# The Role of Determinism in the Prediction of Corrosion Damage

## Abstract

**:**

## 1. Introduction

## 2. Philosophical Basis of Determinism vs. Empiricism

- The definition of determinism versus empiricism;
- Why determinism is so important;
- The structures of deterministic vs. empirical models;
- The concept of the corrosion evolutionary path (CEP).

## 3. The Structure of a Deterministic Model

_{2}]. Accordingly, the AGWH lacks a valid scientific basis [24]. In adopting the AGW hypothesis, the global warming models invariably predict that human activity is responsible for global warming, but they are fatally flawed, and their predictions should not be relied upon. The fault lies in adopting a postulate that foretells the desired result and that also violates causality.

- A theory must be based upon experimental observation [23].
- The model based on that theory contains N “constitutive” equations that describe the relationships between various components and M “constraints”, which are statements of the natural laws (typically the conservation conditions) and which constrain the output to that which is “physically real”.
- M + N must be at least equal to the number of unknowns in the model. If it is not, the system is said to be mathematically underdetermined and deterministic prediction is not possible. If N + M is greater than the number of unknowns, the model is said to be mathematically “overdetermined” and deterministic prediction is unimpeded.
- All equations must be mathematically independent.
- Ad hoc relations cannot be added simply to “make the model work” (Einstein’s famous admonishment to the scientific community!) [23].

## 4. Model Building

^{+}, H

_{2}O, O

_{2}, and H

_{2}O

_{2}), so that charge is conserved in the entire system. This required that the external environment/surfaces be included in formulating the model, something that had not been done in an analytical manner in previous models.

_{I}> K

_{ISCC}, where K

_{ISCC}is the critical value of K

_{I}for crack propagation, which requires the presence of residual or applied tensile stress and/or a defect dimension of a certain minimum magnitudes.

_{I}(25 MPa·m

^{1/2}), and DoS (EPR = 15 C/cm

^{2}), with the lower limit being defined by the creep crack growth rate (1.69 × 10

^{−10}cm/s) and the upper limit being controlled by the mass transport of the cathodic depolarizer (e.g., O

_{2}) to the external surface. Also shown is the positive impact that solution conductivity has on the CGR at a given ECP, a relationship that will be explored later in this paper.

- Collate property data—a valid “global” theory must account for all the known properties of the system;
- Formulate hypotheses, postulates, and assumptions. These must agree with our empirical knowledge or theoretical expectation of the system;
- Specify the “mechanism”, and hence the “constitutive equations”;
- Specify the “constraints” (e.g., conservation equations, Faraday’s law of mass-charge equivalency);
- Solve the equations and predict the output;
- Compare the output with the experimental data and adjust the model to make new predictions that are in better agreement with the experiment;
- The last step is repeated until no amount of valid adjustment can make the model “work” by accounting for new observations within experimental uncertainty. The theory/model is then rejected, a new theory/model is developed, and the process starts over again;
- Finally, it is important to recognize that modeling is always a compromise between complexity and mathematical tractability. After a certain threshold, the modeler must make a compromise by either simplifying the model (e.g., reducing the number of species considered, and hence the number of independent variables) or by invoking assumptions to simplify the mathematics, or both. This is particularly important in the development of analytical models that may require numerical solutions of coupled high order differential equations for which analytical solutions do not exist.

_{I}, EPR) was established with data being taken from laboratory studies and field observation, provided that the independent variables were clearly defined [38,39]. For example, only CGR data reported from fracture mechanics specimens were employed, while those obtained from CERTs (Constant Extension Rate Tests) were rejected because of the poor definition of K

_{I}, the lack of clear differentiation of crack initiation and crack growth, the existence of multiple cracks, and the resultant uncertainty in the CGR.

_{I}, pH, and DOS) that are often hidden in the database. This is best done by using an artificial neural network (ANN) in the pattern recognition mode, using backpropagation and error minimization, resulting in the optimal weights between neurons. The ANN employed in Refs. [38,39] has one input layer, one output layer and three hidden layers, with each neuron having a sigmoid transfer function, which imbues the net with a certain “fuzziness” for handing data of lower accuracy. The ANN chosen for this work was taken from the MATLAB Neural Network Toolbox software.

_{I}, EPR). In doing so, the database must contain sufficient content related to each independent variable that the ANN can converge on a solution. Once the database is established, 70% of the data, selected at random, are used to train the net, 15% are employed to validate the prediction of the net (i.e., they act as the “known” cases), and the remaining 15% are used to further test the predictions of the net (the test set contains no known CGR data) (Figure 5b).

_{3}BO

_{3}/LiOH) was also reported by Shi et al. [39], in which the independent variables gleaned from the database are T, ECP, K

_{I}, conductivity, yield strength, [LiOH], [H

_{3}BO

_{3}] and pH. The concentrations of boric acid and lithium hydroxide typically vary from 2000 ppm to 0 ppm and 0 ppm to 4 ppm, respectively, from the beginning to the end of a fuel cycle during normal power operation. Since the pH is determined by [LiOH] and [H

_{3}BO

_{3}], strictly pH is, again, not an independent variable. However, it was included in the analysis because pH does have a discernible effect on the CGR. As shown in Table 1, the character of IGSCC in Alloy 600 is markedly different from that in Type 304 SS, being equally environmental (T, ECP, conductivity, pH), metallurgical (yield strength), and mechanical (K

_{I}) in character. In both cases, electrochemistry (ECP, conductivity, pH) plays an important role in determining CGR, but electrochemistry was ignored for many years because such studies tended to be carried out in Mechanical Engineering and Nuclear Engineering departments in universities and in research institutes/national laboratories where electrochemical expertise was minimal. In the author’s opinion, this reflected the fact that electrochemistry is seldom included in the teaching curricula in those disciplines. Regardless of the tortured path taken to include electrochemistry in such studies, any viable theory and resulting model must account for the characters identified above, including the electrochemical character.

_{I}). The measured CC and load are plotted in Figure 8 as the specimen was loaded and unloaded to/from various K

_{I}values in dilute Na

_{2}SO

_{4}solution at 252–288 °C under static autoclave conditions.

_{I}= 44 MPa·m

^{1/2}. Since the current is easily measured down to the sub-picoamp level with standard equipment, our initial fears of an immeasurably low CC were unfounded. In any event, the first load increment to K

_{I}= 11 MPa·m

^{1/2}does not produce a CC response, showing that SCC has not activated sufficiently to be detectable, but an increment to 22 MPa·m

^{1/2}does elicit a response. Unloading reduces the CC to zero, which is attributed to crack closure. Further increments in K

_{I}result in larger CC responses, but the response saturates for K

_{I}> 33 MPa·m

^{1/2}at about 500 μA. The CC was measured at an acquisition rate of 1 Hz, which proved to be too low to capture the fine structure in the current record, so that the CC appears to be randomly noisy. A second experiment was performed using uncatalyzed Type 304 SS cathodes in which the CC was measured at 100 Hz, and the CC record is given in Figure 9.

