# Kinetic Analysis of Low-Rank Coal Pyrolysis by Model-Free and Model-Fitting Methods

Authors: Qiuli Zhang; Min Luo; Long Yan; Aiwu Yang; Xiangrong Hui
Journal: Journal of Chemistry
 (2019)

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## Abstract

Coal SJC, coal WJG, coal ZJM, and coal HCG were selected to investigate the pyrolysis kinetics of northern Shaanxi coals. TG and DSC experiments of four coals were carried out with a synchronous thermal analyzer at heating rates 5, 10, 15, and 20 C/min, respectively. The pyrolysis characteristics were described by thermogravimetric data, and the kinetic parameters were calculated by Flynn–Wall–Ozawa (FWO), Kissinger, general integration, and MacCallum–Tanner methods. The results show that coal SJC, coal ZJM, and coal HCG all conform to the reaction series equation, the thermal decomposition reaction rate is controlled by chemical reaction, and coal WJG conforms to Avrami–Erofeev equation. The activation energies of the four coals are 177.53 kJ/mol, 200.34 kJ/mol, 158.59 kJ/mol, and 240.47 kJ/mol, respectively.


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## Body

## 1. Introduction

Coal is the most important primary energy in China at present. The national characteristics such as “rich coal, less gas, and poor oil” determine that the main energy status of coal will not change in a long period of time [1]. The coal resources in northern Shaanxi are rich, and the coal has characteristics low ash, low sulfur, low phosphorus, and high calorific values. Furthermore, its pyrolysis products are ideal materials to replace oil and gas [2]. At the same time, pyrolysis, as the basis of coal thermal conversion (gasification, liquefaction, and oxidation) process, is the first step to develop new coal utilization technology [3]. At present, many studies on coal pyrolysis have been reported at home and abroad. The distributed activation energy model (DAEM) is the most common method to study the dynamic behavior of coal pyrolysis. It has been used to analyze the pyrolysis kinetics of typical long flame coal [4], Russian anthracite and Italian bituminous coal [5], Yunnan, Shandong, and Inner Mongolia coal [6], and Ningxia and Heilongjiang coal [7]. Based on the standard DAEM model, 2-DAEM and 3-DAEM are proposed. In recent years, the model-free method and model-fitting method have been applied to the kinetic analysis of the solid fuel pyrolysis process. The model-free method (i.e., isoconversion method) obtains relatively reliable apparent activation energy without involving the kinetic model function. By comparing the activation energy values of different conversion rates, the consistency of the reaction mechanism of the whole pyrolysis process is verified. The model-fitting method is a method that combines the experimental data with the dynamic model. The model-free methods include the Flynn–Wall–Ozawa (FWO) method, Kissinger method, Starink method, and Friedman method. The Coats–Redfern method, MacCallum–Tanner method, general integration methodm and some others are model-fitting methods. Shi et al. [8] used the FWO, Kissinger, and FR methods to analyze the kinetic characteristics of combustion of moso bamboo and hydrothermal charcoal. The activation energies of the FWO and Kissinger methods were similar. Wang et al. [9] discussed the thermal degradation behavior of nano-CaCO3, BPF, epoxy resin, and IMBPF/epoxy resin composites in nitrogen atmosphere. The average activation energy of IMBPF/epoxy resin composites obtained by FWO and Vyazovkin and Weight (VW) methods was higher than that of the unmodified BPF. Furthermore, the value determined by the FWO method was better than that by the VW method. Peng et al. [10] used the Coats–Redfern method to calculate the global and main product kinetic parameters of coal pyrolysis based on flow-bed hot-carrier gas. It was found that higher initial temperature or velocity of gas heat carrier and smaller particle size or moving speed of coal could accelerate the rate of volatilization and shorten the pyrolysis time. Jain et al. [11] described the devolatilization kinetics of typical Indian coal by means of the model-free method and model-fitting method. Liu et al. [12] determined the pyrolysis kinetic parameters of Inner Mongolia superfine pulverized coal by Coats–Redfern, Starink, and Miura–Maki DAEM methods and found that the activation energies determined by the Starink method were more accurate.However, there are few reports on the kinetics analysis of coal pyrolysis by the FWO method, Kissinger method, general integration method, and MacCallum–Tanner method. Meanwhile, there are few studies on coal in northern Shaanxi. Therefore, in this work, TG and DSC analysis experiments with different heating rates were carried out on coal samples from four different mining areas in northern Shaanxi. The kinetic parameters activation energyE and pre-exponential factor A of each coal sample pyrolysis process were calculated by the FWO method, Kissinger method, general integration method, and MacCallum–Tanner method, respectively. It will lay a foundation for the following kinetics analysis of gas product release and solid coke change in coal pyrolysis.


