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The prediction of corrosion by statistical analysis of corrosion profiles
Corrosion Science, Vol. 25, No. 5, pp. 305-315, 1985 Printed in Great Britain
0010-938X/85 $3.00 + 0.00 Pergamon Press Ltd.
THE PREDICTION OF CORROSION BY STATISTICAL ANALYSIS OF CORROSION PROFILES J. E. STRUTr, J. R. NICHOLLS and B. BARBIER School of Industrial Science, Cranfield Institute of Technology, Cranfield, Bedford, U.K.
Abstract--The corrosion of carbon manganese steel in CO2-acidified sea water has been assessed by a statistical analysis of corrosion profiles. The total population of depth data was multi-modal and could not be fitted to any well known statistical models. However the deepest pits in the profile followed a type I extreme value distribution. The time dependence of the extreme value distribution parameters has been determined and used to predict the rate of penetration of steel components under pitting conditions. INTRODUCTION
AN I~IPORTANTmethod of preventing corrosion failures on a chemical plant is the provision of a corrosion allowance on the thickness of materials used in the plant construction. The size of the corrosion allowance should normally be based on the anticipated service corrosion rate and the required life. Data for such calculations are frequently derived from corrosion tests involving either electrochemical polarization or weight loss measurements. A corrosion allowance may be calculated from such measurements by converting the raw data to equivalent section loss per unit time and extrapolating to the required lifetime. To carry out this conversion, it is necessary to assume that the corrosion is uniform, hence any variation in the depth of metal lost, both across samples and from sample to sample, must be ignored. In many cases this variation is not important, particularly if small and associated with some uncontrolled variability in the environment. However, there are applications where the environment is well defined and yet the surface profile is rough and pitted. The uncertainty in the depth of penetration can then be a major factor in determining the time to failure. Under these conditions, it is difficult to define the exact meaning of corrosion rate, when quoted as section loss per unit time, and experimenters often resort to expressing data as mean or maximum penetration rate, with little consideration of the meaning or statistical significance of these parameters. The analysis described in this paper is an attempt to develop a more rational approach to presenting section loss or penetration rate information, when the corrosion process exhibits a rough or pitted surface texture. The method presented provides a means for assessing corrosion allowances at a given level of risk, given sets of thickness data taken over the area of interest over a period of time, and hence may form a basis for assessing the lifetime of corrosion protection coatings or surface treatments.
Manuscript received 6 July 1984. 305
306
J. E. STRUTT.J. R. NICHOLLSand B. BAaaIER DR I
Remaining metal ( Vu )
Corroded metal ( Vc )
A2
(a)
(b)
°i
/•f(z)
l~e
Roe
FIG. ]. Schematic representation of corrosion in terms of statistical models. (a) Corrosion profile, (b) cumulative distribution function, F(z), plotted against depth z, and (c) probability density function, f(z), and extreme value probability density function, f=(ze), plotted against depth ze.
The statistical assessment of metal loss due to corrosion It is possible to characterize the depth profile of a corroded surface by a probability density function f (z), or its integral, the cumulative distribution function F(z), determined from metal loss measurements. The cumulative distribution function (cdf) represents in physical terms the probability that an observed value of depth is less than or equal to a given depth value (z).
Prediction of corrosion by statistical analysis of corrosion profiles
307
It is possible to show that an exact relationship exists between the cumulative depth distribution function, and the total volume of metal lost, (Vc), due to a corrosion process as illustrated in Fig. 1. By integrating the depth of metal lost, z(A), within an element of area dA, over a total sample area Ao, it can be shown that,
Vc = ( z(A) dA. J Ao Integrating equation (1) by parts results in equation (2),
(1)
where Dw is the original thickness of the corroding section. The volume of metal remaining (11.) is thus given by equations (3) and (4):
V. = AoDw - V~ i.e.
Vu = Ao
f0DwF(z) dZ.
(3) (4)
Equations (2) and (4) can be interpreted graphically as shown in Fig. 1, where Fig. la is an amplified representation of a typical corrosion profile. From this, the corresponding cdf for the profile can be derived and is illustrated in Fig. lb. In practical terms, the cdf is obtained by taking a set of depth data, sorting them into order of increasing depth and then plotting depth (zi) against i/(n + 1), where n is the total number of depths sampled and i is the rank of the depth measurement in the ordered list of increasing depths. The parameter i/(n + 1) is the cdf of the sample and defines the boundary between the metal and the environment in terms of its probability of occurrence. The shaded area A~ is proportional to the volume of metal corroded, Vc whilst area A 2 is proportional to the volume of metal remaining, Vu. Equations (2) and (4) are theoretically valid for a wide range of corrosion morphologies with open pits and generally roughened surfaces, but excludes those morphologies which show subsurface lateral attack, such as intergranular corrosion, for example. Although equations (2) and (4) accurately represent many observed corrosion profiles, their use in practice is often limited by the need to fit