f

distribution, iron, nucleation, pit depth, prediction, pitting

INTRODUCTION

As industrial and infrastructural systems (refineries,power plants, pipelines, etc.,) age, considerableeconomic incentive develops to avoid unscheduledoutages and to extend operation beyond the design

age into the future as a function of various systemvariables. The most effective prediction technologiesare those based on determinism, in which the systembehavior is described in terms of the natural laws.

Corrosion is a major cause of component failureand, hence, in the occurrence of unscheduled down-time in complex industrial systems. In particular, thelifetime. Avoidance of unscheduled outages is of par-ticular interest because the failure of even a minorcomponent can result in the complete shutdown of afacility. For example, the unscheduled shutdown of a1,000-MW nuclear power plant may cost the operatorbetween $1 million and $3 million per day, depend-ing upon the cost of replacement power and otherfactors. However, if component failures could be pre-

various forms of localized corrosion, including pittingcorrosion, crevice corrosion, stress corrosion crack-ing (SCC), and corrosion fatigue, are particularlydeleterious because they frequently occur withoutany outward sign of damage and because they oftenresult in sudden and catastrophic failures. Thus, thedevelopment of effective localized corrosion damageprediction technologies is essential for successfulDeterministic Prediction o

G. Engelhardt and D.D. Macdonald*

ABSTRACT

In this paper, the deterministic prediction of damage viadamage function analysis (DFA), which provides a robusttechnology for estimating the damage function at futuretimes, is described. General analytical expressions for calcu-lating the damage functions that can be used with arbitrarydependencies of pit nucleation rate, pit growth rate, andrepassivation rate on the system parameters have beenobtained. The application of DFA to prediction of pitting dam-age is illustrated by reference to the pitting corrosion of ironin neutral chloride-containing solutions.

KEY WORDS: chloride, damage function analysis,CORROSIONVol. 54, No. 6

Submitted for publication January 1997; in revised form,September 1997.

* Center for Advanced Materials, Pennsylvania State University, 517Dieke Bldg., University Park, PA 16802-2714.

dicted accurately, maintenance could be performedduring scheduled outages, the cost of which alreadyhas been built into the price of the product. With

0010-9312/98/00010 1998, NACE ICORROSION ENGINEERING SECTION

Pit Depth Distribution

regard to life extension, the successful extension ofoperation beyond the design life translates into en-hanced profits and the avoidance of costly licensingand environmental impact assessments associatedwith construction of a new facility. However, in thiscase as well, the key to successful operation is theability to avoid downtime and maintain production.Eventually, the frequency and severity of unsched-uled outages render continued operation uneco-nomic, and at that point, replacement of the facilityis necessary. To develop effective inspection andmaintenance scheduling and life extension technolo-gies, it is necessary to predict the evolution of dam-469

avoidance of unscheduled downtime and successfulimplementation of life extension strategies.

Corrosion damage may be extrapolated to futuretimes using damage tolerance analysis (DTA). In thisstrategy, known damage is surveyed during eachsubsequent inspection, and the damage is extrapo-lated to the next inspection period allowing for asuitable safety margin. It has been argued that this

5/$5.00+$0.50/0nternational

CORROSION ENGINEERING SECTIONstrategy is inaccurate, inefficient, and, in manyinstances, too conservative.1 Instead, it has beenproposed that damage function analysis (DFA) is amore effective method for predicting progression ofdamage, particularly when combined with periodic

FIGURE 1. Pit depth distribution for type 403 SS (After Ishikawa,et al.2): (a) initial distribution and (b) after 5 years.470

(1) UNS numbers are listed in Metals and Alloys in the UnifiedNumbering System, published by the Society of AutomotiveEngineers (SAE) and cosponsored by ASTM.

inspection. DFA is based upon deterministic predic-tion of the rates of nucleation and growth of damage,with particular emphasis on compliance of theembedded models with the natural laws. Althoughcorrosion generally is complicated mechanistically, ahigh level of determinism has been achieved in vari-ous treatments of general and localized corrosion.

