Modified Discrete Random Pore Model Considering Pore Structure

Nov 29, 2017 - The O2/CO2 combustion of coal chars currently considered as one of the promising technologies has a remarkable effect on reducing green...
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Modified Discrete Random Pore Model Considering Pore Structure Evolution to Depict Coal Chars Combustion in O2/CO2 Hua Fei,†,‡ Peisheng Li,*,‡ Qing Jun Gu,† and Yang Liu† †

The Laboratory of Energy and Environment Application Engineering, Jiangxi University of Science and Technology, Ganzhou 341000, China ‡ The Laboratory of Thermal Energy and Power Engineering, University of Nanchang, Nanchang 330000, China ABSTRACT: The O2/CO2 combustion of coal chars currently considered as one of the promising technologies has a remarkable effect on reducing greenhouse gas emissions, and it is of importance to investigate the mathematical model of pore evolution at char reaction process, while discrete random pore model (DRPM) can be usually considered as the mathematic modeling of pore evolution for effective prediction of char combustion at O2/CO2. In this work, pore structure variation of three chars during reaction at O2/CO2 were presented by the N2 isothermal adsorption/desorption method. The universally used assumption of proportional relation of superficial area and combustion rate was improved for application to coal chars, indicating true superficial area associated with the reaction rate should be corresponding to the pore superficial area 2πl(r0 + θΔr/2). Therefore, a new pore structure model (MDRPM) was created to simulate carbon conversion of three coals combustion process in O2/CO2 when chemical reaction is considered. Compared with the Liu model, the Struis model, as well as the Bhatia model (DRPM), carbon conversion calculated by the MDRPM was more consistent to experimental values for three char reaction process. According to these models, the combustion characteristic of coal chars in O2/CO2 was investigated.

1. INTRODUCTION Coal is considered as a major energy used for syngas, thermal power, and electrical energy in China. However, the utilization of coal resources brings about a large number of greenhouse gases (such as CO2, SO2, and NO) that are directly discharged into the atmosphere. The O2/CO2 combustion of coals having an important influence on reducing greenhouse gases emissions can be recognized as one of several promising technologies.1−3 Moreover, some researchers have conducted wide and in-depth investigations to study on the evolution feature of the chemical structure and physical structure of coals during reaction at O2/CO2 and also have made lots of research for the influence factors of the pore characteristic of chars during reaction.4,5 It is of importance to further investigate the characteristic property and pore structure model of coals reaction under O2/CO2 atmosphere.6,7 As is well-known, the problems of depicting the physical/ chemical characteristics of coal chars during combustion are important to rapid development of utilization of the coal resources. Coal chars combustion or gasification, which is affected by some aspects (coal type, reaction temperature, reaction atmosphere, and so on), has recently attracted large attention and a large number of mathematic modeling of char particle reaction were presented in previous work.8−12 Whereas less attention was paid to investigate the pore models of the structure evolution during char reactions. The primal pore structure model, pore volume model, and simple particle model constructed by Bhatia and Perlmutter,13 Ramachandran and Doraiswamy,14 and Lee et al.,15 respectively, are considered as being appropriate for describing the carbon conversion of chars during reaction. The pore characteristics of char particles were investigated by Bhatia et al.16 modeling (DRPM) of the literature, which is the improved pore structure model to simulate superficial area evolution of chars during combustion, and it is taking into account the © XXXX American Chemical Society

influence factors of pore property and superficial area of coal chars during combustion has shown better agreement between experimental results and theoretical values. However, other researchers showed some limitations on the DRPM, which have been widely applied for coal utilization.16−18 The property of two chars reaction under O2/CO2 environment were investigated with help of DRPM, RPM, and fractal random pore model (FRPM),17 which showed DRPM could be well simulated to chars reaction process for lower conversion, but larger errors would appear when the DRPM depicts the conversion curve of two chars combustion at the latter stage. Moreover, some researchers presented the property of char particles reaction at carbon dioxide with the DRPM,18 which was poorly appropriate to describe char reaction when pore diffusion effect occurs. Morimoto et al.19 investigated pore structure evolution of char particles, and the structure parameter ψ in the DRPM was mended on the basis of pores evolution. Gupta and Bhatia20 revised the DRPM in pore structure ψ based on different initial surface reactivity of particle structure, which often was more consistent to experiment results of lower conversion. In this context, a new MDRPM is created to simulate the carbon conversion of three chars during combustion in O2/CO2 when dynamical effect is controlled. The simulation analysis of MDRPM is compared to that of the Liu model, the Struis model, and DRPM developed by Liu et al.,21 Struis et al.,22 and Bhatia et al.,16 respectively, carbon conversion obtained by the MDRPM taking into account the influence factors of pore structure of char particles combustion process is more accurate Received: October 4, 2017 Revised: November 29, 2017 Published: November 29, 2017 A