the experimental depth data sample to a known statistical model such as the normal, log normal or Weibull distribution, before predictive modelling can be undertaken. However, as expected, many corrosion profiles do not fit any of the commonly used statistical models, particularly when a combination of corrosion processes operate at the same time, giving multi-modal distributions. Under such circumstances the penetration probabilities must be calculated graphically or some other approach taken. One alternative which may be adopted when the sampled data do not conform to any of the well known statistical models, is to fit the maximum tail of the pdf using an extreme value distribution. It is possible to fit the fraction of data corresponding to the deepest penetrations to a type I extreme value model, provided that the maximum tail of the depth probability distribution function (pdf) decreases at least as rapidly as an exponential function. 2 The use of the type I extreme value model
308
J.E. STRUTr,J. R. NICHOLLSand B. BARBIER
offers the advantage that the shape of this distribution can be written as an explicit equation, both in the pdf and cdf form, and hence this facilitates the development of statistical equations which can be related to the corrosion rate. The cdf of a type I extreme value function is given by: FI(Ze; /.re, Ore) = exp
(
-exp
tre
]
(5)
where ze is the extreme value of the depth,/~e is the location parameter and tre is the scale parameter of the extreme value distribution. Equation (5) has a physical interpretation; 1 - F(z~) is the probability of obtaining a maximum depth measurement greater than some value of depth (z~). Consequently, 1 - Fi(z~) is a measure of the risk of accepting a given value of z~ as the maximum penetration. The physical significance of equation (5) in terms of corrosion measurements can be more clearly seen by taking a specific case of equation (3). Suppose that it is required to define a depth, DR (see Fig. 1), for which the risk of accepting D R as the maximum penetration is a, where a = 1 - FI(DR). Then from equation (5) one obtains: DR = tzc + treR (6) where R = - I n ( - I n (1 - a)). (7) Equation (6) shows that the depth DR, the maximum penetration depth for a risk of value ot is simply the most probable maximum penetration,/x~, plus a term tr~R. The term tr~R is a safety factor which depends on the shape (width) of the extreme distribution. Thus R can be thought of as the number of ore values which must be added to the most probable maximum depth (/xe) in order to achieved the required level of risk. For example, for a very low risk (a = 10-3), approximately 7o'~ must be added qm to/z~, whilst for a very high risk (or = 0.1), only 2.25tr~ need be added. CorrestJonding R values for typical ot values are shown in Fig. 2. The relationship between the full distribution of depth data and its extreme value distribution is shown schematically in Fig. lc. The depths corresponding to the shaded upper tail of the pdf for the complete population of depths are treated as extreme depth data, and hence are fitted to the extreme value pdf, fl(Ze). If the pdf for the complete population of depth data can be fitted to a normal distribution, then mathematical relationships between the mean (/xN) and standard deviation (O'u) parameters of the normal distribution and the location (/xe) and scale (ore) parameters of the extreme value distribution can be used to link the maximum depth estimate, D R, with the mean metal loss, as shown below. 2 The relationships between the parameters of normal distribution and those of extreme value distribution 3 are given by: /Xe ----//'N + A'OrN o-¢ = B'/z N
(8) (9)
where A' is given by: A' = 2 1 n n - ½ 1 n l n n ~/2 I n n
ln(2V~)
(10)
Prediction of corrosion by statistical analysis of corrosion profiles
309
lO
o!
10-4
i
10-3
t
10-2
i
N
10-1
Risk (a) FIG. 2.
Risk factor R plotted against risk value ~, where R = - I n (1 - a ) , o~ = 1 - F l ( D n ) and D R is a specific value of depth.
and B' by: B' -
1 X/2 In n
(11)
and where n is the number of depth data points sampled. Consequently, equation (6) can be rewritten in terms of the parameters for the normal distribution by substituting equations (8) and (9) into (6) to give: D R = I~N + R n o v N
(12)
Rn = A ' + B ' R .
(13)
where
Values of A ' and B ' a s a function of the number of data points sampled (n), are shown in Fig. 3. For 100 values of depth, A' is approx. 2.4, and B' is approx. 0.33, so for a 1% risk in estimating the extreme corrosion (a = 10 - 2 , R = 4.6), Rn has a value of 3.9. This implies that the estimated extreme thickness loss, DR, at 99% confidence, is the mean depth plus about four standard deviations of the scatter in depth values. Equations (6) and (12) are of more practical value than equations (2) and (4), and can be used to estimate corrosion allowances and remaining life of pipe work or coatings at some specified level of risk of penetration on a rational basis. Industrial uses of these equations could include the analysis of ultrasonic wall thickness surveys4 and caliper surveys.5 However, the techniques can also be used in laboratory corrosion testing, by measuring metallographic sections or more rapidly using surface profiling instruments. 1.6 The application of this method in laboratory corrosion testing is illustrated in the remainder of this paper, for an experimental programme aimed at assessing the corrosion of carbon steels in CO2-acidified sea water.
310
J.E. STRUTT, J. R. NICHOLLS and B. BARBIER 6.0
A ~
5.0
4.0 -
A'
-1.0 0.8
3.0
2.0
B'
=
1.0
0
"-
10
10 2
10 3
-
10 4
0.40"6 B'
~
10 5
°
-
~
10 6
-
~_0.2 0 10 7
Number of data points (n)
FIG. 3.
Values of the statistical parameters A' and B' plotted against the number of depth data points sampled.
