**3. Results and Discussion**

### *3.1. Sectionalized Rules and Peak-Thermal Kinetic Analysis*

### 3.1.1. Characteristics of the Upstream Oily Sludge

Features of the upstream oily sludge are represented in Table 2. Compared with the characteristics of the oily sludge, which was studied by Jing [8] and Xu [9], the upstream oily sludge had higher ash content and heating values, but lower moisture, which was attributed to the quality of the crude oil and the additives that were utilized in the recovery and dehydration processes.



3.1.2. Sectionalized Rules for the Incineration Process of Upstream Oily Sludge

The results of TGA/DSC tests under oxygen-rich conditions for upstream oily sludge are shown in Figure 1 for when the heating rates of *βn* were *βa* = 5 K/min (Figure 1A), *βb* = 10 K/min (Figure 1B), *βc* = 15 K/min (Figure 1C), *βc* = 20 K/min (Figure 1D), and *βd* = 25 K/min (Figure 1E).

**Figure 1.** The DSC/TGA analysis of upstream oily sludge at *βn* (*n* = a, b, c, d, and e); *βa* = 5 K/min (**A**), *βb* = 10 K/min (**B**), *βc* = 15 K/min (**C**), *βc* = 20 K/min (**D**) and *βd* = 25 K/min (**E**), endothermic/exothermic peak analysis (**F**).

As shown in the five TGA curves (Figure 1A–E), the weight loss of upstream oily sludge in the incineration process was 45–47%, and three declining regions were simultaneous obtained. The first weight loss region was obtained from 30 ◦C to 280–350 ◦C, and the weight loss was 15–20%. The second weight loss region of 25–30% was obtained with the operating temperature ranging from 280–350 ◦C to 461.45–553.637 ◦C. The third weight loss region started at 589.248–630.992 ◦C with a slight weight loss of 1.5–5%.

As it was shown in the five DSC curves (Figure 1A–E), two significant exothermic reactions were detected at 237.905–391.764 ◦C and 341.465–553.637 ◦C, sequentially. It was the same as reported in the TGA/DTG test for the pre-dried oily sludge in nitrogen atmosphere [11,12]. Je-Lueng [12] considered that the former exothermic reactions in the pyrolysis process were attributed to the volatilization of volatile contents such as combined water, small hydrocarbons, and small molecular acids, while the latter exothermic reactions were caused by the decomposition of macromolecular compounds, such as for instance, tar, aromatic hydrocarbons, and cycloalkanes. Based on the above-mentioned studies, we inferred that the continuous exothermic reactions in the incineration process were possibly attributed to the volatilization and/or combustion of the volatile contents (in lower temperature region) and macromolecular compounds (in the higher temperature region), respectively. Although the components of volatile contents and macromolecular compounds were not the main objective of this work, the kinetics of each exothermic reaction seemed to be quite different. When the operating temperature exceeded 600 ◦C, with the increase of the heating rate, an endothermic phenomenon was gradually detected, which can probably be attributed to the decomposition of inorganic carbonate [27,28].

When the operating temperature was lower than 237.905 ◦C (Figure 1A–E), no exothermic or endothermic phenomenon occurred in the DSC curves, but a significant weight loss was observed in the TGA curves. When the exothermic or endothermic phenomenon appeared, the weight loss simultaneously accelerated. Both the endothermic and exothermic peak simultaneously shifted to the right with the increase of heating rate from *βa* to *β<sup>e</sup>*, and it was more sensitive and conspicuous in the exothermic regions.

Different types of volatilization decomposition and/or combustion reactions probably occurred in different weight loss regions, which were attributed to the complex components that existed in the upstream oily sludge. Therefore, both the mechanism and kinetic model changed with the increase of operating temperature. Based on whether the endothermic and/or exothermic reactions occurred (or not) in DSC tests, the TGA curves could be sectionalized as four weight loss stages; these are shown in Figure 2A.

**Figure 2.** *Cont.*

**Figure 2.** Four sectionalized stages in TGA. TGA test carried out in different heating rates (**A**); TGA test of stage 1 (**B**); TGA test of stage 2 (**C**); TGA test of stage 3 (**D**); TGA test of stage 4 (**E**).