In the present work, general analytical expres-sions for calculating damage functions, which can beused with any models or algorithms for estimatingthe pit nucleation rate, pit growth rate, and repassi-vation rate as a function of various systemparameters, have been developed. The application ofDFA is illustrated by reference to the development ofdamage resulting from pitting corrosion of iron inneutral sodium chloride (NaCl) solutions as a func-tion of exposure time.

THEORETICAL BASIS OF DFA

The damage function (DF) frequently is definedas the histogram of event (i.e., number of pits perunit area) versus the cumulative depth increment. Toillustrate a typical damage function, Figure 1 showsexperimentally determined DF as reported byIshikawa, et al., for type 403 SS (UNS S40300)(1)

initially and after being buried in the ground for5 years.2 The DF is a distribution and must beinterpreted in terms of nucleation, growth, andrepassivation phenomena. Briefly, in the absence ofa distribution in growth rate, the deepest pit is thatwhich nucleated earliest (but not necessary uponinitial exposure) and, hence, is that which ultimatelymay result in failure. Because a finite nucleation ratedoes not necessary imply the nucleation of a pitwhen conditions become conductive to attack andsince the rate of repassivation of pits may be suchthat those which nucleate first may have a negligibleprobability of survival, it is important that any com-prehensive theory for the development of damageaddress the nucleation, growth, and repassivationphenomena in mechanistic form.

The purpose of the present work was to developthe mathematical basis of DFA in a manner thatretains flexibility with regard to individual models fornucleation, growth, and repassivation and yet isentirely analytical.

Prior to developing the theoretical basis of DFA,it is first convenient to introduce various terms thatare used throughout the analysis. Thus, tobs is theobservation time, (s); t is induction time for a singlepit (s), u = tobs t and is the age of a pit (s); L is thepit depth (cm); Lmax (tobs) = maximum pit depth of pitsobserved at a given observation time, tobs; f (L, tobs) =differential damage function (no./cm3); and:

F L, tobs = *

L

Lmax

f L , tobs dL = integraldamagefunction no./cm2

From the definition of the damage function, f (L, tobs)dL is the number of pits (per cm2) with depths be-tween L and L + dL at an observation time tobs. Theintegral damage function, F (L, tobs), yields the num-ber of pits (per cm2) with depths between L and Lmax(larger than L) at t = tobs. In the initial moment ofobservation (tobs = 0), there are no pits on the metalsurface (i.e., F [L, tobs] = 0 for tobs 0). It also isevident that the differential and integral damagefunctions satisfy the relations:

f L, tobs = F L, tobs

L (1)

and

F 0, tobs = *o

Lmax

f L, tobs dL = N tobs (2)

2CORROSIONJUNE 1998

where N(tobs) is the number of pits (per cm ) that havenucleated prior to tobs.

The main aim of this present task is to calculatehow many active and passive pits (per cm2) havedepths between L1 and L2 (L1 L2) for a given obser-vation time, tobs. This number is denoted by D N(L1, L2,tobs), which is given by:

D N L1, L2, tobs = *L1

L2

f L, tobs dL = F L1, tobs F L2, tobs (3)

Now suppose that the following three functions areknown (can be calculated):

Depth of a Pit of Age u, or L = L(u) It can beassumed that the inverse function:

q L = tobs t (4)

exists and has the derivative:

q

L=

1

v L (5)

where v(L) is the growth rate of a pit that has a depthL. Strictly speaking, just the functions u(L) and v(L)(but not L[u]) are necessary for calculating the dam-age function for the system. The function v(L) iscalculated from the current density in a pit of depth L,and u(L) is calculated as the solution of Equation (5):

q =dL

v L0

L

(6)

In many cases, L(u) can be calculated with sufficientaccuracy by the simple relation:3-4

L q = kq m (7)

and, correspondingly:

q L =Lk

1m

(8)

where k and m are constants. Accordingly:

v L = kmkL

1 mm

(9)

Velocity of Pit Nucleation The second function,which is assumed to be known, is the velocity of pitCORROSIONVol. 54, No. 6

nucleation: n(t) = dN/dt. By definition, n(t)dt is theCORROSION ENGINEERING SECTION

number of pits (per cm2) that nucleate in the timeinterval between t and t + dt. Likewise, the function:

