The next step is to obtain the probability that the PRD will fail to open upon demand in service. API RP 581 provides default failure on demand failure rates developed from industry data. These default values are expressed as default Weibull curves that are modified by several factors based on the following procedure:
1) Determine default Weibull parameters, β and ηdef , based on category of service severity (Section 7.2.4.b), selection of the default POFOD curve (Section 7.2.4.c) and type of PRD (Sections 7.2.4.d through 7.2.4.f).
2) Apply an adjustment factor, Fc, for conventional valves discharging to closed system or to flare (Section 7.2.4.g).
3) Apply an adjustment factor, Fenv, for environmental factors (Section 7.2.4.h).
4) The result of the procedure outlined above will be a modified characteristic life, ηmod, as defined in Equation (1.14).
mod F Fc env def
η = ⋅ ⋅η (1.14)
5) At this point, the modified characteristic life, ηmod, needs to be updated to the updated characteristic life, ηupd, based on the PRD’s specific inspection and testing history (Section 7.2.4.i).
6) This updated characteristic life, ηupd, is then used to calculate the POFOD as a function of time, t ,
for the specific PRD in accordance with Equation (1.15).
1 exp
fod
upd
P t
β
η
= − −
(1.15)
7) The POFOD should be adjusted based on the overpressure scenario with Equation (1.16). The overpressure factor,FOP j,
, is an adjustment for overpressure scenarios higher than 1.3 times the set pressure (Section 7.2.4.j). The subscript jidentifies the specific overpressure and accounts for the fact that each has a different potential overpressure.
, ,
fod j fod OP j
P =P ⋅F (1.16)
b) Categories of Service Severity
The failure rates of PRDs are directly related to the severity of service in which they are installed. Different categories of service are established in the PRD module as a function of the fluid tendency to induce PRD failure due to corrosion, fouling, plugging, or other effects. Temperature has also been found to be a factor in determining the severity of service. The categories of service severity (MILD, MODERATE, or SEVERE) are linked to specific failure tendencies (and default Weibull cumulative failure distribution curves) and are described in Table 7.4.
It is important to note that a fluid that is classified as being a MILD service group for the fail to open failure mode is not necessarily a MILD service for the leakage failure mode. As an example, industry failure data shows that cooling water, which is known to be a dirty/scaling service, has one of the highest failure rates for the FAIL to open case and therefore may be classified as SEVERE for the FAIL case. Conversely, PRDs in cooling water service have not demonstrated a significant amount of leakage failures and therefore may be classified as MILD for the leak case. Another example is steam, where industry data indicates that steam should be classified as MILD for the fail to open case, but classified as SEVERE for the leak case. Steam is known to be a leaking service due to the erosive nature of the high temperature steam.
c) Default Probability of Failure on Demand vs. Time in Service 1) General
Table 7.5 provides the default Weibull parameters for failure to open for conventional spring-loaded pressure relief valves (PRVs), balanced bellows PRVs, pilot-operated PRVs, and rupture disks. These parameters were determined using industry failure rate data. The majority of the available data indicated successful performance during the interval that the PRD was in service. The successful test points are referred to as suspensions and were included with the failure data in determination of the Weibull parameters.
Weibull parameters are provided for the three categories of PRD service severity; MILD, MODERATE, and SEVERE, as discussed in Section 7.2.4.b). These values, when substituted into the Weibull cumulative failure density function,F t( )given by Equation (1.10), provide the default POFOD curves for each of the PRD types listed in the table.
For example, Figure 7.2 provides the default Weibull cumulative failure distribution curves used for spring-loaded conventional PRVs using the Weibull function to describe the three categories of service severity.
Note that the units for the POFOD data presented in Figure 7.2 are failures/demand as this data was established from bench tests of actual PRDs, not from continuous service data. POFOD should not be confused with POF (failures per year) that includes the demands on the PRD (see Section 7.2.3) and the probability that the protected equipment will fail (see Section 7.2.5).
The cumulative failure distribution curves shown in Figure 7.2 and the Weibull parameters presented in Table 7.5 are the default values based on the category of service severity of the PRD being evaluated.
These base values are defaults and should be overridden if the owner-user provides site-specific data as explained in Section 7.2.4.c.3.
