## 4 Case Study
### Failure Modes and Effects Analysis
Thus, the formulation applied for the resolution of the reliability calculation of each configuration present in the subsystems (series, K/N and fractionation) is expressed below:  
* Subsystem 1, HP pumps arranged in K/N configuration: 3/5: \[R(t)=K_{1}+K_{2}+K_{3}\] (1)  
where K\({}_{1}\) represents the probability of failure of two pieces of equipment simultaneously among the five total pieces of equipment that make up the system, expressed by the following:  
\[K_  
Finally, for the term K\({}_{3}\), we consider the probability of operation of the five pieces of equipment without failure; that is:  
\[K_{3}=R_{1}\cdot R_{2}\cdot R_{3}\cdot R_{4}\cdot R_{5} \tag{19}\]  
where:  
\(R_{i}=R_{i}(t)=\) Reliability of Pump HP i at time t  
* Subsystem 2, SW pumps arranged in a K/N: 34 configuration:  
\[R(t)=K_{1}+K_{2} \tag{20}\]  
where K\({}_{1}\) and K\({}_{2}\) are the probability of one equipment failure and no failure, respectively. Thus, the expressions for K\({}_{1}\) and K\({}_{2}\) are as follows:  
\[K_{1}=P_{f1}+P_{f2}+P_{f3}+P_{f4} \tag{21}\]  
\[K_{2}=R_{1}\cdot R_{2}\cdot R_{3}\cdot R_{4} \tag{22}\]  
Developing the terms of K\({}_{1}\), we have the following:  
\[P_{f1}=(1-R_{1})\cdot R_{2}\cdot R_{3}\cdot R_{4} \tag{23}\]  
\[P_{f2}=R_{1}\cdot(1-R_{2})\cdot R_{3}\cdot R_{4} \tag{24}\]  
\[P_{f3}=R_{1}\cdot R_{2}\cdot(1-R_{3})\cdot R_{4} \tag{25}\]  
Figure 6:0  
where:  
\(R_{i}=R_{i}(t)=\) Reliability of SW Pump i at time t.  
* Subsystem 3, ORV and SCV vaporizers arranged as a shared-load based redundancy configuration with 33% of load sharing fraction:  
Figure 6:1
* Subsystem 4, GM gas meters arranged in a K/N: 2/3 configuration:  
Figure 6:2  
where K\({}_{1}\) and K\({}_{2}\) are the probability of one equipment failure and no failure, respectively. Thus, the expressions for K\({}_{1}\) and K\({}_{2}\) are as follows:  
Figure 6:3  
Figure 6:4  
Developing the terms of K\({}_{1}\), we have:  
Figure 6:5  
Figure 6:6  
Figure 6:7  
* Subsystem 5 Pipeline Emergency Valves Figure 6:8 (34)  
The RBD initially proposed in Figure 7 is simplified to an equivalent system with a series configuration composed of the aforementioned five subsystems, resulting in the RBD shown below:A drastic drop in reliability can be seen after approximately 3500 h. This is mainly due to the influence of the ORV and SCV vaporizers, which present an average of two failures per year, influencing or conditioning the drop in system reliability. For the simulation of maintainability, we have chosen to use the negative exponential probabilistic function, which consists of a single parameter \(\mu\) that is equivalent to the inverse value of the MITR obtained from the data provided by the FMEA. **Approximation to estimate availability**  
[PERSON] (2021) defines system availability as the ability of a system to be operating properly when it is required for use [47]. In this work, a tool capable of predicting the availability of the system according to a mission time is developed. According to [PERSON] [47], it becomes complex to obtain the resulting mathematical expressions, so, the use of approximations or numerical solutions becomes the only alternative to obtain the A(t). One of the approximations developed by [36] was used to estimate A(t), which is based on the alternate renewal process, resulting in the equation presented below:  
Figure 6:9  
where:  
* \(\overline{A}(t)\): Unavailability. * t: Time. * \(\lambda(t)\): Failure rate at time t.
* \(\mu\): Reparation rate (constant). Which can be developed in the same way as shown in the following expression: \[R(t)=K_{1}+K_{2}+K_{3}\]0 (36) where: \(\bullet\)\(A(t)\): Availability at time t.  
\[R(t)=K_{1}+K_{2}+K_{3}\]1 System reliability R(t). \[R(t)=K_{1}+K_{2}+K_{3}\]2 Equivalent RBD (Reliability Block Diagram) of each subsystem. This approximation is valid when the failure rate or repair rate is time-dependent and when the value of \(\lambda\) is much smaller than the value of \(\mu\), as is the case for each of the LNG system components described in the FMEA.