N t = n t0

t

dt (10)

is the number of pits (per cm2) that nucleate in thetime interval between 0 and t. In accordance with thepoint defect model for passivity breakdown:5-9

n t = A

exp at

+ b2

t

2 (11)

where:

a =x

2s DB (12)

b =Jm2s DB

D2s D

(13)

and

B =c Fe NAax

c /2

W RTexp

c F b pH + a Ecorr + 2w

2RT (14)

where x is the stoichiome by (MOx/2), F~ is Faradays

constant, e is the electric field strength in the barrierlayer, NA is Avogadros number, ax is the activity ofthe aggressive species that induces breakdown, W isthe volume of the barrier layer per mole of cation, Ris the gas constant, T is the Kelvin temperature, b isthe dependence of the voltage drop across the barrierlayer/solution interface on the pH, a is the polariz-ability of the same interface, Ecorr is the corrosionpotential, and w is an energy related to the absorp-tion of the aggressive species into oxygen vacanciesat the barrier layer/solution interace. Parametervalues used in determining a, b, and B for SS aresummarized in Table 1. A detailed discussion ofthese parameters has been published previously.5-9

For the present purpose, it is important that param-eters a and b depend upon temperature, pH, activityof the halide ion (ax), and the corrosion potential(Ecorr) but not on the induction time (t). Parameter Aalso does not depend on t, so that normalization ofthe nucleated pit population using the condition:

471

N = n0

t dt = N0 (15)

CORROSION ENGINEERING SECTION

LElcu

e (

wn

ondn isn.requires that:

A = N0/

exp at

+ b2

t

2dt =

N02a

p erfc b

0

(16)

Here, N0 is the maximum number of pits (per cm2)

TABValues of Parameters Used in Ca

Parameters

x Barrier layer stoichiometry at metal-film interfacW Mole volume of the oxide per catione Electric field strengthw = D GA1 Gibbs energy function+ D Gs0 F f f0/sJm Critical vacancy fluxz Critical concentration of vacanciesD Mean vacancy diffusion coefficientsD Standard deviation for vacancy diffusivitya Constantb ConstantNA Avogadros number

(A) Corresponds to a current density prior to breakdown but at the breakdovacancies.

(B) Calculated from structural considerations. Thus, if one vacancy cconcentration is 1/(2.866 x 108) cm2 = 1.22 x 1015 cm2, noting that iroformation, z = 1.0 x 1015 cm2 is taken as a reasonable approximatio472

that can exist on the metal surface, and erfc(x) is thecomplementary error function. Accordingly, fromEquations (10), (11), and (16):

N t =N0

erfc berfc

at

+ b (17)

Probability of Survival The last function thatmust be known to determine DF is the probability ofsurvival, P(u). Pit repassivation is assumed to obey afirst-order decay law, which yields:1

P q = e gq = e g tobs t , tobs > t (18)

where g is the repassivation constant. By definition,n(t)P(u)d(t) is the number of pits (per cm2) that nucle-ate in the time interval between t and t + dt and thatare still alive at t = tobs.

The integral damage function, F(L,tobs), then canbe calculated. If pits do not repassivate, only thosethat nucleate with induction times between 0 andt = tobs u(L) have a depth between L and Lmax (for agiven observation time, tobs). This means that theintegral damage function, in the absence of repassi-vation, Fg=0(L, tobs), has the form:

Fg =0 L, tobs = n t dt

0

tobs q L= N tobs q L (19)

From the definition of the function P(u) given above,the probability of a pit not reaching the age u = tobs t(and, hence, achieving the corresponding depth L) is

1lating Velocity of Pit Nucleation

Value Units Source

FeO) 2 Assigned12.61 cm3/mol From density106 V/cm Assigned52 x 103 J/mol Assigned