EXPERIMENTAL METHOD Carbon steels corresponding to BS 4360 50D were used for the experimental work. A sample of steel was first ground on a mechanical grinding machine and then cut into samples of size 10 × 38 × 48 mm 3. The samples were immersed in a flowing sea water cell. The sea water was prepared from Spanish sea salts, de-aerated and saturated with 1 atmosphere pressure of CO2 at 25°C. The flow rate was set at 0.4 m s -~ and CO, gas was bubbled through the reservoir solution continuously throughout the seven week (1176 h) experiment to simulate a sweet corrosion environment] After an initial period of two weeks (336 h), the first sample was removed and subsequently, samples were removed at one week (168 h) intervals. Following exposure to the corrosive environment, samples were cathodically descaled and cleaned ready for surface profiling. The depth of corrosion was measured using a Rank-Taylor-Hobson Talymin type 410, with a stylus tip radius of 3 #m. Reference surfaces were provided by coating the edges of the specimens with a non-conducting epoxy resin. Within the time scales of the experiment, no underfilm or crevice corrosion occurred, leaving a clean flat reference surface from which to estimate metal losses across the samples. The output from the Talymin was displayed on a strip recorder. This data was then measured and fed into a Commodore PET Microcomputer for further data analysis. Five traces were taken across each sample and depths were sampled at 200 ~m intervals, giving 225 depth data points per specimen. EXPERIMENTAL
RESULTS AND
DISCUSSION
The pdf for the complete population of depth data was obtained by first ordering the data into increasing depth and dividing them into cells, to produce the frequency histograms shown in Fig. 4(a) for each specimen. Attempts to fit these data sets to well known statistical models such as normal or log normal distributions were quickly abandoned when no single model was found to be representative of the six samples tested, and bimodai distributions were found for some of the specimens. However, the shape of the maximum tail of all the distributions suggested that an extreme value model might be used to successfully model the deeper pits.
Prediction of corrosion by statistical analysis of corrosion profiles
311
313
30
r = 336 h
t = 336 h 20 10
10 I
0 30
I
f 30 t = 504 h
t = 504 h 20
20
~
10
10
I
A
I
g r.-
t = 672 h
o
=
I
0 30
2
0
~
t = 672 h 'r"
2O
"0
== ~
30-
~
2o-
o
10 F
t = 840 h
10
t = 840 h
t~
20 0
10
8 8
oi
i
u-
t = 1008 h
t = 1008 h
LU
20
2O
10
10
0 30
t = 1176h
0 30
20
20
10
10 1O0
FIG,
0 30
4.
200
0
t = I176h
1O0
200
Depth (z/lO-6m)
Depth (ze/lO-6m)
(a)
(b)
Variation of depth frequency distributions with time during corrosion. (a) Complete depth distribution, (b) extreme depth distribution.
312
J.E. STRU'FI',J. R. NICHOLLSand B. BARBIER 5
/
¢ e-
7
,q
,.../
II ¢:
."
40
1
60
I
80
I
I O0
f
I
120
140
f
l
I
160
180
I
200
I
220
Depth of penetration (ze/10 -s m) FJc. 5.
V a r i a t i o n of the e x t r e m e v a l u e f u n c t i o n - i n
(-in
(F(ze))
with e x t r e m e d e p t h
zc(m). For this extreme value analysis, the hundred deepest pits were selected from the full data set. This data was then ordered and divided into cells to produce the extreme depth frequency histograms shown in Fig. 4b. Fitting the data to the extreme value function by plotting - I n ( - I n Fi(ze)) against ze gives a slope of 1/ore and an intercept at - I n ( - I n (F0) = 0 of/ze. Figure 5 shows the experimental results plotted in this manner, while Fig. 6 shows the theoretical type I extreme value functions calculated using the best fit values o f / ~ and Orefrom Fig. 5. It is interesting to note that not only does the most probable maximum depth (location parameter) increases with time, but also the width of the distribution (shape parameter) increases with time. The time dependence of these parameters is shown most clearly in Fig. 7. In both cases, a linear dependence is observed. The ratio of/ze to Oreis thought to be a characteristic of the morphology of attack, and for the conditions of this experiment is time-independent with the value: /xe
_
13.5.
(14)
Ore
The results of this statistical analysis show that the full distribution of depths of penetration, due to the corrosion process modelled in these experiments, does not fit any of the commonly used statistical distributions. During the early stages of the corrosion process, a bimodal distribution was observed, indicating that two distinct depth levels existed within the corrosion profile.
Prediction of corrosion by statistical analysis of corrosion profiles
313
.o_ c 25 t = 336 h J= "E
t = 504 h
20 "Io
15
=
e-
/ ' ~ = 672 ~ = 8 4 0 h
"O
t = 1008h
E
~
5
h-
50 LU
100
150
200
Extreme corrosion depths (ze/lO-6rn)
FIG. 6. Variation of extreme depth frequency distribution with time, after fitting data to equation (5) of the text. This result, although at first surprising, was not entirely unexpected. Sweet corrosion pits often exhibit a mesa or table-like m o r p h o l o g y . Whilst the mesa pits o b s e r v e d in service are on a m u c h larger scale than those g e n e r a t e d on the laboratory samples, the m o r p h o l o g i e s o f the latter did have a similar a p p e a r a n c e on a microscopic scale when o b s e r v e d using a stereoscopic microscope, T h e results during the initial stages suggest that pits nucleate on the surface, probably at sites where the c a r b o n a t e film has b r o k e n down. E a c h nucleated pit has a particular depth, and p r o p a g a t e s by lateral g r o w t h to expose a new passive surface on which new pits can 200
o
150
2 E
~
8O
100
50
._.1
250
500
750
1000
1250
t(h) FIG. 7.
Variation of extreme value frequency distribution parameters with time of corrosion. • #c, ~ (re.
