The Weibull statistics can be used to predict the optimal replacement frequency for a bundle and determine whether it makes economic sense to inspect or replace a bundle at an upcoming shutdown.
8.7.2 Decision to Inspect or Replace at Upcoming Shutdown
The methodology determines the risk reduction benefit from mitigating actions including various levels of inspection or bundle replacement. The cost benefit calculation includes the cost of the mitigating action to perform the selected activity (inspection or replacement). In addition, an optional hurdle cost or a rate of return,
ROR, may be added to the cost of bundle replacement to encourage an inspection activity versus bundle replacement. A hurdle cost can be used to avoid excessive bundle replacement for borderline risk determinations that require action.
The decision to perform mitigating actions, such as bundle inspection or bundle replacement at an upcoming turnaround, can be made by comparing the incremental risk ($) associated with deferring the action to the cost associated with the action itself.
The expected incremental risk, EIRtt12, associated with deferring the inspection or replacement of a bundle to a subsequent shutdown can be calculated using Equation (1.82).
( ) ( )2
2 1
1
1 1 1
tube t tube f
t f tube
f
P t
EIR C
P t
−
= ⋅ − − (1.82)
In Equation (1.82), t1is the service duration of the bundle at the upcoming shutdown (Turnaround Date 1) and
t2is the service duration of the bundle at the subsequent shutdown (Turnaround Date 2).
The decision to perform an action, whether to inspect or to replace a bundle, can be made by comparing the expected incremental cost of deferral of the action using Equation (1.82) to the cost of the action itself. If the cost of the action (inspect or replace) is greater than the expected incremental risk, the action should be taken.
For example:
( insp maint) (1 ) tt12
If Cost +Cost ⋅ +ROR <EIR then inspect (1.83)
( bundle maint) ( 1 ) tt12
If Cost + Cost ⋅ + ROR < EIR then replace the bundle (1.84)
The equations provided above show a rate of return,ROR, or hurdle rate that adds an economic incentive to the decision process.
The owner-user is responsible for determining the costs that are unique to the operation and strategy. Where possible, the actual inspection costs should be used. Note that the maintenance costs to pull the bundles and make them available for the inspection should be added to the inspection costs to obtain the total cost of inspection, see Equation (1.83).
8.7.3 Decision for Type of Inspection
Once a decision has been made to inspect per Equation (1.83), an economic decision can be made as to the appropriate level of inspection with similar techniques as described in Section 8.7.2 comparing the cost of the various inspection techniques to the reduction in risk expected for the level of inspection.
8.7.4 Optimal Bundle Replacement Frequency a) General
Maintenance optimization helps to strike a balance between cost and reliability. The cost per day of a "run to failure" strategy shows low costs early in the life of the equipment and increasing costs as reliability decreases. By overlaying the costs of an associated preventative maintenance to address the failure mode, initial costs are high, but costs per unit time decrease as time progresses. This optimization occurs at a point where the total cost function (sum of the two cost functions) is at a minimum. The time at which the minimum occurs is the optimum time to perform maintenance [20].
For an exchanger bundle, the optimal replacement frequency can be determined by plotting the costs associated with bundle failure (increases with increasing replacement frequency) to the replacement costs associated with periodic planned shutdowns to replace the bundle (decreases with increasing replacement frequency). The replacement frequency at which these two costs reach a minimum value, when averaged over the expected bundle life, is the optimal replacement frequency.
The methodology in Shultz, 2001 [21] described below is recommended to determine the optimal frequency for replacing bundles.
b) Increasing Risk Cost of Unplanned Outage
If the planned replacement time frequency is defined by the variable, tr, the risk cost associated with an on-line failure (unplanned outage) to replace the bundle including business interruption and bundle replacement costs is calculated using Equation (1.85).
( ) ( )
tube tube tube
f f f
Risk tr = C ⋅ P tr (1.85)
where:
tube
f prod env bundle maint
C = Cost + Cost + Cost + Cost (1.86)
The consequence of an unplanned outage due to a tube bundle failure, Ctubef , is identical to Equation (1.64) and includes any environmental impact, Costenv. The risk cost due to bundle failure increases with time since the POF, Pftube( ) tr , as a function of replacement frequency increases with time.
c) Decreasing Cost of Bundle Replacement
The bundle replacement costs as a function of planned replacement frequency, tr, is calculated using Equation (1.87).
