid models have been developed in previous studies, such as the apparent model 11-12, unreacted shrinking core model 13-14, grain model 15-17, pore model 18-19, and nucleation and growth model

20

, as shown in Figure 1; details of these models can be found in

the literature. The unreacted shrinking core model assumes the solid reactant to be nonporous,


2

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which is not suitable to describe the reaction performance of the CaO particle with large porosity. Among these models, the grain model is widely used to describe the behaviors of the CaO carbonation reaction

21-22

. The grain model assumes CaO to be a spherical porous solid particle

that consists of numerous small grains within, and each of these grains is assumed to be spherical and nonporous. The gas–solid reaction occurs at the interface between the grain surface and the unreacted core; that is, the unreacted shrinking core model is used to describe the grain. The external diffusion (step (1)), internal diffusion (step (2)), and structural change (step (5)) are described at the particle scale, whereas the surface reaction (step (3)) and product layer diffusion (step (4)) are described at the grain scale. (a)缩核模型 pore

固体产物

固体反应物

(a)缩核模型 (a) shrinking core model (a)缩核模型孔

(b)晶粒模型孔

孔

固体产物 product

reactant

固体反应物

(b)(b)晶粒模型 grain model (c) pore model (b)晶粒模型 (c)孔隙模型

(d) (d)成核与和生长模型 nucleation and growth

孔

Figure 1. Summary of some typical gas–solid reaction models 固体产物

For the CaO-CO2 固体产物 carbonation reaction, the formation and growth of the solid product CaCO3 occurring at the grain scale are significantly important, because they involve structural 固体反应物

固体反应物

(c)孔隙模型

(d)成核与和生长模型

(d)成核与和生长模型 change (step (5))(c)孔隙模型 and affect the subsequent mass transfer (steps (2) and (3)) and reaction (step

(3)). Most grain models assume that the solid product forms and grows continuously and uniformly at the interface of the unreacted core and that the morphology of the solid product is a nonporous film that covers the unreacted core 15-16, 21-22. However, this assumption cannot predict the transition of the kinetics from the initial fast stage to the second slower product layer diffusion stage, and it also cannot reasonably predict the temperature effect. Some studies adopted a variable diffusion coefficient to reflect the temperature effect on CaO-CO2 carbonation


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23-26

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from only a mathematical viewpoint; however, experimental evidence demonstrated that the

critical thickness of the nonporous CaCO3 film controls the end of the fast reaction period

27-28

.

The theory of critical product thickness cannot well explain the temperature effect on CaO carbonation; furthermore, it has recently been discovered that the solid product shows three-dimensional island-shaped morphology on the reactant surface

29-31

and that the

temperature affects the size and number of islands and consequently influences the critical conversion 32. A rate equation theory for CaO-CO2 carbonation has been developed to replace the assumption of a critical product layer thickness with a complete calculation of CaCO3 nucleation and growth rates, and it has been demonstrated that the critical size of CaCO3 “product islands” is temperature dependent surface

34-35

33

. However, this previous rate equation theory considered a smooth

and did not consider the steps involved at the particle scale, such as external

diffusion (step (1)), internal diffusion (step (2)), and structural change influencing internal diffusion (step (5)). Furthermore, the rate equation theory cannot reflect some phenomena occurring at the particle scale, for example, pore plugging due to growth of a solid product. The present study proposes a modified grain model for the carbonation of CaO with CO2. The assumption of a uniform CaCO3 film at the grain scale is replaced with the product island morphology, and the rate equation theory is used to describe the growth of solid product islands; this modified grain model is integrated into the particle scale model. The following goals are achieved in this study: (1) simplification of the rate equation theory for CaO grains and development of a rate-equation-based grain model that considers the surface reaction, island growth, and product layer diffusion; (2) integration of the rate-equation-based grain model into Fick’s diffusion equation through consideration of gas diffusion inside CaO particles; (3) development of a numerical algorithm for the nonlinear reaction–diffusion equation and


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validation of the proposed model using experimental data; (4) investigation of the effects of particle structure associated with product layer diffusion and pore plugging on the carbonation reaction and discussion of the effects of structural parameters on the carbonation kinetics; (5) proposal of a nanometer-scale grain design criterion to optimize particle structure and achievement of maximum CaO conversion in the initial fast stage.

2. Theory The proposed model considers CaO as a spherical porous particle composed of a matrix of nonporous and uniform spherical CaO grains having an initial radius r0, as shown in Figure 2. Pores are present between grains. The carbonation reaction occurs when CO2 penetrates and diffuses through the porous areas in the sorbent particle and reaches the surfaces of the CaO grains.

Figure 2. Schematic diagram of CaO particle and grain

The following assumptions are made in the model. (1) The particle size is constant during the reaction process. However, the grain size changes during the reaction because of the different molar volumes of the solid product (CaCO3) with respect to the reactant (CaO). Therefore, the physical properties of the CaO particle, i.e., its porosity and molar density, change with time. (2) The temperature inside the particle is uniform and equal to that of the ambient gas


16

.