DOI: 10.1021/acs.energyfuels.7b02987 Energy Fuels XXXX, XXX, XXX−XXX

Article

Energy & Fuels

According to the DRPM, the reaction rate variation with dimensionless time can be given by

to describe the experimental results, indicating that MDRPM proposed in this work is considered as being appropriate to simulate carbon conversion of char particles during reaction under dynamical reaction control.

⎡ ⎛ ψτ ⎞⎤ ⎟⎥ X = 1 − exp⎢ −τ ⎜1 + ⎣ ⎝ 4 ⎠⎦

(2)

dX = dτ

(3)

2. EXPERIMENTAL SECTION Xiaolongtan lignite (XLTL), Yunfu bituminous (YFB), and Jiaozhuo anthracite (JZA) were used in this work. The properties analysis of three coals were accomplished in TGA2000 (Navas Instruments, Spain) and vario EL-2, and the properties of three coals are presented in Table 1. Thermogravimetric experiments of the three chars reaction in O2/CO2 were accomplished as follows. First, coal samples (10 ± 0.2 mg) coated on the crucible were lowered into the reactor of the TGA system. The purified nitrogen (≥99.999%) was applied to sweep the TGA experiment system for about 40 min in order to establish an inert environment. Second, coal samples were heated up to (fixed at 30 K/min) the predetermined temperature of 800 and 900 °C in N2 environment fixed at 100 mL/min and pumped in reaction gas of O2/CO2 (fixed at 80% carbon dioxide and 100 mL/min) and kept 15−25 min afterward. The thermogravimetric experiments of three chars reaction were repeated for the analytical accuracy of TGA experiments. Average relative errors of thermogravimetric experiments were less than ±2.0%. To investigate the structure evolution of three chars (Xiaolongtan lignite chars, Yunfu bituminous chars, and Jiaozhuo anthracite chars) reaction under O2/CO2 atmosphere and validate the reliability of experimental results above, char samples were prepared in O2/CO2 at different combustion time by a tube furnace. At first, the coal particles coated on the crucible were pyrolyzed in N2 (100 mL/min) up to the desired value of 1000 °C for 30 °C/min, and then char samples were moved into the top zone of the combustion apparatus and cooled in N2. After that, a device of the sample preparation should be preheated to the desired value of 1000 °C and were pumped in O2/CO2 (fixed at 80% carbon dioxide and 100 mL/min) to move out of other gas from combustion apparatus later on. As the carbon dioxide content of outlet of the coal char preparation device is 80% in total gas, the samples coated on the crucible were rapidly moved from the top zone into the heating zone of combustion apparatus, and reached a desired time and then immediately cooled at N2 environment. The most experiments of char prepared by using the tube furnace were repeated three times for the analytical accuracy of three chars. The surface area SBET of three coal chars at 1000 °C under O2/CO2 atmosphere measured by using Micromeritics ASAP2020 was evaluated by a multilayer adsorption model.23

in which τ is dimensionless time, ψ is structure parameter, and can be given by ψ=

4πL0(1 − ε0) S0 2(1 + α)2

(4)

where ε0 is initial porosity, S0 is original surface area of pore, α is discreteness parameter, and L0 is original total length of pore. 3.2. Modified Discrete Random Pore Model. On the basis of DRPM, it assumes that the radial interlining space in coal chars particle is Δr, and combustion or gasification in the coal char pore takes place by the radial erosion of the char particle.16 The changes of the pore volume in char particle according to consumption of a single layer are expressed as Δv = πl[(Δr + r0)2 − r0 2] = 2πlΔr(Δr /2 + r0)

(5)

in which r0 is pore radius. According to eq 1, in the continuum limit, eq 5 can be changed to dv = 2πl(r0 + Δr /2)ksC n dt

(6)

According to eq 6, it indicates that the surface area related to the reaction rate of the pores in coal chars is corresponding to the superficial area of pores in char particles 2πl(r0 + Δr/2) rather than 2πr0l. On the other hand, L(r0) is supposed to be length distribution function of the pore structure during coal chars combustion in O2/CO2, LE is the total length of nonoverlapped system and can be expressed as LE =