314
J . U . STRU'IT, J. R. NICHOLLS and B. BARB1ER
subsequently nucleate and grow. With increasing time, pits nucleate in freshly exposed surfaces and the distribution eventually gives way to a rather broad flat distribution made up of many sub-distributions. The evolution of the distribution with time is interesting. The location parameter for the extreme distribution (bee) increases with time in an approximately linear manner, indicating that the most probable maximum corrosion rate is constant. Furthermore, the scale parameter o-e, which describes the width of the distribution, also increases linearly with time. Further consideration of the nature of electrochemical corrosion processes indicates that a time-dependent a e is not entirely unexpected. An ideally uniform corrosion process, corresponding to a flat surface profile, would result in a very narrow pdf giving a very high value for the ratio of bee to o-e. This distribution would shift to increasing values of ze with no broadening of the distribution, as the corrosion process proceeds for as long as the corrosion morphology remains 'uniform'. However, this can only occur if anodic and cathodic sites are continually exchanging in a random manner with time. In practice, the great majority of corrosion systems acquire a pattern in the spatial distribution of anodic and cathodic sites, which is frequently 'set' for long period of time, resulting in localized metal loss. This can, for example, arise from microstructural factors, such as anodic and cathodic microphases. Alternatively, it may be related to local spatial variations in cathodic constituents which may be slow to change in the time domain. The result, in either case, is that metal is lost at persistent anodic sites at a faster rate than at persistent cathodic sites, causing a broadening of the distribution of depths with time. It is perhaps surprising, however, that both be~ and o-~ were linear with time, indicating that autocatalytic mechanisms were not operative during this particular experiment. Apart from the potential use of the techniques described above for fundamental morphological studies of corrosion processes, it is considered that the main application for statistical analysis of corrosion profiles is to improve the ability to predict corrosion degradation. The material used in this study is not a pipeline steel and the corrosion test falls short of an accurate simulation of the type of sweet corrosion process experienced offshore. Since corrosion failure is associated with the rate of growth of deep pits, the extreme value distribution for deepest pits can be used to predict the likelihood of failure. In order to demonstrate this approach, the data generated in this study will be used to make an assessment of the corrosion allowance that would be required on a new pipeline carrying the commodity used in the laboratory tests. From the results shown in Fig. 7, equation (6) can be rewritten as: DR = (Y + f l R ) t
(15)
DR = kRt
(16)
or
where y and fl are the slope of the (bee, t) and (o-e, t) plots respectively, t is the time, R is the risk factor defined in equation (7). Hence kR is a composite corrosion rate predicting the rate of growth for the expected deepest pit, which includes the most probable maximum penetration rate, the rate of change of the width of the depth distribution and the risk which the pipeline owner or operator wishes to take. If it is
Prediction of corrosion by statistical analysis of corrosion profiles
315
assumed that the lifetime required for the pipeline is 20 y and that the pipeline operator will tolerate a risk of 0.5% chance of failure, corresponding to a R value of 5.3, then using the experimentally determined values of y and/3 of 1.21 m m y-~ and 0.089 m m y-~ respectively, k R is found to be 1.68 m m y-~. If failure is defined as the first penetration greater than the wall corrosion allowance ( D w ) i.e. D R > D w at failure, then failure will occur with a 0.5% probability after 20 y, with a corrosion allowance of 33.6 mm. It is interesting to compare this result with that calculated using the D e W a a r d formula 7 which gives a corrosion rate of I m m y-1 under the same conditions and an equivalent wall thickness of 20 m m for a 20 y life. This analysis indicates that a significantly higher corrosion allowance may be required to achieve a particular level of risk when account is taken for the uncertainty in depths of attack. Finally it must be emphasised that although the above analysis is based on laboratory corrosion tests, the method could easily be used to assess industrial ultrasonic wall thickness surveys 4 or caliper surveys 5 on plant pipework and vessels. By taking samples of thickness data over the area of interest as a function of time, and fitting the smallest values of wall thickness to an extreme value model, it should be possible to determine a 'service' value for the statistical corrosion rate, which could then be used in conjunction with equation (16) to determine the remaining wall thickness, and hence the remaining life of plant. CONCLUSIONS 1. Statistical modelling of corrosion profiles provides a basis for more precisely presenting corrosion rate information when the surface profile is non-uniform. E x t r e m e value models can be used to successfully predict corrosion when the full depth distribution cannot be fitted to any of the well known statistical models. 2. The use of surface profiling instruments, together with statistical analysis, are rarely used in corrosion studies, but could become extremely useful for morphological corrosion studies, adding to the range of techniques available to the corrosion scientist. 3. The experiments in this paper have shown that the time dependence of the statistical p a r a m e t e r s is important for any prediction exercises and can give clues to the mechanism of corrosion. The Type I extreme value model, with linear timedependent parameters, has been shown to model successfully the corrosion of steel in CO2-acidified sea water (sweet corrosion) and is a useful model for predicting the lifetime of steel components in carbonic acid at temperatures of about 20°C. REFERENCES 1. J. R. NICHOLLSand P. HANCOCK,High Temperature Corrosion, NACE-6, pp. 198-210. San Diego (1983). 2. E.J. GUMBEL,Statistics of Extremes. Columbia University Press, New York (1966). 3. K.V. BURY,Statistical Models in Applied Sciences. John Wiley, New York (1975). 4. L. E. W~LDE and T. S~Aw~ ~n-line M~nitoring of C~ntinu~s Pr~cess Plant (ed~ D. W` BUTcHER). El~is
Horwood Ltd, Westergate (1983). 5. G. G. ELDRIDGE, Corrosion 13, 1 (1957).
6. J. E. STRUT~r,L. ANWARand B. S. HOCKENHULL,Proc. Conf. on Control and Exploitation of the Corrosion of Aluminium Alloys, Cranfield. Cheneys Ltd, Banbury (1983). 7. C. DEWAARDand~D. E. WILLIAMS.1st Int. Conf. on Internal and External Corrosion of Pipelines, 9-11 Sept, University of Durham (1975),
0010-938X/85 $3.00 + 0.00 Pergamon Press Ltd.
