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Assessing Probability ofFailure for Pressure
Equipment: Part 1
The Equity Engineering Group, CanadaSherwood Park, Alberta
Boyd McKay, P.Eng.
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Overview
Introduction
Statistics Background
Pressure Equipment Risk Calculation Process
Damage Models and Damage Rates
Probability Of Failure Calculation Pressure Equipment Reliability Model
Examples Summary
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Introduction
Importance and value of RBI concepts in thesafe operation of pressure equipment
Fundamental physical and mathematicalprinciples are valid irrespective of how theProbability Of Failure (POF) is calculated
Most RBI methodologies are generally used forscreening purposes
The general risk process as well as the API RBIProbability Of Failure calculation will also bereviewed along with several applications
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Statistics Background
What are the location, shape and scaleparameters ?
How to test for goodness of fit? How are the distributions ranked for fit? Is the sample representative of the population?
4 questions requireanswers for validanalysis.
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Statistics Background
The calculation of POF can be generally divided
into 3 classes:(1) Non-statistical- Same weight on data points
(2) Statistical can censor data
(3) Hybrid:
- Combination of statistical and non statistical- Expert judgement
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Risk Calculation Process General steps in the process:
Establish time period
Determine damage mechanisms
Determine final risk scenario
Calculate damage rate usingspecified damage model
Select damage rate model(s)
Calculate POF over specified time interval
Calculate CONS over specif ied time interval
Calculate RISK over specified time interval
Compare RISK torisk matrix threshold
Perform mitigationactivitiesDo nothing
RISKacceptable
Calculate RISK overspecified time interval
Yes No
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Risk Calculation Process
Time Period Under Consideration
A time period that is too long (e.g. 15 years) may
overestimate POF (e.g. no clear calculation of whento perform mitigating activities. A short time period(e.g. 1 day) will tend to produce a low POF.
The time period under consideration is important forseveral reasons:(i) The POF is influenced by time.
(ii) The consequence is influenced by time (butusually less so than the POF time dependence).(iii) The stream composition may change as a
function of time
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Risk Calculation Process
Damage Mechanisms
Stream components and conditions are such that areaction or susceptibility can exist
damage
mechanism exists
Assumptions made regarding valve position,isolation, operating pressure, temperature andvelocity parameters
Stream CompositionComponents may either be sampled frequently anddifficult to test or sampled infrequently and easy totest.
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Risk Calculation ProcessDetermining The Final Risk Scenario (FRS)
1. Localized internal corrosion that may requirerepair.
2. A corrosion pinhole resulting in a leak to atm. 3. A corrosion pinhole resulting in leak to atmosphere
with exposure and business loss.
FRS POF COF Risk1 P 1 PMO COF 1 P 1* COF 1 2 P 2
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Risk Calculation Process
Determining The Final Risk Scenario
(a) Decreased confidence in pressure equipmentintegrity program within the organization.
(b) Decrease in funds for inspecting equipment thatmay really need to be inspected and repaired(e.g. unacceptable risk not being addressed).
(c) Increased effort to conduct multiple equipmentreviews, with same data, and arriving at usuallydifferent answers .
The PMO can be used for all three FRS. This
conservative approach can lead to the following:
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Risk Calculation Process
Determining The Final Risk Scenario
An important point to note is that Fitness-For-Service (FFS) can be applied to FRS-1. A FFSanalysis could be run to determine limiting flawsizes prior to inspection (internal or external).
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Risk Calculation Process
Damage Models & Distribution of Damage Rates
Damage mechanisms for different industries arepublished in API 571, WRC 488, 489 and 490.
Deliverable from damage models is usually thelowest wall thickness in the system of interest.
A set of damage rates over the time interval can begenerated.
Determine what statistical distribution fits the data.
Calculate the probability of a damage rate occurringthat would result in a specified thickness (e.g. t min )being present in the system.
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Risk Calculation Process
Conversely, for a given data set of wallthicknesses over the time interval of interest, theuser can determine what statistical distribution (ifany) will fit the data.
Using the distribution parameters, the user cancalculate the probability of a thickness occurring
that would less than a specified thickness (e.g.t min ) being present in the system.
Damage Models & Distribution of Damage Rates
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Risk Calculation Process
The data set may come from one or morepopulations that have different distributions.
Re-sample from data set based on populationcharacteristics.
Characteristics such as no flow region, bottom/topof pipe, extrados/intrados of elbow, etc., are allvalid ways of producing another sample data setthat can be tested against known distributions.
It is not uncommon to calculate that the data doesnot come from any known distributions.
Damage Models & Distribution of Damage Rates
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Risk Calculation Process
Distribution can change type over time.
Re-sample from data set based on populationcharacteristics as well as time (morecomplicated).
Changes in damage drivers/operation can aid inre-sampling.
Damage Models & Distribution of Damage Rates
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Risk Calculation Process
Calculation of Cumulative Probability Of Failure
Integrate continuous random variable to get POF:( ) ( )
t
F x f t dt
=
For a maximum/threshold POF of P cr require:
( ) cr F x P
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Risk Calculation Process
(a) Can theoretically be used for any continuousdistribution and censored data.
(b) Approximate confidence bounds can be calculated.(c) Less suitable for less than 5 (approximately) data
points.
Calculation of Cumulative Probability Of Failure
Maximum Likelihood Estimate (MLE), linearunbiased estimators, method of moments, etc.,procedures used to calculate parameters
The MLE method has the following significant
characteristics:
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Risk Calculation Process
Parameters must be tested using a test statisticsuch as Anderson-Darling, Kolmogorov-Smirnov(KS), etc.