As it was shown in Figure 1A, the ending temperature was 511.05–550.93 K in Stage 1, and no endothermic or exothermic reaction was detected, but a 10% weight loss ratio was obtained. The first and second exothermic reaction occurred in Stage 2 (weight loss = 10%) and Stage 3 (weight loss = 20%), and the ending temperatures were 614.62–664.91 K and 719.15–822.69 K, respectively. As shown in Figure 1B–E, the weight loss in the TGA curve caused by the endothermic phenomenon started at 614.422 °C, 620.927 °C, 627.884 °C, and 630.992 °C, respectively. Meanwhile, the DSC curve of each *βn* (10 K/min, 15 K/min, 20 K/min, and 10 K/min) showed that the endothermic temperature regions were 635.968–684.208 °C, 655.017–726.35 °C, 659.994–776.673 °C, and 664.713–759.575 °C, respectively. Therefore, the temperature regions for the incineration kinetic modeling of Stage 4 should be set as 635.968–900.00 °C. The thermal parameters such as the weight loss ratios and peak temperature at *βn* (*n* = a, b, c, d, and e) were simultaneously obtained in Table 3.


**Table 3.** Thermal characteristics of upstream oily sludge in stages one through four.

Note: 1. Endothermic (Endo)/Exothermic (Exo); 2. Not Detected.

It was not the weight loss, but rather the conversion ratios of the reactant that were employed in the equations of the incineration kinetics model. Therefore, the instantaneous weight of the samples detected in the TGA curves (Figure 1A–E) could be rearranged as instantaneous conversion ratios at certain operating temperature. The results of the conversion ratios versus operating temperature of stages one through four are shown in Figure 2B–E, respectively. In addition, the relationship between the instantaneous conversion ratio and the operating temperature (*dα*/*dT*) was the slope or the first-order derivative of each curve, as shown in Figure 2B–E. For instance, when the conversion rate was 55% and the heating rate was five K/min (in Stage 1), *dα*55%/*dT* equaled the slope (K), which is shown in the enlarged area of Figure 2B.

### 3.1.3. Peak-Thermal Kinetic Analysis of Endothermic/Exothermic Reactions

Based on Equations (1), (3), and (4), the basic differential form for the study of thermal kinetic analysis could also be represented as Equation (9).

$$
\hbar \alpha / dt = A e^{-\mathcal{E}\_a / RT} \times (1 - a)^n \tag{9}
$$

taking the quadratic differential on both sides of Equation (9), the following equations were obtained:

$$\begin{array}{ll} \frac{d}{dt} \left[ \frac{da}{dt} \right] &= A \left( 1 - a \right)^{n} \frac{d e^{-E\_{d}/RT}}{dt} + A e^{-E\_{d}/RT} \frac{d \left( 1 - a \right)^{n}}{dt} \\ &= \frac{d \alpha}{dt} \times \frac{E\_{d}}{RT^{2}} \times \frac{dT}{dt} - A e^{-E\_{d}/RT} \times n \left( 1 - a \right)^{n - 1} \frac{da}{dt} \\ &= \frac{d \alpha}{dt} \left[ \beta E\_{d} / RT^{2} - A e^{-E\_{d}/RT} \times n \left( 1 - a \right)^{n - 1} \right] \end{array} \tag{10}$$

when the quadratic differential *ddt dαdt* equals zero, which means the maximum or minimum value could be obtained. In DSC curves, the exothermal peak was the maximum value, and the endothermic peak was in response to the minimum value. At the limit value, Equation (10) equals zero, and the peak thermal kinetic equation was expressed as Equation (11):

$$
\beta E\_p / RT\_p^2 = A e^{-E\_p / RT\_p} \times n \left(1 - \alpha\_p\right)^{n-1} \tag{11}
$$

where *Ep* and *Tp* are the exothermal/endothermic peak activation energy and exothermal/endothermic peak operating temperature; and *<sup>α</sup>p* is the exothermal/endothermic peak conversion ratio of the reactant;

Kissinger [36] considered that the formula *n*<sup>1</sup> − *<sup>α</sup>p<sup>n</sup>*−<sup>1</sup> equaled one, taking the natural logarithm of Equation (11) and rewriting it as Equation (12):

$$\ln \frac{\beta}{T\_p^2} \cong \ln \frac{AR}{E\_p} - \frac{E\_p}{RT\_p}; \ln \frac{\beta\_n}{T\_{n-p}^2} \sim \frac{1}{T\_{n-p}} \tag{12}$$

where *Tn*−*<sup>p</sup>* was the peak temperature obtained in the DSC curve, and changed with *β<sup>n</sup>*.

