Using Reliability Statistics To Estimate Metal ... - ACS Publications

Chapter 6

Downloaded by UNIV OF CALIFORNIA SANTA BARBARA on May 28, 2018 | https://pubs.acs.org Publication Date: March 30, 1998 | doi: 10.1021/bk-1998-0689.ch006

Using Reliability Statistics To Estimate MetalContainer Failure Levels from Censored Tests

W. Stephen Tait

S. C. Johnson and Son, Inc., 1525 Howe Street, Racine, WI 53403

Tests terminated prior to failure of all test samples are referred to as censored tests, and the samples in a censored test are referred to as censored samples. Failure times for censored samples can be estimated if the pit depth and pitting rate are known, or if the pitting rate can be reasonably estimated from pit depth and exposure time. Reliability statistics are used to estimate failure levels from censored data so that censored tests can be used to estimate population failure levels. Cumulative failure levels are estimated by calculating median rank for each estimated failure time, and failure levels are plotted versus corresponding times on a Weibull probability plot. The Weibull plot typically yields a linear graph that can be extrapolated to find failure levels outside the data set.

The objective of corrosion testing is usually to determine metal service lifetime: that is, when will a metal, exposed to a given environment, fail from corrosion. Failure is a subjective term that is usually defined by the type of service for which the metal is being used. Failure is defined here as when a pit perforates a metal container wall and causes the container to leak. It is also often the objective of a corrosion test to determine what level of failures to expect after a specified amount of time. Failure level is defined here as the total number of metal containers failing by a given time. Failure levels are obvious when a test is allowed to run until all samples (containers) fail. Unfortunately, one does not always have the luxury of waiting long times for results (e.g., years). Time constraints often force one to either prematurely remove all samples from a test prior to failure, and (destructively) inspect them for corrosion, or prematurely remove a 58

©1998 American Chemical Society

Bierwagen; Organic Coatings for Corrosion Control ACS Symposium Series; American Chemical Society: Washington, DC, 1998.

59 fraction of the samples from test at several intermediate times and (destructively) inspect for corrosion. Tests whose samples are removed before test completion (i.e., before all samples fail) are referred to as censored tests, and samples removedfromtest prior to failure are referred to as censored or suspended samples. Data from censored samples are often used to estimate the percentage of a population that will fail by a specified time: for example, ten percent of all metal containers are expected to fail one year after filling. It will be shown that: 1) expected failure time (EFT) can be estimated when a censored sample is pitted, 2) failure levels can be estimated from expected failure times (EFTs) using statistical median ranking, and 3) a Weibull probability plot of failure levels (median ranks) versus corresponding EFTs can be used to determine failure levels outside of the data set.


1

2

3

Experimental A previous publication contains descriptions of both uncoated and internally coated metal containers.^ Filled metal containers were stored in a 21°C (70°F) constant temperature room, and sets of samples were periodically removed and inspected for corrosion inside the containers. Test length was one year. Containers that were inspected for corrosion were not returned to test because they were emptied and cut open in order to locate pits and measure their depths. Pit depths were measured by taking the difference between the point of focus at the pit mouth and the point of focus at the pit bottom. The coefficient of variation^ for this type of depth measurement is approximately ten percent.

Pitting corrosion Pitting corrosion occurs in small localized areas and proceeds through several stages: 1) nucleation, 2) growth to a critical size, and 3) steady state growth.6-12 Pit nucleation. It is not unusual to have samples, in a set of censored samples, with different pit depths. Pits nucleate at different times, thus different pit depths among censored samples are most likely due to different pit nucleation times and not different pitting rates. It is reasonable to hypothesize that the deepest pit found at any given exposure time has been growing for the longest time, and has the smallest times for nucleation and growth to critical size. Pit growth to critical size. Only pits reaching a critical size^-18 continue growing until their aspect ratio decreases below a critical level, ^ or the pit perforates the metal.20-22 censored samples (containers) that are visible to the unaided eye are most likely larger than critical size, because most critical sizes reported in the literature are on the order of 50 μπι (human hair is approximately 25 μχη thick). I'M** Mature pit steady state growth. Figure 1 contains a typical graph of maximum pit depths observed in metal containers after different exposure times. It can be seen in Figure 1 that maximum pit depth increases with time, supporting the hypothesis that mature pits grow at a steady state rate, whose magnitude is determined by a given environment. Other researchers observed the same pit growth trend for o

n


60 23

2


different metals and environments. " ^ A filled metal container is a closed static system, thus a visible pit is most likely larger than critical size and will continue growing until it perforates a container, providing the container is not prematurely removed from test.