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Because of low conversion rate and low heat release during the carbonation reaction of CaO with CO2, the temperature gradient is neglected in both the overall particle and the individual grains. (3) The external diffusion resistance of CO2 from the gas phase to the CaO particle surface is ignored

36

, which is consistent with the typical experimental conditions. Based on theoretical

analysis, the external diffusion of CO2 is much faster than internal diffusion of CO2. The reaction rate is determined by the diffusion of CO2 into the pores, the diffusion of O2- and CO32- ions through the product layer 9-10, and the surface chemical reaction of CO2 with CaO. (4) During the reaction, the CaCO3 product will grow with development of the island’s morphology, and this indicates that a part of the CaO grain surface will be covered with the CaCO3 islands, whereas the remaining part of the CaO surface can still contact CO2 directly. This assumption has already been validated using the rate equation theory 33. 2.1 Model at particle scale According to the mass balance around the porous particle, the CO2 concentration profile in the CaO particle is obtained as follows:

1   2  C  1   0    0  R De M R2  R   R  VCaO t

(1)

The boundary conditions are

C  0  R  0, R   R  R0 , C  C0

(2)

In eq. (1),  denotes the local CaO conversion. The overall CaO conversion () is obtained by integrating all local CaO conversions ():



3 R03



R0

0

R 2  dR


(3)

6

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The effective diffusivity De for CO2 diffusion into the porous CaO particle is calculated as a function of the particle porosity as De  D1  

(4)

where the gas diffusivity through the particle pores, D1, is considered as a combination of the molecular diffusivity DA and Knudsen diffusivity DK. The porosity  is calculated as a function of the initial porosity 0 and the local CaO conversion :

  1  1   0   [1   Z  1   ]

(5)

Here, Z is the stoichiometric molar volume ratio of the solid product (36.9 × 10-6 m3/mol for CaCO3) to the solid reactant (16.9 × 10-6 m3/mol for CaO). 2.2 Model at grain scale 2.2.1 Product island growth The rate equation theory is used to describe the growth of solid product islands at the grain surface. Let δ denote the ratio of the unoccupied CaO area at the grain surface, which will decrease with time. As shown in Figure 2, CO2 contacts two different surfaces: the CaO surface and the CaCO3 surface. When CO2 contacts the CaO surface, the carbonation reaction can occur directly on this surface. However, in the case of CO2 contacting the CaCO3 surface, the diffusion of ions through the CaCO3 product layer is the controlling step.


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Figure 3. Conversion of CaO carbonation with CO2

The initial stage of the carbonation of CaO with CO2 is fast and a high conversion of the grain will be achieved in a relatively short time, as shown in Figure 3. This is followed by a slower stage that is controlled by the diffusion in the solid product layer, where the conversion increases slowly. The typical CaO conversion curve obtained by the grain model is as shown in Figure 3. Due to the assumption of uniform solid product film growth, the grain model, where the unreacted shrinking core model is used to describe the grain, cannot describe the transition behavior of the reaction kinetics. Therefore, it is important to replace the assumption of the uniform product film with the product island morphology, and the key is to describe the growth process of solid product islands. The local CaO conversion  can be written as

 =  I + II   1- 

(6)

where I is the critical grain conversion, which can be calculated as

 I   r03  r2c3  r03

(7)

II is the grain conversion increment during the product layer diffusion stage; it can be calculated as

 II   r2c3  r23  r03

(8)

Here, r2c is the critical unreacted core radius for the CaO grain and r2 is the unreacted core radius for the CaO grain, as shown in Figure 4.


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Figure 4. Schematic diagram of growth of solid product island

The critical reactant layer thickness is given as hc = r0 - r2c, which is temperature dependent and can be determined by calculation using the rate equation theory. For the CaO surface that can directly contact CO2, the carbonation reaction can occur directly on the surface and the amount of consumed CaO is expressed as FCaO  4πr0 2 kCCaO  C  Ce 

n

(9)

where k is the chemical reaction rate constant; CCaO is the CaO active site; n is the reaction order; and Ce is the equilibrium concentration of CO2, which can be calculated as follows 38:

1.826 106  19680  Ce  exp    8.314T T  

(10)

The consumed CaO will form the CaCO3 product, and the CaCO3 product will cover the CaO surface. Therefore, the change in δ satisfies

FCaO  4πr0 2

r0  r2 c d 1    M VCaO dt

(11)

The ratio of the unoccupied CaO area can be calculated by combining eq. (9) and eq. (11), as follows: n M  kCCaOVCaO C  Ce      exp  t r0  r2c  


(12)

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2.2.2 Product layer diffusion In the product layer diffusion stage, the CaCO3 product will cover the CaO surface and hinder the direct contact of CaO with CO2. In this case, the carbonation reaction is controlled by the diffusion of ions through the CaCO3 product layer. It was proposed by Bhatia and Perlmutter 9

and validated by Sun et al.

10

through inert marker experiments that solid reactant ions diffuse

outward from the CaCO3/CaO interface to the CaCO3/gas interface and react with CO2 to form product at the CaCO3/gas interface. Therefore, the chemical reaction occurs at the CaCO3/gas interface. The diffusion flow rate of solid reactant ions can be expressed as

J ion  4πr 2 Ds

dCion dr

(13)

where Ds is the diffusivity of solid reactant ions through the product layer and Cion is the concentration of solid reactant ions. The boundary conditions are given as

r  r2 , Cion  Csi  r  r1 , Cion  Cion,b

(14)

where Csi is considered to remain unchanged during the reaction. Integration of eq. (13) gives

J ion

r1  r2  4πDs  Csi  Cion,b  r1r2

(15)

The diffusion flow rate of solid reactant ions also satisfies eq. (16) and eq. (17):

J ion  4πr12 kCion,b  C  Ce 

J ion  

1 M CaO

V

4 r2 2

dr2 dt


n

(16) (17)

10

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Through combination of eqs. (15)–(17), the change in the unreacted core radius for the CaO grain can be obtained as M VCaO

r12 Csi r2 2 1

dr2  dt r1  r1  r2   n Ds r2 k  C  Ce 

(18)

The expanded CaO grain radius is calculated as

r13  r23  Z  r03  r23 

(19)

2.3 Rate-equation-based grain model Combination of eq. (1), eq. (4), eq. (6), eq. (12), eq. (18), and eq. (19) gives the rate-equation-based grain model as

 1   2 C  1   0    2  R De  M R  VCaO t  R R    1     1  r 3 r 3  2 0  n  M     exp  kCCaOVCaO  C  Ce  t   r0  r2c    2  M r1 VCaO Csi  dr r2 2  2   dt r1  r1  r2  1  n  Ds r2 k  C  Ce   r 3  r 3  Z r 3  r 3 0 2 2 1   De  D1 1  1   0   1   Z  1   



(20)



The unknown variables in the above equations are as follows: the CO2 concentration C; the local conversion ; the ratio of the unoccupied CaO area δ; the grain dimensions r1 and r2; and the effective diffusivity De.