3. MATHEMATIC MODELING 3.1. Discrete Random Pore Model. The DRPM presented by Bhatia et al.16 is based on the assumption that combustion or gasification of coal chars is taken place in the pore surface. The pores size distribution of coal chars consisted of random cylindrical pores, and combustion or gasification rate of the coal char is proportional to the superficial area of pores and is expressed as dv dr = S = SksC n dt dt

1 − ψ ln(1 − X ) ·(1 − X )

∫0

SE = 2π VE = π

∞

L(r0) dr0

∫0

∫0

(7)

∞

∞

r0L(r0) dr0

(8)

r0 2L(r0) dr0

(9)

in which VE is total volume in char particles, and SE is total superficial area in char particles. According to the DRPM and eq 7,

(1)

in which C is gaseous concentration, v is pore volume, S is surface area, n is gaseous reaction order, ks is reaction rate dr constant, ksC n = dt = k , and k is growth rate of pore size.

dS E = 2πksC nL E dt

(10)

dVE = ksC nSE dt

(11)

Table 1. Proximate and Ultimate Analysis of Coal Samples proximate analysis (wt %, air-dried)

ultimate analysis (wt %, air-dried)

sample

moisture

volatile

ash

fixed carbon

C

H

N

S

O

Xiaolongtan Lignite (XLTL) Yunfu Bituminous (YFB) Jiaozuo Anthracite (JZA)

11.82 10.65 1.43

43.92 25.91 10.93

15.98 17.10 20.37

28.28 46.34 67.27

46.87 58.03 71.20

4.90 3.97 3.17

5.04 1.00 1.22

2.01 1.26 0.70

13.38 7.99 1.91

B

DOI: 10.1021/acs.energyfuels.7b02987 Energy Fuels XXXX, XXX, XXX−XXX

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Energy & Fuels

in which ω = dθ/dt. The term of Δr in eq 13 is small enough according to the small increment of Δr, the above equation can be expressed as

The pore structural property of char reaction in CO2 and air were investigated by Feng and Bhatia,24 and it showed that the volume and superficial area in the small microporous chars increase significantly with carbon conversion of coal chars during reaction in CO2, while the volume and superficial area for micropores of pore sizes 10 Å. In addition, it was observed that the pore structural variation of char particles during reaction was related with closed pores and submicropores,19,25 which should be taken into account when using the DRPM and the RPM. It also indicates that the superficial area of pores varies as the rate of chars reaction, which should be mainly affected by some aspects (coal type, reaction atmosphere, reaction temperature, and so on). Therefore, the true surface area related to the reaction rate of the pores should be corresponding to the surface area of the pores in coal chars 2πl(r0 + θΔr/2). On the basis of the above analysis and eqs 6,7, and 10, the nonoverlapped area SE′ is expressed as SE′ = 2π + θπ

∫0

∫0

∞⎛

Δr ⎞ r + θ ⎟L(r0) dr0 = 2π ⎝0 2 ⎠ ⎜

∫0

dSE′ = (2 + θ )πksC nL E + ωπL EΔr ≈ (2 + θ )πksC nL E dt

On the basis of eq 8, the nonoverlapped volume VE′ is expressed as VE′ = π + θπ

∫0

= VE +

∞⎛ ⎜

⎝

Δr ⎞⎟ L(r0) dr0 = π 2 ⎠ 2

r0 + θ

θ 2π Δr 2 ∫

∞

∫0 ∞

0

Δrr0L(r0) dr0 +

∞

r0 2L(r0) dr0

L(r0) dr0

4 2

θ ΔrSE θ π Δr L E + 2 4

(15)

The derivative with respect to time, the following equation is given by θS dr ωθπ Δr 2L E ωSEΔr dVE′ dV = E 0 + E + + dt 2 dt dt 2 2 n 2 r θ S k C d S θ π rL Δ d dV θ Δr E 0 E + + = E s + E + 2 dt 2 dt 2 dt n θπ Δ rL k C ωSEΔr E s + + θ ΔrπksC nL E + 2 2 θS k C n 3θπL EksC n ωθπ Δr 2L E dV + = E + E s + Δr 2 dt 2 2 θS k C n ωθπ Δr 2L E dV + ωSEΔr + = E + E s 2 dt 2 ⎛ 3θπL EksC n ωθπ ΔrL E ⎞ + Δr ⎜ + ωSE + ⎟ ⎝ ⎠ 2 2 (16)

