Barringer-Corrosion-With-Gumbel-Lower[1].pdf

8/14/2019 Barringer-Corrosion-With-Gumbel-Lower[1].pdf

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Barringer & Associates, Inc. 2007 1

Corrosion Problems QuantifiedWith Gumbel Lower Distribution

Paul Barringer, P.E.Barringer & Associates, Inc.P.O. Box 3985Humble, TX 77347-3985

Phone: 281-852-6810

FAX: 281-852-3749

Email: [email protected]: http://www.barringer1.com

Abstract: Several case studies show how to separate general corrosion from accelerated

corrosion and how to predict end of useful life of products.


Gumbel Upper or Gumbel Lower?

The Gumbel upper distribution is used when

you have BIG numbers. Its best know for

flood data (you only record the deepest

[largest] stream gage reading for a single year).

The Gumbel lower distribution is used when

you have LITTLE numbers. Its used whereyouve only recorded the thinnest [smallest]

wall in a single corrosion area.The Gumbel Smallest Extreme Value is considered a model for a system having n elements in a series and

where the failure distributions of components are reasonably uniform and similar (See British Standard BS 5760).


2/22


Whats The Math Difference?The Gumbel largest ex treme value CDF is: The Gumbel smallest extreme value CDF is:F t( ) e

e

t ( )

F t( ) 1 e e

t

Rear anging the equations to read Rear anging the equations to read

F t( ) e e

t ( )

1

ee

t ( )

Or 1

F t( )e

e

t ( )

1 F t( ) e e

t

1

ee

t

Or 1

1 F t( ) e

e

t

Taking t he l og of bot h s ides you ge t: Taking t he log of bot h s ides you ge t:

ln 1

F t( )

e

t ( )

ln

1

1 F t( )

e

t

Again, taking the log of both sides you get: Again, taking the log of both sides you get:

ln ln 1

F t( )

t ( )

t

+ ln ln

1

1 F t( )

t

t

Y = mX + b Y = mX + b

t ln ln 1

F t( )

t l n l n 1

1 F t( )

+

For Monte Carlo modeling: For Monte Carlo modeling:

t ln ln a_random_no( )( ) t ln ln 1 a_random_no( )( )+

Both are alsoknown as the

double

exponential

The Weibull distribution straight line equation

l n l n 1

1 F t( )

ln t( ) l n( )

is a scale factor is a shape factorSmall steeplines for G- & G+

distributions

Observations:

Same Y-axis

Weibull haslog X-axis

Gumbel hasuniform

X-axis


Problem 1: Heat Exchanger Thin Tubes?

We have a shell & tube heat exchanger

Process fluids are inside the tubes and the

tubes are loosing wall thickness with use

Outside the tubes are cooling water

Periodic inspections have recorded theminimum wall thickness in each tubeselected randomly. We have only one wall

thickness for each tube inspected.


3/22


Whats The Issue? How To Resolve? Heat exchanger is 17 years old460 tubes

At turnaround, eddy current wall thickness

inspection occurredWere worried!

Did an IRIS inspection on 10% of tubesNowwere more worriedwhat does the data say?

Retube NOW at 17 years with T/A delays?Retube next turnaround in 3 years at 20 years?

Retube at 2nd turnaround in 6 years at 23 years)?Time Issues


What Are Cost Consequences?

Failure $ is dependent on outside temperatures:

Summer failure = $750,000 lost margins & retube

Fall failure = $500,000 lost margins & retube

Winter failure = $100,000 lost margins & retube

Spring failure = $250,000 lost margins & retube

Another key issue is environmental impact

along with the cost issues if failure occurs

Murphy says: Big Money Issues Will Prevail


4/22


Why Did They Inspect? Rule of thumb for this facility-

Inspect tubes if wall thickness has been

reduced by 1/3, i.e. from 0.083 to 0.055

Consider retubing heat exchangers when tubewall thickness has been reduced to of

original wall thickness, i.e. when wall thickness

has been reduced from 0.083 to 0.0415

This exchanger has environmental concerns


Eddy Current vs IRIS Inspection

Eddy current inspection is the usual quick

and inexpensive inspection of each tube

minimum wall is reported for each tube

IRIS inspection is a more detailed and more

expensive inspection with a rotating head

ultrasonic toolminimum wall is reportedfor each tube and tube IDs must be veryclean for an accurate IRIS inspection.