_{I}for two cathodes (Type 304 SS and titanium) [45]. The plot shows that the MFF is initially zero (no microfracture events) for K

_{I}< 10 MPa·m

^{1/2}, but increases sharply with an increase in K

_{I}to 11 MPa·m

^{1/2}. At higher loads, the MFF increases only modestly corresponding to the Stage II region of the CGR vs. K

_{I}correlation. The MFF is seen to be essentially independent of the type of the cathode (Ti vs. Type 304 SS), which indicates that the kinetics of oxygen reduction on the external surfaces are also little different since it is unlikely that the crack tip strain rate is a function of the kinetics of the oxygen reduction reaction. This seemingly surprising result is understandable when one notes that the exchange current density (${i}_{0}$) and the Tafel constants depend upon the thickness (${L}_{ss}$) but not the identity of the barrier oxide layer on the surface [50,51], because electronic charge carriers (electrons and electron holes) must quantum mechanically tunnel through the layer. The exchange current density on a passive surface may be expressed as ${i}_{0}=\widehat{{i}_{0}}exp\left(-\widehat{\beta}{L}_{ss}\right)$, where $\widehat{{i}_{0}}$ is the exchange current density on the hypothetical bare metal and $\widehat{\beta}$ is the tunneling constant that can be estimated from quantum mechanical tunneling theory (QMT) or measured experimentally [50]. Since $\widehat{{i}_{0}}$ for any given redox reaction appears to be similar on the bare surfaces of many metals of the same group (e.g., transition metals and their alloys), the difference in this parameter on passive surfaces lies in the differences in ${L}_{ss}$. However, at the equilibrium potential of the oxygen electrode reaction for the same T, pH, and [O

_{2}], the barrier layer thicknesses appear to be similar on stainless steels and Ti, although this conclusion is based upon somewhat sparse and equivocal data because no exchange current or Tafel constant information is available for Ti at elevated temperatures. However, modeling by Sutanto and Macdonald [52] of IGSCC in weld-sensitized heat-affected zones in simulated BWR primary coolant circuits suggests that the quantum mechanical correction to the exchange current density is not particularly important, at least in this case.

^{−1}) and B is the width of specimen (1.27 cm). Rearranging Equation (1) yields

^{−7}cm/s [12], we find that r~3 μm. Given a grain size of 20–50 μm and that the distance traveled by the crack during the “slow” advance stage represents about one half of the grain size, there should be roughly four to ten events in each package, which is in good agreement with the experiment. It is important to note that the value of r is too large to be attributed to slip, for which the microfracture dimension (MFD) should be some small multiple of the Burger’s vector or a few nm. We conclude that while slip is responsible for the intermittent events, the MFD is determined by hydrogen-induced cracking (HIC) of the matrix ahead of the crack tip, with that matrix being susceptible to HIC possibly because of the presence of strain-induced martensite. The source of hydrogen is the hydrogen evolution reaction via proton reduction that occurs on the metal at the crack tip in contact with the highly acidic environment that is maintained by differential aeration, assuming a diffusivity for H in Type 304 SS of about 10

^{−12}cm

^{2}/s [53], and noting that, to a first approximation, the diffusion length of hydrogen at 288 °C, x

_{d}> (D/f)

^{1/2}= 7 × 10

^{−6}cm. An embrittlement dimension of 100x

_{d}is not unreasonable, as the MFD likely reflects the momentum of the microfracture event once the crack has initiated in the embrittled phase ahead of the crack tip due to slip, or it may reflect the spacing of voids (possibly at precipitates, such as Cr

_{23}C

_{6}) that nucleate on the grain boundary ahead of the crack tip. The resulting dimension (1.4 μm) is of the same order as the MFD calculated above from the coupling current noise. Finally, in this case, the remarkable conclusion is that the crack advances fracture event by fracture event, with minimal overlap between events. However, this is not the case for IGSCC in sensitized Type 304 SS in thiosulfate solution at ambient temperature, where it is found that extensive overlapping occurs due to microfracture events occurring more-or-less simultaneously at different points on the crack front, resulting in a coupling current comprising structured noise [54].

- The crack growth rate (CGR) increases roughly exponentially with the potential of the metal at sufficiently high potentials. At lower potentials, the CGR is potential-independent, corresponding to mechanical creep fracture (Figure 4);
- In the case of IGSCC, the crack propagates intergranularly, giving rise to the intergranular crack pathway (Figure 3). In other cases, the crack propagates across the grain in a process termed transgranular stress corrosion cracking (TGSCC), and in still other cases mixed mode (IDSCC/TGSCC) may be observed [55];
- The CGR increases with the DOS, the yield strength, the hardness, and the extent of cold work [37];
- SCC only occurs if the stress intensity factor (K
_{I}) exceeds a lower limit, K_{ISCC}. The upper limit of the stress intensity factor, K_{Ic}, is defined by unstable, mechanical fracture. Between these limits, cracking occurs via stress corrosion cracking, with the CGR increasing sharply with K_{I}in a Stage I region and then progressing almost independently of K_{I}in a Stage II region [55]; - A coupling current is observed to flow through the solution (including that in the crack) from the crack tip to the external surfaces, where it is annihilated by the corresponding electron current flowing through the metal via a charge transfer reaction (e.g., oxygen reduction) [7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,45,46]. The environmentally mediated CGR is proportional to the magnitude of the coupling current [56];
- Oscillations appear in the coupling current that are attributed to microfracture events at the crack tip. In the case of IGSCC in sensitized Type 304 SS, as described in Ref. [45], the oscillations come in packages that are separated by brief periods of intense activity;
- Coating the external surfaces with an insulator, and hence inhibiting the reduction of oxygen, causes the coupling current to sharply decrease and the crack to be reduced accordingly [57];
- The character of an SCC model contains contributions from electrochemistry, mechanics and metallurgy, as demonstrated by the ANN analyses reported by Shi et al. [38,39]. The model engine must contain mechanistic concepts and relationships that allow the model to predict the character without the input of any additional information;
- Enhanced mass transfer of oxygen to the external surfaces increases the CGR [58]. For a sufficiently short, open crack, increasing the flow rate may destroy the aggressive conditions that develop within the cavity and hence inhibit crack growth;

## 5. The Coupled Environment Fracture Model

#### 5.1. Default Conditions

#### 5.2. Constitutive Equations and Constraints

_{I}, κ, flow velocity, etc.) [7]. These distributions are obtained by solving a set of Nernst–Planck equations for the flux (${J}_{i}$) of each ionic species in the system having a concentration ${C}_{i}$ and a diffusivity, ${D}_{i}$. The first term on the right side of Equation (3) describes diffusional transport, and the second represent migration in response to the gradient in the electrostatic potential, $\varphi $.

^{+}, Cl

^{−}, SO

_{4}

^{2−}, Fe

^{2+}, Fe(OH)

^{+}, Ni

^{2+}, Ni(OH)

^{+}, Cr

^{3+}, Cr(OH)

^{2+}, H

^{+}, and OH

^{−}. Only the first hydrolysis products of the metal ions are considered, because the low pH at the crack tip (−1 < pH < 2) probably precludes the formation of higher hydrolysis products. The set of 11 flux equations are coupled by Poisson’s equation that describes the variation of the electrostatic potential in the system:

^{−14}F/cm, any small deviation from electroneutrality results in a large gradient in potential, which, of course, activates a large restoring force to reduce the difference in $\rho $ from 0. Nevertheless, to simplify the problem when describing the internal crack environment, we assume that $\rho $ = 0, so that Equation (5) collapses into the one-dimensional Laplace equation that predicts that the electrostatic potential, varies linearly with distance down the crack. A consequence of this assumption is that the distribution in potential is different from that obtained by using Poisson’s equation [64,65]. However, a comparison of the predicted crack growth rates reveals insignificant differences, so the Laplace assumption was maintained because of the mathematical simplicity obtained. This is possibly due to CGR control residing with the kinetics of the cathodic depolarizing reactions (e.g., oxygen reduction) occurring on the external surface.

^{−}.