## 2. Methods

### 2.1. Materials

Four coals with distinctive coalification extent were collected from different mines in northern Shaanxi, China, denoted as Wang Jiagou (WJG), He Caogou (HCG), Sun Jiacha (SJC), and Zhang Jiamao (ZJM). The coal samples were arranged in accordance with increasing carbon content. In order to avoid the effect of mass and heat transfer on the pyrolysis process, the samples were pulverized and sieved into <0.075 mm (200 mesh) of particle size used for TG-DSC experiment. The proximate and ultimate analyses of four coal samples are listed in Table1. It can be seen that coal WJG has the lowest volatile content (25.73 wt.%) and the highest ash content (24.92 wt.%). The volatile content of others was high in the range of 38.76–54.73 wt and the ash was low in the range of 1.04–5.00 wt.%. The contents of carbon and hydrogen of all coal samples ranged from 78.16 to 83.27 wt.% and from 3.87 to 5.42 wt.%, respectively.Table 1
Proximate and ultimate analyses of coal samples.
Property
Coal SJC
Coal WJG
Coal ZJM
Coal HCG
Proximate analysis (wt.%, ad)
Moisture
6.22
4.74
6.71
2.58
Volatiles
52.83
25.73
54.73
38.76
Fixed carbon
35.95
44.61
36.66
57.62
Ash
5.00
24.92
1.90
1.04
Ultimate analysis (wt.%, daf)
Carbon
78.16
78.41
82.88
83.27
Hydrogen
5.33
3.87
5.42
5.41
Oxygen
14.94
15.55
10.08
8.78
Nitrogen
1.10
1.16
0.89
2.07
Sulfur
0.47
1.01
0.73
0.47


### 2.2. TGA Experiment

The pyrolysis experiments were carried out in a simultaneous DSC-TGA (NETZSCH–STA 409) under high-purity N2 with a flow rate of 50 ml/min. In each experiment, about 5 mg coal samples were placed in a platinum crucible and heated at room temperature to 900°C at four different heating rates: 5, 10, 15 and 20°C/min. Each experiment was repeated at least three times to eliminate the vibration error.


### 2.3. Principle of Dynamics

The rate of solid-state chemical reaction is generally expressed as follows [13]:(1)dαdt=kTfα,where fα is the differential mechanism function that is correlated to the extent of conversion α; kT represents the temperature dependence of the rate constant which can be substituted with the Arrhenius equation; and T=T0+βt, T0 is the initial temperature, the heating rate β is a constant, t is the time.Then, equation (1) is transformed into the following forms under nonisothermal conditions adopted in this work:(2)dαdT=Aβfαexp−ERT,where A and E are kinetic parameters that represent the pre-exponential factor and activation energy, respectively. R is referred to as the gas constant.The integration of equation (2) is as follows:(3)Gα=∫0αfα=Aβ∫T0Texp−ERTdT.The conversion rateα can be calculated from TG data of thermogravimetric analysis, and the mechanism function can be selected from 47 function forms in the literature. The formulas for calculating α are as follows:(4)α=m0−mtm0−m∞.where m0 is the initial mass, mt is the residual mass at t time, and m∞ is the mass of the sample at the end of the reaction. The purpose of kinetic analysis is to obtain activation energy E, pre-exponential factor A, and mechanism function of the reaction process fα.


## 2.1. Materials

Four coals with distinctive coalification extent were collected from different mines in northern Shaanxi, China, denoted as Wang Jiagou (WJG), He Caogou (HCG), Sun Jiacha (SJC), and Zhang Jiamao (ZJM). The coal samples were arranged in accordance with increasing carbon content. In order to avoid the effect of mass and heat transfer on the pyrolysis process, the samples were pulverized and sieved into <0.075 mm (200 mesh) of particle size used for TG-DSC experiment. The proximate and ultimate analyses of four coal samples are listed in Table1. It can be seen that coal WJG has the lowest volatile content (25.73 wt.%) and the highest ash content (24.92 wt.%). The volatile content of others was high in the range of 38.76–54.73 wt and the ash was low in the range of 1.04–5.00 wt.%. The contents of carbon and hydrogen of all coal samples ranged from 78.16 to 83.27 wt.% and from 3.87 to 5.42 wt.%, respectively.Table 1
Proximate and ultimate analyses of coal samples.
Property
Coal SJC
Coal WJG
Coal ZJM
Coal HCG
Proximate analysis (wt.%, ad)
Moisture
6.22
4.74
6.71
2.58
Volatiles
52.83
25.73
54.73
38.76
Fixed carbon
35.95
44.61
36.66
57.62
Ash
5.00
24.92
1.90
1.04
Ultimate analysis (wt.%, daf)
Carbon
78.16
78.41
82.88
83.27
Hydrogen
5.33
3.87
5.42
5.41
Oxygen
14.94
15.55
10.08
8.78
Nitrogen
1.10
1.16
0.89
2.07
Sulfur
0.47
1.01
0.73
0.47


## 2.2. TGA Experiment

The pyrolysis experiments were carried out in a simultaneous DSC-TGA (NETZSCH–STA 409) under high-purity N2 with a flow rate of 50 ml/min. In each experiment, about 5 mg coal samples were placed in a platinum crucible and heated at room temperature to 900°C at four different heating rates: 5, 10, 15 and 20°C/min. Each experiment was repeated at least three times to eliminate the vibration error.


## 2.3. Principle of Dynamics

The rate of solid-state chemical reaction is generally expressed as follows [13]:(1)dαdt=kTfα,where fα is the differential mechanism function that is correlated to the extent of conversion α; kT represents the temperature dependence of the rate constant which can be substituted with the Arrhenius equation; and T=T0+βt, T0 is the initial temperature, the heating rate β is a constant, t is the time.Then, equation (1) is transformed into the following forms under nonisothermal conditions adopted in this work:(2)dαdT=Aβfαexp−ERT,where A and E are kinetic parameters that represent the pre-exponential factor and activation energy, respectively. R is referred to as the gas constant.The integration of equation (2) is as follows:(3)Gα=∫0αfα=Aβ∫T0Texp−ERTdT.The conversion rateα can be calculated from TG data of thermogravimetric analysis, and the mechanism function can be selected from 47 function forms in the literature. The formulas for calculating α are as follows:(4)α=m0−mtm0−m∞.where m0 is the initial mass, mt is the residual mass at t time, and m∞ is the mass of the sample at the end of the reaction. The purpose of kinetic analysis is to obtain activation energy E, pre-exponential factor A, and mechanism function of the reaction process fα.