THE PREDICTION OF CORROSION BY STATISTICAL ANALYSIS OF CORROSION PROFILES J. E. STRUTr, J. R. NICHOLLS and B. BARBIER School of Industrial Science, Cranfield Institute of Technology, Cranfield, Bedford, U.K.
Abstract--The corrosion of carbon manganese steel in CO2-acidified sea water has been assessed by a statistical analysis of corrosion profiles. The total population of depth data was multi-modal and could not be fitted to any well known statistical models. However the deepest pits in the profile followed a type I extreme value distribution. The time dependence of the extreme value distribution parameters has been determined and used to predict the rate of penetration of steel components under pitting conditions. INTRODUCTION
AN I~IPORTANTmethod of preventing corrosion failures on a chemical plant is the provision of a corrosion allowance on the thickness of materials used in the plant construction. The size of the corrosion allowance should normally be based on the anticipated service corrosion rate and the required life. Data for such calculations are frequently derived from corrosion tests involving either electrochemical polarization or weight loss measurements. A corrosion allowance may be calculated from such measurements by converting the raw data to equivalent section loss per unit time and extrapolating to the required lifetime. To carry out this conversion, it is necessary to assume that the corrosion is uniform, hence any variation in the depth of metal lost, both across samples and from sample to sample, must be ignored. In many cases this variation is not important, particularly if small and associated with some uncontrolled variability in the environment. However, there are applications where the environment is well defined and yet the surface profile is rough and pitted. The uncertainty in the depth of penetration can then be a major factor in determining the time to failure. Under these conditions, it is difficult to define the exact meaning of corrosion rate, when quoted as section loss per unit time, and experimenters often resort to expressing data as mean or maximum penetration rate, with little consideration of the meaning or statistical significance of these parameters. The analysis described in this paper is an attempt to develop a more rational approach to presenting section loss or penetration rate information, when the corrosion process exhibits a rough or pitted surface texture. The method presented provides a means for assessing corrosion allowances at a given level of risk, given sets of thickness data taken over the area of interest over a period of time, and hence may form a basis for assessing the lifetime of corrosion protection coatings or surface treatments.
Manuscript received 6 July 1984. 305
306
J. E. STRUTT.J. R. NICHOLLSand B. BAaaIER DR I
Remaining metal ( Vu )
Corroded metal ( Vc )
A2
(a)
(b)
°i
/•f(z)
l~e
Roe
FIG. ]. Schematic representation of corrosion in terms of statistical models. (a) Corrosion profile, (b) cumulative distribution function, F(z), plotted against depth z, and (c) probability density function, f(z), and extreme value probability density function, f=(ze), plotted against depth ze.
The statistical assessment of metal loss due to corrosion It is possible to characterize the depth profile of a corroded surface by a probability density function f (z), or its integral, the cumulative distribution function F(z), determined from metal loss measurements. The cumulative distribution function (cdf) represents in physical terms the probability that an observed value of depth is less than or equal to a given depth value (z).
Prediction of corrosion by statistical analysis of corrosion profiles
307
It is possible to show that an exact relationship exists between the cumulative depth distribution function, and the total volume of metal lost, (Vc), due to a corrosion process as illustrated in Fig. 1. By integrating the depth of metal lost, z(A), within an element of area dA, over a total sample area Ao, it can be shown that,
Vc = ( z(A) dA. J Ao Integrating equation (1) by parts results in equation (2),
(1)
where Dw is the original thickness of the corroding section. The volume of metal remaining (11.) is thus given by equations (3) and (4):
V. = AoDw - V~ i.e.
Vu = Ao
f0DwF(z) dZ.
(3) (4)
Equations (2) and (4) can be interpreted graphically as shown in Fig. 1, where Fig. la is an amplified representation of a typical corrosion profile. From this, the corresponding cdf for the profile can be derived and is illustrated in Fig. lb. In practical terms, the cdf is obtained by taking a set of depth data, sorting them into order of increasing depth and then plotting depth (zi) against i/(n + 1), where n is the total number of depths sampled and i is the rank of the depth measurement in the ordered list of increasing depths. The parameter i/(n + 1) is the cdf of the sample and defines the boundary between the metal and the environment in terms of its probability of occurrence. The shaded area A~ is proportional to the volume of metal corroded, Vc whilst area A 2 is proportional to the volume of metal remaining, Vu. Equations (2) and (4) are theoretically valid for a wide range of corrosion morphologies with open pits and generally roughened surfaces, but excludes those morphologies which show subsurface lateral attack, such as intergranular corrosion, for example. Although equations (2) and (4) accurately represent many observed corrosion profiles, their use in practice is often limited by the need to fit the experimental depth data sample to a known statistical model such as the normal, log normal or Weibull distribution, before predictive modelling can be undertaken. However, as expected, many corrosion profiles do not fit any of the commonly used statistical models, particularly when a combination of corrosion processes operate at the same time, giving multi-modal distributions. Under such circumstances the penetration probabilities must be calculated graphically or some other approach taken. One alternative which may be adopted when the sampled data do not conform to any of the well known statistical models, is to fit the maximum tail of the pdf using an extreme value distribution. It is possible to fit the fraction of data corresponding to the deepest penetrations to a type I extreme value model, provided that the maximum tail of the depth probability distribution function (pdf) decreases at least as rapidly as an exponential function. 