(Perforation)

Pit nucleation time (NT plus time to grow to a mature pit (GTCS)

0

60

120 180 240 300 Exposure time (days)

360

420

Figure 1. Maximum pit depth increases with time. Steady state growth rate of mature pits can be mathematically expressed in terms of the three stages of pitting corrosion: Pitting Rate (PR) = ë

V

}

^îldepth ET - (GTCS +NT)

(1)

Nucleation time (NT) is the time it takes for a pit to nucleate after initial exposure of a metal to a corrosive environment (stage 1); growth to critical size (GTCS) is the time it takes after nucleation for a pit to grow to critical size (stage 2); and exposure time (ET) is the total time a metal is exposed to a corrosive environment (stages 1-3). Pit depth in Equation (1) is the total depth measured when the sample is opened for inspection. Failure times (FT) for pitted censored samples can be calculated when pit depths, NT, GTCS, and PR are known: PY

=

(Original thickness - Pit depth) PR

+

£j

(2)

Unfortunately, PR is difficult to determine for metal containers because the magnitudes of NT and GTCS are rarely known. A good estimate for pitting rate can be obtained by applying linear regression to data like that in Figure 1. The slope of the regression line in Figure 1 is the pitting rate. NT plus GTCS can also be estimated from the intersection of the line with the time axis as illustrated in Figure 1. However,



61 it takes a long time (e.g., one year) to generate data like that in Figure 1, and more than three data points are needed to obtain a proper regression line. Figure 1 suggests there is a unique pitting rate for a given metal-environment system. Unfortunately this unique rate must be estimated because there is currently no way to determine NT and GTCS. We found it useful to estimate the system pitting rate as follows: a) set NT and GTCS equal to zero, b) use Equation (1) to calculate pitting rates for all pitted censored samples, c) use the highest rate among all samples as an estimate for the system rate. The highest rate is substituted for PR in Equation (1) to yield: EPT _

(Original thickness - pit depth)

+

ET

(3)

P^max PR in Equation (3) is the highest pitting rate observed among all censored test samples, and Equation (3) is used to calculate the expected failure time (EFT) for each censored pitted sample. It will be shown that Equation (3) yields expected failure levels (EFLs) that are the same order of magnitude as observed failure levels. m a x

Reliability Statistics The procedure for calculating EFLs includes: 1) ranking all samples, and 2) calculating cumulative probabilities according to rank. Table 1 contains a raw data set from a test that was prematurely terminated after 364 days. The table consists of containers (samples) that are: a) perforated (10, 12, 15, 19, and 20), b) pitted censored samples (5-7, 14, 16-18), c) non-pitted censored samples (1-4, 8, 9, 11, and 13), and d) unopened censored containers (21-24).

Table 1. Example failure data Sample 1 2 3 4 5 6 7 8 9 10 11 12

Exposure 200 days 200 days 214 days 217 days 234 days 234 days 234 days 234 days 234 days 302 days 302 days 308 days

EFT (days)

421 401 401

302 308

Sample 13 14 15 16 17 18 19 20 21 22 23 24

Exposure 308 days 347 days 347 days 349 days 349 days 357 days 357 days 364 days unopened unopened unopened unopened

EFT (days) 521 347 549 522 454 357 364

Non-pitted and unopened censored samples are temporarily assigned an EFT equal to their ET, because it is not possible to calculate an EFT when no pits are present. Temporary EFTs in Table 2 are in parentheses. For example, container


62 numbers 1 and 2 are temporarily assigned an EFT equal to their 200 day exposure time, and unopened containers 21-24 are temporarily assigned a 364 day EFT, which is the total length of the test.


Table 2. Failure data with temporary EFT values Sample 1 2 3 4 5 6 7 8 9 10 11 12


EFT (days) (200) (200) (214) (217) 421 401 401 (234) (234) 302 (302) 308

Sample 13 14 15 16 17 18 19 20 21 22 23 24

Exposure 308 days 347 days 347 days 349 days 349 days 357 days 357 days 364 days unopened unopened unopened unopened

EFT (days) (308) 521 347 549 522 454 357 364 (364) (364) (364) (364)

EFT data are arranged in ascending order as illustrated in Table 3. Non-pitted and unopened samples precede pitted samples when their EFTs are equal. For example, no pits were found on sample 11 when it was opened at 302 days and sample 10 perforated at 302 days. Both samples have 302 day EFTs, but sample 11 is a censored sample (with a temporary EFT) so it precedes sample 10.

Table 3. EFT data in ascending order Sample 1 2 3 4 8 9 11 10 13 12 15 19


EFT (days) (200) (200) (214) (217) (234) (234) (302) 302 (308) 308 347 357

Sample 21 22 23 24 20 6 7 5 18 14 17 16


EFT (days) (364) (364) (364) (364) 364 401 401 421 454 521 522 549

EFLs are calculated only for pitted containers (samples), so temporary EFTs are removed from the table, after arranging in ascending order, to give a table like that in Table 4.