_{T}is the distance on the surface at which the effect of the crack is no longer apparent in the potential. This potential is ${\varphi}_{s}^{c}$. Once the potential distribution is calculated (Figure 11), the rates of the redox reactions and the electrodissolution of the steel are calculated as a function of distance across the external surface from the crack mouth, using the generalized Butler–Volmer equation [7,9,10,11,12,40]

_{2}] increases. This is a direct result of more oxygen being available at the surface for reduction as the bulk [O

_{2}] increases, so that a lower oxygen electrode reaction overpotential is required to consume the current from the crack. Upon decreasing the conductivity or stress intensity factor, the distribution is found to flatten for similar reasons [10,11].

#### 5.3. Creep Crack Growth Rate

_{she}, Figure 4), the CGR becomes independent of the electrochemical properties of the system. Under these conditions, the crack propagates mechanically and the fracture morphology changes from IGSCC for ECP > −0.23 V

_{she}to a ductile fracture with evidence of micro-void nucleation and coalescence. The model adopted in the CEFM to describe this limit is the micro-void coalescence model of Wilkinson and Vitek [70], as shown in Figure 14. Briefly, the model proposes that voids nucleate at fixed distances ahead of the crack tip. Because of the local stress field the void nearest the crack tip grows the fastest, and when it reaches a critical size the ligament between it and the crack tip fractures, resulting in crack advance by a distance c, and the process repeats. The frequency of these events can be calculated, resulting in a creep crack growth rate (CCGR) of

_{I}) on the CCGR according to the model of Wilkinson and Vitek [70] are displayed in Figure 15. Figure 15a displays the Arrhenius plot of log(CCGR) vs. reciprocal Kelvin temperature at a stress intensity factor of 27.5 MPa·m

^{1/2}. Note that, below a CCGR of about 1 × 10

^{−11}cm/s, the CCGR is essentially unmeasurable using current techniques. Thus, for T = 250 °C, the CCGR is so low that it has no impact on the accumulation of cracking damage in Type 304 SS in reactor coolant circuits, because this value corresponds to a crack extension of 3.14 μm/a. Figure 15b displays the dependence of log(CCGR) on the stress intensity factor at 288 °C. The plot displays the expected dependence, with log (CCGR) increasing monotonically with K

_{I}. The data are well-represented by log(CCGR) = 2 × 10

^{−13}K

_{I}

^{2.0}.

#### 5.4. CEFM Algorithm

## 6. The Predictions of the CEFM

#### 6.1. External Polarization

#### 6.2. Role of the Reactions on the External Surface

_{2}] for different values of the standard exchange current density multiplier (SECDM) for the OER, HER, and HPER, in a scenario described as general catalysis/inhibition. SECDM = 1 corresponds to uncatalyzed stainless steel, SECDM > 1 describes general catalysis, and SECDM < 1 corresponds to general inhibition. For SECDM = 10

^{−4}, the crack growth is predicted to be completely inhibited and the CGR is calculated to be reduced to the creep crack growth rate (CCGR) limit of 1.69 × 10

^{−10}cm/s. For SECDM = 10

^{−2}complete inhibition is predicted except for [O

_{2}] > 1 ppm, but for SECDM = 1 and 10

^{2}the CGR is predicted to increase (general catalysis) by one and two orders in magnitude, respectively, for low [O

_{2}] and by more than an order of magnitude at high [O

_{2}]. Thus, we conclude that the kinetics of the redox reactions on the external surfaces exert powerful influences on the CGR, and it follows the original dictum for localized corrosion be the case of a “large cathode driving a small anode” [28]. Thus, the important finding is that, under normal operating conditions in a BWR primary coolant circuit, the CGR is controlled by the redox reactions on the external surface. Therefore, previous models that ignored the role played by the external surface arguably missed the most important aspect in CGR control.

_{I}value (33 MPa·m

^{1/2}), the CC from the Pt-catalyzed Ni cathodes (Figure 8) is 500 μA compared with 10 μA for the uncatalyzed cathodes, a difference of a factor of 50. Accordingly, the CGR should be a factor of 50 higher in the former compared with that in the latter. Although we do not know the standard exchange current densities, these results are generally in agreement with the predictions of Figure 18.

_{I}= 25 MPa·m

^{1/2}, with dilute Na

_{2}SO

_{4}solution (conductivity at 25 °C of 220 μS/cm, and [O

_{2}] = 40 ppm (pure O

_{2}). One specimen had been coated with an electrophoretically deposited ZrO

_{2}layer, which was then cured at 250 °C for 48 h resulting in an impervious coating that was about 150 μm thick. The crack opening displacement (COD) of each specimen was monitored over 400 h and the COD was converted into crack length using standard compliance methods, while simultaneously monitoring of the ECP using an Ag/AgCl external pressure-balanced reference electrode. The specific impedance of the coating was measured at ambient temperature using a fast Fe(CN)

_{6}

^{3/4−}redox couple to monitor the coating integrity before and after the experiment [57].

_{6}

^{3/4−}) varies inversely with the specific impedance, we estimated that the exchange current density of the oxygen electrode reaction in the actual experiment (Figure 19a) was reduced by a factor of 10

^{2}to 10

^{3}by the coating. Accordingly, as calculated from the CEFM, the ECP should have been reduced from about 100 mV

_{she}to −200 to −400 mV

_{she}and the crack growth rate should have been reduced by factor 100 to 1000. The CGR calculated from the uncoated specimen (Figure 19a) was 3.4 × 10

^{−7}cm/s, which reflects the high [O

_{2}] and conductivity used in the experiment. Thus, the CGR of the coated specimen should have been reduced to between 3.4 × 10

^{−9}and 3.4 × 10

^{−10}. Since the sensitivity of the experiment to CGR does not allow CGR to be determined about 2 × 10

^{−8}cm/s, the finding that the observed crack growth rate is below this limit was consistent with the calculation. Regarding the ECP, the observed value for the uncoated specimen was 200 mV

_{she}, while that of the coated specimen was −200 ± 100 mV

_{she}; again, in reasonable agreement with calculation (Figure 19b).

#### 6.3. Crack Growth Rate/Coupling Current Relationship

^{−5}cm

^{3}/C and that the slope of the line is 0.17, we calculated the crack tip area as 2 × 10

^{−4}cm

^{2}. While we do not have an independent measurement of the crack tip area, the value obtained from Figure 20 is not unrealistic. For example, from Table 2, the crack tip area is assumed arbitrarily to be 10

^{−3}cm

^{2}, which is within a factor of 5 given by the CGR/CC analysis presented above. Given that the COD and crack width parameters given in Table 2 were arbitrarily chosen, the level of agreement must be judged to be acceptable. However, some equivocation exists regarding how the crack tip area is defined, since it appears that crack advance is not entirely due to electrodissolution at the crack tip, but that hydrogen-induced fracture (HIC) plays a significant role in determining the microfracture dimension (MFD) and, hence, also the CGR. We do not currently have data to resolve this issue.

#### 6.4. Crack Tip Conditions

^{+}, OH

^{−}, Na

^{+}, Cl

^{−}, Fe

^{2+}, Ni

^{2+}, Cr

^{3+}, Fe(OH)

^{+}, Ni(OH)

^{+}, Cr(OH)

^{2+}] allows the concentrations of these species to be calculated at any position within the crack as a function of the various independent variables [10,11,12,64,70]. To illustrate these calculations, the examples given are restricted to pH ([H

^{+}]), [Na

^{+}], and [Cl

^{−}] at the crack tip as a function of the ECP, which controls the CGR (see Figure 4), for two different bulk environment conductivities as set by the bulk [Na

^{+}]

_{b}and [Cl

^{−}]

_{b}. Thus, Figure 21 shows that Na

^{+}is rejected from the crack, while Cl

^{−}concentrates within the crack enclave as the ECP is made more positive, and hence as both the CC and CGR increase.

