Corrosion Science 41 (1999) 2165±2190
Transport processes in steam generator crevicesÐI. General corrosion model George R. Engelhardt a, Digby D. Macdonald a,*, Peter J. Millett b a
Stanford Research Institute, 333 Ravenswood Ave, Menlo Park, CA 94025, USA Electric Power Research Institute, 3412 Hillview Ave, Palo Alto, CA 94304, USA
b
Received 30 January 1998; accepted 26 February 1999
Abstract A new model for describing transport processes in Pressurized Water Reactor (PWR) Steam Generator (SG) tube/tube support plate crevices has been developed. The internal and external systems are coupled using the principle of charge conservation, recognizing that anodic and cathodic reactions may occur at any location. This model includes the in¯uence of convective transport, diusion, and migration of species in the crevice on the evolution of crevice properties. Calculations show that under normal operating conditions after an initial time period, the concentrations of all species and potential of the metal reach limiting values at the center of the crevice. After this period, the space in the cavity can be subdivided into two parts: (1) a region near the crevice mouth where concentration and potential gradients exist, and (2) a region near the crevice center where the concentrations of the various species and metal potential attain constant values. In the latter, the cathodic current density compensates for the corrosion current. The limiting values of the specie concentrations and the metal potential do not depend on the corrosion potential of the metal, but do depend on the available superheat and the bulk composition of the solution. For a suciently negative corrosion potential, the net current in the cavity can be negative, which must be balanced by a positive current on the external surface. # 1999 Published by Elsevier Science Ltd. All rights reserved. Keywords: Steam generator corrosion; Boiling crevice model; Coupled environment fracture model
* Corresponding author. Tel.: +1 650 326 3195; Fax: +1 650 859 3250. E-mail address:
[email protected] (D.D. Macdonald) 0010-938X/99/$ - see front matter # 1999 Published by Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 0 - 9 3 8 X ( 9 9 ) 0 0 0 3 9 - 6
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Nomenclature Ac Ak a b Ck Dk E F i Jk H hfg Kw K1 Kw k krl krv L m n p pc Q R r ri ro S T t v w x z
heated area, cm2 parameters [see Eq. (17)] constant [see Eq. (10)], N/cm2 constant [see Eq. (10)], N/cm2 local concentration of species k, mol/cm3 or M super®cial diusion coecient of species k, cm2/s potential of metal, V (SHE) Faraday's constant, 96487 C/mol current density, A/cm2 super®cial diusion ¯ux density species k, md/(scm2) heat transfer coecient, j/(s cm2 K) latent heat of vaporization, j/g equilibrium constant for Reaction (31), mol2/l2 equilibrium constant for Reaction (32), mol/l equilibrium constant for Reaction (33), mol3/l3 permeability, cm2 liquid relative permeability, cm2 vapor relative permeability, cm2 depth of the crevice or half depth of the hollow annulus, cm constant [see Eq. (11)] constant [see Eq. (12)] pressure, N/cm2 capillary pressure, N/cm2 volumetric heat source, j/(cm3 s) universal gas constant, 8.314 j/(mol K) polar coordinate, cm inner radii of a hollow annulus, cm outer radii of a hollow annulus, cm liquid saturation temperature, K time, s super®cial velocity, cm/s width of the crevice, cm distance along the cavity (x = 0 at the mouth of the cavity), cm charge
Greek ak transfer coecient [see Eq. (17)] e porosity m viscosity, N s/cm2 t tortuosity
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r k j
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density, g/cm3 conductivity, S/cm potential drop in the solution, (j=0 far away from the cavity mouth), V
Subscripts l liquid v vapor S metal surface 0 mouth of the crevice
1. Introduction Steam generator (SG) corrosion is a major problem in the operation of Pressurized Water Nuclear Reactors (PWRs). Among other factors, corrosion is due to the segregation and concentration of impurities in the tube/support plate and tube/tube sheet crevices. Although the SG feedwater contains impurities at extremely low levels, impurities accumulate (along with corrosion products) in cracks, crevices, and sludge piles by a thermo-hydraulic (or ``concentration'') mechanism [1]. In the case of SG crevices, the temperature gradient across the SG tube produces boiling in the crevice, which in turn results in the ejection of steam from, and ¯ow of water into, the cavity. The coupling of diusive and convective ¯uxes within the crevice, and the lower solubility of the impurities in steam compared with water, leads to impurity concentrations in the crevice that may be many orders in magnitude greater than those in the