## 3. Results and Discussions

### 3.1. TG-DSC

Figure1 shows the TG and DTG profiles of the pyrolysis process of SJC, WJG, ZJM, and HCG coals at different heating rates. According to the TG and DTG curves, the pyrolysis process can be roughly divided into three stages. The first stage is the desorption stage, the removal of water and gas adsorbed by coal particles is usually completed before 300°C. The second stage is the thermal decomposition process, the main weight loss stage, the unstable bridge of the coal and the weak bond generate gas, tar, and semicoke, and the temperature range is between 350∼650°C. Finally, the third stage is the polycondensation reaction between the semicoke generated in the thermal decomposition stage, releasing a small amount of gas to form coke. As the heating rate increases, the TG profile shifts to the left, and the final weight loss of the coal becomes larger, and the temperature corresponding to the peak of the DTG becomes larger. This is because the progress of the pyrolysis reactions and the escape of the products require a certain period of time. When the heating rate increases, some products cannot precipitate in time, resulting in a lag phenomenon. The DSC curve shown in Figure 2 describes the endothermic and exothermic phenomena of physical and chemical reactions during coal pyrolysis. It can be seen that the change in enthalpy of pyrolysis of different coals at different heating rates has no obvious regularity because the coal structure and pyrolysis process are quite complex, the structure and composition of different coals are different, and the pyrolysis process is also different.Figure 1
The curves of TG and DTG.Figure 2
DSC profiles.


### 3.2. Kinetic Analysis

In this paper, the kinetics of the second stage of pyrolysis, i.e., the thermal decomposition stage, which accounts for the main weight loss, is analyzed. The analytical methods adopted include the FWO method, Kissinger method, general integral method, and MacCallum–Tanner method. The specific calculation formulas are as follows.


#### 3.2.1. FWO Method

(5)
log
β
i
=
log
A
E
α
R
G
α
i
−
2.315
−
0.4567
E
R
T
i
,
i
=
1,2,3,4
,where βi is the heating rate, Eα is the activation energy, and Ti is the corresponding reaction temperature on TG data. Under different heating rates, with the same conversion, Gα will be a constant value. Then, logβi versus 1/Ti becomes a linear line so that E can be calculated from the slope of the regression line.


#### 3.2.2. Kissinger Method

(6)
ln
β
i
T
p
i
2
=
ln
A
K
R
E
K
−
E
K
R
1
T
p
i
,
i
=
1,2,3,4
,where EK is the activation energy, AK is the pre-exponential factor, and Tpi is the temperature corresponding to the peak on DTG. The linear regression curve of lnβi/Tpi2−Tpi was plotted according to the temperature corresponding to the maximum weight loss rate in four DTG data. The activation energy EK was calculated by slope and the pre-exponential factor AK was calculated by intercept.


#### 3.2.3. General Integral Method

Based on the Coats–Redfern approximate expression, equation (3) is transformed into equation (7):(7)Gα=AβRT2E1−2RTEexp−ERT.In this work, equation (7) can be simplified considering 2RT/E≪1, the logarithmic form is presented as:(8)lnGαT2=lnARβE−ERT.