3. Numerical algorithm


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3.1 Differential equations of rate-equation-based grain model Of the above equations of the rate-equation-based grain model, eq. (1) needs to be solved numerically whereas the other equations can be solved analytically. In eq. (1), the change in local conversion is calculated as 2 M 1 CaO 3 0

3(1   )V

r Csi r 1

M r23  kCCaOVCaO  C  Ce     (1  3 ) t r1 (r1  r2 )  r0 r0  r2c n Ds r2 k  C  Ce 

n

(21)

Combination of eq. (1) and eq. (21) gives

1  C  2C  2 De C 2 2 RD  R  R D    f 01  f 02 e e R2  R R R R 2 

(22)

r12 3(1   0 )(1   ) 3 Csi r0 f 01  r1 (r1  r2 ) 1  n Ds r2 k  C  Ce 

(23)

where f01 is expressed as

and f02 is expressed as

 kCCaO  C  Ce  r3 f 02  (1  23 )(1   0 ) r0 r0  r2c

n

(24)

For obtaining the numerical solution, the differential method is used to transfer each term in eq. (22) separately. Considering that the particle is spherical and isotropic, the center is chosen as the origin of coordinates. The particle is uniformly divided into nR layers along the radial direction, with the thickness of each layer being dR (dR = R0/nR). i (i = 0, 1,…, nR-1) denotes the layer number from the center to the particle surface. When i = 1, 2,…, nR-2, the following are obtained:


12

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f1 (i) 

 2C C (i  1)  2C (i )  C (i  1)  2 R i (dR)2

(25)

C C (i  1)  C (i  1)  R i 2dR

(26)

De D (i  1)  De (i  1)  e R i 2dR

(27)

f 2 (i) 

f3 (i) 

When i = nR-1 and C(i+1) = C0, the following are obtained:

f1 (i ) 

C  2C (i )  C (i  1)  2C  0 2 R i (dR)2

(28)

C  C (i  1) C  0 R i 2dR

(29)

De D (i  1)  De (i  1)  e R i 2dR

(30)

f 2 (i) 

f3 (i)  Therefore, eq. (22) converts to

2 De (i) f 2 (i )  f 2 (i ) f3 (i )  De (i ) f1 (i )  f 01 (i )  f 02 (i) idR

(31)

The following relation applies at the particle center: C (1)  C (0)  0

(32)

The above differential equations can be expressed as nonlinear equations as follows:

  F C (0)  C (1)  C (0)    F C (i )  2 De (i) f 2 (i)  (idR) f 2 (i) f 3 (i)  (idR) De (i) f1 (i)  (idR)  f 01 (i)  f 02 (i)  i =1,2,

, nR 1 (33)

3.2 Numerical algorithm for differential equations


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The nonlinear equations in eq. (33) can be written as Fi(xj) = 0, where Fi represents an equation and xj denotes each unknown variable, i.e., C(j). According to the Newton–Raphson method, the iterative step is given as 39  F xk 1  xk   i  x  j

1

  Fi ( x j ) 

(34)

Transposition gives

Fi  x  Fi ( x j ) x j

(35)

The change in the unknown variable, Δx, can be obtained by the Gauss elimination method. Then, in the next iterative step, the unknown variables can be obtained as xk 1  xk  x

(36)

The key step for the Newton–Raphson method is the calculation of the coefficient matrix or Jacobian matrix. For eq. (33), the Jacobian matrix is given as follows: When i = 0, we have F C (0)  1, C (0)

F C (0)  1, C (1)

F C (0)   0 ( j  0,1) C ( j )

(37)

When i = 1, 2,…, nR-2, we have

 F C (i )  2 D (i )  i  dR  f 3 (i ) i  dR  De (i )  e    C ( i  1) 2 dR (dR) 2   1 n   k C  i   Ce  n 1 i  dR  f 02  i   n 2  i  dR  De (i )  F C (i )     i  dR  f 01  i   2  C (i )   r1 (r1  r2 ) 1 (dR) C  i   Ce   n  Ds r2 k C  i   Ce    F C (i )  2 D (i )  i  dR  f (i ) i  dR  D (i ) e 3 e    2 2dR (dR)  C (i  1)   F C (i )   0 ( j  i  1, i, i  1)  C ( j )


(38)

14

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Energy & Fuels

When i = nR-1, we have F C (i )  2 D (i )  i  dR  f 3 (i ) i  dR  De (i )  e   2dR (dR) 2  C (i  1)  n 1   k C  i   Ce  n 1 i  dR  f 02  i   n 2  i  dR  De (i )  F C (i )      i  dR  f 01  i    2 r1 (r1  r2 ) 1 (dR) C  i   Ce  C (i )  n Ds r2  k C  i   Ce    F C (i )   0 ( j  i  1, i )   C ( j )

(39)

Figure 5 shows the flowchart of the detailed algorithm; this algorithm was implemented in Visual C++. The iterative solving steps of this model are as follows: (1) Initialize all parameters. (2) Let t2 = t + Δt. (3) Update r2 and δ using eq. (18) and eq. (12), respectively. (4) Update r1, , and De using eq. (19), eq. (6), and eq. (4), respectively, with the updated r2 and δ. (5) Calculate the present CO2 concentration profile by the Newton–Raphson method. (6) Go to step (2).