^{+}] increase) with increasing ECP (Figure 22). These are the expected behaviors corresponding to the migration of positively charged ions (Na

^{+}) down the potential gradient and negatively charged ions (Cl

^{−}, OH

^{−}) up the potential gradient from the crack tip to the crack mouth, resulting in the flow of positive current out of the crack through the solution to the external surface. In the case of H

^{+}, protons are produced by the hydrolysis of metal cations at the crack tip at a sufficient rate to overcome the migration out of the crack, so that the pH at the crack tip remains low.

#### 6.5. Effect of Crack Length

_{I}is shown in Figure 23 where the CGR is plotted vs. the ECP for crack lengths varying from 10 μm to 10 cm. It is seen that the crack growth rate varies over almost three orders of magnitude for the assumed range in crack length and is reduced with increasing crack length.

_{I}, and the ECL that determines the electrochemical activity. As we will see below, the dependence of CGR on the ECL has a profound impact on the accumulation of corrosion damage due to SCC in practical systems.

^{−9}μA, and the CCGR (creep crack growth rate) arm in which the CCGR is independent of the CC. A critical CC of 10

^{−9}μA exists below which crack advance is entirely due to creep. Elsewhere in this paper, I propose that the critical CC is more fundamental than the ECP in determining the conditions at which the crack transitions from propagating mechanically (via creep) to propagating by SCC. It is also important to note that the CC is finite within the CCGR arm, so that SCC still occurs but is “outrun” by creep, which controls the observed CGR. The existence of a CC for ECP < −0.23 V

_{she}, the potential at which the CGR becomes dominated by SCC, also accounts for the variations of the crack tip [Na

^{+}], [Cl

^{−}], and pH shown in Figure 21 and Figure 22 under conditions for which CCGR dominates.

^{−9}cm/s under the prevailing conditions. The figure also demonstrates an important issue that arises in the adoption of hydrogen water chemistry (HWC); in this case via the addition of 1 ppm of H

_{2}to the reactor feedwater. Thus, if no HWC is adopted, the crack is predicted to grow by 2.2 cm over the ten-year operating period. If HWC is adopted immediately, the crack is predicted to grow by only 0.6 cm. However, if HWC is adopted after 5 years of operation, the crack will be 1.7 cm long: only 0.5 cm less than that under normal water chemistry (NWC). This clearly follows the law of decreasing returns, in that the longer one waits to introduce HWC the lower the benefit will be. Since the installation of HWC is expensive, because of the need to store large amounts of flammable hydrogen on-site and the need to strengthen and shield the turbine hall to from irradiation from

^{16}N

_{7}and

^{17}N

_{7}in the form of volatile ammonia, a cost/benefit analysis of the type presented above is necessary.

#### 6.6. Microfracture Frequency and Dimension

^{−10}cm/s). At lower CGR, the MFF becomes independent of CGR to yield a correlation that is reminiscent of that between the CC and CGR. A significant difference exists between the crack tip strain rate models of Congleton [68] and Shoji [61,62,63], even though both predict similar forms of the log (f) vs. log (CGR) correlation. In any event, for the conditions relevant to Figure 9, the Congleton CTSR expression predicts f = 1 Hz whereas Shoji’s expression predicts f = 10 Hz. Both are within an order of magnitude of the experimental value of 2 Hz (Figure 8), which is acceptable agreement. Likewise, the MFD (r) is predicted to be about 3 μm for the conditions that are relevant (Figure 8 and Figure 9), which is in excellent agreement with experimentation (2–3 μm). Finally, the two fracture mechanics-based models (Congleton et al. [68] and Shoji et al. [61,62,63]) predict that the MFF increases strongly with increasing CGR, as expected from the dependence of fracture frequency on K

_{I}(Figure 10) and from the relationship CGR = Gfc

^{2}, where G = 2/B, if c (i.e., r) is only weakly dependent on the CGR. Because c is expected to be determined by the diffusion length of hydrogen in the matrix ahead of the crack, or by the spacing of C

_{23}C

_{6}precipitates on the grain boundaries, it is evident that it should not depend on K

_{I}or on any other environmental variable. As shown in Figure 27b, the microfracture dimension (MFD) at low CGR values decreases sharply in the mechanical fracture region, but in the SCC region (CGR > 4 × 10

^{−10}cm/s) the microfracture dimension is predicted to be almost independent of CGR. Note that, in the case of the Ford et al. [37] model, the crack tip strain rate is a function of K

_{I}only so that for a fixed fracture strain the MFF is independent of the CGR.

#### 6.7. Effect of Temperature

^{1/2}, [O

_{2}] = 200 ppb, and a sulfate concentration of 9.62 ppb. The CGR was observed to pass through a maximum at temperatures between 175 °C and 225 °C. The CEFM, with all three CTSR models, reproduced the maximum with a crack tip strain rate activation energy of 1000 kJ/mole [8]. The maximum arose from a competition between the thermal activation of the CTSR, which causes the CGR to increase with increasing temperature, and the decrease in the ECP, that results in a decrease in the CGR, noting the exponential relationship between the CGR and ECP (Figure 3).

^{−8}cm/s, corresponding to a crack extension of 0.63 cm/a, but during start-up and shut-down the CGR is about 2 × 10

^{−7}cm/s (6.3 cm/a) during the time that the reactor is at 125–225 °C. Of course, a reactor undergoing start-up and shutdown spends little time in the susceptible temperature region compared with the time spent under normal operation at 288 °C. Thus, for each hour spent in this 175–225 °C region the crack is predicted to advance by 7.2 μm, whereas for each hour spent at 288 °C (or 25 °C) the crack grows by about 0.72 μm. Nevertheless, Balachov et al. [73,74] predict that a significant fraction of the stress corrosion cracking damage incurred by a BWR under normal operating conditions accumulates during typical start-ups and shutdowns (Figure 29). It should also be noted that operational transients, such as start-ups and shutdowns, are accompanied by transients in conductivity due to hide-out return, which also affects the CGR.

#### 6.8. Effect of Flow Velocity

_{2}) and, hence, should be countered as being part of the external surface. Accordingly, the external area is increased (slightly) but the ECL decreases, resulting in an increase in the CGR. A similar approach was adopted to define the impact of loading frequency on the corrosion fatigue CGR in AA5083 in NaCl solution [75]. In that case, increasing frequency not only increases the rate of fracture at the crack tip but also enhances the exchange of solution between the crack and the external environment, thereby leading to a decrease in the ECL and, hence, to an increase in the CGR.

_{2}or H

_{2}O

_{2}) to the external surface. This has the effect of increasing the ECP (Figure 32a) and in supporting a higher CC, and hence CGR (Figure 32b). Figure 32a,b show that the impact of increased [O

_{2}] becomes increasingly muted as the flow velocity is increased.