3.12 x 1012 no./cm2 s (A)1015 no./cm2 (B)1019 cm2/s (7)0.5 x 1019 cm2/s (7)0.5 Assigned0.01 V (4)6.023 x 1023 no./mol

voltage of 1 m A/cm2, assuming that all of the current is carried by cation

enses on each metal atom at the metal-film interface, the critical body-centered cubic. Allowing for expansion of the lattice due to oxideCORROSIONJUNE 1998

1 P(u[L]). Accordingly, the number of pits (per cm2)that nucleate with an induction time between 0 andt = tobs u(L), and that repassivate before reachingthe depth L, Fp (L, tobs), is:

F p L, tobs = 1 P q L0

tobs q L

n t dt

= 1 P q L n t dt0

tobs q L

(20)

It is evident that the integral damage function,F(L, t) is the difference between Fg=0(L, t) and Fp(L, t),i.e.:

F L, tobs = P q L n t dt = P q L N tobs q L0

tobs q L

(21)

The differential damage function, f (L, tobs), then

can be obtained easily from Equation (21) bydifferentiation:

f L, tobs = F L

= P q L n tobs q L Pq

N tobs l L /v L

(22)

Equations (21) and (22) are the sought-after integraland differential damage functions, respectively.

It is possible to define differential and integraldamage functions for the active and passivated pitsseparately. Thus, the integral damage function forthe active pits, Fa(L, t), is given by:

Fa L, tobs = P tobs t n t dt0

tobs q L (23)

and for passivated pits is:

Fp L, tobs = F L, tobs Fa L, tobs (24)

Accordingly, the differential damage function foractive pits is:

fa L, tobs = Fa L, tobs

L= n tobs q L P q L /v L (25)

and for passivated pits, the differential damagefunction is:

fp L, tobs = f L, tobs fa L, tobs (26)

The usefulness of Equations (24) and (26) at firstmight be questioned because, from experimentalmeasurements, the total number of pits is deter-mined without subdividing this number into activeand passivated entities. However, the ability to exam-ine the active and passivated population separately iscrucial for devising damage mitigation strategies.

The equations for the damage functions simplifyif the pits cannot repassivate. In this case, P = l(since g = 0) and, for the integral damage function:

F L, tobs = Fg =0 L, tobs = N tobs q L (27)

Correspondingly, for the differential damage function:CORROSIONVol. 54, No. 6

j L, t = j a L, tobs = n tobs q L /v L (28)CORROSION ENGINEERING SECTION

It is important to note that, in this case, the inte-gral damage function, F(L, t), does not depend on thevariables L and tobs separately, but depends on thesingle variable tobs u(L). This means that pit distri-butions do not have to be calculated for differentobservation times. Instead, all pit distributions for agiven set of conditions can be obtained from a singlecurve F(tobs u[L]).

As follows from the theory described above, calcu-lation of the pit distribution reduces to estimating theintegral damage function from Equation (21). If thedepth of a pit, as a function of pit age, is known in ana-lytical form, the damage function and the pit distributionalso can be expressed in analytical form. Thus, if Equa-tion (7) holds, from Equations (8), (17), (18), and (21):

F L, tobs =

N0

erfc bexp g L/k

1/merfc

a

tobs L/k1/m

+ b

(29)

Correspondingly, for the differential damage function:

f L, tobs = F

tobs=

N02a

p erfc b

exp g L/k1/m

exp a

tobs L/k1/m

+ b

2

tobs L/k1/m

2

1

km

L

k

1 m

m

+ g exp g L/k1/m 1

km

L

k

1 m

m

N0

erfc berfc

a

tobs L/k1/m

+ b

(30)

It is important to note that the direct use of Equa-tions (29) and (30) is impossible for large values of b,because of erfc(b) fi 0 at b fi . In this case, it isconvenient to use the approximation:

erfc x e x

2

p xfor x fi (31)

By substituting this approximation into Equations(29) and (30), the factor eb2 disappears. Hence, theseequations can be used for large values of b.

