## 4 Data Analysis and Results
### Unloading System Reliability  
Since the components and the whole system are assumed to have constant failure rates (random failures), the reliability figures must be investigated in multiple time intervals. For each component, the reliability of the system is calculated at the end of the first year, the fifth year, the tenth year, and the twentieth year using Equation (3). Table 5 shows the reliability of each component in the system. \[R_{i}(t) = e^{-\lambda t} \tag{3}\] \[R_{p}(t) = 1-F(t) = 1-\prod_{i=1}^{n_{p}}[1-R_{i}(t)] \tag{4}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Component** & **Failure Mode** \\ \hline \multirow{2}{*}{Gate Valve} & Internal leakage \\ \cline{2-3}  & External leakage \\ \hline \multirow{4}{*}{Pressure Relief Valve} & External leakage \\ \cline{2-3}  & Fail to close on demand \\ \cline{2-3}  & Fail to open on demand \\ \cline{2-3}  & Spurious operation \\ \cline{2-3}  & Valve leakage in closed position \\ \cline{2-3}  & Delayed operation \\ \hline \multirow{4}{*}{Positive Displacement Flow Meter} & Parameter deviation \\ \cline{2-3}  & Low output \\ \cline{2-3}  & Vibration \\ \hline Plug Valve & External \& Internal leakage \\ \hline \multirow{4}{*}{Flow Control Valve} & Leakage in closed position \\ \cline{2-3}  & Internal leakage \\ \cline{2-3}  & External leakage \\ \cline{2-3}  & External leakage \\ \cline{2-3}  & Low output \\ \cline{2-3}  & Structural deficiency \\ \hline Differential Pressure Transmitter & Leakage \\ \hline \multirow{4}{*}{Level Switch} & High output \\ \cline{2-3}  & Low output \\ \cline{2-3}  & \multicolumn{2}{c}{Erroric output} \\ \cline{2-3}  & Fail to function on demand \\ \cline{2-3}  & Spurious operation \\ \hline Pressure Transmitter & Leakage \\ \hline \multirow{4}{*}{Air Eliminator—Separator} & Parameter deviation \\ \cline{2-3}  & Plugged/Choked \\ \cline{2-3}  & Structural deficiency \\ \hline \multirow{4}{*}{Strainer} & External leakage \\ \cline{2-3}  & Plugging \\ \hline \hline \end{tabular}
\end{table} 
Table 4: _Cont._where \(R_{i}(t)\) is the component \(i\) reliability, \(R_{p}(t)\) is the reliability of components connected in parallel, \(F(t)\) is the cumulative probability distribution, and \(n_{p}\) is the number of components connected in parallel. For the other components connected in series, individual reliability was calculated using Equation (3). Considering each sub-system with its redundant as a single component that has a single reliability value as illustrated in the streams 1 and 2 combined row as given in Table 3 will validate the concept of a complete series system. Streams 1 and 2 combined reliability was calculated using Equation (4), considering stream 1 and stream 2 in parallel connection. As a result, the system's components shown in Figure 2 become one line of series components from start to end. Consequently, the total system reliability can be calculated using Equation (5). It is important to note that neither the unloading arms nor the ball valves were directly considered in the system reliability calculations, instead, their subsystem (stream 1 and 2 combined) was directly considered in the system reliability calculations and this will account for both unloading arms and the ball valves connected to them. \[R_{s}(t)=\ r_{1}(t)r_{2}(t)\ldots\ r_{n}(t)=\ \prod_{i=1}^{n}r_{i}(t) \tag{5}\]  
where \(r_{i}(t)\) is the reliability of component \(i\) and \(n\) is the number of components connected in series. Figure 4 shows the total system reliability at different time periods (e.g., t = 0, 1, 5, 10, and 20 years) during the normal operating phase. During this phase, it is certain that the system is currently fully functional; therefore, the reliability equals 1 at time 0. Based on the system reliability plot, it is noticed that the reliability decreases dramatically with the highest slope between 0 and 2 years. Then, the slope (2-5 years) tends to be much smaller.