#### 6.9. Effect of Solution Conductivity

#### 6.10. Development of Semi-Elliptical Cracks

_{I}at 90° from the origin is greater than K

_{I}at 0° and 180°, and if the creep crack growth rate increases with K

_{I}(Figure 15), the crack develops as a prolate ellipse with the major axis perpendicular to the surface (Figure 37). On the other hand, if K

_{I}at 90° from the origin is less than K

_{I}at 0° and 180°, the crack develops as an oblate ellipse with the major axis coincident with the surface. While both forms are predicted theoretically from mechanics, stress corrosion cracks are invariably oblate in form and this author is not aware of any reports of prolate surface stress corrosion cracks. An important point in this regard is that cracks grow at constant load; the CGR for purely mechanical creep is so low for Type 304 SS (1.69 × 10

^{−10}cm/s), as indicated in Figure 4 and Figure 15, that the crack would advance by only 2.1 mm over forty years of operation. Of course, under fatigue loading conditions the crack can advance much more rapidly. According to James and Schwenk [79], the fatigue crack growth rate (FCGR) in air at 600 °F (315 °C) for Type 304 SS can be written as $\frac{da}{dt}=9.802\times 10-16xfx{\left(\Delta {K}_{I}\right)}^{2.237}$, where $\Delta {K}_{I}$ is the stress intensity range in psi.in

^{1/2}(note that 1 psi·in

^{1/2}= 0.9100477 × 10

^{−3}MPa·m

^{1/2}). Choosing $\Delta {K}_{I}$ = 1.2 × 10

^{4}psi.in

^{1/2}(10.92 MPa.m

^{1/2}), the cycle-based FCGR is 7.62 × 10

^{−6}cm/cycle or for a loading frequency of 1000 Hz, a FCGR is estimated to be 7.62 × 10

^{−3}cm/s. After 40 years of service, such a crack is predicted to grow to an impossible length of more than 9 m! Thus, while fatigue loading may well account for the growth of oblate and prolate surface cracks, creep does not appear to be a viable mechanism for producing cracks that threaten the integrity of the coolant piping system in a BWR or the correlation indicated above in inapplicable. Of course, in assessing the fatigue loading case, the comparison should be made between FCGR and the corrosion fatigue crack growth rate (CFCGR), but that comparison is beyond the scope of this paper.

_{j+}

_{1}− t

_{j}). Because of the existence of the IR potential drop down the crack, ${\left(\frac{dL}{dt}\right)}_{L={L}^{0}}$ > ${\left(\frac{dL}{dt}\right)}_{L=L\left(j\right)}$ and becomes increasingly so as time proceeds (i.e., as the crack length normal to the surface increases). Note that ${\left(\frac{dL}{dt}\right)}_{L={L}^{0}}$ is a constant because the ECL (L

_{0}), which is the length of the least resistant path from the bottom of the crack nucleus at the crack edge (intersection of the crack with the surface) to the external surface, is very small for all times and remains approximately invariant with time. The equation for the oblate case is given by

_{I}at the edge and center of a surface crack are

_{I}(black lines) or were plotted without updating the stress intensity factor (red line); one sees that updating KI has virtually no effect on the crack shape or dimension. The lack of impact of increasing K

_{I}on the evolution in the shape and dimension of surface stress corrosion cracks is a general finding in our work, but this finding should not be extended to cracks that grow by purely mechanical means (i.e., under creep conditions) where it is also possible that prolate creep cracks might develop according to the solution for the stress intensity factors along the major and minor axes that are adopted. The lack of the dependence of shape crack on the increase in K

_{I}is a manifestation of the weak dependence of the CGR on K

_{I}in the Stage II region of the CGR vs. K

_{I}correlation [80].

_{2}], and hence ECP, are displayed in Figure 40. The corresponding ECP for [O

_{2}] = 1 ppb, 10 ppb, 100 ppb, 1 ppm and 10 ppm are estimated using the mixed potential model (MPM) to be −0.6025, −0.1987, −0.0818, −0.024 and 0.1259 V

_{she}, respectively, for sensitized Type 304 SS in BWR primary coolant (water) at 288 °C [81]. Note that the stress intensity factor was updated with the progression of the crack, and note also the different scales on the axes for each [O

_{2}]/ECP combination. The crack nucleus was assumed to be semi-circular with a 10 μm radius, corresponding to a small pit, for example. The nucleus grows by creep alone, because at 1 ppb O

_{2}the ECP < E

_{crit}(−0.23 V

_{she}) and SCC is inactive. For all other [O

_{2}], the ECP > E

_{crit}, so that the crack advances by stress corrosion cracking, the rate of which depends on the ECL.

#### 6.11. Global Assessment of the Accuracy of the CEFM

_{I}, conductivity, flow velocity, etc.) in the experimental database that was used to train the ANN (Figure 7). This “CEFM” dataset was then used to train the ANN in the same manner that was used in training the net on the experimental database. A plot was then made of the CEFM-calculated CGR vs the CGR calculated using the net trained on the experimental database for the same set of independent variables and the plot is shown in Figure 42.

## 7. Corrosion Evolutionary Path

- Assume that the system will behave in the future as it has in the recent past for which a record exists on the evolution of the independent variables. This is a viable approach for modeling nuclear reactors because of the wealth of information that is recorded during operation, although the data are not always of the type or in the form that are readily incorporated into predictive models. For example, power plants record ambient temperature conductivity, not the conductivity at the operating temperature that is employed in the CEFM. Likewise, to the author’s knowledge no nuclear plants regularly monitor the ECP at any point in the coolant circuit. Fortunately, these parameters can be calculated with sufficient accuracy to permit their inclusion in the models.
- Assume a future operating history (CEP) that is designed to probe the impact of specific operating issues, such as HWC, reduced conductivity, stress relief, etc. These “what if” scenarios are one of the great benefits of the modeling described in this paper, because they allow issues to be addressed in a computer that are not practical in an operating reactor. For example, if one sought to define the cost/benefit of operating a reactor with ultralow conductivity, that is more easily done, and at much lower cost, with programs developed by the author and his colleagues, such as REMAIN, ALERT, FOCUS and MASTER_BWR, than by installing the additional ion-exchange columns in the RWCU system that would be required to achieve the desired low conductivity.

^{−}.aq, O

_{2}

^{−}, etc.), which then react amongst themselves to produce even more radiolysis species (e.g., O

_{2}, H

_{2}, H

_{2}O

_{2}; 11 in all). These species are transported by convection, simultaneously with the undergoing reaction, throughout the coolant system by the flowing coolant. The solution of the set of stiff differential equations yields the concentrations of all species at 1 cm increments around the circuit, except in the recirculation system where the increment is 10 cm. These concentrations are then used in the CEFM to calculate the ECP and, hence, the CGR, with temperature, conductivity, stress intensity factor and flow velocity also being employed in the calculation. Thus, the codes yield the ECP and CGR at 1 cm spaced points around the entire coolant circuit (10 cm in the recirculation system). The CGR is then integrated over time along the CEP at each location of the coolant circuit to yield the crack length at any point along the path.

^{1/2}. Of course, this standard crack may be replaced by an actual crack having characteristics selected by the use, as is shown below. Accordingly, the current codes are particularly useful for predicting the damage that may evolve from a crack that has already been detected by an inspection during a planned outage. However, crack initiation models have now been developed [34] and these models will be inserted into BWR_MASTER soon. An important feature of the codes developed later than but including ALERT is the inclusion of the impact on CGR of increasing ECL, because without that feature all CGR codes greatly over-predict the evolution of IGSCC damage (Figure 27). Thus, in Figure 27, if the impact of ECL on the CGR was not included, the predicted crack length after 20 months would be about 2.6 cm rather than about 1.4 cm, which could have important consequences for any cost/benefit analysis. This is a matter of the utmost importance because the over-prediction of damage may have serious economic impact for a plant operator who might be tempted to install expensive remedial measures that are not warranted by the actual evolution of the damage.