PREDICTION OF DAMAGE

In accordance with the theory developed, calcu-473

lation of the damage function requires the

CORROSION ENGINEERING SECTION

determination of three independent functions: the

FIGURE 2. Structure of the algorithm for prediction of DF.474

rate of pit nucleation, n(t), the pit growth rate, v(L),and the probability of survival, P(u). Accordingly, thealgorithm for estimating the damage function musthave at least three modules for handling these func-tions, but the operation of the algorithm is quiteindependent of the form of each function. Of course,additional modules are required for determining un-known parameters, such as the composition of thebulk electrolyte, pH, corrosion potential, and soforth.

Thus, the algorithm that was developed previ-ously for estimating the damage functions forcondensing heat exchanger10 contains five modules,as outlined in Figure 2. Also indicated are the param-eters that propagate from one module to the next.The output of the algorithm can be specified in threeforms:

For a specified probability of failure, the algo-rithm estimates the damage function as a function ofexposure time and computes the number of pits withlengths exceeding the condenser wall thickness topredict the service life;

For a specified probability of failure anddesign life, the algorithm calculates the wall thick-ness to ensure acceptable performance; and

For a specified wall thickness and design life,the algorithm calculates the failure probability.

In the present work, only the principles of themodules used in this algorithm are outlined.Chemistry ModuleThe goal of the chemistry module (CM) is to cal-

culate the pH, composition, and conductivity of theexternal environment as a function of the constraints(e.g., gas pressure) and temperature. The pH andchloride activity are key parameters in controlling therates of pit nucleation and pit growth. Also, the con-centrations of all ionic species in the liquid layer([Ck]) determine the ionic conductivity of the solution,which has a great impact on pit growth rate. Themodule uses an equilibrium model, along with massbalance and charge balance constraints, to estimatespecies concentrations and computes ion activitycoefficients using the extended Debye-Huckel theory.Details of a thin-film version of this model have beendescribed in the literature.11

Mixed-Potential ModuleThe mixed-potential module (MPM), which is

based on the Wagner-Traud hypothesis for free corro-sion processes,11 was developed originally to calculateEcorr for SS components in the heat-transport circuitsof boiling water reactors (BWR).12 The theory is basedon the physical condition that charge must be con-served in the system. Because electrochemicalreactions transfer charge across a metal-solutioninterface at a rate measured by the partial currents,charge conservation demands that:

S

j

iR/O,j E + icorr E = 0 (32)

where iR/O,j is the partial current density due to thej-th redox couple in the system, and icorr is the corro-sion current density of the substrate. The currentsare written as functions of the potential (E) toemphasize the fact that the partial currents dependon the potential drop across the metal-solution inter-face. Indeed, the solution to Equation (32) providesthe quantity of Ecorr. In deriving Equation (32), it isassume that the surface of the alloy is equally acces-sible to all charge-transfer reactions in the system.

Pit Nucleation ModuleAs a result of an intensive effort over the past

decade to understand of the breakdown of passivefilms, theoretical distribution functions have beenderived for passivity breakdown. These distributionfunctions are in good agreement with experimentalresults9 and have been derived from the point-defectmodel for the growth and breakdown of passive filmsby assuming that breakdown occurs when a criticalconcentration of cation vacancies accumulates locallyat the metal-passive film interface, such thatdecohesion occurs between the barrier layer and themetal substrate. Subsequently, localized dissolutionof the barrier layer, and/or mechanical instability,CORROSIONJUNE 1998

leads to rupture of the film. It is assumed that break-

down sites normally are distributed with respect tothe diffusivity of cation vacancies within the film. Thefull derivation of these distribution functions can befound in the literature.5-9 The application of thistheory allows for calculation of the rate of pit nucle-ation n(t) as a function of temperature, pH, and theactivity of damaging species (e.g., halide ions) inaccordance with Equations (9) through (13).

Pit Growth ModuleThe pit growth module (PGM) computes the pit

growth rate, v(L), of an individual pit as a functionof its depth. The transport equations are solvedseparately for processes that occur inside a pit ofcylindrical geometry and for the external environ-ment in the form of a thin electrolyte film coveringthe metal surface. Thereupon, these solutions arecombined with the charge conservation constraint toestimate the current and potential distributions and,hence, to estimate the pit growth rate. Details of thetheoretical approach can be found in the literature.10

Pit Repassivation ModuleThis module (PRM) computes the probability of

survival of a pit as a function of its age, P(u). In thesimplest case, when pit repassivation obeys a first-order decay law, this module reduces to theapplication of Equation (18).

