# Impacts of Thermal Inertia Factor on Adiabatic Decomposition of 40% Mass Content DCP in Ethyl Benzene

Authors: Xiaojuan Wu; Liping Chen; Guoning Rao; Wanghua Chen; Ruili Yin
Journal: Journal of Chemistry
 (2020)

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## Abstract

To reduce the fire and explosion accident of dicumyl peroxide (DCP) in experiment and production, the thermal hazards of DCP and 40% mass content DCP in ethyl benzene (40% DCP) have been studied by the differential scanning calorimeter (DSC) and the accelerating rate calorimeter (ARC) in this paper. DSC experiment showed that ethyl benzene has no effect on the characteristic parameters of thermal decomposition of DCP, such as the temperature of the exothermic peak (Tpeak) and the decomposition energy (Ea), and the thermal decomposition reaction of 40% DCP followed the one-step reaction principle. ARC experiment showed that with the increase of inertia factor (Φ), the measured initial decomposition temperature (Ton) would be higher and the caculated Ea and pre-exponential factor (A) would be greater. It was also proved that after modification of Φ, TD24 was relatively consistent near Ton, but different at higher temperatures. Fisher's correction method was used to verify the necessity of consistency between experimental conditions and prediction conditions.


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## Body

## 1. Introduction

With fast development of industries and wide usage of organic peroxides, the hazard assessment of organic peroxides has become more and more important in many countries. As a kind of common organic peroxides, dicumyl peroxide (DCP) has been widely employed as a vulcanizing agent of rubber and a crosslinking agent of plastic in the chemical industry. Because of the thermal sensitivity and high antipyretic effect of DCP, great damages would be brought into the production process, workers, and environment once the reaction runaway occurs [1, 2].Thermal analyses have been carried out on organic peroxides by different thermal analysis apparatuses. Ding [3] has investigated the thermal decomposition of DCP by the thermogravimetry-differential thermogravimetry (TG-DTG) method. One endothermic peak and one exothermic peak were found in the decomposition process, and the apparent activation energy was calculated. Jiang [4] has studied the influence on the heating rates of DCP by C80. The advanced kinetics and technology solutions (AKTS) method has been used to calculate the thermodynamic parameters. Guo [5] has used an accelerating rate calorimeter (ARC) to determine the adiabatic decomposition rule of DCP. “Townsend–Tou” method has been used in the correction of the experiment data.Although the sample is small, the maximum self-heating rate can exceed the maximum tracking rate of ARC for the large decomposition heat of DCP. It will result in some errors in the experiment [6]. In order to reduce the decomposition heat of DCP, ethyl benzene is chosen as the solvent for its stability within the test scope in this paper. Thermodynamic and kinetic parameters of pure DCP and 40% mass content DCP in ethyl benzene can be obtained by a differential scanning calorimeter (DSC) and ARC. Based on these thermal data, safety studies can be done to distinguish the dangers caused by the two material systems. Furthermore, with “Fisher’s” method, the best experiment condition of ARC will be discussed.


## 2. Materials and Methods

DCP of 99.5% mass content was purchased from Aladdin. It is in the solid state at room temperature. Ethyl benzene of 98.5 % mass content is bought from Sinopharm Chemical Reagent Co., Ltd. It is in the liquid state at room temperature.40% mass content DCP in ethyl benzene (40% DCP) was prepared by assuming both DCP and ethyl benzene were at 100% purity. The configured solution was placed in a refrigerator. It is in the liquid state, and the density is 1.044 g·cm−3 at room temperature.The heat capacities of DCP and 40% DCP at 90°C were 1.94 and 1.95 J·g−1·K−1, respectively, which were obtained from uRC (manufactured by THT).


### 2.1. DSC Experiments

The DSC used in this paper was manufactured by Mettler Toledo (DSC-1). Stainless steel high-pressure crucibles (15 MPa) were employed in tests, with extra pure nitrogen purging (30 mL·min−1). Samples were tested at a heating rate of 2, 4, 8, and 1 K·min−1 for rapid screening. The temperature range of all measurements was from 25 to 250°C.


### 2.2. ARC Experiments

ARC supplied by Thermal Hazard Technology Company was performed in the heat-wait-search mode. The experimental procedure was sealed. The temperature step was 5°C, the wait time was 10 minutes, and the slop sensitivity was 0.02°C·min−1. The trackable heating rate was up to 15°C·min−1 (full power). Sample bomb was made of titanium with a thick wall of 0.5 mm, and the heat capacity and density are 0.53 J·g−1·K−1 and 4.53 g·cm−3, respectively. The ARC test temperature range is from the room temperature to 450°C.The mass of each sample is listed in Tables1 and 2.Table 1
Mass in DSC tests.
Samples
Essential components (%)
Sample mass (mg)
Heating rate (K·min−1)
DCP
Ethyl benzene
#1
100
0
1.26
8
#2
40
60
1.98
2
40
60
1.98
4
40
60
1.98
8
40
60
1.98
10Table 2
Mass in ARC tests.
Samples
Essential components (%)
Mass (mg)
DCP
Ethyl benzene
No. 1
No. 2
No. 3
No. 4
No. 5
No. 6
No. 7
#2
40
60
0.510
0.593
1.005
1.501
2.316
2.551
4.283