^{1/2}rather than as specified in the standard crack. As given in the original publication, a table of this type is developed for each region of the coolant circuit, as specified in Figure 43.

_{2}] indicates NWC, 1 ppm indicates a specified level of HWC); the flow rate (assumed to be proportional to the power level); the % of rated full-power temperature; and the concentrations of the three common anionic impurities (Cl

^{−}, SO

_{4}

^{2−}and CO

_{3}

^{2−}) with the counter cation assumed to be Na

^{+}. CEP contains data for the full 120 months or for whatever operating period is desired. Note that the data specifying the CEP displays power-downs and power-ups and that during these periods HWC may be discontinued. It is CEPs of this kind that may be used to explore operational “what-if” scenarios, which is a particularly valuable facility of the codes.

_{2}O

_{2}, HO

_{2}, HO

_{2}

^{−}, O

_{2}, O

_{2}

^{−}and H

_{2}[73,74]. It is found that H

_{2}, H

_{2}O

_{2}and O

_{2}are the dominant species and that, according to the Mixed Potential Model (MPM) [40], they are the only species that need to be considered in establishing the ECP and hence the CGR.

_{2}], [H

_{2}], and [H

_{2}O

_{2}]; ECP; and CGR, coincide with outages of the reactor and will appreciate that these outages make a significant contribution to the damage incurred (see also Figure 29). The temporal increase in CGR during the outages is due to excursions in temperature ECP, conductivity and flow velocity during cool-down and subsequent heat-up, some of which might be controlled to mitigate the damage. For example, the implementation of HWC during outages might significantly mute the excursion in ECP, which is important because of the exponential relationship between the CGR and ECP. As noted above, the control of coolant conductivity by full-flow deionization of the coolant during excursions in the reactor power might be an option, although an expensive one.

^{−8}cm/s (198 pm/s), which is in reasonable accord with the calculated value of approximately 50 pm/s (Figure 51). It is important to note that only the comparison at Month 20 has probative value as the lack of a crack initiation time forced us to adjust the calculated curve of crack length (equivalent to adjusting the crack initiation time) to coincide with the measured crack depth at Month 10 (Figure 54), so that the comparison is between the calculated and measured increase in crack length from Month 10 to Month 20.

## 8. Summary and Conclusions

- All theories and models are inherently incorrect because they are figments of the modeler’s imagination as conceived via imperfect senses and intellect, so that they can never describe “reality”. However, they are nudged toward that ideal goal by the “scientific method” of cyclical prediction and evaluation. Because of the inherent defects, models ultimately fail (i.e., are “falsified”) and must be replaced by a new model that addresses the shortcomings of the old model.
- The models and theories themselves must be based upon empirical observations and must be employ postulates that are consistent with those observation and upon assumptions that, while not necessarily being demonstratably true, are reasonable expectations of current knowledge.
- In the “scientific method”, the model must not be evaluated against the same data and postulates that were used in formulating the theory and calibrating the model;
- The theory itself should be “global”, in that it accounts for all known observations about the system. “Local” theories are discouraged because they are based upon incomplete information, often being only based upon observations made by a single researcher;
- Importantly, all deterministic models must possess a theoretical basis but not all theories need to calculate;
- All deterministic models must contain a feedback loop that facilitates the enactment of the “scientific method”, in which the model is continually tested against new observations. If discrepancies are observed, the model is modified within the bounds of observation and the prediction is repeated;
- A deterministic model generally comprises constitutive equations that describe the operation of the model and constraints, the latter commonly being the natural conservation laws. The number of constitutive equations and the number of constraints must be at least equal to the number of unknown parameters in the model;
- If no amount of change within the bounds of observation can resolve the problem, the model and the theory must be discarded (“i.e., the model is “falsified”).
- It is important to note that no amount of successful prediction can “prove” a model, and its underlying theory to be correct, but only one instance of disagreement is necessary to prove the model and theory incorrect.

^{−}]) at the crack tip with increasing CGR, and successfully predicts the micro-fracture frequency (MFF) and dimension (MFD). Furthermore, as the result of training an artificial neural network (ANN) on a large body of experimental data, the CEFM successfully accounts for the mechanical/metallurgical/electrochemical character of IGSCC in sensitized Type 304 SS, with the fracture process being primarily electrochemical in nature, augmented by mechanics and metallurgy. The CEFM also successfully accounts for the development of oblate, semi-elliptical surface cracks that are found to develop in components in the primary coolant circuits of water-cooled nuclear power reactors.

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Schematic of the origin of the coupling current in stress corrosion cracking, according to the CEFM. The coupling current is required by the differential aeration hypothesis for localized corrosion, and the conservation of charge requires that the electron current flowing from the crack to the external surface must be equal to the positive ionic current flowing through the solution from the crack to the external surface.

**Figure 3.**Micrographs of IGSCC in Type 304 SS in high-temperature water: (

**a**) annealed; (

**b**,

**c**) sensitized [35]. This figure was published in Corrosion Science, 146, Z. Zhang, X. Wu, and J. Tan, “In-situ monitoring of stress corrosion cracking of 304 stainless steel in high-temperature water by analyzing acoustic emission waveform”, pp. 90–99, Copyright Elsevier 2019.

**Figure 4.**Compilation of crack growth rate (CGR) data for intergranular stress corrosion cracking on sensitized Type 304 SS in Boiling Water (Nuclear) Reactor (BWR) primary coolant at 288 °C as a function of the electrochemical potential (ECP) and the system parameters, as follows: 25 mm C(T) specimen, K

_{I}= 27.5 MPa.m

^{1/2}, ambient (25 °C) solution conductivity = 0.1 to 0.3 μS/cm, DoS (EPR) = 15 C/cm

^{2}. Selected data from Ford et al. [37] (open circles). Solid dark lines are CGR calculated using the CEFM, other symbols/lines are from Ford et al. [37].

**Figure 6.**Raw normalized CGR for IGSCC in Type 304 SS in BWR primary coolant plotted as a function of ECP (

**a**), and the division of the data into training, validation, and testing subsets (

**b**) [39].

**Figure 8.**Plot CC and load for crack propagation in sensitized Type 304 SS in simulated Boiling Water (Nuclear) Reactor (BWR) primary coolant at 252–288 °C [45].

**Figure 9.**CC record for crack propagation in sensitized Type 304 SS in simulated Boiling Water (Nuclear) Reactor (BWR) primary coolant. Data acquisition frequency = 100 Hz. After Manahan et al. [45].

**Figure 10.**Microfracture frequency vas K

_{I}for IGSCC in sensitized Type 304 SS in simulated Boiling Water (Nuclear) Reactor (BWR) primary coolant [45].

**Figure 11.**(

**a**) Schematic of a crack in a metal, defining the coordinate system employed in developing the CEFM [7,9,10,11,12]. (

**b**) CEFM showing the redox reactions occurring on the surface external to the crack and the flow of the electron current through the metal (the “Coupling Current”), and the ionic current through the solution from the crack tip to the external surfaces where they are annihilated by the charge transfer redox reactions.

**Figure 12.**(

**a**) Calculated distribution of the potential in the solution on the external surface as a function of distance from the crack mouth as a function of the concentration of oxygen in the solution at 288 °C. Other parameter values employed in the calculation are given by Macdonald and Urquidi-Macdonald [7]. (

**b**) Schematic of the current density across the crack at charge conservation (dark and light shaded area are equal). After [10,11].

**Figure 14.**(

**a**–

**c**) Creep crack growth model of Wilkinson and Vitek illustrating crack advance via the nucleation and linking of voids ahead of the crack tip. Adapted from [70].