Alternatively, a correlation coefficient (CC) can beused.
Graphically viewing the probability plots is also
used to validate the correlation coefficient and tosee if there may be different failure modes presentin the data.
Calculation of Cumulative Probability Of Failure
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Risk Calculation Process
Correlation Coefficient and Scale
Parameter plot
Calculation of Cumulative Probability Of Failure
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Risk Calculation Process
Limit State Function
Creates regions of acceptable and unacceptableresults
( ) Resistance( ) - Load( )G t t t =
There are 3 cases to consider:
( ) 0
( ) 0( ) 0
G t
G t G t
Unacceptable (loaded beyond capacity)
Acceptable
Acceptable
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Risk Calculation Process
Limit State Function
is the variable of interest in calculating the POF.
-3 -2 -1 0 321
Load L
-3
-2
-1
0
1
2
3
R = L +
G*i i iCOV =2
1i
V
G vi i
dGS S dv=
=
G
GS
=
NORM(- )POF =
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Risk Calculation Process
Limit State Function
The preceding discussion is the core of the API
RBI POF methodology. The POF calculation shown here is the seed for
the POF to be used in a thinning risk calculation.
There are other considerations such as acomparison to a Generic Failure Frequency (GFF)for equipment type, etc. These considerationscan be modeled numerically and applied to give afinal POF result.
The same techniques can be applied to othervessel failures modes such as cracking.
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Pressure Vessel Example
Vessel Data
Vessel: Atmospheric Overhead AccumulatorMaterial: SA 285 GR C Corrosion Allowance: 3/16Design Pressure: 50 psig Prior Inspection Data: None
Diameter: 6-6 COV (pressure): 20% Age: 6 years COV (CR): 30%
Thickness: 3/8 COV ( UTS ): 10%
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Pressure Vessel Example
Analysis Equations
( ) 000
0.25, , , , ln
0.277
n
G UTS UTS G
e R R R t d n RSF
n n R t d
= + +
( )applied applied L P P=
*Gd time CR=0.25
ln0.277 *
n
oUTS applied
o init
e RG RSF Pn n R t CR time
= + +
( )0.250.277 *
n
UTS o init
dG e time RSF
dCR n n R t CR time
= +
( )0.25 ln0.277 *
no
UTS o init
dG e R RSF d n n R t CR time
= +
1applied
dGdP
=
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Pressure Vessel Example
Analysis Equations2 22
applied UTS G P CRapplied UTS
dG dG dGS S S S
dP dCR d
= + +
NORM(- )=NORMG
GPOF
S
=
ResultsVessel: Atmospheric Overhead Accumulator
Material: SA 285 GR C Corrosion Allowance: 3/16Design Pressure: 50 psig Prior Inspection Data: None
Diameter: 6-6 COV (pressure): 20%
Age: 6 years COV (CR): 30%Thickness: 3/8 COV ( UTS ): 10%Damage State Corrosion Rate POF
Corrosion Rate 1: 0.030/year 1.85388E-07Corrosion Rate 2: 0.035/year 1.05330E-05Corrosion Rate 3: 0.040/year 2.16402E-04
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Pressure Relief Valve Example
PRV Data
Failure Mode: Failure to Open on Demand (FOD)
Service Interval(Years)
FODPass Fail
Intermediate Hydrocarbon 36 2 1
Intermediate Hydrocarbon 48 2 1Intermediate Hydrocarbon 54 4 2Intermediate Hydrocarbon 60 2 1Intermediate Hydrocarbon 66 3 3
Intermediate Hydrocarbon 72 3 4Intermediate Hydrocarbon 78 1 5Intermediate Hydrocarbon 84 0 4Intermediate Hydrocarbon 90 - 1
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Pressure Relief Valve Example
Analysis Equations
Use Maximum Likelihood Estimate of parametersDistrib ution Parameters Type Random
VariableDomain
Probability Density Function Variable Maximum Likelihood Estimates For Parameters
Normal location= shape
=
x < < + 21exp
2( )
2
x
f x
=
1
1 N
j j
x x N
=
= =
( )2
2 2
1
1 N
x j j
S x x N
=
= =
Standard x < < +( )21exp
2( )2
Z f x
=
x Z
=
Weibull(2P)
0= shape = scale
=
( )1
( ) exp x x
f x
=
( ){ }{ }
1
1
11
1 1
1
1 log
N
j j
N N
j j j j j j
x N
x x x x N
=
= =
=
=
Standard ( )( ) ( )1( ) exp f x Z Z
=
x Z
=
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Pressure Relief Valve Example
Results
The PRV POF calculation is the seed for the POF
to be used in an API RBI risk calculation for aPRV.
There are other considerations such as a
comparison to a GFF for PRVs, PRV dischargelocation, chattering. These considerations can bemodeled numerically and applied to give a finalPOF result.
Same techniques can be applied to other PRVfailures modes such as leaking, fail partiallyopen, etc.
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Summary
The API RBI POF general POF calculation methodhas been demonstrated for thinning damagemechanisms and a PRV.
The methodology is in the public domain and isdefensible from a technical perspective.
The API RBI methodology can be of significantbenefit in obtaining more information frominspection data.
The methodology generates reproducible anddefensible inputs to a risk analysis for pressureequipment.
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P.O. Box 3648
Sherwood Park, Alberta CanadaPhone: 780-449-4180 Fax: 866-775-1528Cell: 780-722-3246