## 2.1. DSC Experiments

The DSC used in this paper was manufactured by Mettler Toledo (DSC-1). Stainless steel high-pressure crucibles (15 MPa) were employed in tests, with extra pure nitrogen purging (30 mL·min−1). Samples were tested at a heating rate of 2, 4, 8, and 1 K·min−1 for rapid screening. The temperature range of all measurements was from 25 to 250°C.


## 2.2. ARC Experiments

ARC supplied by Thermal Hazard Technology Company was performed in the heat-wait-search mode. The experimental procedure was sealed. The temperature step was 5°C, the wait time was 10 minutes, and the slop sensitivity was 0.02°C·min−1. The trackable heating rate was up to 15°C·min−1 (full power). Sample bomb was made of titanium with a thick wall of 0.5 mm, and the heat capacity and density are 0.53 J·g−1·K−1 and 4.53 g·cm−3, respectively. The ARC test temperature range is from the room temperature to 450°C.The mass of each sample is listed in Tables1 and 2.Table 1
Mass in DSC tests.
Samples
Essential components (%)
Sample mass (mg)
Heating rate (K·min−1)
DCP
Ethyl benzene
#1
100
0
1.26
8
#2
40
60
1.98
2
40
60
1.98
4
40
60
1.98
8
40
60
1.98
10Table 2
Mass in ARC tests.
Samples
Essential components (%)
Mass (mg)
DCP
Ethyl benzene
No. 1
No. 2
No. 3
No. 4
No. 5
No. 6
No. 7
#2
40
60
0.510
0.593
1.005
1.501
2.316
2.551
4.283


## 3. Results and Discussion

### 3.1. Analysis of DSC Measurement

#### 3.1.1. Comparison of #1 and #2 with the DSC Test

The DSC results of samples #1 and #2 at 8 K·min−1 are shown in Figure 1 and Table 3.Figure 1
DSC curve of samples #1 and #2 at 8 K·min−1.Table 3
DSC results of samples #1 and #2 at 8 K·min−1.
Samples
Mass (mg)
T
onset of exo
T
peak of exo
ΔHa (J·g−1)
Δ
H
b
d (J·g−1)
#1
1.26
115.7
177.4
886.58
886.58
#2
1.98
114.7
181
362.88
907.22
aDividing total heat by total mass of the sample, ΔH. bDividing total heat by total mass of DCP, ΔHd.According to Figure1, it is found that thermal decomposition of DCP contains one endothermic peak at 43.12°C and one exothermic peak at 177.40°C. Since ethyl benzene does not decompose in the test scope, DCP is dissolved into ethyl benzene and decomposed without the influence of phase transition, and this is why there is only one exothermic peak for 40% DCP. The onset exothermic temperature of DCP and 40% DCP is approaching 115°C. There is a big difference of decomposition heat (ΔH) between DCP and 40% DCP in Table 3, and they are 886.58 and 362.88 J·g−1, respectively. However, when we divide 649.56 (the total heat of 40% DCP) by 0.792 (mass of DCP in sample #2), the value is 907.22 J·g−1, which is close to the ΔH of sample #1. So, it can be concluded that ethyl benzene has little impact on the decomposition of DCP.


### 3.2. Test Results of Sample # 2 under Different Heating Rates with the DSC Test

A typical run for sample #2 at different heating rates is shown in Figure2. The characteristic parameters are given in Table 4.Figure 2
Heatflow of sample #2 under different heating rates.Table 4
Results for 40% DCP with the DSC test.
40% DCP
β/K·min−1
2
4
8
10
T
0 (°C)
138.36
144.72
153.69
156.96
T
p (°C)
163.75
171.90
178.99
185.79
Q/J·g−1
293.80
365.81
362.94
302.93The Friedman isoconversional method was employed to calculate the kinetics in this paper. As a “model-free” dynamic analysis method, it does not need to use concrete mechanism function, and the result has high precision and wide application [7]. We get a Friedman analysis diagram which shows the relationship between Ea and α, as in Figure 3 and Table 5.Figure 3
E
a(α) and ln A(α) vs α of sample #2.Table 5
lnA and Ea values of 40% DCP obtained by the Friedman method.
α
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
40% DCP
E
a/kJ·mol−1
110.60
110.42
110.08
109.95
108.90
106.91
104.16
98.14
ln (A/s−1)
22.93
23.00
22.98
22.99
22.72
22.19
21.42
19.75As given in Table5, between α = 0.2 and 0.9, Ea is between 110.60 and 98.14 kJ·mol−1 and the variation range is nearly 10%, indicating that it follows a reaction mechanism and can be expressed by the same mechanism function for 40% DCP.