5/22


What Did IRIS Inspection Find? The minimum wall thickness report shows:

Minimum allowed wall thickness is 0.036for structural integrity.

Wall*qty0.050*1 0.063*90.055*1 0.064*90.056*2 0.065*40.058*2 0.066*50.059*1 0.067*20.061*6 0.069*4

Wall thickness

measured

in inches

Rule of thumb triggers

inspection at 0.050


Stacks Of DataUse SherwinsInspection Option

Wall Thickness Discovered At Inspection

(low)ProbabilityOf

Occurrence(

high)

Discovery

Age/Thickness

Failur

eages

Use top

of stack for

regression

We have stacks of data from the heat exchanger inspection because

the IRIS data have been rounded to three significant digits.

Benign failure

occurred here?

Benign failure

discovered here


6/22


Competing Models:

Weibull? or Gumbel Distributions?

Weibull Distribution

with rank regression

& inspection option

Data stacks from

course measurements

use inspection optionfor regression

R= Coefficient of regression

ccc= critical correlation coefficient

Small risk of wall thickness

less than min allowed


Competing Models:Weibull or Gumbel Distributions?

Gumbel- Distribution

with rank regression

R2= (Coefficient of regression)2

(ccc)2= (critical correlation coefficient)2Higher risk of wall thickness less than

min allowed more conservative

Bigger than for

Weibull distribution

use Gumble-


7/22


PDF Curves

Note the Gumbel- distribution says to

expect more occurrences with thinner wallsx

~2*x


PDF Details

0.04 0.060

50

100

1

e

t

e

t ( )

t t0( )

1

e

t t0

t

Gumbel Lower PDF

Weibull PDF

0.04 0.050

2

4

1

e

t

e

t

t t0( )

1

e

t t0

t

(0.050459, 3.933565)


8/22


The SMALLEST value is recorded for each

tube thickness which motivates use of the

Gumbel smallest distribution. Just as for

flood data (the largest yearly value) motivates

the use of the Gumbel largest distribution.

The Gumbel smallest distribution is a better

curve fit and shows greater % potentialfailure than Weibull, thus more conservative.

Why Gumbel Lower Distribution?


1

5

2

10

20

30

40

5060

7080

9095

99

.03 .04 .05 .06 .07 .08 .09 .1 .11

Heat Exchanger IRIS Inspection Data

Tube Wall Thickness (inches)

0.06427 0.00316 0.989 46/0

Xi Del r^2 n/s

G-/rr/insp1

Year 17

OccurrencesCDF%

Area is 1% high

by 0.01 wide

note the

magnification!

Area is 1% high

by 0.01 wide

=0.06427

Structuralminimumis0.036

Parameters:

Location

Slope/Shape

Considerretubeiflessthan0.0

415

Inspectiflessthan0.0

55

Small steep line slopeLarge flat line slope.

Heres Where We Are At Year 17. Can We Make Year 20?


9/22


General Corrosion

Wall Thickness

Probabilityof

Occurrence

Start = datum

General Deterioration

Note Parallel Lines

min

t=3yrst=6t=9

Low probability

of thin wall below

minimum!

63.2%

Dont exceed thisprobability of thin wall

t=?This becomes acritical value!


General Corrosion Trend Line

Characteristic

WallThickness,

Time

Critical Value, ,For Wall Thickness

End of life!An easy decision.


10/22


Accelerated Corrosion

Wall Thickness

Probabilityof

Occurrence

start

General Deterioration

min

Accelerated

Deterioration

Breaks The

Min Wall

Limits!

!

t=3t=6t=9

Dont exceed this

probability of thin wall

You must know when toaccept the risk of failureand when to accept the

risk of failure!

$Risk = pof*$Consequence

99.9%


Accelerated Corrosion Trend Line

Wall

ThicknessAtA

SpecifiedRisksay0.1%

Time

Minimum Wall Thickness At

Acceptable Risk Level.

End of life!Difficult decision.

Wall loss from general corrosion

Wall loss from accelerated corrosion

Wall loss fromboth general + accelerated corrosion


11/22


.1

.5

.2

1

5

2

10

20

3040

5060

7080 90

9599

99.9

.03 .04 .05 .06 .07 .08 .09 .1 .11

Heat Exchanger IRIS Inspection Data


0.09706 0.0020362

0.06427 0.0031573 0.989 46/0

= Xi = Del r^2 n/s

G-/rr/insp1

Year 17

Year 0

OccurrenceCDF%

MinAllowedWall=0.036

Year ??