Damage Function ModuleThis damage function module (DFM) combines

the PNM with the PGM and the PRM to calculate thedamage function, which is presented mathematicallyin the form of a histogram of the number of pits perunit area versus the pit depth. The module then cal-culates the populations of active (surviving) pits andrepassivated pits, and the distribution of each isderived as a function of the observation time.

RESULTS AND DISCUSSION

The modular structure of the algorithm describedabove renders it very convenient for application to awide variety of systems. Thus, in transforming from onesystem to another, only the component codes need tobe modified, and not the algorithm as a whole. In thiswork, the damage function is calculated for the local-ized corrosion of a metal (iron) in a large volume of theelectrolyte in equilibrium with a specified gas phase.

It has been shown that, if the single electro-chemical reaction of metal dissolution takes placewithin a pit, the growth rate of sufficiently deep pitsis described with great accuracy as a function of timeby Equation (7), with:13

a + 1CORROSIONVol. 54, No. 6

m =2a + 1

(33)CORROSION ENGINEERING SECTION

and

k =2a + 1a + 1

Mm

r m2F

a + 1

2a + 1

3FDmCb

a + 1

2a + 1 ia + 1

2a + 1 (34)

where a^ is the anodic transfer coefficient; and Dm,Mm, and rm, are the diffusion coefficient, the atomicweight, and the density of the metal, respectively,Cb is the total bulk concentration of anions, and i

is

the current density calculated in the absence of apotential drop in the cell (the maximum possiblecurrent density on the electrode surface at the givenpotential of the metal). Accordingly, in this case, thePGM is reduced to the application of Equations (9),(33), and (34).

For generality, normalized, dimensionless distri-butions ( D N*) are calculated instead of the usualdimensioned distribution ( D N). The relationshipbetween these distributions is given by:

D N* =D NN0

(35)

where, as before, N0 is the total number of (potential)breakdown sites on the metal surface. By definition,D N* (Ll, L2, tobs) is the probability that a given pitexamined at tobs has a depth between L1 and L2. Byanalogy, normalized differential and integral damagefunctions are defined, as are the normalized rate ofpit nucleation, and so forth. In any event, the staras the upper index means that the correspondingvalue is normalized, (i.e., divided by N0). All normal-ized values have the probability sense. Thus, it isevident that:

D N* L1, L2, t = F* L1, t F* L2, t (36)

where:

F* L1, tobs = f* L1, L2, tobs =

L1

L2

P q L n* t dt = P q L N* t q L ,0

t q L

(37)

n* t =2a

exp at

+ b2

, (38)475

p erfc b t 2

CORROSION ENGINEERING SECTION

(a)476

and

N* t =1

erfc berfc

at

+ b (39)

For example, from the definition of the normalizeddifferential damage function, f*(L, tobs) dL is the prob-ability that an arbitrary pit, which is examined at thetime of observation tobs, has a depth between L andL + dL.

Figure 3 displays normalized pit distributions forthe corrosion of iron in NaCl solution with the bulkconcentration CNaCl = 0.571 x 103 mol/L, withEcorr = 0.334 V, pH = 7, and T = 25 C, as a functionof the repassivation constant. The parameter valuesused in deriving the damage functions are summa-rized in Table 1. As evident from these plots,introduction of pit repassivation and the value of thepit repassivation constant (g) play an important rolein determining the shape of the damage function. In

(c)FIGURE 3. Influence of the value of g on normalized pit dispH = 7, T = 25 C, tobs = 1 year, and d = 0.025 cm.(b)CORROSIONJUNE 1998

the present treatment of pit repassivation, the totalnumber of pits is a constant (i.e., the number of pitscalculated when considering pit repassivation isequal to the number of pits calculated for the case inwhich pit repassivation is not considered). The pitnumber conservation comes from the understandingthat a pit, once nucleated, either will repassivateprior to, or survive until, the time of observation, andthus always will be counted in deriving the damagefunction, regardless of its fate. Thus, the conserva-tion in pit number also implies that the sum ofsurviving pits and repassivated pits is a constantfor a given set of conditions (potential, pH, [Cl], etc.)and is equal to the total number in the final damagefunction. This total number should not be confusedwith N0, because not all of the potential breakdownsites may have broken down at the time of observa-tion.