### 3.3. Analysis of ARC Measurement

Seven experiments with differentΦ are carried out according to the above conditions. The ARC curves of temperature-time (T-t) and pressure-time (P-t) of sample #2 are shown in Figures 4 and 5, and all measured data are listed in Table 6.Figure 4
T-t of exothermic section for sample #2.Figure 5
P-t of exothermic section for sample #2.Table 6
Measured adiabatic test results of sample #2 with differentΦ values.
No.
Ton,m (°C)
ΔTad,m (°C)
(dT/dt)max,m
Tmax,m (°C)
Hm (J·g−1)
Pend,m (bar)
1
120.43
First part: 4.00Second part: 2.01
0.024
131.67
49.34
3.01
2
115.54
36.28
0.36
143.74
266.00
5.42
3
115.60
47.94
1.21
155.35
245.86
7.52
4
110.42
59.89
2.65
162.94
244.08
10.22
5
110.49
71.65
10.38
166.51
238.92
15.39
6
110.50
75.81
12.97
175.86
242.44
17.43
7
105.49
87.88
166.52
175.56
238.00
35.77
Ton,m, measured onset decomposition temperature, °C. ΔTad,m, measured adiabatic temperature rise, °C. (dT/dt)max,m, measured maximum self-heating rate, K·min−1. Tmax,m, measured temperature at maximum rate, °C. Pend,m, measured final pressure of decomposition, bar.As shown in Figure4, the ARC curve of the first test is different from others, and the exothermal reaction is divided into two parts. Table 6 shows that the maximum self-heating rate ((dT/dt)max,m), the measured adiabatic temperature rise (ΔTad,m), and the decomposition heat (Hm) of the first test are much smaller than other tests. The small sample mass results in high Φ of the first test, and it makes part of the (dT/dt)m less than the detection sensitivity 0.02°C·min−1 and part of the decomposition heat is missed. Obviously, too large Φ can change the temperature curve entirely and result in the distortion of the tested results.As shown in Table6, Ton,m is rising with the increase of Φ. The reason is that the small amount of the sample needs a higher temperature to make the (dT/dt)m of the sample and the bomb system reach the ARC detection sensitivity (0.02°C·min−1). It is also shown that ΔTM and (dT/dt)max,m decrease with the increase in Φ. In addition, since the material and energy balance of the sample and the bomb system were complicated, the (dT/dt)max,m with Φ was less linear.