A straight line with slope −*Ep*/*<sup>R</sup>* could be obtained by plotting *ln βn <sup>T</sup>*2*n*−*<sup>p</sup>* versus 1 *Tn*−*<sup>p</sup>* at every endothermic or exothermic peak parameter. This method was called as "Kissinger approach" in this study. The results of the linear fittings for the endothermic peak and the two exothermic peaks were shown in Figure 1F and Table 2. The peak activation value (*Ep*) and the coefficient of determination (R2) of stages two, three, and four were 64.43 ± 5.34 KJ/mol (0.9798), 90.71 ± 13.35 KJ/mol (0.9389), and 102.11 ± 28.93 KJ/mol (0.8616), respectively.

### *3.2. The Reasoning Process of the Modeling Method Applied for Oily Sludge Incineration*

### 3.2.1. The Reasoning Process of Differential Methods

By rearranging and taking the natural logarithm in Equation (6), the activation value obtained in the differential method was shown in Equation (13). Friedman [37] considered that the activation value could be solved in spite of both the nth-order reaction rate equation and the pre-exponential factor. At each heating rate, a straight line with slope −*Ea*/*R* could be obtained by plotting *lnβn*(*dα*/*dT*) versus 1*T* :

$$\text{l.}\,\beta \times da/dT = A \exp(-E\_a/RT)f(a) \Rightarrow \ln \beta (da/dT) = \ln A + \ln f(a) - E\_a/RT \tag{13}$$

If the activation value had been solved and the nth-order reaction rate equation was fitted to Equation (3), simultaneously, the intercept (*lnA* + *lnf*(*α*)) of the straight lines could be applied to solve the reaction order (*n*) and pre-exponential factor (A) from the equation in two unknowns established under different heating rates. Based on Friedman methods (Equation 13), another pathway to gain the reaction order (*n*) and the pre-exponential factor (A) was shown in Equation (14):

$$\beta \times da / dT = A \exp(-E\_d / RT) f(a) \Rightarrow \ln \frac{\beta (da / dT)}{\exp(-E\_d / RT)} = \ln A + n \ln(1 - a) \tag{14}$$

Repeating the method of plotting the straight line by *ln β*(*dα*/*dT*) *exp*(−*Ea*/*RT*) versus *ln*(1 − *<sup>α</sup>*), the slope of the line was the reaction order, and the intercept was *lnA*. In addition, for this study, if all of the activation values obtained by the Friedman methods were shown with a high coefficient of determinations at a variety of heating rates, Equation (14) was fit for the solution of the reaction order and the pre-exponential factor. Otherwise, two intercepts (*lnA* + *lnf*(*α*)) in Equation (13) obtained with a higher coefficient of determination at a certain conversation ratio would be applied, and two linear equations in two unknowns were simultaneously established for the solution of the reaction order and the pre-exponential factor.

### 3.2.2. The Reasoning Process of Integral Methods

⎪⎪⎨

Flynn [38] confirmed that the *<sup>G</sup>*(*α*) in the integral method (Equation 8) could be rearranged by the temperature integral ( *<sup>P</sup>*(*μ*)). The solution of *<sup>P</sup>*(*μ*) was shown in Figure S1 (Supplementary Materials).