2) Presence of an Upstream Rupture Disk
Rupture disks are often installed in combination with PRVs to isolate the valve from process conditions and corrosive or fouling fluids that can reduce the probability that the valve will open upon demand. API RP 520 Parts 1 and 2 provide additional information related to the use and installation of rupture disks upstream of PRVs.
The presence of upstream rupture disks is accounted for by using the POFOD curve for MILD service regardless of what fluid severity is selected. This assumes that the space between the rupture disk and the PRV is vented and monitored for leakage as required by Code and as recommended by API RP 520. If this is not the case, the upstream rupture disk should not be considered in the analysis (i.e., assume the disk is not present).
3) Use of Plant-Specific Failure Data
Data collected from specific plant testing programs can also be used to obtain POFOD and probability of leakage values. Different measures such as MTTF or failure per million operating hours may be converted into the desired form via simple conversion routines.
d) Default Data for Balanced Bellows Pressure Relief Valves
A balanced spring-loaded PRV uses a bellows to isolate the back side of the disk from the effects of superimposed and built-up back pressure. The bellows also isolates the internals of the PRD from the corrosive effects of the fluid in the discharge system.
An analysis of industry failure rate data shows that balanced bellows PRVs have the same POFOD rates as their conventional PRD counterparts, even though they typically discharge to dirty/corrosive closed systems. This is due to the isolation of the valve internals from the discharge fluid and the effects of corrosion and fouling. As shown in Table 7.5, the characteristic life (Weibull η parameter) is the same for bellows PRVs as it is for conventional PRVs.
e) Default Weibull Parameters for Pilot-Operated Pressure Relief Valves
To date, there is little failure rate data in the industry available for pilot-operated PRVs. One source [15]
indicates that pilot-operated PRVs are 20 times more likely to fail than their spring-loaded counterparts.
The Weibull parameters for the POFOD curves for conventional PRVs as shown in Table 7.5 are used as the basis for pilot-operated PRVs with adjustment factors applied to the characteristic life (η parameter).
For MILD service, the η parameter for pilot-operated PRVs is reduced by a factor of 1.5; for MODERATE service, the reduction factor is 3.0; and for SEVERE service, the reduction factor is 5.0.
f) Default Weibull Parameters for Rupture Disks
To date, there is little failure rate data in the industry available for rupture disks. Rupture disks are simple, reliable devices that are not likely to fail to open at pressures significantly over their burst pressure (unless inlet or outlet plugging is a problem, or unless they are installed improperly). Typically, if a rupture disk were to fail, it would burst early. Therefore, the Weibull parameters for the failure to open upon demand case for rupture disks are based on the MILD severity curve for conventional PRVs. This makes the assumption that a rupture disk is at least as reliable as a conventional PRV. It also assumes that the rupture disk material has been properly selected to withstand the corrosive potential of the operating fluid.
Where it is known that the rupture disk material is not properly selected for the corrosive service, the disk Weibull parameters should be adjusted accordingly.
g) Adjustment for Conventional PRVs Discharging to Closed System
An adjustment is made to the base Weibull parameters for conventional valves that discharge to a closed system or to flare. Since a conventional valve does not have a bellows to protect the bonnet housing from the corrosive fluids in the discharge system, the characteristic life (represented by the η parameter) is reduced by 25%, using an adjustment factor of 0.75.
F = 0.75c for conventional valves discharging to closed system or flare
F = 1.0 c for all other cases
h) Adjustment for Environmental Factors
There are several environmental and installation factors that can affect the reliability of PRDs. These include the existence of vibration in the installed piping, a history of chatter, and whether or not the device is located in pulsing flow or cyclical service, such as when the device is installed downstream of reciprocating rotating equipment. Other environmental factors that can significantly affect leakage probability are operating temperature and operating ratio.
The operating ratio of a PRD is the ratio of maximum system operating pressure to the set pressure. When the operating ratio is greater than 90% for spring-loaded PRVs, the system pressure is close to overcoming the closing force provided by the spring on the seating surface, and the PRV will be more likely to leak (simmer). This increased potential for leakage is taken into account by applying an environmental factor to the default leakage curve. Similarly, an environmental factor is applied when the operating margin is greater than 95% for pilot-operated PRVs. Note that some pilot-operated PRVs can function at operating ratios up to 98% (see API STD 520 for guidance on operation margin).