#### 3.2.4. MacCallum–Tanner Method

Whenu=E/RT, equation (3) can be rewritten as(9)Gα=Aβ∫T0Texp−ERTdT=AEβR∫E/RT+∞e−uu2du=AEβRPu,where the function Pu is the temperature integral that has no analytic solutions. Different from the general integral, MacCallum–Tanner does not require approximation of Pu. The relationship between Pu and E is as follows:(10)−logPMTu=0.4828E0.4357+0.449+0.217E0.001T.Thus, the MacCallum–Tanner equation can be obtained through substituting equation (10) into equation (9):(11)logGα=logAEβR−0.4828E0.4357−0.449+0.217E0.0011T.When using the general integral method and MacCallum–Tanner method, it is necessary to assume the function form ofGα and to fit lnGα/T2−1/T and lnGα−1/T, respectively. When the correlation coefficient of fitting curve approaches 1, it is the correct mechanism function. The three principles for determining the correct mechanism function are high linear correlation coefficient, close activation energy to that of FWO and Kissinger methods, and clear physical meaning of the mechanism function.The data used in the FWO method, general integration method, and MacCallum–Tanner method areα−T of the TG profile. When calculating, firstly, the data points are collected from the TG curve and the αi is obtained by equation (4). Then, the corresponding Ti data of αi in the range of 0−1.0 are collected by the step size of 0.025. The Kissinger method is based on four different heating rates and the temperature corresponding maximum weight loss rates. Figure 3 reveals the activation energy of four coals in the thermal decomposition stage calculated by the FWO method.Figure 3
E − α.Figure3 indicates that the activation energies of SJC, WJG, ZJM, and HCG coals during pyrolysis have been changing, with the activation energies ranging from 114.99 to 630.31 kJ/mol, 80.13 to 302.15 kJ/mol, 74.32 to 164.97 kJ/mol, and 123.51 to 404.49 kJ/mol, respectively. The variation tendency of activation energy of coal SJC can be divided into three parts: when 0 < α < 0.150, E increases continuously; when 0.150 < α < 0.800, E decreases steadily; and when 0.800 < α < 1.0, E decreases rapidly. The change in activation energy is the manifestation of the diversification of internal reaction mechanism. Because the activation energy of the initial and termination stages of the reaction changes dramatically, it is generally considered that there is no stable mechanism function, and the whole reaction mechanism is determined by the mechanism of the dominant process [14]. Therefore, the range of the mechanism function of SJC coal thermal decomposition reaction is determined to be from 0.150 to 0.800. The activation energies of WJG, ZJM, and HCG are similar to those of SJC and can be divided into three steps. Similarly, the range of mechanism functions of WJG, ZJM, and HCG coal is 0.050–0.800, 0.100–0.800, and 0.100–0.920, respectively.According to 47 mechanism functions in the literature, the general integration method and MacCallum–Tanner method fitα−T data of four coals at different heating rates in the light of equations (5) and (8) respectively, and determine the correct mechanism function of coal thermal decomposition reaction according to three principles. The results show that the most probable mechanism function of thermal decomposition reaction of coal SJC, coal HCG, and coal ZJM is function No.37, and the most probable mechanism function of coal WJG is function No.17. The differential form, integral form, and physical meaning of the most probable mechanism function are shown in Table 2. The kinetic parameters of the thermal decomposition stage for all coals calculated by four algorithms are shown in Tables 3–6. In these calculations, except for the poor linear correlation coefficients calculated by the Kissinger method for SJC, the Kissinger method for WJG, and the FWO method for ZJM (correlation coefficients are 0.9689, 0.9700, and 0.9751, respectively), the other correlation coefficients are all above 0.98, the overall fitting accuracy is high, and the kinetic calculation results are reliable.Table 2
The differential form and integral form of mechanism function.
Coal
Number
Function name
f
α
G
α
Function mechanism
SJC
37
Reaction series
(1−α)2
(1−α)−1–1
Chemical reaction
WJG
17
Avrami–Erofeev
2
/
3
1
−
α
−
ln
1
−
α
−
1
/
2
−
ln
1
−
α
3
/
2
Randomly nucleating and nucleus growth
HCG
37
Reaction series
(1−α)2
(1−α)−1–1
Chemical reaction
ZJM
37
Reaction series
(1−α)2
(1−α)−1–1
Chemical reactionTable 3
Calculation results of coal SJC.
Method
β (°C/min)
E (kJ/mol)
A (1/S)
R
2
The general integral
5
223.657
1.16 × 1014
0.9997
10
160.223
9.85 × 108
0.9985
15
163.37
1.65 × 109
0.9984
20
153.127
5.09 × 108
0.9956
MacCallum–Tanner
5
228.401
3.13 × 1014
0.9997
10
165.168
2.41 × 109
0.9998
15
168.254
4.04 × 109
0.9986
20
158.027
1.22 × 109
0.9963
Average
177.53
2.75 × 1010
0.9983
Kissinger
—
247.84
5.19 × 1015
0.9689
FWO
—
329.526
—
0.9856Table 4
Calculation results of coal WJG.
Method
β (°C/min)
E (kJ/mol)
A (1/S)
R
2
The general integral
5
217.04
9.55 × 1012
0.9940
10
207.72
1.78 × 1012
0.9960
15
187.76
6.92 × 1010
0.9919
20
187.23
6.03 × 1010
0.9874
MacCallum–Tanner
5
221.86
2.57 × 1013
0.9947
10
212.71
4.79 × 1012
0.9965
15
192.74
1.78 × 1011
0.9930
20
192.35
1.62 × 1011
0.9891
Average
200.34
8.51 × 1011
0.9928
Kissinger
—
203.05
5.99 × 1011
0.9700
FWO
—
192.83
—
0.9920Table 5
Calculation results of coal ZJM.
Method
β (°C/min)
E (kJ/mol)
A (1/S)
R
2
The general integral
5
172.95
1.32 × 1010
0.9987
10
152.63
4.57 × 108
0.9999
15
153.55
4.17 × 108
0.9994
20
145.77
1.23 × 108
0.9990
MacCallum–Tanner
5
177.46
3.09 × 1010
0.9990
10
157.20
1.05 × 109
0.9999
15
158.43
1.02 × 109
0.9995
20
150.72
2.88 × 108
0.9992
Average
158.59
2.88 × 108
0.9992
Kissinger
—
153.90
2.46 × 108
0.9909
FWO
—
140.58
—
0.9751Table 6
Calculation results of coal HCG.
Method
β (°C/min)
E (kJ/mol)
A (1/S)
R
2
The general integral
5
276.67
3.98 × 1017
0.9995
10
252.23
5.13 × 1015
0.9997
15
211.63
8.71 × 1012
0.9991
20
210.43
6.76 × 1012
0.9998
MacCallum–Tanner
5
282.17
1.41 × 1018
0.9996
10
257.84
1.70 × 1016
0.9997
15
216.94
2.51 × 1013
0.9992
20
215.86
1.95 × 1013
0.9998
Average
240.47
1.05 × 1015
0.9995
Kissinger
—
223.67
1.53 × 1013
0.9976
FWO
—
264.33
—
0.9900The calculated results show that the kinetic parameters of four coals obtained by two kinds of the model-free method are different, and the order of activation energy is ZJM < WJG < HCG < SJC. This is due to the different structure and composition of different coals and the different reactions during pyrolysis. The average activation energies of coals SCC, WJG, ZJM, and HCG calculated by model-fitting methods are 177.53 kJ/mol, 200.34 kJ/mol, 158.59 kJ/mol, and 240.47 kJ/mol, respectively. Coals SJC, HCG, and ZJM conform to the reaction series equation. The rates of thermal decomposition reaction are controlled by chemical reaction, but their coefficients and exponents of the specific mechanism function expression are different. Coal WJG conforms to the Avrami–Erofeev equation. The expressions of average activation energy, pre-exponential factor, and reaction mechanism function of different coals are substituted into equation (2), respectively, and the kinetic equations of their thermal decomposition reactions are obtained. Through the corresponding kinetic equation, the thermal decomposition reaction process of the same kind of coal can be simulated at any heating rate.


