API 581 Probability of Failure (POF) Using Weibull analysis for Pressure Relief Devices (PRDs).

The two-parameter Weibull distribution method

The two-parameter Weibull distribution method is a more flexible approach to calculating the Probability of Failure (POF) for equipment like Pressure Relief Devices (PRDs) and heat exchanger bundles. This method relies on historical failure data, similar to the Generic Failure Frequency (GFF) approach. However, the Weibull method employs a probability distribution that is characterized by two parameters: the shape parameter (β) and the scale parameter (η).

Where the Weibull Shape Parameter, β, is unit-less, the Weibull characteristic life parameter, η, in years, and t is the independent variable time in years.

In the two-parameter Weibull distribution method, you start by collecting historical failure data for the asset in question. Statistical analysis is then performed to estimate the shape and scale parameters of the Weibull distribution. The shape parameter β indicates the nature of the failure rate. A β less than 1 indicates that the failure rate decreases over time, β equal to 1 indicates a constant failure rate, and β greater than 1 indicates that the failure rate increases with time. The scale parameter ηη is a scaling factor that stretches or compresses the distribution.

Once these parameters are estimated, they are used to calculate the POF for the asset over a specific time period. The formula for calculating POF in this case is:

POF=1?e?(t/η)β

Here, t represents the time period of interest, and e is the base of the natural logarithm. By using this formula, the Weibull method provides a versatile way to model different kinds of failure behaviours, making it particularly useful for assets where the failure rate is not constant.

This method offers a more nuanced representation of failure probability, capable of capturing increasing, decreasing, or constant failure rates. It is computationally more intensive than the GFF method but offers a more detailed and accurate assessment, especially for complex assets with variable operating conditions.

In the context of the Weibull distribution, two important parameters define its shape and scale: the shape parameter (β) and the scale parameter (η). These parameters help in modelling different types of failure behaviours and provide a way to quantify the reliability of systems or components.

Shape Parameter (β)

The shape parameter, often denoted by β, is crucial for determining the nature of the failure rate over time.

• β<1: Indicates that the failure rate decreases over time, which is often the case for "infant mortality" failures, where components are more likely to fail when they are new but become more reliable as they survive the initial period.
• β=1: Signifies a constant failure rate over time, akin to the exponential distribution. This is often applicable to random failures that are memoryless, meaning the likelihood of failure doesn't change over time.
• β>1: Suggests that the failure rate increases with time, often seen in "wear-out" phases where components become increasingly likely to fail as they age.

Scale Parameter (ηη)

The scale parameter, often represented by ηη, essentially stretches or compresses the distribution along the time axis. It gives a measure of how "spread out" the failures are and is sometimes considered a characteristic life parameter. Specifically, when t=η, about 63.2% of the population will have failed if β=1.

The scale parameter can be interpreted as the time at which 63.2% of the population would have failed (again assuming β=1). For other values of β, ηη can be seen as a sort of 'shift' in the lifespan of the component, moving the failure curve left or right along the time axis.

Combined Role

Together, the shape and scale parameters allow the Weibull distribution to model a wide range of failure behaviours. They can be estimated using statistical methods based on historical failure data or sometimes using maximum likelihood estimation methods. Once these parameters are known, they can be plugged into the Weibull formula to predict the probability of failure at any given time, providing a powerful tool for reliability engineering and risk-based inspection analysis.

Weibull Characteristic Life

The η parameter is defined as the time at which 63.2% of the units have failed. For β = 1 , the MTTF and η are equal. This is true for all Weibull distributions regardless of the shape factor. Adjustments are made to the characteristic life parameter to increase or decrease the POF as a result of environmental factors, and asset types, or as a result of actual inspection data. These adjustments may be viewed as an adjustment to the mean time to failure (MTTF).

Using Weibull curves in calculating failure probabilities for Pressure Relief Devices (PRDs), specifically the Probability of Failure to Open on Demand (POFOD) and the Probability of Leakage (Pl). In the Weibull distribution, the η parameter, known as the characteristic life, is especially important. When the shape parameter β is 1, the η parameter is equivalent to the Mean Time to Failure (MTTF), essentially serving as an average life expectancy for the device.

Adjustments to the η parameter are often made to fit specific conditions such as the environmental factors, the type of PRD, and even actual inspection data. By modifying this parameter, you're effectively changing the MTTF for the device, which in turn alters the calculated probabilities for both POFOD and leakage.

If device-specific failure data is lacking, industry-wide data can serve as a default setting. The underlying assumption here is that PRDs in similar types of service will have similar failure behaviours, including POFOD and leakage probabilities.

So, even in the absence of large sets of failure data for a specific device, the Weibull distribution still serves as a useful starting point.

In addition to these statistical considerations, various operational and environmental factors play a role in the failure rate of PRDs. These include not just the conditions under which the PRD operates—like temperature and corrosivity—but also external conditions like how the device is handled during transport and installation. Other factors like excessive piping vibration and increased demand rates can also significantly impact the failure rates, and these are considered when adjusting the η parameter.

Overall, the Weibull distribution provides a versatile and adaptable framework for risk-based inspection and planning. By allowing for adjustments to the scale parameter η, it accommodates real-world variables, making it particularly useful in industries where PRDs are a common component.