API RP 581 - 3rd Ed.2016 - Add.2-2020 - Risk-Based Inspection Methodology
5-36 API RECOMMENDED PRACTICE 581c) Optimization of Total CostThe total cost as a function of replacement time frequency averaged over the service bundle life iscalculated using Equation (5.84).Costtotal( tr )n( ) + ( )tubeRisk tr Cost tr=365.25⋅ESLf n pbr nn(5.84)The estimated service life as a function of replacement time interval may be approximated using anintegration technique using Equation (5.85).ESL = ESL + ESL(5.85)n f, n pn ,Where the average life of the bundles that would have been expected to fail prior to the plannedreplacement time, ESL , and the average life of the bundles that would not have been expected tof , nfail prior to the planned replacement time, ESL are summed, ESLpn ,n.The average life of the bundles that would have been expected to fail prior to the planned replacementtime is:tube tube( − )ESL = ESL + tr ⋅ P − P(5.86)f , n f , n−1 n f , n f , n 1The average life of the bundles that would have been expected to not fail prior to the plannedreplacement time is:tube( 1 )ESL = tr ⋅ − P(5.87)pn , n f,nA planned replacement frequency is selected, and the costs associated with the frequency is calculatedto allow optimization of the total cost. The frequency is incrementally increased and the costs arecalculated for each incremental step, n ( n= n+ 1 ). The point where the costs reach a minimum is theoptimum replacement frequency:STEP 2.1⎯Select an appropriate time step, t s, in days. (A value for t sof 7 to 14 days should be sufficient)and an increment of n = 1 . Subsequent calculations will increase the increment by 1 ( n= n+ 1 ).STEP 2.2—Calculate the planned replacement frequency, trn, by multiplying the increment number, n , bythe time step, t sas follows:tstrn= n⋅ (5.88)365.25P tr , usingtubeSTEP 2.3—Calculate the POF at the planned replacement frequency at increment n , ( )Equation (5.55), the updated Weibull parameters based on the latest inspection of the bundle and the timevalue to use in Equation (5.56) is tr obtained in STEP 2.2. Note that the time unit is in years.nSTEP 2.4—Calculate the average life of the bundles that would have been expected to fail prior to theplanned replacement time, ESL , using Equation (5.86).f , nSTEP 2.5—Calculate the average life of the bundles that would have not been expected to fail prior to theplanned replacement time, ESL , using Equation (5.87).pn ,STEP 2.6—Calculate the estimated service life, ESLn, using Equation (5.83).STEP 2.7—Calculate the risk cost associated with bundle failure at the replacement frequency, Riskf ( trn),using Equation (5.80).STEP 2.8—Calculate the bundle replacement cost at the replacement frequency, Costpbr ( trn ) , usingEquation (5.82).f , nn
RISK-BASED INSPECTION METHODOLOGY, PART 5—SPECIAL EQUIPMENT 5-37STEP 2.9—Calculate the total costs at the replacement frequency averaged over the expected life of theCost tr , using Equation (5.84).bundle, ( )totalnSTEP 2.10—Increase the increment number by 1 ( n n 1Cost tr in STEP 2.9 is obtained.minimum value of ( )totaln= + ) and repeat STEPs 2.2 through 2.9 until aSTEP 2.11—The optimal bundle replacement frequency, t opt, is where the tr nis at the minimumtotal( )Cost tr .βΓηnNomenclatureis the Weibull shape parameter that represents the slope of the line on a POF vs. time plotis the Gamma functionis the Weibull characteristic life parameter that represents the time at which 62.3% of thebundles are expected to fail, yearsη is the Weibull characteristic life parameter at the plan date after inspection, yearsinspηmodis the Weibull modified characteristic life parameter modified with inspection history, yearsη is the Weibull target characteristic life parameter based on the risk target, yearstgtAU % is the percent additional uncertainty, %AUw/ insp% is the additional inspection uncertainty at the plan date after inspection, %AUw/ outinsp% is the additional inspection uncertainty at the plan date before inspection, %tubeAU is the additional inspection uncertainty required to remain below the Pf , tgtat the