feedwater. Review articles on these concentrating mechanisms can be found in the literature [2±4]. The most complete model [boiling crevice model (BCM)] of Millett and Fenton [1,5] computes a concentrating factor for species in a porous crevice by solving the appropriate heat, mass, and momentum transfer equations. The model yields sensible predictions for concentration factors, the hydrodynamic velocity, temperature, and wetted length as a function of available superheat. However, the driving forces in this model are purely thermal and mechanical in nature, and no electrochemical eects are considered. Accordingly, the model cannot yield complete information on the chemical composition of the crevice or on its electrochemical properties, including the concentrations of corrosion products, pH, conductivity, potential distribution, and the rate of corrosion. In the present work, a crevice model is proposed that combines the attributes and essential features of the BCM with those of the coupled environment fracture model (CEFM) [6,7], which was developed by Macdonald and Urquidi± Macdonald to describe stress corrosion cracking. This new Coupled Environment Boiling Crevice Model (CECBM) includes the in¯uence of convection on the
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transport of species in the crevice, in addition to the diusion and migration driving forces that were previously considered in the CEFM. The equations describing heat and momentum transfer are taken from the BCM. Because the species ¯uxes depend on the hydrodynamic and temperature ®elds, the equations describing energy, mass, momentum, and charge transfer in the system must be solved simultaneously. First, the transport equations for the multicomponent SG crevice will be formulated. Then, the application of this model will be demonstrated using the example of iron corrosion in NaCl solutions. Due to the mathematical complexity, the practical application of this (and the other models) for quantitatively describing corrosion processes in SG crevices is not a simple task. For example, even in the 1-D case, only the solution of the hydrodynamic problem (without temperature and concentration gradients) for two-phase ¯ow in the porous media requires the determination of no less than ten parameters [8]. Because of the very low bulk concentrations of impurities, an extremely high electric ®eld is predicted to exist at the crevice mouth, forcing us to use special numerical algorithms with adaptive stepsize control. However, the main diculties in the quantitative description of corrosion processes in SG crevices are connected not with the mathematical complexity of the problem, but with the lack of experimental data for physical parameters, especially at elevated temperatures. For a system with a single species (NaCl) in the crevice, detailed experimental data and correlations of the thermodynamic and transport properties are available [5] whereas, in the general case, the solution in the corrosion crevice is multi-component. Under these
Fig. 1. Schematic illustration of the model.
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latter conditions, the solution of the system of transport equations with great accuracy is generally very dicult. The results that are obtained in this work help elucidate the principal features of the impurity accumulation and corrosion mechanisms in boiling, multicomponent systems contained within crevices via the hydraulic mechanism. The ®ndings are used to develop accurate, mathematically tractable methods for describing the accumulation of corrosion damage, when only meagre information is available for transport coecients and kinetic parameters (see Ref. 9).
2. Theory 2.1. General model The physico-electrochemical basis of the model, as applied to the case of a tube/ support plate crevice, is shown in Fig. 1. The crevice is assumed to be a ®nite hollow annulus whose inner and outer radii are ri and ro, respectively, and whose length is 2L. The section shown in Fig. 1 represents only half the length of the crevice. We assume that our system is axisymmetric (meaning that the solution does not depend on the polar angle), and that the width of the crevice (w=roÿri) is much smaller than both the inner or outer radii and the length of the crevice, i.e., wLlim) incorporating the plateau where the concentrations attain constant values, Ck,lim. Determination of the limiting length that separates the cavity space into these two regions, Llim, as a function of time is very important in the quantitative description of the corrosion process.
Fig. 5. Distribution of species down the crevice. Ecorr=ÿ0.2 V, tobs=1.8 year.
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Fig. 6. Distribution of species down the crevice. Ecorr=ÿ0.5 V, tobs=112 s.