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Page 16 of 45

Figure 5. Flowchart of detailed algorithm of rate-equation-based grain model

4. Results 4.1 Model validation In order to validate the proposed model, the overall CaO conversion at different temperatures was calculated and then compared with the results of a thermogravimetric analysis (TGA) experiment, as shown in Figure 6. According to the study of Alonso et al. 36, the external diffusional limitations can be avoided in thermogravimetric analyzers using sufficiently small sample mass. However, too small sample mass will result in poor repeatability of experiments. Considering both avoiding the external diffusional limitations and having good reproducibility of


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TGA experiments, an 8–10 mg sample of analytical pure CaO with an average particle radius of 10 μm was used as the sorbent and loaded in the TGA (Q500) instrument. In the initial period, pure N2 was fed into the instrument at a flow rate of 100 mL/min, and the temperature was increased at a rate of 20 °C/min to the carbonation temperature and maintained for 10 min. When the temperature was stable, a CO2 + N2 gas mixture was introduced to provide 14 vol% CO2 at a flow rate of 100 mL/min. During the carbonation reaction, the weight of the sorbent was monitored and recorded continuously, and the conversion of CaO was calculated from the weight changes and used as an indicator of the CO2 capture performance.

Figure 6. Effects of temperature on carbonation of CaO with CO2 (R0: 10 μm; r0: 137 nm; ε0: 0.52; CO2: 14 vol%; N2: 86 vol%. Dots: experimental data; lines: calculated results)

In this work, the carbonation reaction was considered as a reversible first-order chemical reaction. This assumption was verified in the work of Sedghkerdar 40. According to the study of Alvarez D. et. al. 28, the critical product layer thickness in the reaction of CaO with CO2 is 49 nm at 650 °C, which means

hc  r1c  r2c  49 nm

(40)

And the relationship between r1c and r2c is


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r1c3 - r2c3 Z r03 - r2c3

Page 18 of 45

(41)

It can be derived from the experimental data in Figure 6 that the critical particle conversion at 650 °C is 0.46. Considering that the particle radius is rather small, it is assumed that there is a uniform reaction occurring through the CaO particle. That is to say,

  1

r2c3 =0.46 r03

(42)

Through combination of eqs. (40)–(42), the value of initial radius, r0, is obtained as 137 nm, which was used to obtain the calculated results in the model. The initial porosity, ε0, was measured using the pressure mercury meter (AutoPore IV9510) in the experiment, and the value was equal to 0.52. The following were the adjustable parameters included in the rate-equation-based grain model: chemical reaction rate constant k; CaO active site CCaO; diffusivity of solid reactant ions through the product layer, Ds; concentration of solid reactant ions at the CaCO3/CaO interface, Csi; and critical reactant layer thickness hc. The parameter kCCaO determines the chemical reaction rate in the initial fast reaction stage, and DsCsi determines the reaction behavior of the product layer diffusion stage. Among these parameters, kCCaO can be obtained by fitting the calculated results with the experimental data for the initial fast stage; DsCsi can be obtained by fitting the calculated results with the experimental data for the product layer diffusion stage; and hc can be obtained by fitting the critical CaO conversion, as proposed by Alvarez et al. 28, or by performing a theoretical calculation using the detailed rate equation theory, as proposed by Li et al.

33

. A summary of the values of the parameters used in the proposed

model is given in Table 1. Table 1. Values of Parameters Used in Model


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Energy & Fuels

T

CO2

C0

Ce

kCCaO

DsCsi

hc

°C

vol%

mol/cm3

mol/cm3

cm/s

mol/(cm·s)

cm

515

2.165 × 10-6

3.97 × 10-9

1.5 ×10-3

1.4 ×10-16

1.37 × 10-6

590

1.977 × 10-6

3.18 × 10-8

2.1 ×10-3

2.1 ×10-16

2.01 ×10-6

640

1.869 × 10-6

1.05 × 10-7

3.1 ×10-3

3.4 ×10-16

2.41 × 10-6

690

1.772 × 10-6

3.04 × 10-7

4.0 ×10-3

5.3 ×10-16

2.82 × 10-6

720

1.718 × 10-6

5.46 × 10-7

3.0 ×10-3

5.4 ×10-16

2.95 ×0-6

743

1.679 × 10-6

8.37 × 10-7

1.1 ×10-4

5.9 ×10-16

3.04 × 10-6

14

From Figure 6 and Table 1, it can be seen that the reaction temperature has a significant effect on the carbonation of CaO with CO2. The value of kCCaO generally increases with increasing temperature when the reaction temperature is below 690 °C. However, it decreases when the temperature is higher than 690 °C, and the carbonation rate becomes very low at 743 °C without any obvious critical CaO conversion before 20 min. This is because of the chemical equilibrium limitation, which means the decomposition rate of CaCO3 increases pretty faster than the carbonation rate when the temperature is higher than 720 °C and the overall carbonation reaction rate will decrease

33

. At 743 °C, the product grows with larger islands

morphology than that at lower temperature, but the number of product islands on each grain surface is much smaller due to the slow carbonation reaction rate. Therefore, the CaO conversion at 743 °C is much lower than that at 720 °C. Furthermore, the value of hc generally increases with increasing temperature, which agrees well with the conclusion of Li et al. 33. As can be seen in Figure 6, the conversion profiles obtained using the model agree reasonably well with the experimental data, implying that the model developed in this work well predicts the carbonation performance.