An analysis of the industry failure rate data shows that PRDs installed in vibratory or cyclical service tend to have higher leakage rates. The analysis showed, however, that the fail to open failure rates remain about the same when a PRD is installed in these services.
If a PRV has a history of chattering, the installation should be modified or re-designed as soon as possible to eliminate the chatter, since the effects of chatter may be very detrimental to the protection provided by the PRD. An assumed adjustment factor of 0.5 is applied to the Weibull η parameters for the POFOD and probability of leak (POL) curves of a PRD that has a history of chattering in service.
Table 7.6 provides the environmental adjustment factors applied to the default POFOD and POL Weibull curves.
The environmental factor, Fenv, is used to increase the POFOD or leakage by reducing the curve’s characteristic life (Weibull η parameter). As shown in Figure 7.5, the modifier effectively shifts the probability curves to the left.
i) Updating Failure on Demand Based on PRD-Specific Testing Data 1) Tracking Historical Inspection and Testing Data
An inspection program should track each PRD’s testing and inspection history from its initial installation.
From this history, adjustments can be made to each device’s Pfod and Pl curves to take advantage of the knowledge gained by the testing of a particular relief device in a specific service.
After actual testing data is obtained for a PRD, the probability functions of that device are adjusted up or down (modifying the Weibull parameters) depending upon the results (pass/fail/leak) of the device’s specific inspection tests and the length of service since the last inspection. In this way, an increase or reduction in the recommended interval is obtained based on historical test data.
In general, the adjustment of the POFOD is based on the results of the inspection of the PRD itself (e.g., bench test results). This could lead to non-conservative results if the inlet or outlet piping plugs during operation and could affect the reliability of the PRD system. For each inspection date entered, the inspected condition of the piping should be documented. If the piping is determined to be plugged, the methodology assumes the inspection/test to be a FAIL, regardless of the results of the bench test or inspection method used on the PRD. Good engineering practice would suggest that if the piping is plugged by more than 25%, the piping should be defined as being plugged, since this would then drive down the inspection interval. This methodology adjusts the reliability of the PRD system to reflect excessive pipe plugging.
2) Effectiveness of Inspection Programs in Confirming Failure Rates
Inspection programs vary in their effectiveness for determining failure rates. The definitions for inspection and testing effectiveness for PRDs are provided in Part 2, Annex 2.C, Table 2.C.3.1. The inspection’s effectiveness is based on its ability to adequately predict the failure (or pass) state of the PRD being inspected. Limitations in the ability of the program to improve confidence in the failure rate result from the inability of some test methods to detect and quantify damage.
For PRDs, an inspection and testing program should track the effectiveness of the inspection and the testing performed for each PRD. The concept of inspection effectiveness is similar to the concept that is described in Section 4.4.3 of this document for fixed equipment. For inspection effectiveness of PRDs, a measure of confidence in the pass/fail/leak result of the inspection effort is used.
Table 7. 8 provides default confidence values based on expert opinion. The confidence values are an indication that the inspection will result in an accurate representation of actual PRD performance during an overpressure demand case. For example, the 90% effectiveness associated with passing a “highly effective” bench test means that there is a 90% probability the device would have opened upon demand in its installed service. Therefore, it also carries a 10% probability that the PRD would have failed upon demand during operation. The values shown in Table 7. 8 are called conditional probabilities.
The conditional probabilities listed reflect the confidence that an inspection result will predict the device’s performance upon demand. For passing PRDs, the highest confidence is assigned when the PRD is bench tested without any prior cleaning (i.e., as-received condition). Bench testing where the devices are cleaned prior to testing, in-situ testing, and visual inspections provide some information about PRD performance, but are not considered as reliable as the as-received bench test.
The philosophy is different for PRDs that fail an inspection. In the case of a “highly effective” bench test failure, the 95% confidence translates to a 95% chance that the PRD would have failed upon demand in service. Unlike the passing test case, the “usually effective” in-situ test, or bench test where the device has been steamed out prior to testing, is assumed to have the same 95% confidence for failure upon demand in actual service.
An ineffective test does not provide any information to predict PRD performance upon demand and therefore the PRD does not receive any credit for the test/inspection date. The inspection still will get some credit if an overhaul was performed in that the device is assumed to be returned to service in like- new condition, and the in-service duration is calculated from the ineffective inspection date.