For comparison purposes, concentration pro®les corresponding to those given in Figs. 2±5, were calculated for quite dierent electrochemical conditions [at Ecorr=ÿ0.5 V (SHE)]. For example, Figs. 6 and 7 shows the results for observation times of 112 s and 1 year, respectively. The values of all the other parameters that were used for deriving these two sets of ®gures are the same in both cases. Calculations show that no signi®cant dierence exists between these
Fig. 7. Distribution of species down the crevice. Ecorr=ÿ0.5 V, tobs=1 year.
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Fig. 8. Current density down the crevice. Ecorr=ÿ0.2 V, tobs=1 year.
two sets of calculations (only very small dierences could be detected for the shortest observation time, i.e., at tobs=112 s). The fact that impurity accumulation in the cavity does not depend on the electrochemical parameters is important, because it allows us to treat the thermal hydraulic and electrochemical problems separately in quantitatively describing the corrosion process. On the other hand, Figs. 8 and 9 show the current distributions for Ecorr=ÿ0.2 V, and Ecorr=ÿ0.5 V for an observation time of one year. As one would expect, the corrosion current density, as the concentration, exhibits a step pro®le, because it is considered in this work to depend only on pH [see Eq. (38)]. At the same time, the cathodic current density depends on the corrosion potential [see Eq. (17)], but only in the entry region. At x>Llim, the cathodic current density compensates for the corrosion current and correspondingly becomes
Fig. 9. Current density down the crevice. Ecorr=ÿ0.5 V, tobs=1 year.
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constant. This means that in the region x>Llim, general corrosion takes place. Accordingly, the potential of the metal in the region x>Llim also becomes constant (see Fig. 10) and does not depend on the (external) corrosion potential, Ecorr. This fact is the simple consequence of the independence of concentration distribution on the potential and of the condition is=0 at x>Llim. Actually, the total current density on the crevice sides, iS, depends only on the local pH and the metal potential, E. Because, in the region x>Llim, the value of pH does not depend on the Ecorr, the corresponding value of metal potential, Elim, also does not depend on Ecorr and is determined from the equation is
pHlim ,Elim 0
42
It is important to note, that in general case iS=iS,i+iS,o is the sum of the current densities that ¯ow through the inner and outer walls. The polarization curves can be dierent for the dierent walls. Thus, in our case, we assumed that the inner wall is inert, i.e., iS,i=0. The ``wave'' shape of the metal potential distribution near the plateau can be explained by the in¯uence of the diusion potential in the solution. As it follows from Eqs. (3)±(5) and (18) the distribution of the metal potential in the crevice, E, is described by the equation ! @ @E @ X @Ck is
43 zk D k , k F @x w @x @ x @x k where k is the local conductivity. The ®rst term in the right side of Eq. (43) is the diusion potential. It plays the signi®cant role in the region where concentration gradients, @Ck/@x, are signi®cant. Figs. 4 and 7 show that especially signi®cant gradients @Ck/@x are observed near the plateau region. Of course, due to the condition of electroneutrality the diusion potential disappears if the species in
Fig. 10. Distributions of metal potential, E, down the crevice.
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Fig. 11. Model calculation of distributions of metal potential, E, down the crevice. All diusion coecients are equal to the diusion coecient of OHÿ species.
the system have equal diusion coecients. Fig. 11 shows the distribution of the metal potential for the model system where all diusion coecients are equal to the diusion coecient of OHÿ species. It is interesting to note that, for a suciently negative Ecorr, the whole current in the cavity can be negative (see Fig. 12). The cathodic processes in the cavity nearly compensate for the corrosion current. However, the total current (that is, the dierence between the corrosion and cathodic current values, which are positive for high corrosion potentials and negative for low corrosion potentials) is still signi®cant. The total current leads to the potential drop in the external environment, a drop that can be a 0.1 V in magnitude (see Fig. 10). The existence of the potential drop in the external environment means that it is necessary, in
Fig. 12. Total net current, Itot, that ¯ows into (Itot < 0) or out of the crevice (Itot>0) for dierent corrosion potentials, Ecorr, on the external surface.