## 3.1. TG-DSC

Figure1 shows the TG and DTG profiles of the pyrolysis process of SJC, WJG, ZJM, and HCG coals at different heating rates. According to the TG and DTG curves, the pyrolysis process can be roughly divided into three stages. The first stage is the desorption stage, the removal of water and gas adsorbed by coal particles is usually completed before 300°C. The second stage is the thermal decomposition process, the main weight loss stage, the unstable bridge of the coal and the weak bond generate gas, tar, and semicoke, and the temperature range is between 350∼650°C. Finally, the third stage is the polycondensation reaction between the semicoke generated in the thermal decomposition stage, releasing a small amount of gas to form coke. As the heating rate increases, the TG profile shifts to the left, and the final weight loss of the coal becomes larger, and the temperature corresponding to the peak of the DTG becomes larger. This is because the progress of the pyrolysis reactions and the escape of the products require a certain period of time. When the heating rate increases, some products cannot precipitate in time, resulting in a lag phenomenon. The DSC curve shown in Figure 2 describes the endothermic and exothermic phenomena of physical and chemical reactions during coal pyrolysis. It can be seen that the change in enthalpy of pyrolysis of different coals at different heating rates has no obvious regularity because the coal structure and pyrolysis process are quite complex, the structure and composition of different coals are different, and the pyrolysis process is also different.Figure 1
The curves of TG and DTG.Figure 2
DSC profiles.


## 3.2. Kinetic Analysis

In this paper, the kinetics of the second stage of pyrolysis, i.e., the thermal decomposition stage, which accounts for the main weight loss, is analyzed. The analytical methods adopted include the FWO method, Kissinger method, general integral method, and MacCallum–Tanner method. The specific calculation formulas are as follows.


### 3.2.1. FWO Method

(5)
log
β
i
=
log
A
E
α
R
G
α
i
−
2.315
−
0.4567
E
R
T
i
,
i
=
1,2,3,4
,where βi is the heating rate, Eα is the activation energy, and Ti is the corresponding reaction temperature on TG data. Under different heating rates, with the same conversion, Gα will be a constant value. Then, logβi versus 1/Ti becomes a linear line so that E can be calculated from the slope of the regression line.


### 3.2.2. Kissinger Method

(6)
ln
β
i
T
p
i
2
=
ln
A
K
R
E
K
−
E
K
R
1
T
p
i
,
i
=
1,2,3,4
,where EK is the activation energy, AK is the pre-exponential factor, and Tpi is the temperature corresponding to the peak on DTG. The linear regression curve of lnβi/Tpi2−Tpi was plotted according to the temperature corresponding to the maximum weight loss rate in four DTG data. The activation energy EK was calculated by slope and the pre-exponential factor AK was calculated by intercept.


### 3.2.3. General Integral Method

Based on the Coats–Redfern approximate expression, equation (3) is transformed into equation (7):(7)Gα=AβRT2E1−2RTEexp−ERT.In this work, equation (7) can be simplified considering 2RT/E≪1, the logarithmic form is presented as:(8)lnGαT2=lnARβE−ERT.


