API RP 581 - 3rd Ed.2016 - Add.2-2020 - Risk-Based Inspection Methodology

5-56 API RECOMMENDED PRACTICE 581
η =
upd
t
( ) 1
prd
− ⎡1−
⎤
f ,wgt
ln P β
⎣ ⎦
(5.99)
6) Example—Modified Bayesian Updating Calculation
Consider a conventional PRV that is in a severe fluid service. The default Weibull parameters in
accordance with Table 6.6 are as follows:
β = 1.8 (5.100)
η
= η = 17 6
(5.101)
def mod .
For this example, there are assumed to be no other adjustments, so that the modified
characteristic life value is equal to the default value.
prd
The Pf ,prior
may be determined using Equation (5.102) for any in-service duration. Assuming an
prd
inspection is conducted at an in-service duration of 6 years, the POF at 6 years, P f ,prior
, is:
⎡ 18 .
prd
⎛ 6 ⎞ ⎤
Pf ,prior = 1−exp ⎢− ⎜ ⎟ ⎥ = 01342 .
⎢ ⎝17.
6⎠
⎣ ⎥⎦
(5.102)
The probability that the device will pass a bench test may be determined using Equation (5.103):
P prd
prd
p,prior = 1− P f ,prior = 1− 01342 . = 0 . 8658
(5.103)
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 6.9, 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 using Equation (5.104):
( pass )
P prd
prd
f ,cond = 1− CF ⋅ P p,prior = 01 . ⋅ 0 . 8658 = 0 . 087
(5.104)
For a highly effective pass, the weighted probability is calculated (see Table 5.10 for equations):
prd prd prd ⎛ t ⎞ prd ⎛ t ⎞
Pf ,wgt = P f ,prior −02 . ⋅P f ,prior ⋅ ⎜ + 02 . ⋅Pf ,cond
⎝η
⎟
mod ⎠
⎜
⎝η
⎟
mod ⎠
(5.105)
or
prd
⎛ 6 ⎞ ⎛ 6 ⎞
P f ,wgt = 01342 . −0. 2⋅01342 . ⋅ ⎜ + 0. 2⋅( 0. 087)
⋅ = 01310 .
⎝
⎟
17. 6⎠ ⎜
⎝
⎟
17.
6⎠
(5.106)
Finally, using the prior β and the calculated weighted probability,
prd
P f ,wgt
parameter is calculated for the in-service duration using Equation (5.107).
, an updated value for the η