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Page 20 of 45

Figure 7. The comparison of the calculated results from rate-equation-based grain model and traditional grain model with experimental data at 590 °C (R0: 10 μm; r0: 137 nm; ε0: 0.52; CO2: 14 vol%; N2: 86 vol%. Dots: experimental data; lines: calculated results)

Moreover, the calculated results from rate-equation-based grain model and traditional grain model were compared with the experimental data. Taking the condition at 590 °C as an example, as shown in Figure 7, it is obvious that the rate-equation-based grain model can describe the transition of the reaction kinetics from the initial fast stage to the second slower product layer diffusion stage successfully, while the traditional grain models cannot predict this transition behavior. When the diffusivity in the traditional grain model equals to that in the rate-equation-based grain model, there is an obvious difference of the initial reaction performance described by grain model and that shown from experimental data. To reduce this difference in initial reaction stage, an effective way is to increase the diffusivity of the reactant ions through product layer in the grain model. However, it can be seen from Figure 7 that the reduction of the initial reaction performance difference will result in a significant difference of the second stage reaction performance. The main reason is that grain models assume that the solid product forms and grows continuously and uniformly at the grain surface. Therefore, it is necessary to replace the assumption of a uniform CaCO3 film at the grain scale with the product


20

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island morphology, and the rate-equation-based grain model developed in this work is more suitable to describe the carbonation reaction behaviors. 4.2 Effects of structural change on carbonation mechanism The carbonation reaction will terminate when the effective diffusivity De decreases to 0. It can be seen from eq. (4) that De will decrease to 0 when the porosity is equal to 0. From eq. (5), when the porosity is equal to 0, we have

 0,opti =1 

1 1   Z  1   opti

(43)

Under the assumption that the optimal conversion approaches the maximum value, i.e., opti = 1, we have

 0,opti  1 

1 =0.54 Z

(44)

One important necessary condition for De = 0 is that the initial porosity must be no higher than 0.54, as shown in eq. (44). Another necessary condition for De = 0 is that the local CaO conversion must be no smaller than opti, which is calculated using eq. (43). The local CaO conversion depends on several parameters, e.g., the initial grain radius, reaction temperature, and reaction time. The particle structure is associated with two different types of controlling mechanisms for carbonation: the first type is the product-layer-diffusion-based controlling mechanism, which occurs when  ≪ opti though De＞0 during the reaction stage; the second type is pore-plugging-based controlling mechanism, which occurs when ε0 ≤ 0.54 and  ≥ opti, i.e., De = 0. 4.2.1 Product-layer-diffusion-based controlling mechanism When the pore size is large enough for the growth of solid product islands, the initial grain


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Page 22 of 45

size has a strong effect on CaO conversion. As shown in Figure 8, if the grain size is too large, the product layer prevents further contact between the gas and the unreacted CaO core. In this work, the microscopic reaction mechanism at the grain scale is discussed in detail using the developed rate-equation-based grain model.

Figure 8. Carbonation of CaO grain with CO2 under influence of product-layer-diffusion-based controlling mechanism

To demonstrate the effect of a large grain size with a large pore size on the carbonation of CaO with CO2, the initial grain radius is increased to 685 nm while keeping the other structural parameters unchanged. As shown in Figure 9(a), as expected, increasing the grain radius will greatly decrease the initial reaction rate and the overall CaO conversion. This phenomenon can be explained by comparing Figures 9(b)–(d). During the reaction process, the radial distribution of the CO2 concentration is as shown in Figure 9(b), which demonstrates that the diffusion of CO2 through pores is not the controlling resistance. Figure 9(c) shows that the ratio of the unoccupied CaO area decreases rapidly from 1.0 to 0 in the fast reaction stage, which means that the product layer has completely encapsulated the grain. Meanwhile, the porosity decreases from 0.52 to 0.45 in the fast reaction stage and remains at 0.45 in the product layer diffusion stage, which means that the pore size is enough for CO2 diffusion. The slight change in the overall CaO conversion beginning from 3 min is attributed to the fact that product layer diffusion finally becomes the dominant stage in the sorbent and therefore fully controls the carbonation reaction.


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(a)

(b)

(c)

(d) Figure 9. Effects of large grain size with large pore size on (a) CaO conversion, (b) radial distribution of CO2 concentration, (c) change in radial distribution of ratio of unoccupied CaO area with time, and (d) change in radial distribution of porosity with time at 590 °C (R0: 10 μm; r0: 685 nm; ε0: 0.52; CO2: 14 vol%; N2: 86 vol%. Dots: experimental data; lines: calculated results)

An interesting phenomenon observed in Figure 9 is that a uniform reaction occurs through


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Page 24 of 45

the CaO particle when the particle radius is 10 μm; this is because the particle radius is so small that the diffusion resistance for CO2 is negligible. However, the carbonation of a typical CaO particle in a fluidized bed will result in a nonuniform reaction along the radial direction inside the particle. This will occur under the condition of a larger particle radius, e.g., 500 μm, as shown in Figure 10, which is within the typical range of 80–2000 μm reported in the literature 41. The main differences between the results depicted in Figure 10 and Figure 9 pertain to the nonuniform radial distributions of the CO2 concentration, the ratio of the unoccupied CaO area, and porosity. When the particle radius is 500 μm, the diffusion of CO2 through the pores and the pore size are also not the controlling resistance, as shown in Figures 10(b) and (d). The reaction is still controlled by product layer diffusion, which causes a flat radial distribution of the overall CaO conversion beginning from 3 min, as shown in Figures 10(c) and (d). To demonstrate the typical carbonation of CaO with CO2, the particle radius is set as 500 μm by default for the subsequent part of this work.