### 3.2.4. MacCallum–Tanner Method

Whenu=E/RT, equation (3) can be rewritten as(9)Gα=Aβ∫T0Texp−ERTdT=AEβR∫E/RT+∞e−uu2du=AEβRPu,where the function Pu is the temperature integral that has no analytic solutions. Different from the general integral, MacCallum–Tanner does not require approximation of Pu. The relationship between Pu and E is as follows:(10)−logPMTu=0.4828E0.4357+0.449+0.217E0.001T.Thus, the MacCallum–Tanner equation can be obtained through substituting equation (10) into equation (9):(11)logGα=logAEβR−0.4828E0.4357−0.449+0.217E0.0011T.When using the general integral method and MacCallum–Tanner method, it is necessary to assume the function form ofGα and to fit lnGα/T2−1/T and lnGα−1/T, respectively. When the correlation coefficient of fitting curve approaches 1, it is the correct mechanism function. The three principles for determining the correct mechanism function are high linear correlation coefficient, close activation energy to that of FWO and Kissinger methods, and clear physical meaning of the mechanism function.The data used in the FWO method, general integration method, and MacCallum–Tanner method areα−T of the TG profile. When calculating, firstly, the data points are collected from the TG curve and the αi is obtained by equation (4). Then, the corresponding Ti data of αi in the range of 0−1.0 are collected by the step size of 0.025. The Kissinger method is based on four different heating rates and the temperature corresponding maximum weight loss rates. Figure 3 reveals the activation energy of four coals in the thermal decomposition stage calculated by the FWO method.Figure 3
E − α.Figure3 indicates that the activation energies of SJC, WJG, ZJM, and HCG coals during pyrolysis have been changing, with the activation energies ranging from 114.99 to 630.31 kJ/mol, 80.13 to 302.15 kJ/mol, 74.32 to 164.97 kJ/mol, and 123.51 to 404.49 kJ/mol, respectively. The variation tendency of activation energy of coal SJC can be divided into three parts: when 0 < α < 0.150, E increases continuously; when 0.150 < α < 0.800, E decreases steadily; and when 0.800 < α < 1.0, E decreases rapidly. The change in activation energy is the manifestation of the diversification of internal reaction mechanism. Because the activation energy of the initial and termination stages of the reaction changes dramatically, it is generally considered that there is no stable mechanism function, and the whole reaction mechanism is determined by the mechanism of the dominant process [14]. Therefore, the range of the mechanism function of SJC coal thermal decomposition reaction is determined to be from 0.150 to 0.800. The activation energies of WJG, ZJM, and HCG are similar to those of SJC and can be divided into three steps. Similarly, the range of mechanism functions of WJG, ZJM, and HCG coal is 0.050–0.800, 0.100–0.800, and 0.100–0.920, respectively.According to 47 mechanism functions in the literature, the general integration method and MacCallum–Tanner method fitα−T data of four coals at different heating rates in the light of equations (5) and (8) respectively, and determine the correct mechanism function of coal thermal decomposition reaction according to three principles. The results show that the most probable mechanism function of thermal decomposition reaction of coal SJC, coal HCG, and coal ZJM is function No.37, and the most probable mechanism function of coal WJG is function No.17. The differential form, integral form, and physical meaning of the most probable mechanism function are shown in Table 2. The kinetic parameters of the thermal decomposition stage for all coals calculated by four algorithms are shown in Tables 3–6. In these calculations, except for the poor linear correlation coefficients calculated by the Kissinger method for SJC, the Kissinger method for WJG, and the FWO method for ZJM (correlation coefficients are 0.9689, 0.9700, and 0.9751, respectively), the other correlation coefficients are all above 0.98, the overall fitting accuracy is high, and the kinetic calculation results are reliable.Table 2
The differential form and integral form of mechanism function.
Coal
Number
Function name
f
α
G
α
Function mechanism
SJC
37
Reaction series
(1−α)2
(1−α)−1–1
Chemical reaction
WJG
17
Avrami–Erofeev
2
/
3
1
−
α
−
ln
1
−
α
−
1
/
2
−
ln
1
−
α
3
/
2
Randomly nucleating and nucleus growth
HCG
37
Reaction series
(1−α)2
(1−α)−1–1
Chemical reaction
ZJM
37
Reaction series
(1−α)2
(1−α)−1–1
Chemical reactionTable 3
Calculation results of coal SJC.
Method
β (°C/min)
E (kJ/mol)
A (1/S)
R
2
The general integral
5
223.657
1.16 × 1014
0.9997
10
160.223
9.85 × 108
0.9985
15
163.37
1.65 × 109
0.9984
20
153.127
5.09 × 108
0.9956
MacCallum–Tanner
5
228.401
3.13 × 1014
0.9997
10
165.168
2.41 × 109
0.9998
15
168.254
4.04 × 109
0.9986
20
158.027
1.22 × 109
0.9963
Average
177.53
2.75 × 1010
0.9983
Kissinger
—
247.84
5.19 × 1015
0.9689
FWO
—
329.526
—
0.9856Table 4
Calculation results of coal WJG.
Method
β (°C/min)
E (kJ/mol)
A (1/S)
R
2
The general integral
5
217.04
9.55 × 1012
0.9940
10
207.72
1.78 × 1012
0.9960
15
187.76
6.92 × 1010
0.9919
20
187.23
6.03 × 1010
0.9874
MacCallum–Tanner
5
221.86
2.57 × 1013
0.9947
10
212.71
4.79 × 1012
0.9965
15
192.74
1.78 × 1011
0.9930
20
192.35
1.62 × 1011
0.9891
Average
200.34
8.51 × 1011
0.9928
Kissinger
—
203.05
5.99 × 1011
0.9700
FWO
—
192.83
—
0.9920Table 5
Calculation results of coal ZJM.
Method
β (°C/min)
E (kJ/mol)
A (1/S)
R
2
The general integral
5
172.95
1.32 × 1010
0.9987
10
152.63
4.57 × 108
0.9999
15
153.55
4.17 × 108
0.9994
20
145.77
1.23 × 108
0.9990
MacCallum–Tanner
5
177.46
3.09 × 1010
0.9990
10
157.20
1.05 × 109
0.9999
15
158.43
1.02 × 109
0.9995
20
150.72
2.88 × 108
0.9992
Average
158.59
2.88 × 108
0.9992
Kissinger
—
153.90
2.46 × 108
0.9909
FWO
—
140.58
—
0.9751Table 6
Calculation results of coal HCG.
Method
β (°C/min)
E (kJ/mol)
A (1/S)
R
2
The general integral
5
276.67
3.98 × 1017
0.9995
10
252.23
5.13 × 1015
0.9997
15
211.63
8.71 × 1012
0.9991
20
210.43
6.76 × 1012
0.9998
MacCallum–Tanner
5
282.17
1.41 × 1018
0.9996
10
257.84
1.70 × 1016
0.9997
15
216.94
2.51 × 1013
0.9992
20
215.86
1.95 × 1013
0.9998
Average
240.47
1.05 × 1015
0.9995
Kissinger
—
223.67
1.53 × 1013
0.9976
FWO
—
264.33
—
0.9900The calculated results show that the kinetic parameters of four coals obtained by two kinds of the model-free method are different, and the order of activation energy is ZJM < WJG < HCG < SJC. This is due to the different structure and composition of different coals and the different reactions during pyrolysis. The average activation energies of coals SCC, WJG, ZJM, and HCG calculated by model-fitting methods are 177.53 kJ/mol, 200.34 kJ/mol, 158.59 kJ/mol, and 240.47 kJ/mol, respectively. Coals SJC, HCG, and ZJM conform to the reaction series equation. The rates of thermal decomposition reaction are controlled by chemical reaction, but their coefficients and exponents of the specific mechanism function expression are different. Coal WJG conforms to the Avrami–Erofeev equation. The expressions of average activation energy, pre-exponential factor, and reaction mechanism function of different coals are substituted into equation (2), respectively, and the kinetic equations of their thermal decomposition reactions are obtained. Through the corresponding kinetic equation, the thermal decomposition reaction process of the same kind of coal can be simulated at any heating rate.