Assumes new tubewith tmin = 0.083

and tmax = 0.101

for ~6* = 99.8

Typical Corrosion rate = (0.09706-0.06427)/17 = ~0.002/yr

Note the flatter slope

with larger meansmore wall thk. scatter!

You Must Know Wall Thickness At Time Zero


Wall Thickness @ 99.9%

0.06496 @ 20 years

0.101 @ 0 years

0.07034 @ 17 years

0.05956 @ 23 years

Data needed for constructionof trendlines on next pagewith as new slopes.


12/22


.1

.5

.2

1

5

2

10

20

3040

5060

7080 90

9599

99.9

.03 .04 .05 .06 .07 .08 .09 .1 .11

Heat Exchanger Construction Lines


G-/rr/insp1

Year 17

Year 0

OccurrenceCDF%

MinAllowedWall=0.036

Year 20

0.05237 0.083

0.04246

As New Slope

Accelerated

CorrosionEffects

0.1010.070370.06496

0.03531

General Corrosion

Year 20 Forecasted Line: = 0.05848, = 0.0033541 with 0.1228% occurrence at 0.036 wall.

Year 23


Wall Thickness at 0.1% Risk vs Time

0.05237

0.083

0.04246

17

Y=0.083-0.0018017t

Y=0.083-0.0023847t

0.03531 @ 20 years

0.02815 @ 23 years General + AccelerateCorrosion Rate


13/22


Retube Or Not Retube Now? At year 20 (next turnaround) the minimum

wall thickness will decline to just under 0.036

The risk for falling below 0.036 min wall is

0.1228%

$risk = (prob. of failure)*$Consequence, $risk exposure = 0.1228%*$750,000 = $921

take the risk for running 3 more yearsDo not retube now. Run to TA at yr 20.

Time & Money Issues Converge


Tube Exchanger Summary

Avoided the recently discovered and recently

expected turnaround delay for accelerated delivery of

heat exchanger ($750,000 expenditure avoided)

based on use of one day analysis of data.

Pressing on toward the next turnaround three years

into the future

At year 20, install a new tube bundle. Whats the risk for continuing to year 23?

0.91%*$750,000 = $6,825if risk adverse, reject.

If risk acceptingmaybe, but very doubtful.


14/22


Problem 2: Column Corrosion A column is rapidly loosing wall thickness.

Fluids/gasses within the column are violent.

Frequent Inspectionsdata is all over the map!

Loss of containment will impact personnel and

environment issues withbig $s

What should we do:

--Run?if so, for how long?--Shut down?if so, how to persuade themanagement team?


Developed Outer Surface Of Tower

Height

Circumference

Inspection

Grid

Over Bad

Spots

Data collection

on the grid

will contain both

good walls and

bad walls!


15/22


Raw Data UT Inspections

Thin worry!

Thick

ignore!Rapid

Deterioration

In WallThickness

Remaining Wall Thickness (Mils/10)


Truncated DataThin Data Only

Remaining Wall Thickness (Mils/10)


16/22


End Points For Corrosion Curve

49

51

32.3

33.09

31.09

25.17

25.56

25.17

Gen + Accel Cor.

@ 99.9%Days Thickness

0 51

906 33.09

966 32.3

1105 27.561127 27.17

UT Wall Thickness Construction Lines

Gen + Accel Cor.

@ 0.1%

Days Thickness

0 49

906 25.56966 23.61

1105 19.26

1127 19.53

General Corros.@ 0.1%

Days Thickness

0 49

906 31.09

966 25.17

1105 25.56

1127 25.17


End Of Life Clearly Shown

Y=49.065-0.02128*X General Corrosion

949

General + Accelerated Corrosion

1176

1178

1460

End Of Life


17/22


Summary ASME minimum wall was violated at 949 days

API fitness for service will be violated at 1176 daysand we are 1127 days into service

Plan an immediate orderly shutdown for replacement

Outage + planned replacement =$10,000,000

Emergency outage + emergency replacement =$20,000,000 because of safety hazards

Risk is too high! 0.1%*$10,000,000 = $10,000 andclimbing toward $20,000,000. Take action now!