(a)

(b)


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Energy & Fuels

(c)

(d) Figure 10. Effects of large grain size with large pore size on (a) CaO conversion, (b) radial distribution of CO2 concentration, (c) change in radial distribution of ratio of unoccupied CaO area with time, and (d) change in radial distribution of porosity with time at 590 °C (R0: 500 μm; r0: 685 nm; ε0: 0.52; CO2: 14 vol%; N2: 86 vol%)

4.2.2 Pore-plugging-based controlling mechanism When the grain size is small enough—which means that the grain has the potential to achieve complete carbonation—the pore size will have a strong effect on the CaO conversion. As shown in Figure 11, if the pore size is too small, the pores are not large enough for further growth of the solid product CaCO3, and CO2 cannot diffuse into the particle interior when the pores are plugging.


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Page 26 of 45

Figure 11. Carbonation of CaO grain with CO2 under influence of pore-plugging-based controlling mechanism

To demonstrate the effects of a small pore size with a small grain size on the carbonation of CaO with CO2, the initial grain radius is decreased to 34.25 nm while keeping the other structural parameters unchanged. As shown in Figure 12(a), the smaller grain radius increases the initial reaction rate, whereas the reaction terminates at about 1.5 min and the overall CaO conversion decreases, as expected. It can be seen from Figure 12(d) that there is a vast difference between the porosity at the particle center and that at the particle surface before 1.5 min. The radial distribution of the CO2 concentration is as shown in Figure 12(b). There is also a vast difference between the CO2 concentration at the particle center and that at the particle surface, which reveals the resistance offered by pore plugging upon CO2 diffusion through the particle pores. Furthermore, it can be seen from Figure 12(d) that the porosity near the particle surface decreases to 0 beginning from 1.5 min. Correspondingly, the overall CaO conversion and the radial distribution of the ratio of the unoccupied CaO area remain unchanged beginning from 1.5 min, as shown in Figures 12(a) and (c); this means that pore plugging prevents further growth of the product layer and the reaction terminates. It has to be mentioned that when pore plugging occurs, turning points shown in Figure 12(a) does not take place in practice. However, the reaction only happens on the outer surface of the sorbent particle after pore plugging happens, and thus the reaction rate of this stage is pretty slow, which means the reaction process after pore plugging is not important to improve CaO conversion. Therefore, this reaction process after pore plugging happens is neglected in the model, and the carbonation reaction will stop as soon as the


26

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Energy & Fuels

pore plugging happens.

(a)

(b)

(c)

(d) Figure 12. Effects of small pore size with small grain size on (a) CaO conversion, (b) radial distribution of CO2 concentration, (c) change in radial distribution of ratio of unoccupied CaO area with time, and (d) change in radial distribution of porosity with time at 590 °C (R0: 500 μm; r0: 34.25 nm; ε0: 0.52; CO2: 14 vol%; N2: 86 vol%)


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Page 28 of 45

4.3 Effects of particle parameters on CaO carbonation with CO2 Particle parameters such as the particle size, grain size, and initial porosity have a significant impact on the carbonation performance of CaO with CO2. Therefore, it is necessary to investigate the effects of particle structure on sorbent kinetic behaviors, which depend on the competition between the chemical reaction and diffusion. Sorbent particle size may affect the carbonation reaction through CO2 diffusion into the pores inside the particle. Figure 13 shows the overall CaO conversion for different particle sizes. As can be seen from this figure, the overall CaO conversion increases as the particle size decreases. However, when the particle radius is smaller than 1000 μm, a further decrease in the particle size will not significantly affect the CaO conversion in the product layer diffusion stage, but it will still greatly affect the CaO conversion in the initial fast reaction stage. Furthermore, when the particle radius is smaller than 200 μm, a change in the particle size has no significant effect on the carbonation reaction; Bhatia et al. 9, Grasa et al. 18, Alonso et al. 36 and Sedghkerdar et al.

40

also drew a similar conclusion based on their experimental results. This is due to the

occurrence of a uniform reaction through the solid sorbent, as shown in Figure 9.

Figure 13. Effects of particle size on carbonation of CaO with CO2 at 590 °C (r0: 137 nm; ε0: 0.52; CO2: 14 vol%; N2: 86 vol%)


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The grain size determines the reaction interface in the carbonation process. Smaller grains provide larger reaction interface areas, and this will enhance the reaction inside the sorbent particle and consequently contribute to higher overall conversion in the same time period. However, smaller grains are more difficult to prepare and more likely to sinter in reaction. Figure 14 shows the overall CaO conversion with a variation in grain radius as the carbonation reaction proceeds. When the grain radius is larger than 75 nm, as expected, an increase in the grain size will result in a lower overall conversion in both the initial fast reaction stage and the product layer diffusion stage. However, when the grain radius is smaller than 75 nm, a decrease in the grain radius will result in a lower overall conversion because the pore size is so small that pore plugging occurs as the carbonation reaction proceeds. Pore plugging will lead to high resistance to product layer growth and CO2 diffusion, and the carbonation reaction will stop progressing in advance, perhaps in the product layer diffusion stage, e.g., r0 = 60 nm, or even in the initial fast reaction stage, e.g., r0 = 45 nm. Therefore, a smaller grain size may not lead to a higher overall conversion.