$$\begin{cases} \begin{array}{l} G(\mu) = \frac{A}{\overline{\mu}} \int\_{-T\_{\sigma}}^{T} e^{-E\_{\mu}/RT} dT \cong \frac{AE\_{\mu}}{\overline{\rho}\mathbb{R}} \int\_{\infty}^{\mu} -e^{-\mu}\mu^{-2} d\mu\\ \qquad = \frac{AE\_{\mu}}{\overline{\rho}\mathbb{R}} P(\mu); \mu = \frac{E\_{\mu}}{\overline{\kappa}\mathbb{T}} \end{array} \\\ P(\mu) = \int\_{\infty}^{\mu} -e^{-\mu}\mu^{-2} d\mu \Rightarrow \frac{e^{-\mu}}{\mu^{2}} \left(1 - \frac{2!}{\mu} + \frac{3!}{\mu^{2}} - \frac{4!}{\mu^{2}} \cdot \cdots \right) \\\ \qquad = \frac{e^{-\mu}}{\overline{\mu^{2}}} \times \sum\_{N=1}^{\infty} (-1)^{N-1} \frac{N!}{\mu^{N-1}}; \ N \ge 1 \end{array} \tag{15}$$

where *<sup>G</sup>*(*α*) is the integral equation of *f*(*α*), *<sup>P</sup>*(*μ*) is the temperature integral; and N is positive integer, which is greater or equal to one.

In the integral method, *dα*/*dt* disappeared, and the noise effect was avoided. However, the activation energy at certain heating rate equations (Equation (15)) was extremely intractable to acquire, which was attributed to *<sup>P</sup>*(*μ*). Thus, some methods were provided to simplify the solution of Equation (15). Akahira-Sunose [39] deduced that if the *N* in Equation (15) was equal to one, the *<sup>G</sup>*(*α*) could be simplified and expressed as:

$$\begin{cases} P(\mu) \approx \frac{\varepsilon^{-\mu}}{\mu^2}; N = 1\\ G(a) \approx \frac{AE\_a}{\beta R} \times \frac{\varepsilon^{-\mu}}{\mu^2} = \frac{T^2}{\beta^2} \frac{AR}{E} e^{-E\_a/RT} \\ \ln \frac{\beta}{T^2} = \ln \frac{AR}{G(a)E} - E\_a/RT \end{cases} \tag{16}$$

Equation (16) was similar to Kissinger approach (Equation (12)), and the activation energy could be solved by plotting *ln β T*<sup>2</sup> VS. 1 *T* . This type of integral method was utilized in the following sections and was labeled as the Kissinger-Akahira-Sunose method (abbreviated as the KAS (Kissinger-Akahira-Sunose) method) in the following studies. Similarly, if the *N* in Equation (15) was equal to two, combined with the Doyle approach [33], another type of integral method (Equation (17)) was acquired and labeled as the Flynn–Wall–Ozawa method (abbreviated as the FWO (Flynn–Wall–Ozawa) method) [40]. The slope ( −1.0516 *Ea*/*R*) of the straight line that was plotted by *lnβ* VS. 1 *T*was more convenient to solve the acquired energy:

$$\begin{cases} \quad P(\mu) \approx \frac{e^{-\mu}}{\mu^2} \left(1 - \frac{2!}{\mu} \right) \approx 0.00484e^{-1.0516\mu}; N = 2\\ \quad \beta \approx \frac{AE}{RG(a)} \times 0.00484e^{-1.0516\mu} \Rightarrow \ln \beta \approx \ln \frac{AE}{RG(a)} - 5.311 - 1.0516E/RT \end{cases} \tag{17}$$

### *3.3. The Incineration Kinetic Analysis and Model of Stage One*

No endothermic or exothermic phenomenon was obtained in Stage 1. Therefore, the TGA curves were utilized for the incineration kinetic analysis. Both *dα*/*dT* and *T* under various conversion rates *αn* (*n* = 10%, 30%, 50%, 70%, and 90%) and heating rates *βn* (*n* = five K/min, 10 K/min, 15 K/min, 20 K/min, and 25 K/min) were shown in Table 4. Based on equations (13), (16), and (17), the results of linear fitting under the KAS method (Figure 3A), the Friedman method (Figure 3B), and the FWO method (Figure 3C) are shown in Figure 3. The activation energies under various conversion rates are shown in Table 5.