∞

r0L(r0) dr0 (12)

in which θ is the impact factor of the pore surface area, and it is assumed that there is a linear characteristic between the impact factor of the pore surface area and reaction time. The derivative with respect to time, the above equation can be transformed to dr dSE′ dS = E + θπL E 0 + ωπL EΔr dt dt dt = (2 + θ )πksC nL E + ωπL EΔr

∫0

2

∞

ΔrL(r0) dr0 = SE + θπ ΔrL E

(14)

(13)

Similarly, the term of Δr in eq 16 can be small enough to other terms for a small increment of Δr and can be excluded from the equation. The above equation can be transformed to θk C nS dVE′ dV = E + s E = (1 + θ /2)ksC nSE = ksC nSE′ dt dt 2

(17)

Combining eqs 11 and 15−17, the following equation is given by SE′ =

SE0′ 2 + (1 + θ /2)4πL E0(VE′ − VE0′)

VE′ = VE0′ + SE0′ksC nt + (1 + θ /2)πL E0(ksC nt )2

(18) (19)

The porosity ε and superficial area S′ according to RPM are given by 13

ε = 1 − exp(− VE′)

(20)

S′ = SE′(1 − ε)

(21)

The carbon conversion X = (m0 − m)/(m0 − mash) which, according to the porosities, can be transformed to X = (ε − ε0)/ (1 − ε0). The mash is ash mass of coal, m0 and m is original mass, instantaneous mass of chars, respectively. On the basis of eqs 17−21, the following equations are given by

Figure 1. Relationship between pore properties: (a) pore volume and (b) specific surface area and carbon conversions for three coal chars. C

⎡ ⎛ ⎞⎤ 1 + θ /2 X = 1 − exp⎢ −A 0t ⎜1 + ψ A 0 t ⎟⎥ ⎝ ⎠⎦ ⎣ 4

(22)

dX = A 0 1 − (1 + θ /2)ψ ln(1 − X ) ·(1 − X ) dt

(23)

DOI: 10.1021/acs.energyfuels.7b02987 Energy Fuels XXXX, XXX, XXX−XXX

Article

Energy & Fuels in which A0 = ksCnS0/(1 − ε0), ψ is a structural parameter, eqs 22 and 23 are appropriate to simulate carbon conversion and reaction rate of coal char combustion process in O2/CO2 under dynamical effect control.

where φ is the Thiele modulus, which can be obtained by H ksρ0 /DO2 /CO2,eff .28 The DO2/CO2,eff of coal char particles should fully be associated with the tortuosity τp, the temperature T and the influence of the porosity ε28 ε DO2 /CO2,eff = Dmol,O2 /CO2, T0(T /T0)1.75 τp (25)

4. RESULTS AND DISCUSSION 4.1. Experimental Analyses of the Diffusional Factor. The combustion experimental analyses of three coal chars in O2/CO2 were accomplished on thermogravimetric balance (NETZSCH STA 409C). At purified nitrogen of 100 mL/min, the coal char particles coated on the crucible composed of geometric construction of diameter d (5.5 mm) and height H (4 mm) should be construed as one pellet, which should be heated up to predetermined value for 30 K/min and then pumped in reaction gas of O2/CO2 (fixed at 80% carbon dioxide and 100 mL/min) for about 25 min. For investigating the main influencing factors of diffusion on thermogravimetric experiments, a μpore of pore effective factor of char particles could be obtained by using an equation of the following Thiele modulus.26,27 3⎡ 1 1⎤ − ⎥ μpore = ⎢ φ ⎣ tanh φ φ⎦ (24)

where ks is combustion rate constant, ρ0 is initial bulk density of coal chars, Dmol is molecular diffusion coefficient, Deff is effective diffusion coefficient, T is the temperature of coal chars combustion. Under atmospheric pressure, the value of ε/τp is estimated as 0.2, and the Dmol value of O2/CO2 in coal char particles is 1.387 × 10−5 m2 s−1.28,29 The μpore of effectiveness factor should not be more than 5.1% for three coal chars combustion in O2/CO2 at the temperature range from 800 to 1000 °C. Therefore, three coal chars combustion in O2/CO2 should be controlled mainly by dynamic reactions at temperature of T < 1000 °C. It shows that the MDRPM and DRPM can be employed for simulating three coal chars reaction. 4.2. Characteristics of Pore Structure and Model Simulation. The volumes of porous structure in three char particles and specifical surface areas (SSA) in three char particles were obtained by isothermal adsorption/desorption (N2) and BET method. Figure 1a shows the changes of pore volumes of three coals combustion in O2/CO2 with carbon conversion, which can be determined by changing the amount of N2 adsorbed.30 At the X = 0.2−0.5, it shows a maximum, which is consistent with Balci et al.31 investigating results. Similar to the variation of pore volume of XLTL at 1000 °C,