Figure 14. Effects of grain size on carbonation of CaO with CO2 at 590 °C (R0: 500 μm; ε0: 0.52; CO2: 14 vol%; N2: 86 vol%)

The porosity of the sorbent particle plays an important role in the carbonation performance


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Page 30 of 45

of CaO. An optimum porosity will simultaneously provide good mechanical properties, e.g., high strength and attrition resistance, and good chemical properties, e.g., low resistance to diffusion. Figure 15 shows the overall conversion of the CaO particle at different initial porosities. As expected, a higher initial porosity induces a higher overall conversion within 20 min because a particle with high porosity permits the reactant gas to approach active sites more easily. However, when the initial porosity is higher than 0.40, a higher initial porosity does not lead to higher overall conversion in the product layer diffusion stage but it still significantly affects the CaO conversion in the initial fast reaction stage. Furthermore, when the initial porosity is higher than 0.55, the change in the initial porosity does not cause improvement of the carbonation reaction, because pore plugging does not occur under this condition. This critical value of 0.55 agrees well with the value calculated using eq. (44).

Figure 15. Effects of porosity on carbonation of CaO with CO2 at 590 °C (R0: 500 μm; r0: 137 nm; CO2: 14 vol%; N2: 86 vol%)

5. Discussion As discussed above, particle structure has a strong effect on the overall CaO conversion. Recently, experimental methods for modification of particle structure with the aim of improving the sorbent performance have attracted considerable research interest


42-52

. However, such

30

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previous studies could not clarify the mechanism of the effect of particle structure modification. In this study, we propose a nanometer-scale grain design criterion to optimize the modification of particle structure.

Figure 16. Optimal design of grain inside particle

As shown in Figure 16, let r0 be the initial radius of the CaO grain, r2 be the radius of the shrinking core of CaO, r1 be the radius of the CaCO3 shell, and l be the distance between two CaO grains. Then, CaO conversion can be expressed as

r23   1 3 r0

(45)

The relationship between the CaCO3 shell and the CaO core satisfies the following relation:

Z

r13  r23 r03  r23

(46)

From the expression of the reactant layer thickness, the following relation is obtained: h  r0  r2

(47)

From the viewpoints of the chemical reaction and solid product growth, the reactant CaO is expected to be utilized to the maximum extent possible; therefore,  needs to be maximized. If  approaches the maximum, r2 will approach zero; that is, in this case, r2,opti = 0, opti = 1, and r0,opti = h. From eq. (46), the following can be derived:

r1,opti  Z 1/3r0,opti  Z 1/3h


(48)

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Page 32 of 45

As discussed above, the initial stage of CaO carbonation is fast and the later stage is slower, and therefore, the initial stage is of interest for practical applications; the initial stage corresponds to the critical thickness of the reactant layer, hc. The critical thickness of the reactant layer is dependent on temperature; therefore, the optimal initial radius of a grain is also dependent on temperature, i.e., r0,opti ∝ T. After obtaining the optimal grain radius, we focus on the optimal distance between two grains. The distance between two grains (l) affects both solid product growth and gas molecule diffusion. On the one hand, the distance between two grains provides space for the growth of the solid product; this growth space should be large enough for solid product growth, else insufficient space will be available for solid product growth and this will result in termination of the carbonation reaction. The relationship between the optimal conversion of CaO and the distance (l) can be expressed as

 opti  1 

3 r2,opti 3 r0,opti

r2,opti

0, l  2Z 1/3hc    Zh3  l 3 / 8 c )1/3 , 2hc  l  2Z 1/3hc (  Z 1

(49)

On the other hand, gas molecules will diffuse through the space between two grains, and the Knudsen diffusivity depends on the distance l 17:

DK  97rpore T / M  97(l  2r1,opti ) T / M

(50)

It can be seen clearly that both the optimal conversion of CaO and the Knudsen diffusivity depend on the distance l between two CaO grains; therefore, this distance is an important parameter for CaO sorbent design. Figure 17 shows the changes in the profiles of the optimal conversion of CaO and the Knudsen diffusion coefficient with the distance l between two CaO grains. If l < 2Z1/3hc, insufficient space is available for CaCO3 growth and the CaO conversion increases linearly with l. The distance l should be larger than 2Z1/3hc in order to achieve


32

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maximum CaO conversion. However, from the viewpoint of gas molecule diffusion, the Knudsen diffusivity depends on the void distance between grains (l - 2r1,opti), and the void distance should be large enough for gas molecule diffusion to occur.

Figure 17. Effects of distance between grains on CaO conversion and diffusion

From Figure 17, it can be seen that D1, calculated using eq. (51), increases with increasing l:

D1 =

1 1 1 + DK DA

(51)

However, the slope of D1, calculated using eq. (52), decreases with increasing l:

D1 D 1  97 T / M  1 =97 T / M  2 l DK  DK  1+   DA 

(52)

The optimal distance lopti corresponds to

D1 l

lopti

=x 

D1 l

l  2 Z 1/3hc

=x  97 T / M

(53)

where x is a constant smaller than 1. As shown in Figure 17, a higher value of x will result in a smaller value of l and thus a smaller value of D1, which will further lead to a relatively slow reaction rate. However, it doesn’t mean the smaller value of x, the better the reaction performance. The pretty small value of x will result in a much larger value of l, which will surely


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correspond to the poor mechanical strength of particle. Therefore, the value of x should be set after considering various factors, and it is up to the researcher based on their requirement. In this study, the value of x is set as 0.25. From eq. (52), we obtain DK = DA, and this distance should be



l  2r1,opti  DA / 97 T / M



(54)

Finally, the optimal distance is expressed as







lopti  2r1,opti  DA / 97 T / M  2Z 1/3hc  DA / 97 T / M



(55)

Figure 17 shows the size necessary to be used in the nanometer-scale grain optimal design. The nanometer-scale grain optimal design criterion is given as follows: f  l  2r1  0