Now, For Grins

Consider the Gumbel larger distribution

Houston flood

Aircraft gust loads

Space shuttle rocket motor O-ring burnsDiscussions about the

Gumbel lower distribution

always raise questions

about the Gumbel upper

distribution


18/22


It Rained A Little

On June 9, 200123 Inches!

Cars are

submerged

on US 59

highway!


.1.5.2

15

2

1020

3040

5060

70

80

90

95

99

99.9

0 10 20 30 40 50 60 70 80

Peak Annual Stream Flows-Gage Height (feet)

USGS 08074000 Buffalo Bayou at Houston, Texas

Peak Gage Height (feet)

17.2 5.99 0.978 67/0

Xi Del r^2 n/s

G+/rr

June9,2001

OccurrenceCDF(%)

F

orecastedOneHundred

Y

earFloodGageHeight

44.76

Depth for 100 yr flood

comes from the return

period, RP = 1/(1-p).

When RP = 100 years,

then p = 99%

The flood was bad but not the worst

recorded near downtown Houston!


19/22


0

1

2

3

4

5

10 50

Assumed Houston Flood Cost In June 2001

Gage Height (feet)

20 30 40

AssumedFloodCostUS

($Billion)

June9,2001

100yearflood

willbea2X

$problem

Flood Cost Estimates In June 2002


Aircraft Positive Gust Loads


20/22


Space Shuttle Burned O-Rings

Calculated Joint Temperature, oF45o 50o 55o 60o 65o 70o 75o 80o

Numberof

Incidents

3

2

1

0

Field Joint

STS 51-C

41B

61C

41C

61A

41D

STS-2

Field Joint

Calculated Joint Temperature, oF45o 50o 55o 60o 65o 70o 75o 80o

Numberof

Incidents

3

2

1

0

STS 51-C

41B

61C

41C

61A

41D

STS-2

Flightswith noincidents

Source: Engineering Ethics, Gail D. Baura,

Elsevier, ISBN 13:978-0-088531-2, 2006, Page 73.

Data from the Rogers Commission 1986

Data53*3

57*158*1

63*1

70*2

75*2

Data

53*3

57*158*1

63*1-66*1

-67*3

-68*1

-69*1

70*2-70*2

-72*1

-73*175*2

-76*2

-78*1

-79*1-80*1

-81*1

FailuresOnly

Failures

And

Successes


Good Practice AdviceWatch Out!

Gumbel upper & lower distributions allow the use

of negative numbers on the X-axis

When using suspensions (as a sign) make sure

you turn on display of the suspensions (under

magnifying glass) so you can view they are in the

correct locations AND (under the Method icon)

make sure to turn the negative sign to indicatesuspension!

Else, youll get misleading results.


21/22


Which Plot?

Poor curve fit

Suspended data

shown on plot as >


Gumble Upper Slightly Better-ButNot Every Data Fits A Plot!

?

Better but not good curve fit

Failures were resolved by rocket joint/O-ring redesign

Failures demonstrated to exist

If you fail to turn on - is a suspensionyou will conclude this is a good fit!!


22/22


Gumbel Upper Summary Works well when you have the largest

recorded data such as flood data, fatiguedata, etc.

Watch for traps with suspensions when usedwithout good practices can result in badconclusions.

If Weibull, Lognormal, etc. dont work thendont expect automatic success with all data

by use of the Gumbel upper distribution.


Want More Details? Got to http://www.barringer1.com/problem.htm Look at WinSMITH Weibull software (which also includes Gumbel large

and small distributions)

See biographies at http://www.barringer1.comof Dr. Weibull and Dr.Abernethy who is the worlds leading expert in Weibull analysis

Dr. Weibull got many of his ideas on extreme values while working atBofors Steel in Swedenyou can see Bofors antiaircraft guns at theMuseum of the Pacific in Fredricksburg, TX.

See Gumbel, E. J., Statistics of Extremes, Columbia University Press, NewYork, 1958

See Statistical Theory of Extreme Values And Some PracticalApplications, A Series of Lectures, PB 175818, 12 Feb 1954 by Emil J.Gumbel, National Bureau of Standards, U.S. Dept of Commerce, NTIS

See A New Method Of Analyzing Extreme-Value Data, NACATN 3053, Jan 1954, U.S. Dept of Commerce, NTIS, byJulius Lieblein National Bureau of Standards

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Barringer-Corrosion-With-Gumbel-Lower[1].pdf