**Figure 15.**Plots of log(CCGR) vs. 1/T (

**a**) and vs. K

_{I}(

**b**) for Type 304 SS as calculated by the Wilkinson and Vitek model [70], using the Creep Crack Growth Rate algorithm.

**Figure 17.**Effect of temperature and pH on the electrostatic potential at the crack tip, crack mouth, and at an effectively infinite distance on the external surface away from the crack mouth (-ECP), together with the variation of the pH for IGSCC in sensitized Type 304 SS in dilute H

_{2}SO

_{4}solution having an ambient temperature conductivity of 0.27 μS/cm and a dissolved oxygen concentration of 200 ppb. Adapted from [13].

**Figure 18.**Impact of exchange current density multiplier on the IGSCC in sensitized Type 304 SS in water at 288 °C as a function of oxygen concentration. The number next to each curve is the multiplier of the standard exchange current density (MSECD). Parameter values are given by Lu et al. [13]. Adapted from [13].

**Figure 19.**(

**a**) Crack length vs. time for the ZrO

_{2}-coated and uncoated specimens. (

**b**) CGR and ECP as a function of the SECDM. IGSCC in sensitized Type 304 SS in dilute Na

_{2}SO

_{4}solution at 250 °C (conductivity at 25 °C of 220 μS/cm, and [O

_{2}] = 40 ppm (pure O

_{2})). Adapted from [57].

**Figure 20.**Relationship between CGR and CC for IGSCC in sensitized Type 304 SS in dilute NaCl solution (50 ppm), [O

_{2}] = 7.6 ppm at 250 °C. From [56].

**Figure 23.**Dependence of CGR in sensitized Type 304 SS in water at 288 °C as a function of ECL. K

_{I}= 27.5 MPa·m

^{1/2}. Adapted from [12].

**Figure 25.**Predicted coupling current vs. crack growth rate for the growth of a standard crack (Table 3) for IGSCC in sensitized Type 304 SS in dilute H

_{2}SO

_{4}/NaCl solution at 288 °C for different values of ECL. COD = 5 × 10

^{−4}cm, crack width = 1.0 cm, K

_{I}= 27.5 MPa.m

^{1/2}, V

_{f}= 100 cm/s, d = 50 cm, T = 288 °C, [H

_{2}] = 10

^{−4}ppb, [H

_{2}O

_{2}] = 10

^{−4}ppb, [Na

^{+}] = 1.35 ppb, [H

_{2}SO

_{4}] 10

^{−6}ppbS, [O

_{2}] = 1 ppb − 2.33 × 10

^{6}ppb, κ

_{288}= 2.69 uS/cm, κ

_{25}= 0.0618 μS/cm, creep crack growth rate = 1.61 × 10

^{−10}cm/s. Adapted from [12].

**Figure 26.**Predicted damage accumulation (crack length) in the heat affected zone adjacent to the H3 weld in a BWR core shroud over 10 years of operation for different hydrogen water chemistry (HWC) regimes. Note that the discontinuities in the CGR are due to reactor outages. The broken line represents the crack length in the case where no dependence of the CGR on crack length is recognized. Adapted from [73].

**Figure 27.**Calculated MFF (

**a**) for three CTSR models and MFD (

**b**) for IGSCC in Type 304 SS in water at 288 °C.

**Figure 28.**Plot of CGR for constant [O

_{2}], K

_{I}, and sulfate concentration as a function of temperature calculated using different crack tip strain rate models by Ford [37], Congleton [68], and Shoji [61,62,63]. The ambient temperature conductivity was 0.27 μS/cm. The experimental data are from Andresen [59].

**Figure 29.**Predicted contribution of start-ups and shut downs to the total stress corrosion cracking damage incurred in the outer surface of the core shroud during normal operation of a BWR. Adapted from [74]. A soft restart corresponds to low conductivity excursion, whereas a hard restart is characterized by a large conductivity excursion.

**Figure 31.**Experimental illustration of flow-induced rotating eddies in a cavity of AR = 3. [75], ©1966 ASME. This image is hereby used courtesy of the ASME.

**Figure 34.**Plots of calculated CGR vs. ambient temperature conductivity as a function of [O

_{2}] (

**a**) and [H

_{2}O

_{2}] (

**b**) for IGSCC CGR in sensitized Type 304 SS as a function of ambient temperature conductivity in simulated BWR coolant at 288 °C [12]. Other parameters as in Table 2. Adapted from [12].

**Figure 35.**Dependence of solution conductivities at 25 °C and 288 °C as a function of [NaCl] (

**a**) and correlation of conductivities at 25 °C and 288 °C (

**b**). Adapted from [12].

**Figure 36.**Semi-elliptical crack in the surface of a pipe [76].

**Figure 38.**Semi-elliptical oblate surface crack in a plate. Adapted from [80].

**Figure 39.**Evolution of an oblate semi-elliptical stress corrosion crack for the non-default parameter values given in the figure. (

**a**) Minor and major axes vs. time. (

**b**) Oblate surface stress corrosion cracks vs. time. Adapted from [80].

**Figure 40.**Predicted crack shapes vs. time as a function of [O

_{2}], and hence ECP, that the ECP for [O

_{2}] = 1 ppb, 10 ppb, 100 ppb, 1 ppm and 10 ppm are estimated to be −0.6025, −0.1987, −0.0818, −0.024 and 0.1259 V

_{she}, respectively, for sensitized Type 304 SS in BWR primary coolant (water) at 288 °C. Adapted from [81]. Note that the stress intensity factor was updated with the progression of the crack. Note also the different scales on the axes for each [O

_{2}]/ECP combination.

**Figure 41.**Evolution of an oblate semi-elliptical surface stress corrosion crack in sensitized Type 304 SS in BWR primary coolant (water at 288 °C) as a function of solution conductivity (

**a**) and stress intensity factor (

**b**). Adapted from [81].

**Figure 45.**CEP of the Chinshan BWR with respect to reactor power [82].

**Figure 46.**CEP of the Chinshan BWR with respect to feedwater hydrogen injection [82].

**Figure 47.**CEP with respect to hydrogen in the coolant at the HAZ adjacent to the H3 weld in the outer core shroud surface of the Chinshan BWR in Taiwan [82].

**Figure 48.**CEP with respect to oxygen in the coolant at the HAZ adjacent to the H3 weld in the outer core shroud surface of the Chinshan BWR in Taiwan [82].

**Figure 49.**CEP with respect to hydrogen peroxide in the coolant at the HAZ adjacent to the H3 weld in the outer core shroud surface of the Chinshan BWR in Taiwan [82].

**Figure 50.**CEP of the Chinshan BWR with respect to ECP at the HAZ adjacent to the H3 weld in the outer core shroud surface of the Chinshan BWR in Taiwan [82].

**Figure 51.**CEP of a crack in the HAZ adjacent to the H3 weld in the outer core shroud surface of the Chinshan BWR in Taiwan with respect to the CGR [82].

**Figure 52.**CEP of the Chinshan BWR with respect to stress intensity factor of a crack in the HAZ adjacent to the H3 weld in the outer core shroud surface of the Chinshan BWR in Taiwan [82].

**Figure 53.**CEP of the Chinshan BWR with respect to the integrated damage (crack depth) of a crack in the HAZ adjacent to the H3 weld in the outer core shroud surface of the Chinshan BWR in Taiwan [82].

**Figure 54.**Comparison of predicted crack length in the heat-affected zone of the H3 weld in a BWR core shroud as a function of time after Outage 11. ♦ Tang et al. [82].