Figure 3 also shows that, in the absence ofrepassivation (at g = 0), the damage function consistsof a single increment in depth. This occurs because,

(d)tribution for CNaCl, = 0.571 x 103 mol/L, Ecorr = 0.334 V,

for the selected parameters, all pits nucleate duringan induction time that is much smaller than the timeof observation. The point-defect model yields clearcriteria when such a situation occurs. Thus, asfollows from Equation (38), the maximum value ofthe pit nucleation rate occurs at the induction time:

t max =a b + b2 + 4

2 (40)

Accordingly, if:

t max < < tobs (41)

then practically all of the pits nucleate during a veryshort period of time at the beginning of the experi-ment. For the parameter values given in Table 1,tmax = 162 s, and Criteria (41) holds very well.However, tmax increases very sharply with decreasingNaCl bulk concentration. Thus, for CNaCl = 106 M,tmax equals 0.26 x 108 s and Criteria (41) clearly doesnot hold.

When Criteria (41) is satisfied, it is possible tosimplify the expressions for damage functions. Thus,in this case, the rate of pit nucleation (n*) can beapproximated by the asymmetrical impulse functiond + (t) and, correspondingly, the function N* can beapproximated by the asymmetrical unit function,U + (t):

N* t = U+ t =0 at t 01 at t > 0

(42)

Accordingly, from Equations (36), (37), and (42), itfollows that:

D N* L1, L2, t = exp gq L1 exp gq L2 (43)

if L1 L2 < Lmax, and

D N* L1, L2, t = exp g tmax (44)

for L1, < Lmax < L2, if the pit depth increment,d = L1 L2 satisfies the condition:

L t max 5 yearsand for g 0), all pits repassivate, and furtherincreases in tobs do not change the pit distribution.

It is important to note that, in answering somedesign questions, the pit distribution does not needto be calculated, but only the normalized integraldamage function F*(L, tobs) (Equation [37]) needs tobe estimated. Thus, assuming that Lw is the wallthickness and td is the design life, the failure prob-ability (Pf) can be calculated by setting:

Pf = F* Lw, td (46)

For a specified Pf and td, Lw can be calculated toensure acceptable performance by solving theEquation (46) for Lw. In a similar manner, for a speci-fied Pf and Lw, the service life (ts) can be calculated bysolving Equation (46) for time.

SUMMARY AND CONCLUSIONS477

An algorithm has been developed for predictinglocalized damage functions for corroding systems.

on

(bCORROSION ENGINEERING SECTION

FIGURE 4. Influence of CNaCI on normalized pit distributit = 1 year.

(a)478

The model incorporates methods for defining thechemistry of the bulk environment, for estimating theelectrochemical corrosion potential of the metal orthe alloy, and for mechanistically treating the nucle-ation and growth of pits. General analyticalexpressions for calculating the damage functions,which can be used with any deterministic or empiri-cal models for the pit nucleation rate, pit growth rate,and repassivation rate, have been obtained. Usingthe point-defect model for passivity breakdown, ananalytical model for pit growth, and a first-orderdecay law for repassivation, the influence of the bulkconcentration of the electrolyte, Ecorr, and the repassi-vation constant on the DF has been investigated.Calculations show that incorporation of repassivationinto the model significantly modifies DF.

Finally, the theories developed in this workrepresent a radical departure from traditional

obs

FIGURE 5. Influence of Ecorr on normalized pit distribution for Ctobs = 1 year.

(a) (bfor Ecorr = 0.334 V, pH = 7, T = 25 C, g = 1 year1, and

) (c)CORROSIONJUNE 1998

treatments of pitting corrosion, which are largelyempirical or phenomenological in nature. Accord-ingly, future work must aim at testing thepredictions of these theories, because if the accurate,deterministic prediction of pitting damage can bedemonstrated, and if the models are found to berobust, the techniques outlined in this paper wouldrepresent a major advance in corrosion science andtechnology.