**Table 4.** Parameters for incineration kinetic modeling in Stage 1.

**Table 5.** The activation energies of Stage 1 obtained under the KAS, FWO, and Friedman methods.


In Figure 3 and Table 4, when the conversion ratios are over 50% (Table 4), the corresponding coefficient of determinations under three methods were dramatically higher than 0.9555. Only 5% weight loss was detected in Stage 1 before *α*50%, which was attributed to the dehydration process of oily sludge. When the conversion ratios exceed 50%, the volatilization (not combustion) process of volatiles was performed and confirmed in the thermal kinetic analysis of Stage 2 (Section 3.3). Compared with the coefficient of determinations obtained in the three methods, the Friedman method (Figure 3B) was more fitting for the kinetic modeling of volatilization. Furthermore, the activation energy changed with the conversion ratios in Stage 1, which means that neither the dehydration process nor the volatilization of volatiles was an elementary reaction [39,40]. For the volatilization process, the average activation energy *<sup>E</sup>*0*<sup>α</sup>*−(50%−100%) that was acquired under the Friedman method was 60.87 ± 5.27 KJ/mol. Then, the reaction orders and pre-exponential factors under various heating rates could be solved via Equation (14), as shown in Figure S2 and Table 6. In addition, the average reaction order (or the average pre-exponential factor) was not the arithmetic mean value obtained at a variety of heating rates, but rather the slope (or intercept) of the straight line plotting by the *ln βn*(*dα*/*dT*) *exp*(−*Ea*/*RT*) versus *ln*(1 − *<sup>α</sup>*).

**Figure 3.** The linear fitting results for Stage 1 under the KAS, Friedman, and FWO methods ((**A**)—KAS method, (**B**)—Friedman method, and (**C**)—FWO method).


**Table 6.** The reaction order and pre-exponential factor of Stage 1 at a variety of heating rates.

The average reaction order was *n* = 0.82 ± 0.30, and the average pre-exponential factor was *lnA* = 13.32 ± 0.45. The volatilization kinetic model expressed in differential form and integral form were shown in Equation (18), respectively. Due to the simplification of the temperature integral (*P*(*μ*)), the differential form was better to state the volatilization kinetic model in Stage 1.

$$\text{rad/dt} = \exp(13.32 - 60870/RT)(1 - a)^{0.82}; \text{ } a \in [0.5, 1], T \in (435K, 511K) \tag{18}$$

*3.4. The Incineration Kinetic Analysis and Model of Stages Two, Three, and Four*

Based on Figure 2C–E, the *dα*/*dT* and *T* values for stages two, three, and four under various conversion rates *αn* (*n* = 10%, 30%, 50%, 70%, and 90%) and heating rates *βn* (*n* = five K/min, 10 K/min, 15 K/min, 20 K/min, and 25 K/min) are shown in Table 7.

**Table 7.** Parameters for incineration kinetic modeling in stages two, three, and four.


The linear fitting results of stage two, three, and four under the KAS method (following Equation (13)), Friedman method (following Equation (16)), and FWO method (following Equation (17)) were shown in Figures 4–6, respectively. The activation energies of each stage obtained under the three methods were shown in Table 8.

Both Stage 2 and Stage 3 were exothermic stages, and the *E*0−*α<sup>n</sup>* (Table 8) values that were separately obtained by the KAS method, Friedman method, and FWO method at *αn* (*n* = 10%, 30%, 50%, 70%, and 90%) were quite different. Meanwhile, *E*0−*α<sup>n</sup>* apparently changed with *αn* in each method. In Table 8, the average *E*0 of Stage 2 (Stage 3) obtained under the KAS method, Friedman method, and FWO method were 78.11 KJ/mol (98.82 KJ/mol), 58.07 KJ/mol (77.68 KJ/mol), and 65.63 KJ/mol (81.10 KJ/mol), respectively. In stages two and three (Table 8), the R<sup>2</sup> at a variety of heating rates in the KAS method were all higher than that in the Friedman method and FWO method. Therefore, *E*0 = 78.11 KJ/mol and 98.82 KJ/mol were appropriate for the incineration kinetic modeling of Stage 2 and Stage 3, respectively.