### 3.4. Kinetic Calculation

The heat released by the sample in the ARC test not only heats the sample itself, but also raises the temperature of the bomb. The smaller the sample mass, the higher proportion of heat will be used for heating the bomb and the corresponding temperature parameters (such as temperature curves and heating rate curves) will be affected too. So, it is very important to introduce the thermal inertia factorΦ, which can be estimated by the following equation:(1)Φ=msCs+mbCbmsCs.The Townsend–Tou method [8–10] and the Fisher method [11–13] have been widely used. Literature [3] shows that thermal decomposition of DCP is an nth-order reaction, and the Fisher method is adopted to modify the ARC data in this paper.The corrected equations for the Fisher method are listed as follows:(2)1Ton,a=1Ton,m+REalnΦ,Tad,a=Ton,a+ΦTm−Ton,m,dTdtad,a=ΦdTdtad,mexpEaR1Ton,a−1Tad,a,where R is the gas constant; Ea is the apparent activation energy, kJ·mol−1; Ton,m is the corrected onset temperature by the Fisher method, °C; Tad,a is the corrected temperature by the Fisher method, °C; (dT/dt)ad,a is the corrected self-heating rate by the Fisher method, K·min−1; and (dT/dt)ad,m is the measured self-heating rate, K·min−1.The corrected results are listed in Table7.Table 7
Adiabatic test results of sample #2 corrected by the Fisher method.
No.
2
3
4
5
6
7
Φ
3.76
2.63
2.09
1.71
1.64
1.39
Ton,a (°C)
106.86
108.84
105.45
106.79
107.03
103.17
ΔTad,a (°C)
136.41
126.08
125.17
122.52
124.33
122.15
(dT/dt)max,a
6964.1
1491.9
1184.4
928.8
913.3
3292.3
Ton,a, corrected onset decomposition temperature, °C. ΔTad,a, corrected adiabatic temperature rise, °C. (dT/dt)max,a, corrected maximum self-heating rate, K·min−1.Kinetic parameters are important for thermal hazard prediction. For the decomposition reaction in the adiabatic system, the relationship between temperature and the self-heating rate is [14](3)dTdta=ATf,a−Tad,aΔTad,anΔTad,ac0n−1exp−EaRTad,a,where Tf,a is the corrected final temperature of decomposition, °C; A is the pre-exponential factor, min−1; n is the reaction order; and c0 is the initial concentration of sample, mol·L−1.Suppose(4)k∗=kc0n−1=c0n−1Aexp−EaRTa=dTdtaΔTad,aTf,a−Tad,anΔTad,a−1where k is the rate constant.Logarithmic form of equation (4)is(5)lnk∗=lnAc0n−1−EaR1Tad,a.The reaction order which can make the plot of ln(k∗) vs. 1/Tad,a to be a straight line is analysed correctly. Then, Ea and A can be calculated from the slope and the intercept.The kinetic analysis is not applied to the first test for its self-heating rate is discontinuous. For other samples, the reaction ordern = 1 is assumed and substituted into equation (4), and then ln(k∗) is plotted against 1/Tad,a. Linear fitting of ln(k∗) and 1/Tad,a is done, and the fitting results and the values of Ea and A are shown in Table 8.Table 8
The linear fitting results.
No.
2
3
4
5
6
7
Φ
3.76
2.63
2.09
1.71
1.64
1.39
n
1
1
1
1
1
1
E
a (kJ·mol−1)
187.30
176.51
178.92
175.70
172.76
168.37
A (min−1)
1.38 × 1022
5.00 × 1020
1.01 × 1021
3.84 × 1020
1.58 × 1020
5.00 × 1019
R
2
0.9964
0.9992
0.9981
0.9969
0.9982
0.9968
R
2: linear fitting correlation coefficient.The high correlation coefficient indicates that the linear fitting results are reliable whenn = 1. It means that the DCP decomposition is a very typical n-order reaction, and the Fisher method which is selected for correction before is appropriate.In addition, a good linear regression of lnA vs. Ea is found with different Φ, showing a typical kinetic compensation effect [15], as shown in Figure 6. This means that the kinetic compensation effect exists only if the reaction mechanism is the same or similar [16].Figure 6
lnA vs Ea with different values of Φ.Comparing DSC and ARC experiments, we can find that the calculatedEa in ARC is generally higher than that in DSC and higher with the increase in Φ. It is mainly because these calculated kinetic parameters obtained by ARC data depend on the adiabatic temperature rise rate of equation (3).With the influence of the big Φ values, the measured adiabatic temperature rise curve of the sample is not the decomposition curve of itself, and it is the curve which is formed after the sample ball “passivation.” So, this analysis of kinetic parameters is not incidental to the sample decomposition, but to the decomposition of composite samples, sample ball, heat transfer, and heat transfer information such as the comprehensive parameter.


### 3.5. Effect ofΦ on TD24

The corresponding initial temperature when the time to maximum rate under adiabatic condition (TMRad) equals 24 hours (TD24) is an important parameter to evaluate the risk of samples. In order to obtain the effects of different Φ on the measured TD24,m and the modified TD24, the experimental data before and after the correction are analysed in this paper. It can be seen from Table 6 that (dT/dt)max,m measured in No. 1 and No. 2 is less than 1, while in No. 7 is 166.52°C·min−1, indicating an unsatisfactory adiabatic environment. Therefore, the data of 40%DCP solution in No. 1, No. 2, and No. 7 are not analysed in this part and only comparing the data of the other four groups is done.The measuredTMRad can be obtained by the following method:(6)TMRad,m=tm−t=∫ttmdt=∫TTmdTAexp−Ea/RTTf−T/ΔTadnΔTadC0n−1,where TMRad,m is the measured arrival time of the maximum reaction rate, min; Tm is the temperature corresponding to the maximum temperature rise rate, °C; tm is the time corresponding to Tm, min; Tf is the exothermic termination temperature,°C; T is the adiabatic test temperature, °C; ΔTad is the measured adiabatic temperature rise, °C; and C0 is the initial molality of the sample, mol·L−1.Equation (6) can be mathematically integrated to obtain TMRad,m, which is the corresponding temperature TD24 for 24 h. If the effect of Φ is taken into account, the correction of Φ needs to be introduced by the following equation:(7)TMRad0=TMRad0,mΦ.The relationship between the measured temperature andTMRad,m under different Φ for 40% DCP is shown in Figure 7. Extrapolated from the measured value to the low temperature, the obtained TD24 is also shown in Figure 7. According to the measured results, equation (7) is used to modify the modified TD24, as shown in Figure 8.Figure 7
The relationship curve between uncorrected solution temperature andTMRad,m for 40% DCP.Figure 8
The relationship curve between corrected solution temperature andTMRad for 40% DCP.It can be seen from Figure7 that under different Φ, the TD24 of the sample obtained from the test data is different and increases with the increase in Φ. This is because the larger the Φ, the slower the heat release rate of the sample in the initial stage, the lower the temperature rise rate, and the longer the time to reach the maximum temperature rise rate. Figure 8 shows that after the correction of the TMRad near the initial decomposition temperature, the relationship between the different Φ and TMRad is relatively consistent and the TD24 of 40% DCP obtained by extrapolation of the modified sample sphere is about 87.20°C. It also shows that after the correction with Φ, the relationship between temperature and corrected TMRad is still quite different at a high temperature. However, near the initial decomposition temperature, the results are almost the same. Therefore, it is verified that equation (7) is only applicable to TMRad near the modified initial decomposition temperature.