3) Inspection Updating
As previously discussed, Weibull parameters for the failure on demand curves have been determined based on the analysis of a sample set of data. Initially, these values are default (suggested) parameters for the listed fluid services. As inspection data is collected for each PRD, these parameters may be adjusted for each device based on the inspection results.
Applying a Bayesian updating approach to problems of this type is common to adjust probabilities as new information is collected. This approach assumes that the Weibull shape parameter (β parameter) remains constant based on the historical data, and adjusts the characteristic life (η parameter), as inspection data is collected. This is analogous to evaluating a one-parameter Weibull to update the PRD performance. Bayes’ Theorem works best when the error rates for a test are very small. This is not the case for PRDs. Test effectiveness, shown in Table 7. 8, range from 50 to 90%. This uncertainty using Bayes’ Theorem results in an unrealistically high adjusted POF, particularly for a passed bench test.
Therefore, a similar inspection updating method was devised to provide reasonable adjustments of characteristic life.
Since the default Weibull parameters for a given PRD provide the probability of a failure on demand versus time, a default POFOD (modified as per Section 7.2.4) may be obtained for the device based on its in-service duration at the time of inspection. This is known as the prior POFOD and is calculated using Equation (1.17) as follows:
, 1 exp
prd f prior
mod
P t
β
η
= − −
(1.17)
The prior probability that the device will pass on demand is:
, 1 ,
prd prd
p prior f prior
P = − P (1.18)
After the inspection, a second POFOD is calculated based upon the conditional probability factor, or confidence factor (CF) for the effectiveness of the inspection performed (see Table 7. 8). This second, calculated probability is called the conditional POFOD and is calculated using Equations (1.19) or (1.20) depending on the result of the inspection:
When the pressure relief device passed the inspection the conditional POFOD is calculated as follows:
( )
, 1 ,
prd prd
f cond pass p prior
P = − CF ⋅ P (1.19)
With a failed inspection, the conditional POFOD is calculated as follows:
( )
, , 1 ,
prd prd prd
f cond fail f prior pass p prior
P = CF ⋅ P + − CF ⋅ P (1.20)
A weighted POF, Pf wgtprd, , is then calculated, where the weighting factors have been formulated to give more credit to tests conducted later in the characteristic life. Using the prior and conditional probabilities and the weighting factors, an updated or posterior POFOD is calculated using the equations provided in Table 7. 9.
A revised characteristic life may be obtained using Equation (1.21) based on the in-service duration of the PRD, the known β parameter, and the posterior probability.
( ln 1 , )1
upd
prd f wgt
t
P β
η =
− −
(1.21)
4) Example – Bayesian Updating Calculation
Consider a conventional pressure relief valve that is in a severe fluid service. The default Weibull parameters in accordance with Table 7.5 are as follows:
β = 1.8 (1.22)
def mod 17.6
η = η = (1.23)
For this example, there are assumed to be no other adjustments, so that the modified characteristic life value is equal to the default value.
The Pfprdmay be determined using Equation (1.19) for any in-service duration. Assuming an inspection is conducted at an in-service duration of 6 years, the POF at 6 years, Pf priorprd, , is:
1.8 ,
1 exp 6 0.1342
17.6
prd f prior
P = − − = (1.24)
The probability that the device will pass a bench test is:
, 1 , 1 0.1342 0.8658
prd prd
p prior f prior
P = − P = − = (1.25)
These probabilities are defined as the Prior Probabilities.
At the 6-year in-service inspection, a highly effective bench test is performed and the device passes the test. Using Table 7.8, a 90% confidence factor has been estimated for this type of inspection, meaning that 10% of devices that pass this type of test would fail on demand in service. An adjusted or conditional POF is calculated:
( )
, 1 , 0.1 0.8658 0.087
prd prd
f cond pass p prior
P = − CF ⋅ P = ⋅ = (1.26)
For a highly effective pass, the weighted Probability is calculated (see Table 7. 9 for equations):
, , 0.2 , 0.2 ,
prd prd prd prd
f wgt f prior f prior f cond
mod mod
t t
P P P P
η η
= − ⋅ ⋅ + ⋅
(1.27)
or