63 Table 4. Ranked EFT data without temporary EFTs Sample Exposure Exposure EFT (days) Sample 1 200 days 21 364 days 2 200 days 22 364 days 3 214 days 23 364 days 4 24 364 days 217 days 8 234 days 20 364 days 9 234 days 6 234 days 11 302 days 7 234 days 10 5 234 days 302 days 302 13 18 357 days 308 days 12 14 308 days 308 347 days 15 347 days 17 349 days 347 19 357 days 16 349 days 357

EFT (days)

364 401 401 421 454 521 522 549

Median statistical ranks are calculated for EFTs like those in Table 4, and these ranks are used to calculate cumulative probabilities (EFLs) for each E F T . There are other statistical ranking methods, ^ but they all give approximately the same probability (EFL) as illustrated in Figure 2. Statistical medians are at the center of a data set, thus more than one EFT is needed to use median ranking because a single EFT will always have a 50% EFL. 2 6

2

80 Φ >

60

I

40

ο

Φ

s η»

20 (12% Mean Rank)

u. •o υ 10 ο α. χ 8 LU 150

^-

.(9.7% Median Rank) >£(9.2% Normal Rank) >~(8.4% Middle Rank)

200

250

300

350

400

450

500

Expected Failure Time (days) Figure 2. Different statistical ranking methods yield approximately the same estimates for cumulative failure at a given expected failure time.


64 Failure data are typically not a continuous function of time as can be seen in Figure 3. A continuous linear function is needed to extrapolate data outside of a data set. Fortunately cumulative failures are a continuous function of time as seen in Figure 4.28 Cumulative failure curves typically have an S-shape like that in Figure 4 , ' and can be made linear by plotting the data on probability paper:


2 8

3 0

2 3

I •5

No failures occurred

It.

φ

80 120 160 Exposure time (days) Figure 3. Number of failures is not a continuous function of time.

110 c

Φ

90

a

Φ

α

70 50

Φ >

3 Ε 3 Ο

30 10 -10 -20

20

60

100

140

180

220

260

Exposure time in days Figure 4. Cumulative number of failures are a continuous function of time.


65 Weibull probability plots were used for EFL and EFT data like that in Table 4 (or Figure 4) because the Weibull function can be transformed into a linear form. The cumulative Weibull equation is:


F(t)

=

1 -

P

e-(t/0)

(4)

F(t) is the cumulative number of failures at a given time, t, and θ is the point on a Weibull curve that divides the curve area into 63.2% and 36.8% (Θ is analogous to the mean for a normal distribution curve), and β is the shape parameter for the Weibull probability density function. The cumulative expected failure level (EFL) can be substituted for F(t) and EFT for t in Equation (4) to yield: EFL

=

1

-

e-0EFT/9)

P (

5

)

Rearranging Equation (5) and taking the natural log twice yields an equation that has the linear form shown in Equation (6): Ln(EFT) =

(l/p)LnLn

1

+ LnO

(6)

(1- EFL) Figure 5 contains a Weibull plot of EFL and EFT data like that in Table 4. The resulting linear plot is continuous and can be extrapolated (within reason) outside the data set as shown in Figure 5, where a two percent failure level is expected to occur by 124 days (or 100% failures at 509 days). Using Weibull probability paper allows EFT and EFL data to be plotted without having to mathematically transform the data in order to obtain a linear graph. 31

100 80 60

(100% failure at 509 days)

I Φ > Φ

ε 3

EFL is expected to be 2% by 124 days

κ UJ CM

Ο CD

CO CM

CM CO

if

Ο Ο

ooc

00 CM CO Ο ' ' CO CM ID ID CDC >CDI^

Expected failure time (days) Figure 5. A Weibull probability plot of EFLs versus EFTs.


66 Results and Discussion


Several comparisons between EFLs obtained from Weibull probability plots and observed failure levels are contained in Table 5. It can be seen from Table 5 that EFLs are typically the same order of magnitude as observed failure levels. It can also be seen in Table 5 that approximately half of the EFLs are lower than observed levels.

Table 5. Comparisons between estimated and observed failure levels Percent EFL 5 18.5 17.7 9.2 8.1 36.8 55.9 64.4 5.6 12.6 13.5 8.3 6.7 41.3 23.8 23 29.8

Observed cumulative % failures 11.5 16.7 33.3 22.2 8.3 30 55.6 63.6 8.3 8.3 16.7 12.5 10 41.7 25 14.3 33.3

Underestimation of failure levels arises from underestimating pitting rates. Rate underestimation comes from assuming NT and GTCS are zero, as was done when total exposure time (ET) was used to calculate censored sample pitting rates to find the maximum pitting rate. However, the comparative data in Table 5 show that the error in pit rate estimation is small enough to yield EFLs that are within the same order of magnitude as actual failure levels. The accuracy of EFTs (and thus EFLs) could be improved with a method that measures (without having to destroy samples) either: a) actual pitting rate (PR), or b) GTCS and NT. Equations (1) and (2) can be used to calculate failure times when PR, NT and GTCS magnitudes are known. However, without these quantities the method discussed here provides a reasonable approximation of failure times and levels.