### 3.6. Checking Analysis with the Fisher Method

As mentioned in Section3.2, test results are affected by Φ significantly. A normal Fisher method is to amend the test results to the ideal condition Φ = 1, which cannot be realized by the ARC experiment. The experiment data will be corrected to Φ = 1.64 in this article and be compared with the measured data.The corrected equations are as follows:(8)1Ton,a=1Ton,m+REalnΦΦ1,Tad,a=Ton,a+ΦΦ1T−Ton,dTdtad,a=ΦΦ1dTdtmexpEaR1Ton,a−1Tad,a.The corrected results are shown in Figure9 and Table 9.Figure 9
(dT/dt)-T corrected to Φ1 = 1.64 by the Fisher method.Table 9
The parameters corrected toΦ1 = 1.64 by the Fisher method.
No.
2
3
4
5
6
7
Φ
3.76
2.63
2.09
1.71
1.64
1.39
T
on,a (°C)
110.05
112.27
108.77
110.20
110.5∗
106.66
(dT/dt)max,a
49.3
19.8
13.4
12.4
12.97∗
53.6
The values marked with ∗ are measured values.Figure9 and Table 9 show that when Φ is close to 1.64, the corrected results such as (dT/dt)-T and Ton,a are close to the measured results at Φ = 1.64. However, if the test results based on Φ are far away from the target or predicted condition, the results are very unbelievable. Thus, the weight of the sample has great effect on the result of the ARC test. The study demonstrates that under the condition of safety, Φ will be close to 1 with the increasing dosage of sample, and the revised ideal adiabatic data will be more reliable.


## 3.1. Analysis of DSC Measurement

### 3.1.1. Comparison of #1 and #2 with the DSC Test

The DSC results of samples #1 and #2 at 8 K·min−1 are shown in Figure 1 and Table 3.Figure 1
DSC curve of samples #1 and #2 at 8 K·min−1.Table 3
DSC results of samples #1 and #2 at 8 K·min−1.
Samples
Mass (mg)
T
onset of exo
T
peak of exo
ΔHa (J·g−1)
Δ
H
b
d (J·g−1)
#1
1.26
115.7
177.4
886.58
886.58
#2
1.98
114.7
181
362.88
907.22
aDividing total heat by total mass of the sample, ΔH. bDividing total heat by total mass of DCP, ΔHd.According to Figure1, it is found that thermal decomposition of DCP contains one endothermic peak at 43.12°C and one exothermic peak at 177.40°C. Since ethyl benzene does not decompose in the test scope, DCP is dissolved into ethyl benzene and decomposed without the influence of phase transition, and this is why there is only one exothermic peak for 40% DCP. The onset exothermic temperature of DCP and 40% DCP is approaching 115°C. There is a big difference of decomposition heat (ΔH) between DCP and 40% DCP in Table 3, and they are 886.58 and 362.88 J·g−1, respectively. However, when we divide 649.56 (the total heat of 40% DCP) by 0.792 (mass of DCP in sample #2), the value is 907.22 J·g−1, which is close to the ΔH of sample #1. So, it can be concluded that ethyl benzene has little impact on the decomposition of DCP.


## 3.1.1. Comparison of #1 and #2 with the DSC Test

The DSC results of samples #1 and #2 at 8 K·min−1 are shown in Figure 1 and Table 3.Figure 1
DSC curve of samples #1 and #2 at 8 K·min−1.Table 3
DSC results of samples #1 and #2 at 8 K·min−1.
Samples
Mass (mg)
T
onset of exo
T
peak of exo
ΔHa (J·g−1)
Δ
H
b
d (J·g−1)
#1
1.26
115.7
177.4
886.58
886.58
#2
1.98
114.7
181
362.88
907.22
aDividing total heat by total mass of the sample, ΔH. bDividing total heat by total mass of DCP, ΔHd.According to Figure1, it is found that thermal decomposition of DCP contains one endothermic peak at 43.12°C and one exothermic peak at 177.40°C. Since ethyl benzene does not decompose in the test scope, DCP is dissolved into ethyl benzene and decomposed without the influence of phase transition, and this is why there is only one exothermic peak for 40% DCP. The onset exothermic temperature of DCP and 40% DCP is approaching 115°C. There is a big difference of decomposition heat (ΔH) between DCP and 40% DCP in Table 3, and they are 886.58 and 362.88 J·g−1, respectively. However, when we divide 649.56 (the total heat of 40% DCP) by 0.792 (mass of DCP in sample #2), the value is 907.22 J·g−1, which is close to the ΔH of sample #1. So, it can be concluded that ethyl benzene has little impact on the decomposition of DCP.