Independent Variable | Range | Type 304 SS in BWR Primary Coolant | Range | Alloy 600 in PWR Primary Coolant |
---|---|---|---|---|

Temperature (°C) | 25–292 | 17.8 | 290–360 | 18.6 |

ECP (V_{she}) | −0.575–0.496 | 43.6 | −1.096 to −0.610 | 14.1 |

Stress Intensity Factor (MPa∙√m) | 10.4–67.78 | 10.8 | 4.6–101 | 15.2 |

Conductivity (μS/cm) | 0.52–5.72 | 14.0 | 1.7–116 | 14.1 |

Degree of Sensitization (DoS) (C/cm^{2}) | 0–33.79 | 13.8 | - | - |

Yield Strength (MPa) | N/I | - | 211–500 | 12.0 |

[LiOH] ppm | N/A | - | 0–10 | 4.0 |

[H_{3}BO_{3}] ppm | N/A | - | 0–1800 | 7.6 |

pH | N/A | - | 5.52–9.19 | 14.5 |

**Table 2.**Default system parameter values for calculating CGR in sensitized Type 304 SS in simulated BWR primary coolant.

Parameter | Value | Comments |
---|---|---|

T | 288 °C | Operating temperature of a BWR. |

COD | 0.001 cm | Typical of a tight crack |

Crack width | 1.0 cm | Assumed |

Crack length | 0.5 cm | Assumed for a “standard crack” |

Pipe hydrodynamic diameter | 50 cm | Typical of BWR recirculation system |

Flow velocity | 100 cm/s | Assumed |

Stress intensity factor | 27.5 MPa.m^{1/2} | Assumed |

O_{2} concentration | 100 ppb | Typical of BWR under Normal Water Chemistry (NWC) |

H_{2} concentration | 1 ppb | Assumed |

H_{2}O_{2} concentration | 1 ppb | Assumed |

Degree of sensitization (EPR) | 15 C/cm^{2} | Typical of weld sensitization of Type 304 |

**Table 3.**Default parameter values for the CEFM as used for calculating CGR in sensitized Type 304 SS in simulated BWR primary coolant (T = 288 °C).

Atomic volume | 1.18 × 10^{−23} cm^{3} | Fundamental |

Fracture strain at the crack tip | 8 × 10^{−4} | Assumed |

Young’s Modulus (E) | 2 × 10^{5} MPa | Typical of Type 304 SS |

Dimensionless constant (β) | 5.08 | Refs. [61,62,63] |

Density of the steel (ρ) | 8 g/cm^{3} | Typical of Type 304 SS |

Yield strength (${\sigma}_{y}$) | 215 MPa | Typical of Type 304 SS |

Hwang–Gao strain hardening exponent (${n}_{HG}$) | 1.7 | Refs. [61,62,63] |

Ramberg–Osgood strain hardening exponent (${n}_{HG}$) | xxx | Refs. [61,62,63] |

Dimensionless constant (λ) | 0.11 | Refs. [61,62,63] |

Shear modulus (G) | 7.31 × 10^{10} Pa | Typical of Type 304 SS |

Grain boundary self-diffusion constant (D_{b0}) | 2.5 × 10^{−4} m^{2}/s | Typical of stainless steels |

Activation energy for diffusion (E_{a,D}) | 168 kJ/mol | Refs. [61,62,63] |

Grain boundary diffusion width | 5 × 10^{−10} m | Typical of stainless steels |

Tafel constant for the HER | 0.065/V | Typical of Type 304 SS [40] |

Exchange current density (i_{0}) for HER | 5 × 10^{−4} A/cm^{2} | Typical of Type 304 SS [40] |

Tafel constant for the OER | 0.071/V | Typical of Type 304 SS [40] |

Exchange current density (i_{0}) for OER | 5.05 × 10^{−3} A/cm^{2} | Typical of Type 304 SS [40] |

Passive current density for the steel | 2.6 × 10^{−3} A/cm^{2} | Typical of Type 304 SS [40] |

Standard potential (E^{0}) for steel electrodissolution at the crack tip. | −0.47 V_{she} | Calculated from thermodynamics for Fe^{2+}/Fe |

Model | Equation |
---|---|

Ford [37] | ${\dot{\epsilon}}_{ct}=4.11\times {10}^{-11}{K}_{I}^{4}$ |

Congelton [68] | ${\dot{\epsilon}}_{ct}=\frac{\dot{a}}{r}\left[63.65\pi \alpha \frac{\left(1-{\upsilon}^{2}\right)}{{\sigma}_{yE}}+\beta \left(\frac{{\sigma}_{y}}{E}\right)ln\left(\frac{{R}_{p}}{r}\right)\right]$ |

Shoji [61,62,63] | ${\dot{\epsilon}}_{ct}={\beta}_{1}\left(\frac{{\sigma}_{y}}{E}\right)\left(\frac{{n}_{HG}}{{n}_{HG}-1}\right){\left\{ln\left[\left(\frac{\lambda}{r}\right){\left(\frac{{K}_{I}}{{\sigma}_{y}}\right)}^{2}\right]\right\}}^{\frac{1}{{n}_{CH}-1}}\left(\frac{\dot{a}}{r}\right)$ |

Hall [69] | ${\dot{\epsilon}}_{ct}=\frac{{n}_{R}}{{n}_{R}+1}\frac{{\sigma}_{y}}{E}{\left(\frac{{\alpha}_{ct}EJ}{r{\sigma}_{y}^{2}}\right)}^{\frac{n}{n+1}}\frac{\dot{J}}{J}+\frac{{n}_{R}}{{n}_{R}-1}{\beta}_{ct}\frac{{\sigma}_{y}}{E}\frac{\dot{a}}{r}{\left\{\mathit{ln}\left[\left(\frac{\lambda}{r}\right){\left(\frac{K}{{\sigma}_{y}}\right)}^{2}\right]\right\}}^{\frac{{n}_{R}+1}{{n}_{R}-1}}$ |

Temperature Dependence | ${\dot{\epsilon}}_{ct}\left(T\right)={\dot{\epsilon}}_{ct}\left({288}^{\mathrm{o}}\mathrm{C}\right)exp\left[\frac{Q}{R}\left(\frac{1}{T}-\frac{1}{561.15}\right)\right]$ |

_{1}, λ = dimensionless constants in plastic strain calculation; r distance from growing crack tip; Rp = plastic zone size; ${\sigma}_{y}$ = yield strength; E = Elastic (Young’s) modulus; ${n}_{HG}$ = strain hardening exponent of Hwang and Gao ((see Refs. [61,62,63]); Q = activation energy; ${K}_{I}$ = stress intensity factor; $\dot{a}$ = crack growth rate; ${n}_{R}$ = strain hardening exponent of Ramberg and Osgood (also see [61,62,63]); J is the J-integral; $\dot{J}$ is the rate of change in J with time.

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**MDPI and ACS Style**

Macdonald, D.D. The Role of Determinism in the Prediction of Corrosion Damage. *Corros. Mater. Degrad.* **2023**, *4*, 212-273.
https://doi.org/10.3390/cmd4020013

**AMA Style**

Macdonald DD. The Role of Determinism in the Prediction of Corrosion Damage. *Corrosion and Materials Degradation*. 2023; 4(2):212-273.
https://doi.org/10.3390/cmd4020013

**Chicago/Turabian Style**

Macdonald, Digby D. 2023. "The Role of Determinism in the Prediction of Corrosion Damage" *Corrosion and Materials Degradation* 4, no. 2: 212-273.
https://doi.org/10.3390/cmd4020013