# Aging Performance and an Improved Evaluation Method for PE80 and PE100 Pipelines for Urban Gas
## 4 Aging Rate Calculation Model  
### Aging Rate Calculation Results  
The OIT data of aged SDR11 DN63 PE80, SDR11 DN110 PE80, and SDR11 DN63 PE100 gas pipes were obtained from the experiments above. According to the Arrhenius equation, the aging rates of PE pipes can be calculated as follows [22]. \[Q=f(q)=A\exp(-Kt) \tag{1}\]  
where \(Q,f(q)\) is the rate constant (\(F/F_{0}\)); \(F\) is the property of materials at failure; \(F_{0}\) is the current property; \(t\) is the aging time; \(K\) is the reaction rate constant; and \(A\) is the material constant. Equation (1) can be changed into Equation (2). \[F=AF_{0}\exp(-Kt) \tag{2}\]  
The relationship between \(K\) and the temperature follows the Arrhenius equation (Equation (3)):  
\[K=Ze^{\frac{-E}{Kt}} \tag{3}\]  
where \(T\) is the absolute temperature (K); \(E\) is the apparent activation energy (J\(\cdot\)mol\({}^{-1}\)); \(Z\) is the frequency factor (d\({}^{-1}\)); and \(R\) is the gas constant (8.31 J\(\cdot\)K\({}^{-1}\cdot\)mol\({}^{-1}\)). Let \(X=t\), \(Y=\) In\(F\), \(a=\) In\(AF_{0}\), and \(b=-K\); then, Equation (2) can be changed into a linear equation. \[Y=a+bX \tag{4}\]  
Let \(X_{1}=1/T\), \(Y_{1}=\) In\(K\), \(a_{1}=\) In\(Z\), and \(b_{1}=(-E)/R\); then, Equation (3) can be changed as follows:  
\[Y_{1}=a_{1}+b_{1}X_{1}\] \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Temp/\({}^{\circ}\)C**} & \multirow{2}{*}{**Time/h**} & \multicolumn{8}{c}{**OIT/min**} \\ \cline{3-10}  & & \multicolumn{3}{c}{**PE80 DN63**} & \multicolumn{3}{c}{**PE100 N63**} & \multicolumn{3}{c}{**PE80 DN110**} \\ \cline{3-10}  & & **0 MPa** & **0.2 MPa** & **0.4 MPa** & **0 MPa** & **0.2 MPa** & **0.4 MPa** & **0 MPa** & **0.2 MPa** & **0.4 MPa** \\ \hline \multirow{8}{*}{80} & 0 & 88.5 & 88.5 & 87.3 & 87.3 & 87.3 & 88.5 & 88.5 & 88.5 \\  & 144 & 82.3 & 79.9 & 78.4 & 82 & 79.8 & 79.2 & 84.4 & 84 & 83.9 \\  & 288 & 78.1 & 76.7 & 74.6 & 79.7 & 78.2 & 77.8 & 82.3 & 82.1 & 82 \\  & 576 & 76 & 74 & 71.2 & 78.2 & 75.8 & 74.6 & 80.1 & 80 & 78.1 \\  & 864 & 73.8 & 70.1 & 65.4 & 76.6 & 74 & 71 & 78.9 & 76.8 & 75.4 \\ \hline \multirow{8}{*}{90} & 0 & 88.5 & 88.5 & 88.5 & 87.3 & 87.3 & 87.3 & 88.5 & 88.5 & 88.5 \\  & 96 & 81.2 & 80 & 79 & 83.2 & 82.7 & 82.6 & 85.1 & 84.4 & 83.8 \\  & 192 & 80.3 & 78.1 & 75.7 & 82.3 & 78.6 & 77.8 & 83 & 82.5 & 82.2 \\  & 288 & 77.9 & 76.7 & 74.8 & 80.5 & 77.4 & 75.8 & 81.4 & 80.2 & 79.5 \\  & 384 & 76.7 & 74.7 & 73.3 & 77.1 & 74.8 & 72 & 79.4 & 78 & 76.5 \\ \hline \multirow{8}{*}{100} & 0 & 88.5 & 88.5 & 88.5 & 87.3 & 87.3 & 87.3 & 88.5 & 88.5 & 88.5 \\  & 24 & 86.1 & 84.3 & 83.3 & 84.8 & 82.8 & 81.9 & 85 & 84.5 & 83.7 \\  & 48 & 83.3 & 82.4 & 79.9 & 82 & 79.2 & 80.3 & 83.8 & 83 & 82.5 \\  & 96 & 80.2 & 77.2 & 76.3 & 79.8 & 77.2 & 77 & 83.2 & 82.3 & 80.5 \\  & 192 & 78.5 & 76.3 & 73.4 & 77.8 & 75 & 73.8 & 81 & 79.4 & 78 \\ \hline \multirow{8}{*}{110} & 0 & 88.5 & 88.5 & 88.5 & 87.3 & 87.3 & 87.3 & 88.5 & 88.5 & 88.5 \\  & 8 & 85.8 & 85.6 & 85.4 & 84.6 & 83.9 & 83.6 & 85.6 & 84.7 & 83.9 \\ \cline{1-1}  & 16 & 85.6 & 83.1 & 82 & 83.1 & 81.5 & 80.1 & 85.2 & 84.2 & 83.3 \\ \cline{1-1}  & 32 & 83.5 & 81.9 & 78.6 & 82.2 & 79.8 & 78.5 & 83.5 & 83.4 & 82.3 \\ \cline{1-1}  & 72 & 80.6 & 79.1 & 76.3 & 80.1 & 77.8 & 74.5 & 82 & 80.1 & 78.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Oxidation induction periods under various experimental conditions. The relationship between time and ln(OIT) was plotted to determine the reaction rate \(K\) (Figures 4-6,Table 4). According to the fitting results of ln(OIT), the reaction rate constant \(K\) obtained under different conditions is shown in Table 4.  
Figure 4: Relationship between time and ln(OIT) of PE80 DN63. Figure 5: Relationship between time and ln(OIT) of PE100 DN63. Figure 6: Relationship between time and ln(OIT) of PE80 DN110. When the inner pressure is high, the diffusion and consumption of antioxidants in PE pipes will be accelerated and the aging rate is higher. The reaction rates of PE100 are lower than those of PE80 under the same service conditions.