In Stage 2, R<sup>2</sup> was apparently changed with the heating rates, and relatively high R<sup>2</sup> values by Friedman method (Table 8 and Figure 4B) were obtained at *α*10% (R<sup>2</sup> = 0.9693) and *α*30% (R<sup>2</sup> = 0.9774). The intercepts of *α*10% and *α*30% linear fitting curves (Figure 4B) plotted by *ln*(*βn* × *dα*/*dT*) versus 1*T*

were 19.33 ± 1.24 and 18.45 ± 1.39, respectively. Thus, two linear equations in two unknowns were simultaneously established and expressed in Equation (19):

$$\begin{cases} \ln A + n \ln(1 - a\_{10\%}) = 19.33 \pm 1.24\\ \ln A + n \ln(1 - a\_{30\%}) = 18.45 \pm 1.39 \end{cases} \tag{19}$$

The reaction order and the pre-exponential factor for Stage 2 were *lnA* = 19.69 and *n* = 3.50, respectively. The reaction rate equation of Stage 2 was *f*(*α*) = (1 − *α*)3.5. The kinetic model for the combustion of volatile components was expressed as:

$$da/dt = \exp(19.69 - 78110/RT)(1 - a)^{3.5}; a \in [0, 1], \ T \in (511K, 658K) \tag{20}$$

Similarly for Stage 3, relatively high R<sup>2</sup> values (Table 8 and Figure 5B) were obtained at *α*10% and *α*50% by the Friedman method. The *ln*(*βn* × *dα*/*dT*) intercepts and R<sup>2</sup> values of the *α*10% and *α*50% linear fitting curves (Figure 5B) were 20.74 ± 0.74, 0.9942 and 19.24 ± 1.32, 0.9602, respectively. The reaction order and the pre-exponential factor were solved from the followed equations:

$$\begin{cases} \ln A + n \ln(1 - \alpha\_{10\%}) = 20.74 \pm 0.74\\ \ln A + n \ln(1 - \alpha\_{50\%}) = 19.24 \pm 1.32 \end{cases} \tag{21}$$

The reaction order, the pre-exponential factor, and the reaction rate equation for Stage 3 were *lnA* = 21.00, *n* = 2.50, and *f*(*α*) = (1 − *α*)2.5, respectively. The kinetic model was expressed as:

$$
\ln \text{/dt} = \exp(21.00 - 98820 / \text{RT})(1 - a)^{2.5}; \ a \in [0, 1], \ T \in (658 \text{K}, 793 \text{K})\tag{22}
$$


**Table 8.** The activation energies of stages two, three, and four obtained from the KAS, FWO, and Friedman methods.

The endothermic phenomenon that existed in Stage 4 and the R<sup>2</sup> of the linear fitting curves in the KAS method were dramatically higher than those in the Friedman method and FWO method. Therefore, *E*0 = 15.96 KJ/mol was the optimum parameter for the incineration kinetic modeling of Stage 4. As it was shown in Figure 6B, significant errors appeared in the linear fitting curves of Stage 4 under the Friedman method, and a relatively low R<sup>2</sup> value was obtained in each heating rate. Thus, the reaction order and the pre-exponential factor could not be obtained in Equation (13) or Equation (14). We inferred that the probe that was utilized to detect the weight of the reactant was significantly affected by the operating temperatures and caused the apparent errors of *dα*/*dt* or *dα*/*dT*.

**Figure 5.** The linear fitting results for Stage 3 under the KAS, Friedman, and FWO methods ((**A**)—KAS method, (**B**)—Friedman method, and (**C**)—FWO method).

### *3.5. The Judgement of Sectionalized Modeling in Differential/Integral Method*

For the incineration kinetics modeling of upstream oily sludge, the *Ep* values of Stage 2 (or Stage 3, Table 1) obtained under the Kissinger approach were lower than the *E*0 values from the KAS method, which means that the peak operating temperature was more appropriate as the incineration temperature in engineering use. Attributed to the background noise or sensitivity of the probe, the activation energy (*Ea*) that was obtained in differential methods (the Friedman method) was imprecise, and the value of R<sup>2</sup> also demonstrated that the integral method was more suitable than the differential method.

In comparison with the previous reports in Table 1, the differential method (Friedman method) was more convenient to obtain the pre-exponential factor (A) and the representation of the nth-order reaction rate equation *f*(*α*) or the reaction order (*n*). It seems that the comprehensive differential integral method was more reasonable to solve the basic three elements for the incineration kinetics analysis of upstream oily sludge. However, both the approximate solution of temperature integral *<sup>P</sup>*(*μ*) that existed in the integral method and the model that was utilized in engineering use should be evaluated and adjusted.