plan date, %%tgttubeC is the consequence of bundle failure, $fC is the consequence of bundle failure based on a planned bundle replacement, $tubef , planC is the consequence of bundle failure during an unplanned bundle replacement, $tubef , unplanCostbundleis the replacement cost of the tube bundle, $Costenvis the environmental costs due to a bundle leak, $Costinspis the cost to perform the inspection, $Costmaintis the cost of maintenance for bundle inspection or replacement, $pbr( )Cost tr is the cost per year of bundle replacement at a planned frequency, trn, $/yearnCost is the production losses as a result of shutting down to repair or replace a tube bundle, $prodtotal( )Cost tr is the total cost of a bundle replacement program at a planned frequency, trn, $/yearnDsdis the number of days required to shut a unit down to repair a bundle during an unplannedshutdown, daysD is the number of days required to shut a unit down to repair a bundle during a plannedsd , planshutdown, daysD is the number of days required to shut a unit down to repair a bundle during an unplannedsd , unplanshutdown, days
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- Page 541 and 542: CONTENTS1 GENERAL………………
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5-36 API RECOMMENDED PRACTICE 581
c) Optimization of Total Cost
The total cost as a function of replacement time frequency averaged over the service bundle life is
calculated using Equation (5.84).
Cost
total
( tr )
n
( ) + ( )
tube
Risk tr Cost tr
=
365.25⋅
ESL
f n pbr n
n
(5.84)
The estimated service life as a function of replacement time interval may be approximated using an
integration technique using Equation (5.85).
ESL = ESL + ESL
(5.85)
n f, n pn ,
Where the average life of the bundles that would have been expected to fail prior to the planned
replacement time, ESL , and the average life of the bundles that would not have been expected to
f , n
fail prior to the planned replacement time, ESL are summed, ESL
pn ,
n
.
The average life of the bundles that would have been expected to fail prior to the planned replacement
time is:
tube tube
( − )
ESL = ESL + tr ⋅ P − P
(5.86)
f , n f , n−1 n f , n f , n 1
The average life of the bundles that would have been expected to not fail prior to the planned
replacement time is:
tube
( 1 )
ESL = tr ⋅ − P
(5.87)
pn , n f,
n
A planned replacement frequency is selected, and the costs associated with the frequency is calculated
to allow optimization of the total cost. The frequency is incrementally increased and the costs are
calculated for each incremental step, n ( n= n+ 1 ). The point where the costs reach a minimum is the
optimum replacement frequency:
STEP 2.1⎯Select an appropriate time step, t s
, in days. (A value for t s
of 7 to 14 days should be sufficient)
and an increment of n = 1 . Subsequent calculations will increase the increment by 1 ( n= n+ 1 ).
STEP 2.2—Calculate the planned replacement frequency, tr
n
, by multiplying the increment number, n , by
the time step, t s
as follows:
ts
trn
= n⋅ (5.88)
365.25
P tr , using
tube
STEP 2.3—Calculate the POF at the planned replacement frequency at increment n , ( )
Equation (5.55), the updated Weibull parameters based on the latest inspection of the bundle and the time
value to use in Equation (5.56) is tr obtained in STEP 2.2. Note that the time unit is in years.
n
STEP 2.4—Calculate the average life of the bundles that would have been expected to fail prior to the
planned replacement time, ESL , using Equation (5.86).
f , n
STEP 2.5—Calculate the average life of the bundles that would have not been expected to fail prior to the
planned replacement time, ESL , using Equation (5.87).
pn ,
STEP 2.6—Calculate the estimated service life, ESL
n
, using Equation (5.83).
STEP 2.7—Calculate the risk cost associated with bundle failure at the replacement frequency, Risk
f ( tr
n)
,
using Equation (5.80).
STEP 2.8—Calculate the bundle replacement cost at the replacement frequency, Cost
pbr ( tr
n ) , using
Equation (5.82).
f , n
n
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