Figure 2. Simulation results of the MDRPM and DRPM for three coal chars combustion in O2/CO2.

Figure 3. Comparison of predicted results by the MDRPM and other models for XLTL combustion in O2/CO2. D

DOI: 10.1021/acs.energyfuels.7b02987 Energy Fuels XXXX, XXX, XXX−XXX

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Energy & Fuels

For investigating the structure changes of coal chars during reaction at O2/CO2, MDRPM of this work and DRPM presented by Bhatia et al.16 can be considered as being appropriate for simulating SSA of three chars O2/CO2 reaction process. The blank points in Figure 2 are the calculated values of the SSA obtained by the MDRPM and DRPM. As can be seen, the MDRPM and DRPM are in better agreement with the results of SSA, indicating that the MDRPM and DRPM can be used to depict three chars reaction in O2/CO2. However, some minor errors occur in the latter stage of reaction when the MDRPM and DRPM simulate the experimental values of the SSA, especially to DRPM, which can be brought about by the hypothesis for the structural parameter of DRPM.16 4.3. Prediction of the Models. The three coal chars combustion in the O2/CO2 environment are controlled prevailingly by the dynamical region for temperature being not greater than 1000 °C. This means that the parameter C in the MDRPM is constant because the O2/CO2 concentration within the pore structure of the coal chars is uniform, indicating that the MDRPM and DRPM can be employed in the present work. Figure 3 presents compared results of experiment values and carbon conversions of coal chars O2/CO2 reaction calculated by the MDRPM, DRPM, and the others established by Liu et al.21 and Struis et al.22 for 800 and 900 °C. Parameters used in models are shown in Table 2. It showed that the errors appear for MDRPM, DRPM, the Liu model and the Struis model predicting carbon conversions of the XLTL combustion in O2/CO2, and conversion rate estimated by MDRPM and the Struis model has less errors from the experimental results than those obtained by the DRPM and the Liu model. Variation curve

the volume of YFB and JZA increases with the conversions of char at initial stage before decreasing in the latter stages of reaction, which can be prevailingly connected with pore combination and pore enlargement of coal chars during reaction.19,32 The SSA of coal chars may strongly influence the reactivity of char combustion in O2/CO2, but the SSA of chars are largely impacted by initial temperature of pyrolysis, coal type, and reaction temperature. The interrelation between the conversion rate and SSA of three coals is presented in Figure 1b. It is observed that the SSA of XLTL, YFB, and JZA increases at the preceding stage of chars O2/CO2 reaction. This investigation is consistent to the reported results of the literature for some coals, which is caused by many micropore enlargement as well as new pores occurrence of coal chars during combustion at O2/CO219,25 and then decreases for X > 0.5, which is prevailingly brought about by the collapse of pore during chars combustion. Table 2. Parameters Used in MDRPM and Other Models sample

XLTL

YFB

JZA

parametera

800 °C

900 °C

800 °C

900 °C

800 °C

900 °C

ψ ω α P b

3.3 0.25 0.032 2.3 0.078

3.6 0.15 0.032 3.6 0.13

3.9 0.15 0.032 3.9 0.081

3.7 0.15 0.032 2.5 0.035

3.0 0.15 0.032 3.2 0.051

2.1 0.05 0.032 2.9 0.031

ω is a constant of dimensionless in MDRPM; b is a constant of dimensionless in the Struis model; P is a dimensionless power law constant in the Struis model.

a

Figure 4. Comparison of predicted results by the MDRPM and other models for YFB combustion in O2/CO2.