## 3.2. Test Results of Sample # 2 under Different Heating Rates with the DSC Test

A typical run for sample #2 at different heating rates is shown in Figure2. The characteristic parameters are given in Table 4.Figure 2
Heatflow of sample #2 under different heating rates.Table 4
Results for 40% DCP with the DSC test.
40% DCP
β/K·min−1
2
4
8
10
T
0 (°C)
138.36
144.72
153.69
156.96
T
p (°C)
163.75
171.90
178.99
185.79
Q/J·g−1
293.80
365.81
362.94
302.93The Friedman isoconversional method was employed to calculate the kinetics in this paper. As a “model-free” dynamic analysis method, it does not need to use concrete mechanism function, and the result has high precision and wide application [7]. We get a Friedman analysis diagram which shows the relationship between Ea and α, as in Figure 3 and Table 5.Figure 3
E
a(α) and ln A(α) vs α of sample #2.Table 5
lnA and Ea values of 40% DCP obtained by the Friedman method.
α
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
40% DCP
E
a/kJ·mol−1
110.60
110.42
110.08
109.95
108.90
106.91
104.16
98.14
ln (A/s−1)
22.93
23.00
22.98
22.99
22.72
22.19
21.42
19.75As given in Table5, between α = 0.2 and 0.9, Ea is between 110.60 and 98.14 kJ·mol−1 and the variation range is nearly 10%, indicating that it follows a reaction mechanism and can be expressed by the same mechanism function for 40% DCP.


## 3.3. Analysis of ARC Measurement

Seven experiments with differentΦ are carried out according to the above conditions. The ARC curves of temperature-time (T-t) and pressure-time (P-t) of sample #2 are shown in Figures 4 and 5, and all measured data are listed in Table 6.Figure 4
T-t of exothermic section for sample #2.Figure 5
P-t of exothermic section for sample #2.Table 6
Measured adiabatic test results of sample #2 with differentΦ values.
No.
Ton,m (°C)
ΔTad,m (°C)
(dT/dt)max,m
Tmax,m (°C)
Hm (J·g−1)
Pend,m (bar)
1
120.43
First part: 4.00Second part: 2.01
0.024
131.67
49.34
3.01
2
115.54
36.28
0.36
143.74
266.00
5.42
3
115.60
47.94
1.21
155.35
245.86
7.52
4
110.42
59.89
2.65
162.94
244.08
10.22
5
110.49
71.65
10.38
166.51
238.92
15.39
6
110.50
75.81
12.97
175.86
242.44
17.43
7
105.49
87.88
166.52
175.56
238.00
35.77
Ton,m, measured onset decomposition temperature, °C. ΔTad,m, measured adiabatic temperature rise, °C. (dT/dt)max,m, measured maximum self-heating rate, K·min−1. Tmax,m, measured temperature at maximum rate, °C. Pend,m, measured final pressure of decomposition, bar.As shown in Figure4, the ARC curve of the first test is different from others, and the exothermal reaction is divided into two parts. Table 6 shows that the maximum self-heating rate ((dT/dt)max,m), the measured adiabatic temperature rise (ΔTad,m), and the decomposition heat (Hm) of the first test are much smaller than other tests. The small sample mass results in high Φ of the first test, and it makes part of the (dT/dt)m less than the detection sensitivity 0.02°C·min−1 and part of the decomposition heat is missed. Obviously, too large Φ can change the temperature curve entirely and result in the distortion of the tested results.As shown in Table6, Ton,m is rising with the increase of Φ. The reason is that the small amount of the sample needs a higher temperature to make the (dT/dt)m of the sample and the bomb system reach the ARC detection sensitivity (0.02°C·min−1). It is also shown that ΔTM and (dT/dt)max,m decrease with the increase in Φ. In addition, since the material and energy balance of the sample and the bomb system were complicated, the (dT/dt)max,m with Φ was less linear.