2 The use of the type I extreme value model
308
J.E. STRUTr,J. R. NICHOLLSand B. BARBIER
offers the advantage that the shape of this distribution can be written as an explicit equation, both in the pdf and cdf form, and hence this facilitates the development of statistical equations which can be related to the corrosion rate. The cdf of a type I extreme value function is given by: FI(Ze; /.re, Ore) = exp
(
-exp
tre
]
(5)
where ze is the extreme value of the depth,/~e is the location parameter and tre is the scale parameter of the extreme value distribution. Equation (5) has a physical interpretation; 1 - F(z~) is the probability of obtaining a maximum depth measurement greater than some value of depth (z~). Consequently, 1 - Fi(z~) is a measure of the risk of accepting a given value of z~ as the maximum penetration. The physical significance of equation (5) in terms of corrosion measurements can be more clearly seen by taking a specific case of equation (3). Suppose that it is required to define a depth, DR (see Fig. 1), for which the risk of accepting D R as the maximum penetration is a, where a = 1 - FI(DR). Then from equation (5) one obtains: DR = tzc + treR (6) where R = - I n ( - I n (1 - a)). (7) Equation (6) shows that the depth DR, the maximum penetration depth for a risk of value ot is simply the most probable maximum penetration,/x~, plus a term tr~R. The term tr~R is a safety factor which depends on the shape (width) of the extreme distribution. Thus R can be thought of as the number of ore values which must be added to the most probable maximum depth (/xe) in order to achieved the required level of risk. For example, for a very low risk (a = 10-3), approximately 7o'~ must be added qm to/z~, whilst for a very high risk (or = 0.1), only 2.25tr~ need be added. CorrestJonding R values for typical ot values are shown in Fig. 2. The relationship between the full distribution of depth data and its extreme value distribution is shown schematically in Fig. lc. The depths corresponding to the shaded upper tail of the pdf for the complete population of depths are treated as extreme depth data, and hence are fitted to the extreme value pdf, fl(Ze). If the pdf for the complete population of depth data can be fitted to a normal distribution, then mathematical relationships between the mean (/xN) and standard deviation (O'u) parameters of the normal distribution and the location (/xe) and scale (ore) parameters of the extreme value distribution can be used to link the maximum depth estimate, D R, with the mean metal loss, as shown below. 2 The relationships between the parameters of normal distribution and those of extreme value distribution 3 are given by: /Xe ----//'N + A'OrN o-¢ = B'/z N
(8) (9)
where A' is given by: A' = 2 1 n n - ½ 1 n l n n ~/2 I n n
ln(2V~)
(10)
Prediction of corrosion by statistical analysis of corrosion profiles
309
lO
o!
10-4
i
10-3
t
10-2
i
N
10-1
Risk (a) FIG. 2.
Risk factor R plotted against risk value ~, where R = - I n (1 - a ) , o~ = 1 - F l ( D n ) and D R is a specific value of depth.
and B' by: B' -
1 X/2 In n
(11)
and where n is the number of depth data points sampled. Consequently, equation (6) can be rewritten in terms of the parameters for the normal distribution by substituting equations (8) and (9) into (6) to give: D R = I~N + R n o v N
(12)
Rn = A ' + B ' R .
(13)
where
Values of A ' and B ' a s a function of the number of data points sampled (n), are shown in Fig. 3. For 100 values of depth, A' is approx. 2.4, and B' is approx. 0.33, so for a 1% risk in estimating the extreme corrosion (a = 10 - 2 , R = 4.6), Rn has a value of 3.9. This implies that the estimated extreme thickness loss, DR, at 99% confidence, is the mean depth plus about four standard deviations of the scatter in depth values. Equations (6) and (12) are of more practical value than equations (2) and (4), and can be used to estimate corrosion allowances and remaining life of pipe work or coatings at some specified level of risk of penetration on a rational basis. Industrial uses of these equations could include the analysis of ultrasonic wall thickness surveys4 and caliper surveys.5 However, the techniques can also be used in laboratory corrosion testing, by measuring metallographic sections or more rapidly using surface profiling instruments. 1.6 The application of this method in laboratory corrosion testing is illustrated in the remainder of this paper, for an experimental programme aimed at assessing the corrosion of carbon steels in CO2-acidified sea water.
310
J.E. STRUTT, J. R. NICHOLLS and B. BARBIER 6.0
A ~
5.0
4.0 -
A'
-1.0 0.8
3.0
2.0
B'
=
1.0
0
"-
10
10 2
10 3
-
10 4
0.40"6 B'
~
10 5
°
-
~
10 6
-
~_0.2 0 10 7
Number of data points (n)
FIG. 3.
Values of the statistical parameters A' and B' plotted against the number of depth data points sampled.