( ) tube, 1 tube( )
pbr f plan f
Cost tr = C ⋅ − P tr (1.87)
where:
, 100 ,
tube red
f plan prod sd plan env bundle maint
C = Unit ⋅ Rate ⋅ D + Cost + Cost + Cost (1.88)
Note that the cost of bundle replacement at a planned frequency includes downtime and business interruption, however, the number of days for a planned outage, Dsd plan, , may be much less than if the outage were unplanned due to a bundle failure. An unplanned outage would require some additional lead time to get a replacement bundle on-site.
d) Optimization of Total Cost
The total cost as a function of replacement time interval averaged over the service life of the bundle is computed using Equation (1.89).
( ) ( ) ( )
365.24
tube
f pbr
total
n
Risk tr Cost tr Cost tr
ESL
= +
⋅ (1.89)
The estimated service life as a function of replacement time interval may be approximated using an integration technique using Equation (1.90).
, ,
n fail n pass n
ESL =ESL +ESL (1.90)
This approach adds the average life of the bundles that would have been expected to fail prior to the planned replacement time, ESLfail, to the average life of the bundles that would not have been expected to fail prior to the planned replacement time, ESLpass.
The average life of the bundles that would have been expected to fail prior to the planned replacement time is:
( )
, , 1 , , 1
tube tube
f n f n n f n f n
ESL = ESL − + tr P − P − (1.91)
The average life of the bundles that would not have been expected to fail prior to the planned replacement time is:
( )
, 1 tube,
p n n f n
ESL = t − P (1.92)
To optimize the total cost, a low planned replacement frequency is initially chosen, and the costs associated with this frequency are calculated. This frequency is increased in small increments and the costs are calculated for each incremental step (n n = + 1). At some point, the costs will reach a minimum, indicating that an optimal replacement frequency has been found. The following steps are recommended:
1) STEP 1 – Select an appropriate time step, ts, in days. A value for ts of 7 days should be sufficient.
The initial calculation will be at increment n = 1. Subsequent calculations will increase the increment by 1 or, (n n = + 1).
2) STEP 2 – Determine the planned replacement frequency, trn, by multiplying the increment number,
n, by the time step, ts as follows:
n s
tr = ⋅ n t (1.93)
3) STEP 3 – Calculate the POF at the planned replacement frequency at increment n, Pf ntube, ( ) trn , using
Equation (1.60) and the updated Weibull parameters based on the latest inspection of the bundle.
Note that the time unit in Equation (1.60) is years. The time value to use in Equation (1.60) is trn
obtained in STEP 2.
4) STEP 4 – Calculate the average life of the bundles that would have been expected to fail prior to the planned replacement time, ESLf n, , using Equation (1.91).
5) STEP 5 – Calculate the average life of the bundles that would have not been expected to fail prior to the planned replacement time, ESLp n, , using Equation (1.92).
6) STEP 6 – Calculate the estimated service life, ESLn, using Equation (1.90). Note that for the initial increment (n = 1), when tr=ts, ESLf n, −1 = 0.0.
7) STEP 7 – Calculate the risk cost associated with bundle failure at the replacement frequency,
( )
Risk trf , using Equation (1.85). This value will increase with increasing replacement frequency, tr. 8) STEP 8 – Calculate the bundle replacement cost at the replacement frequency, Costpbr( ) tr , using
Equation (1.87). This value will decrease with increasing replacement frequency, tr.
9) STEP 9 – Calculate the total costs at the replacement frequency averaged over the expected life of the bundle, Costtotal( ) tr , using Equation (1.89).
10) STEP 10 – Increase the increment number by 1 (n n = + 1) and repeat STEPs 2 through 9 until a minimum value of Costtotal( ) tr in STEP 9 is obtained.
11) STEP 11 – The optimal bundle replacement frequency, topt, will be that value of tr that minimizes
( )
total