## 3.4. Kinetic Calculation

The heat released by the sample in the ARC test not only heats the sample itself, but also raises the temperature of the bomb. The smaller the sample mass, the higher proportion of heat will be used for heating the bomb and the corresponding temperature parameters (such as temperature curves and heating rate curves) will be affected too. So, it is very important to introduce the thermal inertia factorΦ, which can be estimated by the following equation:(1)Φ=msCs+mbCbmsCs.The Townsend–Tou method [8–10] and the Fisher method [11–13] have been widely used. Literature [3] shows that thermal decomposition of DCP is an nth-order reaction, and the Fisher method is adopted to modify the ARC data in this paper.The corrected equations for the Fisher method are listed as follows:(2)1Ton,a=1Ton,m+REalnΦ,Tad,a=Ton,a+ΦTm−Ton,m,dTdtad,a=ΦdTdtad,mexpEaR1Ton,a−1Tad,a,where R is the gas constant; Ea is the apparent activation energy, kJ·mol−1; Ton,m is the corrected onset temperature by the Fisher method, °C; Tad,a is the corrected temperature by the Fisher method, °C; (dT/dt)ad,a is the corrected self-heating rate by the Fisher method, K·min−1; and (dT/dt)ad,m is the measured self-heating rate, K·min−1.The corrected results are listed in Table7.Table 7
Adiabatic test results of sample #2 corrected by the Fisher method.
No.
2
3
4
5
6
7
Φ
3.76
2.63
2.09
1.71
1.64
1.39
Ton,a (°C)
106.86
108.84
105.45
106.79
107.03
103.17
ΔTad,a (°C)
136.41
126.08
125.17
122.52
124.33
122.15
(dT/dt)max,a
6964.1
1491.9
1184.4
928.8
913.3
3292.3
Ton,a, corrected onset decomposition temperature, °C. ΔTad,a, corrected adiabatic temperature rise, °C. (dT/dt)max,a, corrected maximum self-heating rate, K·min−1.Kinetic parameters are important for thermal hazard prediction. For the decomposition reaction in the adiabatic system, the relationship between temperature and the self-heating rate is [14](3)dTdta=ATf,a−Tad,aΔTad,anΔTad,ac0n−1exp−EaRTad,a,where Tf,a is the corrected final temperature of decomposition, °C; A is the pre-exponential factor, min−1; n is the reaction order; and c0 is the initial concentration of sample, mol·L−1.Suppose(4)k∗=kc0n−1=c0n−1Aexp−EaRTa=dTdtaΔTad,aTf,a−Tad,anΔTad,a−1where k is the rate constant.Logarithmic form of equation (4)is(5)lnk∗=lnAc0n−1−EaR1Tad,a.The reaction order which can make the plot of ln(k∗) vs. 1/Tad,a to be a straight line is analysed correctly. Then, Ea and A can be calculated from the slope and the intercept.The kinetic analysis is not applied to the first test for its self-heating rate is discontinuous. For other samples, the reaction ordern = 1 is assumed and substituted into equation (4), and then ln(k∗) is plotted against 1/Tad,a. Linear fitting of ln(k∗) and 1/Tad,a is done, and the fitting results and the values of Ea and A are shown in Table 8.Table 8
The linear fitting results.
No.
2
3
4
5
6
7
Φ
3.76
2.63
2.09
1.71
1.64
1.39
n
1
1
1
1
1
1
E
a (kJ·mol−1)
187.30
176.51
178.92
175.70
172.76
168.37
A (min−1)
1.38 × 1022
5.00 × 1020
1.01 × 1021
3.84 × 1020
1.58 × 1020
5.00 × 1019
R
2
0.9964
0.9992
0.9981
0.9969
0.9982
0.9968
R
2: linear fitting correlation coefficient.The high correlation coefficient indicates that the linear fitting results are reliable whenn = 1. It means that the DCP decomposition is a very typical n-order reaction, and the Fisher method which is selected for correction before is appropriate.In addition, a good linear regression of lnA vs. Ea is found with different Φ, showing a typical kinetic compensation effect [15], as shown in Figure 6. This means that the kinetic compensation effect exists only if the reaction mechanism is the same or similar [16].Figure 6
lnA vs Ea with different values of Φ.Comparing DSC and ARC experiments, we can find that the calculatedEa in ARC is generally higher than that in DSC and higher with the increase in Φ. It is mainly because these calculated kinetic parameters obtained by ARC data depend on the adiabatic temperature rise rate of equation (3).With the influence of the big Φ values, the measured adiabatic temperature rise curve of the sample is not the decomposition curve of itself, and it is the curve which is formed after the sample ball “passivation.” So, this analysis of kinetic parameters is not incidental to the sample decomposition, but to the decomposition of composite samples, sample ball, heat transfer, and heat transfer information such as the comprehensive parameter.