Conclusions The three stages of pit growth were used to develop an algorithm for estimating when pitting corrosion will perforate metal containers. The highest pitting rate observed in all censored test containers is used to calculate expected failure times (EFTs) for all censored pitted samples. Expected failure levels (EFLs) obtained from these EFTs are



67 the same order of magnitude as actual failure levels. More accurate EFLs could be obtained if the actual pitting rate could be measured without having to destroy the sample to locate and measure pit depth. The method described in this paper can be used for both internally coated and uncoated metal containers, and for other metal-environment systems where failure is defined as metal thinning or perforation. The algorithm described here can also be used for any coating system where failure can be defined quantitatively, and failure time can be estimated for censored samples. Failure time and level data like that in Figure 5 can be used to estimate when a given level of failure will occur, and whether or not a given product meets product warranty or lifetime expectations.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.

W. Nelson, Journal of Quality Technol., 1970, 2(3), p. 126 K. C. Kapur and L. R. Lamberson, Reliability in Engineering Design, John Wiley and Sons, NY, 1977, Chapter 11, p.297, K. C. Kapur and L. R. Lamberson, Op. Cit., p. 303 W. S. Tait, Corrosion Protection by OrganicCoatings,The Electrochemical Society, Pennington, NJ, 1987, 87-2, p. 229 I. Miller and J. E. Freund, Probability and Statistics for Engineers, third edition, Prentice-Hall, Englewood Cliffs, NJ, 1977, Chapter 5, p. 156 W. S. Tait, An Introduction to Electrochemical Corrosion Testing for Practicing Engineers and Scientists. PairODocs Publications, Racine WI, 1994, Chapter 3, pp. 37-42 N. D. Greene and M. G. Fontana, Corrosion, 1959, 15, p. 25t P. M. Aziz and H. P. Godard, Ind. Eng. Chem., 1952,44, p. 1791 P. M. Aziz, Corrosion, 1953, 9, p. 85t R. May, J. Inst. Metals, 1953, 32, p. 65 M. A. Streicher, J. Electrochem.Soc.,1956, 103, p. 375 W. S. Tait, Corrosion, 1979, 35(7), p. 296 J. R. Scully, S. T. Pride, H. S. Scully, and J. L. Hudson, 1995 (October 8-13), abstract number 105, Electrochemical Society Fall meeting, Chicago IL D. E. Williams, C. Westcott, and M. Fleischmann, HAREWELL publication AERE-M3371, United Kingdom Atomic Energy Authority, 1984 P. M. Aziz, Corrosion, 1956, 12, p. 495t G. T. Burstein and S. P. Mattin, 1995 (October 8-13), abstract number 104, Electrochemical Society Fall meeting, Chicago IL M. Kowaka, Introduction to Life Prediction ofIndustrial Plant Materials, Allerton Press, Ν.Y., 1994, Chapter 3, p. 22 Y. Hisamatsu, Corrosion Engineering, 1983, 32, p. 9 M. Wang and H. W. Pickering, J. Electrochem.Soc.,1995, 142 (9), pp. 2986 2995 C. C. Nathan and J. R. Schieber, 1977 (March 14-18), paper number 177, CORROSION/77, San Francisco, CA Z. Szklarska-Smialowska, Pitting Corrosion ofMetals,National Association of Corrosion Engineers, Houston, TX, 1986, Chapter 6, pp. 121-125 M. Janik-Czachor, Br. Corr. Journ., 1971, 6, p. 57 J. E. Strutt, J. R. Nichols, and B. Barbier, Corrosion Science, 1985, 25(5), pp. 305-315 J. W. Provan and E. S. Rodriguez IIΙ, Corrosion, 1989, 45(3), pp. 178-192


68 25.


26. 27. 28. 29. 30. 31.

D. L. Crews, Galvanic and Pitting Corrosion - Field and Laboratory Studies, ASTM STP 576, American Society for Testing and Materials, Philadelphia, PA, 1976, pp. 217-230, K. C. Kapur and L. R. Lamberson, Op. Cit., pp. 314-322 B. F. Kimball, J. American Statistical Assoc., 1960, 55, p. 546 I. Miller and J. E. Freund, Op. Cit., p. 143, W. S. Tait, Corrosion, 1994, 50(5), pp. 373-377 J. E. Strutt, J. R. Nicholls, and B. Barbier, Corrosion Science, 1985, 25(5), p. 305 W. Nelson, AcceleratedTesting,John Wiley and Sons, NY, 1990, Chapter 3, p. 129


Using Reliability Statistics To Estimate Metal ... - ACS Publications

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