Figure 5. Comparison of predicted results by the MDRPM and other models for JZA combustion in O2/CO2. E

DOI: 10.1021/acs.energyfuels.7b02987 Energy Fuels XXXX, XXX, XXX−XXX

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Energy & Fuels

Figure 6. Calculation error results of three coal chars combustion in O2/CO2 by the different models.

expression that is appropriate for simulating coal chars combustion in O2/CO2 where chemical reaction is considered.

of carbon conversion obtained by MDRPM as well as the others and experimental results for YFB and JZA combustion in O2/CO2 environment are shown in Figures 4 and 5, which also show the average errors of experimental results and carbon conversion calculated by several models. As can be seen, the calculated results of the MDRPM proposed in this study are more satisfactory than the calculated results of Liu model and DRPM, while the Liu model provided the least satisfactory calculated results, especially to higher conversions. It indicates that the Liu model and DRPM are only appropriate to dynamical control region to middle conversion rates of three chars during combustion. 4.4. Error Analysis. The imitative effects of MDRPM, DRPM and the others are validated according to relative errors. The relative errors of experimental results and carbon conversions calculated by using several models are shown in Figure 6. For the temperature is at 800 and 900 °C, the carbon conversion predicted by the MDRPM and Struis model has less deviation than the predicted results of the DRPM and Liu model. At the conversion of X > 0.7, the larger departure from experiment data to the values estimated by the DRPM and Liu model is observed from Figure 6, especially to Liu model, but the MDRPM proposed in this work predicts carbon conversion more accurately than that of DRPM and Liu model. This phenomenon relates with the assumption in the DRPM that SSA is proportional to the rate of char reaction, indicating that the DRPM can be applied only to moderate carbon conversions. Therefore, the MDRPM of this work presents simplified mathematic

5. CONCLUSIONS The Xiaolongtan Lignite, Yunfu Bituminous, and Jiaozhuo Anthracite are the major coal in China. In this context, the characteristics of the pore structure of three chars O2/CO2 reaction were investigated. Higher temperature increases coal chars combustion rate. The SSA and pore volume have a remarkable impact on the overall process of coal char combustion, and the maximum of char pore volume occurs for X = 0.2−0.5. The SSA increases in the initial phases on account of some micropores enlargement as well as new pores occurrence for chars during reaction in O2/CO2 but then decreases at X > 0.5, which is primarily caused by the combination and collapse of pores. In addition, a new pore structure model (MDRPM) is created to describe the conversion rate of chars reaction at O2/CO2 when the dynamical effect is considered. Comparison of estimated values of MDRPM with the Struis model, the Liu model, and DRPM has been presented in the previous section, the estimated values of MDRPM of this work are more satisfactory, while the Liu model and DRPM are only appropriate for dynamical control region to middle conversion rates of coal chars combustion process. Moreover, this method offers a simple mathematical expression that can be handled with small calculations. Therefore, it can be used as an effective F

DOI: 10.1021/acs.energyfuels.7b02987 Energy Fuels XXXX, XXX, XXX−XXX

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Energy & Fuels

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rapid prediction for experimental results of chars combustion process.

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AUTHOR INFORMATION

Corresponding Author

*Phone: 86-791-83969634. Fax: 86-791-83969625. E-mail: [email protected]. ORCID

Peisheng Li: 0000-0003-1800-8900 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors are thankful for financial assistance from the Natural Science Foundation (Grants 51666004 and 20171BAB206041) and the Program of Qingjiang Excellent Talents.

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NOMENCLATURE b = dimensionless constant C = gaseous concentration Deff = the coefficient of effective diffusion in char particles H = height ks = reaction rate constant k = growth rate of the pore size LE = total pore length of coal chars at any time L(r0) = distribution function of pore length L0 = original total pore length mash = ash mass of coal char m0 = original mass of coal chars m = instantaneous mass of coal chars n = gaseous reaction order P = dimensionless constant of Struis modeling r0 = pore radius S = surface area S0 = original total superficial area in char particles SE = total superficial area in char particles at any time S E ′ = corrected superficial area corresponding to 2πl(r0 + θΔr/2) T = temperature t = time v = the pore volume of chars VE = total volume of chars VE′ = corrected total volume of char particles X = conversion

Greek Letters

α = a discreteness parameter ε = porosity ε0 = initial porosity ψ = structure parameter φ = Thiele modulus ω = a constant of dimensionless in MDRPM μpore = the pore effectiveness factor τ = dimensionless time τp = tortuosity ρ0 = initial bulk density of the solid θ = impact factor of the pore surface area

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DOI: 10.1021/acs.energyfuels.7b02987 Energy Fuels XXXX, XXX, XXX−XXX