(56)

where l depends on r0 and ε0; and r1 depends on r0, the product layer thickness (temperature), and the reaction time. Therefore, the criterion function f is dependent on the initial particle structure parameters, temperature, and reaction time. The modeling results of CaO conversion with the particle structure designed using the optimal design criterion at different temperatures are shown in Figure 18(e). It is obvious that the overall CaO conversion will increase to 1.0 in the fast reaction stage, which means that the particle can achieve complete carbonation in a short time when the particle structure is designed optimally, as shown in Figures 18(a)–(c). A higher temperature will result in a higher reaction rate, which is in agreement with the results in Figure 6; therefore, the optimal particle structure depends on the reaction temperature. As shown in Figure 18(d), if the particle is not optimally designed, the distance l between two CaO grains is smaller than the optimal distance lopti; then, the distance between two grains is not large enough for solid product growth and pore plugging occurs, and consequently, the CaO conversion cannot approach the maximum (see Figure 18(e)). Therefore, particle structure optimized using the nanometer-scale grain design criterion can


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Energy & Fuels

greatly improve the overall CaO conversion; consequently, the developed criterion has good application prospects in the experimental modification of sorbent particle structure.

(a)

(b)

(c)

(d)

(e) Figure 18. Modeling results of CaO conversion with different particle structures

6. Conclusions A rate-equation-based grain model was developed for the carbonation of CaO with CO2. In this model, the assumption of a uniform CaCO3 film at the grain scale was replaced with the product island morphology, and the rate equation theory was used to describe the growth of product islands. The developed model was integrated into the particle scale model, and a numerical algorithm was developed for solving the nonlinear reaction–diffusion equation. The carbonation reaction of CaO with CO2, which included an initial fast stage and a subsequent


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product layer diffusion stage, could be calculated using the developed theory, and the calculated results agreed well with experimental data. The results obtained using the model revealed that the controlling step in the case of a large grain size with a large pore size is product layer diffusion whereas the controlling step in the case of a small pore size with a small grain size is pore plugging. The structural parameters, i.e., particle size, grain size, and initial porosity, were demonstrated to play an important role in the carbonation process. The developed model was used to investigate the effects of these parameters on the carbonation reaction at the grain scale in order to determine the optimal particle structure. In this study, a nanometer-scale grain design criterion was proposed to optimize the modification of particle structure. The proposed rate-equation-based grain model provides a link between microscopic mechanisms at the grain level and the macroscopic kinetics of a gas–solid reaction, which is not limited to the carbonation of CaO with CO2. Moreover, the developed nanometer-scale grain design criterion is expected to be useful for the experimental modification of sorbent particle structure.

* Corresponding author. Mailing address: Department of Thermal Engineering, Tsinghua University, Beijing 100084, China Telephone: 86-10-62789955 Fax: 86-10-62770209 E-mail: [email protected]

Acknowledgments This research was supported by National Key R&D Program of China (2016YFB0600801) and National Natural Science Funds of China (No. 51376105, 91434124), and the Shandong Natural Science Foundation (ZR2015YL001) and Shandong Academy of Science Basic Foundation. This


36

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Energy & Fuels

research was supported by EU H2020 Project.

Nomenclature C

concentration of CO2 in the particle, mol/cm3

C0

concentration of CO2 in the ambient gas, mol/cm3

CCaO

CaO active site, dimensionless

Ce

equilibrium concentration of CO2, mol/cm3

Cion

concentration of solid reactant ions, mol/cm3

Cion,b

concentration of solid reactant ions at CaCO3/gas interface, mol/cm3

Csi

concentration of solid reactant ions at CaCO3/CaO interface, mol/cm3

D1

diffusivity of gas through the particle pores, cm2/s

DA

molecular diffusivity, cm2/s

De

effective diffusivity of gas through the particle pores, cm2/s

DK

Knudsen diffusivity, cm2/s

Ds

diffusivity of solid reactant ions through product layer, cm2/s

FCaO

amount of consumed CaO, mol/s

h

reactant layer thickness, cm

hc

critical reactant layer thickness, cm

Jion

diffusion flow rate of solid reactant ions, mol/s

k

chemical reaction rate constant, mol1-n/(cm2-3n·s)

l

distance between two CaO grains, cm

lopti

optimal distance between two CaO grains, cm

M

molar mass of CO2, g/mol

n

reaction order, dimensionless


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Energy & Fuels 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

r

grain radius, cm

r0

initial grain radius, cm

r1

grain radius of CaCO3/gas interface, cm

r1,opti

optimal grain radius of CaCO3/gas interface, cm

r2

unreacted core radius for the CaO grain, cm

r2c

critical unreacted core radius for the CaO grain, cm

r2,opti

optimal unreacted core radius for the CaO grain, cm

rpore

pore size, cm

R

particle radius, cm

R0

initial particle radius, cm

t

reaction time, s

T

temperature, K

VM CaO

molar volume of CaO, cm3/mol

Page 38 of 45

3 VM CaCO3 molar volume of CaCO3, cm /mol

Z

stoichiometric molar volume ratio of the solid product to the solid reactant,

dimensionless α

local CaO conversion, dimensionless

α̅

overall CaO conversion, dimensionless

αopti

optimal local CaO conversion, dimensionless

αI

critical grain conversion, dimensionless

αII

grain conversion increment during the product layer diffusion stage, dimensionless

δ

ratio of unoccupied CaO area, dimensionless



particle porosity, dimensionless


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Energy & Fuels

0

initial particle porosity, dimensionless

0,opti

optimal initial particle porosity, dimensionless

Subscripts 0

initial

c

critical

e

equilibrium or effective

K

Knudsen

s

solid

opti

optimal

I

the initial fast stage

II

the product layer diffusion stage

Superscript M

molar

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Rate-equation-based Grain Model for Carbonation of CaO with CO2

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