## 3.2.1. FWO Method

(5)
log
β
i
=
log
A
E
α
R
G
α
i
−
2.315
−
0.4567
E
R
T
i
,
i
=
1,2,3,4
,where βi is the heating rate, Eα is the activation energy, and Ti is the corresponding reaction temperature on TG data. Under different heating rates, with the same conversion, Gα will be a constant value. Then, logβi versus 1/Ti becomes a linear line so that E can be calculated from the slope of the regression line.


## 3.2.2. Kissinger Method

(6)
ln
β
i
T
p
i
2
=
ln
A
K
R
E
K
−
E
K
R
1
T
p
i
,
i
=
1,2,3,4
,where EK is the activation energy, AK is the pre-exponential factor, and Tpi is the temperature corresponding to the peak on DTG. The linear regression curve of lnβi/Tpi2−Tpi was plotted according to the temperature corresponding to the maximum weight loss rate in four DTG data. The activation energy EK was calculated by slope and the pre-exponential factor AK was calculated by intercept.


## 3.2.3. General Integral Method

Based on the Coats–Redfern approximate expression, equation (3) is transformed into equation (7):(7)Gα=AβRT2E1−2RTEexp−ERT.In this work, equation (7) can be simplified considering 2RT/E≪1, the logarithmic form is presented as:(8)lnGαT2=lnARβE−ERT.