EXPERIMENTAL METHOD Carbon steels corresponding to BS 4360 50D were used for the experimental work. A sample of steel was first ground on a mechanical grinding machine and then cut into samples of size 10 × 38 × 48 mm 3. The samples were immersed in a flowing sea water cell. The sea water was prepared from Spanish sea salts, de-aerated and saturated with 1 atmosphere pressure of CO2 at 25°C. The flow rate was set at 0.4 m s -~ and CO, gas was bubbled through the reservoir solution continuously throughout the seven week (1176 h) experiment to simulate a sweet corrosion environment] After an initial period of two weeks (336 h), the first sample was removed and subsequently, samples were removed at one week (168 h) intervals. Following exposure to the corrosive environment, samples were cathodically descaled and cleaned ready for surface profiling. The depth of corrosion was measured using a Rank-Taylor-Hobson Talymin type 410, with a stylus tip radius of 3 #m. Reference surfaces were provided by coating the edges of the specimens with a non-conducting epoxy resin. Within the time scales of the experiment, no underfilm or crevice corrosion occurred, leaving a clean flat reference surface from which to estimate metal losses across the samples. The output from the Talymin was displayed on a strip recorder. This data was then measured and fed into a Commodore PET Microcomputer for further data analysis. Five traces were taken across each sample and depths were sampled at 200 ~m intervals, giving 225 depth data points per specimen. EXPERIMENTAL
RESULTS AND
DISCUSSION
The pdf for the complete population of depth data was obtained by first ordering the data into increasing depth and dividing them into cells, to produce the frequency histograms shown in Fig. 4(a) for each specimen. Attempts to fit these data sets to well known statistical models such as normal or log normal distributions were quickly abandoned when no single model was found to be representative of the six samples tested, and bimodai distributions were found for some of the specimens. However, the shape of the maximum tail of all the distributions suggested that an extreme value model might be used to successfully model the deeper pits.
Prediction of corrosion by statistical analysis of corrosion profiles
311
313
30
r = 336 h
t = 336 h 20 10
10 I
0 30
I
f 30 t = 504 h
t = 504 h 20
20
~
10
10
I
A
I
g r.-
t = 672 h
o
=
I
0 30
2
0
~
t = 672 h 'r"
2O
"0
== ~
30-
~
2o-
o
10 F
t = 840 h
10
t = 840 h
t~
20 0
10
8 8
oi
i
u-
t = 1008 h
t = 1008 h
LU
20
2O
10
10
0 30
t = 1176h
0 30
20
20
10
10 1O0
FIG,
0 30
4.
200
0
t = I176h
1O0
200
Depth (z/lO-6m)
Depth (ze/lO-6m)
(a)
(b)
Variation of depth frequency distributions with time during corrosion. (a) Complete depth distribution, (b) extreme depth distribution.
312
J.E. STRU'FI',J. R. NICHOLLSand B. BARBIER 5
/
¢ e-
7
,q
,.../
II ¢:
."
40
1
60
I
80
I
I O0
f
I
120
140
f
l
I
160
180
I
200
I
220
Depth of penetration (ze/10 -s m) FJc. 5.
V a r i a t i o n of the e x t r e m e v a l u e f u n c t i o n - i n
(-in
(F(ze))
with e x t r e m e d e p t h
zc(m). For this extreme value analysis, the hundred deepest pits were selected from the full data set. This data was then ordered and divided into cells to produce the extreme depth frequency histograms shown in Fig. 4b. Fitting the data to the extreme value function by plotting - I n ( - I n Fi(ze)) against ze gives a slope of 1/ore and an intercept at - I n ( - I n (F0) = 0 of/ze. Figure 5 shows the experimental results plotted in this manner, while Fig. 6 shows the theoretical type I extreme value functions calculated using the best fit values o f / ~ and Orefrom Fig. 5. It is interesting to note that not only does the most probable maximum depth (location parameter) increases with time, but also the width of the distribution (shape parameter) increases with time. The time dependence of these parameters is shown most clearly in Fig. 7. In both cases, a linear dependence is observed. The ratio of/ze to Oreis thought to be a characteristic of the morphology of attack, and for the conditions of this experiment is time-independent with the value: /xe
_
13.5.
(14)
Ore
The results of this statistical analysis show that the full distribution of depths of penetration, due to the corrosion process modelled in these experiments, does not fit any of the commonly used statistical distributions. During the early stages of the corrosion process, a bimodal distribution was observed, indicating that two distinct depth levels existed within the corrosion profile.
Prediction of corrosion by statistical analysis of corrosion profiles
313
.o_ c 25 t = 336 h J= "E
t = 504 h
20 "Io
15
=
e-
/ ' ~ = 672 ~ = 8 4 0 h
"O
t = 1008h
E
~
5
h-
50 LU
100
150
200
Extreme corrosion depths (ze/lO-6rn)
FIG. 6. Variation of extreme depth frequency distribution with time, after fitting data to equation (5) of the text. This result, although at first surprising, was not entirely unexpected. Sweet corrosion pits often exhibit a mesa or table-like m o r p h o l o g y . Whilst the mesa pits o b s e r v e d in service are on a m u c h larger scale than those g e n e r a t e d on the laboratory samples, the m o r p h o l o g i e s o f the latter did have a similar a p p e a r a n c e on a microscopic scale when o b s e r v e d using a stereoscopic microscope, T h e results during the initial stages suggest that pits nucleate on the surface, probably at sites where the c a r b o n a t e film has b r o k e n down. E a c h nucleated pit has a particular depth, and p r o p a g a t e s by lateral g r o w t h to expose a new passive surface on which new pits can 200
o
150
2 E
~
8O
100
50
._.1
250
500
750
1000
1250
t(h) FIG. 7.
Variation of extreme value frequency distribution parameters with time of corrosion. • #c, ~ (re.