## 3.5. Effect ofΦ on TD24

The corresponding initial temperature when the time to maximum rate under adiabatic condition (TMRad) equals 24 hours (TD24) is an important parameter to evaluate the risk of samples. In order to obtain the effects of different Φ on the measured TD24,m and the modified TD24, the experimental data before and after the correction are analysed in this paper. It can be seen from Table 6 that (dT/dt)max,m measured in No. 1 and No. 2 is less than 1, while in No. 7 is 166.52°C·min−1, indicating an unsatisfactory adiabatic environment. Therefore, the data of 40%DCP solution in No. 1, No. 2, and No. 7 are not analysed in this part and only comparing the data of the other four groups is done.The measuredTMRad can be obtained by the following method:(6)TMRad,m=tm−t=∫ttmdt=∫TTmdTAexp−Ea/RTTf−T/ΔTadnΔTadC0n−1,where TMRad,m is the measured arrival time of the maximum reaction rate, min; Tm is the temperature corresponding to the maximum temperature rise rate, °C; tm is the time corresponding to Tm, min; Tf is the exothermic termination temperature,°C; T is the adiabatic test temperature, °C; ΔTad is the measured adiabatic temperature rise, °C; and C0 is the initial molality of the sample, mol·L−1.Equation (6) can be mathematically integrated to obtain TMRad,m, which is the corresponding temperature TD24 for 24 h. If the effect of Φ is taken into account, the correction of Φ needs to be introduced by the following equation:(7)TMRad0=TMRad0,mΦ.The relationship between the measured temperature andTMRad,m under different Φ for 40% DCP is shown in Figure 7. Extrapolated from the measured value to the low temperature, the obtained TD24 is also shown in Figure 7. According to the measured results, equation (7) is used to modify the modified TD24, as shown in Figure 8.Figure 7
The relationship curve between uncorrected solution temperature andTMRad,m for 40% DCP.Figure 8
The relationship curve between corrected solution temperature andTMRad for 40% DCP.It can be seen from Figure7 that under different Φ, the TD24 of the sample obtained from the test data is different and increases with the increase in Φ. This is because the larger the Φ, the slower the heat release rate of the sample in the initial stage, the lower the temperature rise rate, and the longer the time to reach the maximum temperature rise rate. Figure 8 shows that after the correction of the TMRad near the initial decomposition temperature, the relationship between the different Φ and TMRad is relatively consistent and the TD24 of 40% DCP obtained by extrapolation of the modified sample sphere is about 87.20°C. It also shows that after the correction with Φ, the relationship between temperature and corrected TMRad is still quite different at a high temperature. However, near the initial decomposition temperature, the results are almost the same. Therefore, it is verified that equation (7) is only applicable to TMRad near the modified initial decomposition temperature.


## 3.6. Checking Analysis with the Fisher Method

As mentioned in Section3.2, test results are affected by Φ significantly. A normal Fisher method is to amend the test results to the ideal condition Φ = 1, which cannot be realized by the ARC experiment. The experiment data will be corrected to Φ = 1.64 in this article and be compared with the measured data.The corrected equations are as follows:(8)1Ton,a=1Ton,m+REalnΦΦ1,Tad,a=Ton,a+ΦΦ1T−Ton,dTdtad,a=ΦΦ1dTdtmexpEaR1Ton,a−1Tad,a.The corrected results are shown in Figure9 and Table 9.Figure 9
(dT/dt)-T corrected to Φ1 = 1.64 by the Fisher method.Table 9
The parameters corrected toΦ1 = 1.64 by the Fisher method.
No.
2
3
4
5
6
7
Φ
3.76
2.63
2.09
1.71
1.64
1.39
T
on,a (°C)
110.05
112.27
108.77
110.20
110.5∗
106.66
(dT/dt)max,a
49.3
19.8
13.4
12.4
12.97∗
53.6
The values marked with ∗ are measured values.Figure9 and Table 9 show that when Φ is close to 1.64, the corrected results such as (dT/dt)-T and Ton,a are close to the measured results at Φ = 1.64. However, if the test results based on Φ are far away from the target or predicted condition, the results are very unbelievable. Thus, the weight of the sample has great effect on the result of the ARC test. The study demonstrates that under the condition of safety, Φ will be close to 1 with the increasing dosage of sample, and the revised ideal adiabatic data will be more reliable.


## 4. Conclusions

40% DCP with different masses have been studied by using DSC and ARC to obtain its adiabatic decomposition parameters. Then, kinetic analysis is carried out, and the onset temperature and self-heating rate are corrected by the Fisher method. The results indicate that(1)
The calculatedEa value of 40% DCP is stable under the experimental conditions in the DSC test, and the decomposition followed the one-step reaction mechanism.(2)
The values ofEa and lnA obtained from the ARC tests show obvious kinetic compensation effect, it means that even under different Φ conditions, the reaction still follows the same mechanism. The difference from the DSC experiment is that the larger the Φ value is, the greater the calculated Ea and A values are. It indicates that the value of Φ will affect the experimental accuracy.(3)
After modification,Φ has little effect on TD24 at the initial decomposition temperature, but has greater effect at a high temperature (4) After comparing the results corrected with Φ = 1.64 by the Fisher method, it is suggested to perform the test on the condition close to the process condition which is to be predicted, and then it can provide reliable results.



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*Source: 9845894-2020-05-13.xml*