## 3.2.4. MacCallum–Tanner Method

Whenu=E/RT, equation (3) can be rewritten as(9)Gα=Aβ∫T0Texp−ERTdT=AEβR∫E/RT+∞e−uu2du=AEβRPu,where the function Pu is the temperature integral that has no analytic solutions. Different from the general integral, MacCallum–Tanner does not require approximation of Pu. The relationship between Pu and E is as follows:(10)−logPMTu=0.4828E0.4357+0.449+0.217E0.001T.Thus, the MacCallum–Tanner equation can be obtained through substituting equation (10) into equation (9):(11)logGα=logAEβR−0.4828E0.4357−0.449+0.217E0.0011T.When using the general integral method and MacCallum–Tanner method, it is necessary to assume the function form ofGα and to fit lnGα/T2−1/T and lnGα−1/T, respectively. When the correlation coefficient of fitting curve approaches 1, it is the correct mechanism function. The three principles for determining the correct mechanism function are high linear correlation coefficient, close activation energy to that of FWO and Kissinger methods, and clear physical meaning of the mechanism function.The data used in the FWO method, general integration method, and MacCallum–Tanner method areα−T of the TG profile. When calculating, firstly, the data points are collected from the TG curve and the αi is obtained by equation (4). Then, the corresponding Ti data of αi in the range of 0−1.0 are collected by the step size of 0.025. The Kissinger method is based on four different heating rates and the temperature corresponding maximum weight loss rates. Figure 3 reveals the activation energy of four coals in the thermal decomposition stage calculated by the FWO method.Figure 3
E − α.Figure3 indicates that the activation energies of SJC, WJG, ZJM, and HCG coals during pyrolysis have been changing, with the activation energies ranging from 114.99 to 630.31 kJ/mol, 80.13 to 302.15 kJ/mol, 74.32 to 164.97 kJ/mol, and 123.51 to 404.49 kJ/mol, respectively. The variation tendency of activation energy of coal SJC can be divided into three parts: when 0 < α < 0.150, E increases continuously; when 0.150 < α < 0.800, E decreases steadily; and when 0.800 < α < 1.0, E decreases rapidly. The change in activation energy is the manifestation of the diversification of internal reaction mechanism. Because the activation energy of the initial and termination stages of the reaction changes dramatically, it is generally considered that there is no stable mechanism function, and the whole reaction mechanism is determined by the mechanism of the dominant process [14]. Therefore, the range of the mechanism function of SJC coal thermal decomposition reaction is determined to be from 0.150 to 0.800. The activation energies of WJG, ZJM, and HCG are similar to those of SJC and can be divided into three steps. Similarly, the range of mechanism functions of WJG, ZJM, and HCG coal is 0.050–0.800, 0.100–0.800, and 0.100–0.920, respectively.According to 47 mechanism functions in the literature, the general integration method and MacCallum–Tanner method fitα−T data of four coals at different heating rates in the light of equations (5) and (8) respectively, and determine the correct mechanism function of coal thermal decomposition reaction according to three principles. The results show that the most probable mechanism function of thermal decomposition reaction of coal SJC, coal HCG, and coal ZJM is function No.37, and the most probable mechanism function of coal WJG is function No.17. The differential form, integral form, and physical meaning of the most probable mechanism function are shown in Table 2. The kinetic parameters of the thermal decomposition stage for all coals calculated by four algorithms are shown in Tables 3–6. In these calculations, except for the poor linear correlation coefficients calculated by the Kissinger method for SJC, the Kissinger method for WJG, and the FWO method for ZJM (correlation coefficients are 0.9689, 0.9700, and 0.9751, respectively), the other correlation coefficients are all above 0.98, the overall fitting accuracy is high, and the kinetic calculation results are reliable.Table 2
The differential form and integral form of mechanism function.
Coal
Number
Function name
f
α
G
α
Function mechanism
SJC
37
Reaction series
(1−α)2
(1−α)−1–1
Chemical reaction
WJG
17
Avrami–Erofeev
2
/
3
1
−
α
−
ln
1
−
α
−
1
/
2
−
ln
1
−
α
3
/
2
Randomly nucleating and nucleus growth
HCG
37
Reaction series
(1−α)2
(1−α)−1–1
Chemical reaction
ZJM
37
Reaction series
(1−α)2
(1−α)−1–1
Chemical reactionTable 3
Calculation results of coal SJC.
Method
β (°C/min)
E (kJ/mol)
A (1/S)
R
2
The general integral
5
223.657
1.16 × 1014
0.9997
10
160.223
9.85 × 108
0.9985
15
163.37
1.65 × 109
0.9984
20
153.127
5.09 × 108
0.9956
MacCallum–Tanner
5
228.401
3.13 × 1014
0.9997
10
165.168
2.41 × 109
0.9998
15
168.254
4.04 × 109
0.9986
20
158.027
1.22 × 109
0.9963
Average
177.53
2.75 × 1010
0.9983
Kissinger
—
247.84
5.19 × 1015
0.9689
FWO
—
329.526
—
0.9856Table 4
Calculation results of coal WJG.
Method
β (°C/min)
E (kJ/mol)
A (1/S)
R
2
The general integral
5
217.04
9.55 × 1012
0.9940
10
207.72
1.78 × 1012
0.9960
15
187.76
6.92 × 1010
0.9919
20
187.23
6.03 × 1010
0.9874
MacCallum–Tanner
5
221.86
2.57 × 1013
0.9947
10
212.71
4.79 × 1012
0.9965
15
192.74
1.78 × 1011
0.9930
20
192.35
1.62 × 1011
0.9891
Average
200.34
8.51 × 1011
0.9928
Kissinger
—
203.05
5.99 × 1011
0.9700
FWO
—
192.83
—
0.9920Table 5
Calculation results of coal ZJM.
Method
β (°C/min)
E (kJ/mol)
A (1/S)
R
2
The general integral
5
172.95
1.32 × 1010
0.9987
10
152.63
4.57 × 108
0.9999
15
153.55
4.17 × 108
0.9994
20
145.77
1.23 × 108
0.9990
MacCallum–Tanner
5
177.46
3.09 × 1010
0.9990
10
157.20
1.05 × 109
0.9999
15
158.43
1.02 × 109
0.9995
20
150.72
2.88 × 108
0.9992
Average
158.59
2.88 × 108
0.9992
Kissinger
—
153.90
2.46 × 108
0.9909
FWO
—
140.58
—
0.9751Table 6
Calculation results of coal HCG.
Method
β (°C/min)
E (kJ/mol)
A (1/S)
R
2
The general integral
5
276.67
3.98 × 1017
0.9995
10
252.23
5.13 × 1015
0.9997
15
211.63
8.71 × 1012
0.9991
20
210.43
6.76 × 1012
0.9998
MacCallum–Tanner
5
282.17
1.41 × 1018
0.9996
10
257.84
1.70 × 1016
0.9997
15
216.94
2.51 × 1013
0.9992
20
215.86
1.95 × 1013
0.9998
Average
240.47
1.05 × 1015
0.9995
Kissinger
—
223.67
1.53 × 1013
0.9976
FWO
—
264.33
—
0.9900The calculated results show that the kinetic parameters of four coals obtained by two kinds of the model-free method are different, and the order of activation energy is ZJM < WJG < HCG < SJC. This is due to the different structure and composition of different coals and the different reactions during pyrolysis. The average activation energies of coals SCC, WJG, ZJM, and HCG calculated by model-fitting methods are 177.53 kJ/mol, 200.34 kJ/mol, 158.59 kJ/mol, and 240.47 kJ/mol, respectively. Coals SJC, HCG, and ZJM conform to the reaction series equation. The rates of thermal decomposition reaction are controlled by chemical reaction, but their coefficients and exponents of the specific mechanism function expression are different. Coal WJG conforms to the Avrami–Erofeev equation. The expressions of average activation energy, pre-exponential factor, and reaction mechanism function of different coals are substituted into equation (2), respectively, and the kinetic equations of their thermal decomposition reactions are obtained. Through the corresponding kinetic equation, the thermal decomposition reaction process of the same kind of coal can be simulated at any heating rate.


## 4. Conclusion

(1)
The pyrolysis process of four coals in northern Shaanxi can be divided into three stages: desorption stage, thermal decomposition stage, and polycondensation reaction stage. With the increase in heating rate, the temperature corresponding to the maximum devolatilization rate increases and the final devolatilization of coal increases.
(2)
The most probable mechanism function of coal SJC, ZJM, and HCG in the thermal decomposition reaction stage is No. 37. The reaction rate is controlled by chemical reaction. The most probable mechanism function of coal WJG is No. 17. The function mechanism is random nucleation and subsequent growth. The activation energies of coals SJC, WJG, ZJM, and HCG are 177.53 kJ/mol, 200.34 kJ/mol, 158.59 kJ/mol, and 240.47 kJ/mol, respectively.



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*Source: 9075862-2019-11-11.xml*