ACKNOWLEDGMENTS

The authors acknowledge financial support ofthe Electric Power Research Institute, Palo-Alto, CA,and Kansai Electric Power Co., Osaka, Japan,through contract RPS 520-13, and the Gas ResearchInstitute, Chicago, IL, through contract no. 5092-260-2353.

NaCl = 0.571 x 103 mol/L, pH = 7, T = 25 C, g = 1 year1 , and

) (c)

CORROSION ENGINEERING SECTION

r C

(b)LIST OF SYMBOLS

a parameter defined by Equation (12) sax activity of halide ion mol/cm3

b parameter defined by Equation (13)Cb total bulk concentration of anions mol/cm3

d pit depth increment cmDm diffusion coefficient cm2/sEcorr corrosion potential VSHEf differential damage function no./cm3

F integral damage function no./cm2

F~

Faradays constant coul/equiv.k coefficient defined by Equation (8) cm/sm

L pit depth cmLmax maximum depth of pits observed

at a given observation time cmm exponent in Equation (8)

FIGURE 6. Influence of tobs on normalized pit distribution foand g = 1 year1.

(a)CORROSIONVol. 54, No. 6

Mm atomic weight of the metal g/moln velocity of pit nucleation no./cm2 sN density of pits that have nucleated

prior to the time of observation no./cm2

N0 maximum density of pits thatcan exist on the metal surface no./cm2

P probability of survivalT temperature Ktobs observation time sv pit growth rateu age of a pit srm density of the metal g/cm3

t induction time sD N pit distribution (Equation [3]) no./cm2

Upper index* normalized value (divided by N0)Symbols that are connected with the velocity of

pit nucleation can be found in Table 1.

REFERENCES

1. D.D. Macdonald, M. Urquidi-Macdonald, Corros. Sci. 48 (1992):p. 354.

2. Y. Ishikawa, T. Ozaki, N. Hosaka, O. Nishida, Trans. ISIJ 22(1982): p. 977.

3. Z. Szklarska-Smialowska, Pitting Corrosion of Metals (Houston,TX: NACE, 1986).

4. A. Turnbull, Brit. Corros. J. 28 (1993): p. 297.5. L.F. Lin, C.Y. Chao, D.D. Macdonald, J. Electrochem. Soc. 128

(1981): p. 1,194.6. D.D. Macdonald, M. Urquidi-Macdonald, Electrochim. Acta 31

NaCl = 0.571 x 103 mol/L, Ecorr = 0.334 V, pH = 7, T = 25 C,

(c)479

(1986): p. 1,079.7. D.D. Macdonald, M. Urquidi-Macdonald, J. Electrochem. Soc.

134 (1987): p. 41.8. D.D. Macdonald, M. Urquidi-Macdonald, J. Electrochem. Soc.

136 (1989): p. 961.9. D.D. Macdonald, M. Urquidi-Macdonald, J. Electrochem. Soc.

139 (1992): p. 3,434.10. D.D. Macdonald, C. Liu, M. Urquidi-Macdonald, G.H. Sickford,

B. Hindin, A.K. Agrawal, K. Krist, Corrosion 50 (1994): p. 761.11. C. Wagner, W. Traud, Z. Electrochem. 44 (1938): p. 391.12. D.D. Macdonald, Corrosion 48 (1992): p. 194.13. G. Engelhardt, M. Urquidi-Macdonald, D.D. Macdonald, Corros.

Sci. 39 (1997): p. 419.

MAIN MENUHELP & HINTSSEARCH KNOWLEDGEBASESEARCH THIS DOCUMENTKNOWLEDGEBASECORROSION ONLINECOR-LIT INDEXMP-LIT INDEXMP ABSTRACTSCONFERENCE PAPERS INDEXGLOSSARYPRODUCTS GUIDEREPRINTSSTANDARDS

JUNE 1998SCIENCEENGINEERINGMP ABSTRACTS

1998 INDEXES