314
J . U . STRU'IT, J. R. NICHOLLS and B. BARB1ER
subsequently nucleate and grow. With increasing time, pits nucleate in freshly exposed surfaces and the distribution eventually gives way to a rather broad flat distribution made up of many sub-distributions. The evolution of the distribution with time is interesting. The location parameter for the extreme distribution (bee) increases with time in an approximately linear manner, indicating that the most probable maximum corrosion rate is constant. Furthermore, the scale parameter o-e, which describes the width of the distribution, also increases linearly with time. Further consideration of the nature of electrochemical corrosion processes indicates that a time-dependent a e is not entirely unexpected. An ideally uniform corrosion process, corresponding to a flat surface profile, would result in a very narrow pdf giving a very high value for the ratio of bee to o-e. This distribution would shift to increasing values of ze with no broadening of the distribution, as the corrosion process proceeds for as long as the corrosion morphology remains 'uniform'. However, this can only occur if anodic and cathodic sites are continually exchanging in a random manner with time. In practice, the great majority of corrosion systems acquire a pattern in the spatial distribution of anodic and cathodic sites, which is frequently 'set' for long period of time, resulting in localized metal loss. This can, for example, arise from microstructural factors, such as anodic and cathodic microphases. Alternatively, it may be related to local spatial variations in cathodic constituents which may be slow to change in the time domain. The result, in either case, is that metal is lost at persistent anodic sites at a faster rate than at persistent cathodic sites, causing a broadening of the distribution of depths with time. It is perhaps surprising, however, that both be~ and o-~ were linear with time, indicating that autocatalytic mechanisms were not operative during this particular experiment. Apart from the potential use of the techniques described above for fundamental morphological studies of corrosion processes, it is considered that the main application for statistical analysis of corrosion profiles is to improve the ability to predict corrosion degradation. The material used in this study is not a pipeline steel and the corrosion test falls short of an accurate simulation of the type of sweet corrosion process experienced offshore. Since corrosion failure is associated with the rate of growth of deep pits, the extreme value distribution for deepest pits can be used to predict the likelihood of failure. In order to demonstrate this approach, the data generated in this study will be used to make an assessment of the corrosion allowance that would be required on a new pipeline carrying the commodity used in the laboratory tests. From the results shown in Fig. 7, equation (6) can be rewritten as: DR = (Y + f l R ) t
(15)
DR = kRt
(16)
or
where y and fl are the slope of the (bee, t) and (o-e, t) plots respectively, t is the time, R is the risk factor defined in equation (7). Hence kR is a composite corrosion rate predicting the rate of growth for the expected deepest pit, which includes the most probable maximum penetration rate, the rate of change of the width of the depth distribution and the risk which the pipeline owner or operator wishes to take. If it is
Prediction of corrosion by statistical analysis of corrosion profiles
315
assumed that the lifetime required for the pipeline is 20 y and that the pipeline operator will tolerate a risk of 0.5% chance of failure, corresponding to a R value of 5.3, then using the experimentally determined values of y and/3 of 1.21 m m y-~ and 0.089 m m y-~ respectively, k R is found to be 1.68 m m y-~. If failure is defined as the first penetration greater than the wall corrosion allowance ( D w ) i.e. D R > D w at failure, then failure will occur with a 0.5% probability after 20 y, with a corrosion allowance of 33.6 mm. It is interesting to compare this result with that calculated using the D e W a a r d formula 7 which gives a corrosion rate of I m m y-1 under the same conditions and an equivalent wall thickness of 20 m m for a 20 y life. This analysis indicates that a significantly higher corrosion allowance may be required to achieve a particular level of risk when account is taken for the uncertainty in depths of attack. Finally it must be emphasised that although the above analysis is based on laboratory corrosion tests, the method could easily be used to assess industrial ultrasonic wall thickness surveys 4 or caliper surveys 5 on plant pipework and vessels. By taking samples of thickness data over the area of interest as a function of time, and fitting the smallest values of wall thickness to an extreme value model, it should be possible to determine a 'service' value for the statistical corrosion rate, which could then be used in conjunction with equation (16) to determine the remaining wall thickness, and hence the remaining life of plant. CONCLUSIONS 1. Statistical modelling of corrosion profiles provides a basis for more precisely presenting corrosion rate information when the surface profile is non-uniform. E x t r e m e value models can be used to successfully predict corrosion when the full depth distribution cannot be fitted to any of the well known statistical models. 2. The use of surface profiling instruments, together with statistical analysis, are rarely used in corrosion studies, but could become extremely useful for morphological corrosion studies, adding to the range of techniques available to the corrosion scientist. 3. The experiments in this paper have shown that the time dependence of the statistical p a r a m e t e r s is important for any prediction exercises and can give clues to the mechanism of corrosion. The Type I extreme value model, with linear timedependent parameters, has been shown to model successfully the corrosion of steel in CO2-acidified sea water (sweet corrosion) and is a useful model for predicting the lifetime of steel components in carbonic acid at temperatures of about 20°C. REFERENCES 1. J. R. NICHOLLSand P. HANCOCK,High Temperature Corrosion, NACE-6, pp. 198-210. San Diego (1983). 2. E.J. GUMBEL,Statistics of Extremes. Columbia University Press, New York (1966). 3. K.V. BURY,Statistical Models in Applied Sciences. John Wiley, New York (1975). 4. L. E. W~LDE and T. S~Aw~ ~n-line M~nitoring of C~ntinu~s Pr~cess Plant (ed~ D. W` BUTcHER). El~is
Horwood Ltd, Westergate (1983). 5. G. G. ELDRIDGE, Corrosion 13, 1 (1957).
6. J. E. STRUT~r,L. ANWARand B. S. HOCKENHULL,Proc. Conf. on Control and Exploitation of the Corrosion of Aluminium Alloys, Cranfield. Cheneys Ltd, Banbury (1983). 7. C. DEWAARDand~D. E. WILLIAMS.1st Int. Conf. on Internal and External Corrosion of Pipelines, 9-11 Sept, University of Durham (1975),