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general, to take into account the coupling between the internal and external environments when quantitatively describing the transport processes in SG cavities. We also see that the total current changes very slightly over time, a fact that will be explained later (Part II). The fact that the concentrations of the impurities are constant in that part of the cavity where x>Llim allows us to simplify the description of the accumulation process and to reduce the number of parameters needed in the calculation. In Ref. [9], we will explain this ®nding and describe a very simple (but quite accurate) method for quantitatively analyzing transport phenomena in SG crevices without the need for detailed information on the thermodynamic, transport, and electrochemical properties of the multicomponent (and multiphase) systems. 5. Summary and conclusions A Coupled Environment Boiling Crevice Model (CEBCM) model for describing transport processes in SG tube/tube support plate cavities has been developed. The transport processes in the internal environment (in the crevice) are described in a one dimensional approximation, whereas the charge transfer in the external environment (outside the cavity) is described in a two dimensional (axisymmetric) approximation. The internal and external systems are coupled using the principle of charge conservation, recognizing that anodic and cathodic reactions may occur at any location. The equations describing heat and momentum transfer are taken from the Boiling Crevice Model (BCM) of Millett and Fenton [1]. The CEBCM model includes the in¯uence of convective transport of species in the crevice, diusion, and migration on the evolution of the crevice properties. Because the species ¯uxes depend on the hydrodynamic and temperature ®elds, the equations describing energy, mass, momentum, and charge transfer in the system are solved simultaneously. The model takes into account the porosity and tortuosity of the corrosion product deposit(s) in the crevice. Homogeneous chemical reactions (hydrolysis of the metal ion product of the anodic reaction, saturation reactions, and the autoprotolysis of water) and electrochemical reactions (metal dissolution, hydrogen evolution, oxygen reduction, and water reduction) are included in the model. A numerical code that has been developed on the basis of this model allows one to predict the solution chemistry, potential distribution, and corrosion rate inside the cavity. Calculations show that under normal operating conditions after an initial time period, tin, the concentrations of all species reach limiting values, Ck,lim, in the center of the crevice that do not depend on the corrosion potential of the metal in the external environment. After this period, the space in the cavity can be subdivided into two parts: (1) a region where concentration gradients exist, x < Llim; and (2) a region (x>Llim) where the concentrations of the various species attain constant values, Ck,lim. For x>Llim, the sum of cathodic currents from the both sides of the crevice compensate the sum of the corrosion (anodic)
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currents (i.e. the condition iS=0 holds). It means, that if the crevice consists of the one electrochemically active metal and ane electrochemically passive metal, or, if the crevice is formed from the same active metal, then general corrosion in this region occurs. The metal potential in the solution in the region x>Llim also becomes constant, with a value that does not depend on the corrosion potential. It is evident that such a behavior of the system is the result of the thermohydraulic (or ``concentration'') mechanism. Thus, our calculations show that in the absence of the convection caused by heat transfer, no acidi®cation is observed in crevices for suciently low corrosion potential at the external surface [19]. In contrast, in the presence of hydraulic eects, the concentrations of species on the plateau does not depend on the corrosion potential of the metal but does depend only on the bulk species concentration, available superheat, and chemical equilibrium constants (see Ref. 9). Accordingly, it is dicult to prevent corrosion in systems with intensive thermal hydraulic concentration mechanisms by controlling the corrosion potential of the metal external to the crevice. Apparently, it is only possible to in¯uence the condition that develop in the crevice and, hence, to impact corrosion, by changing the composition of the solution or the available superheat. This situation is discussed in Ref. [9] more detailed.
Acknowledgements The authors gratefully acknowledge the support of this work by the Electric Power Research Institute, Palo-Alto, CA and Kansai Electric Power Co., Osaka, Japan, through contract RPS 520-13.
Appendix A1. Estimation of the potential drop outside the cavity As mentioned above, the potential distribution in the environment outside the cavity is described by Laplace's equation, which, in the polar coordinate system (x, r ) (with the origin of the polar coordinate placed at the center of the SG tube at the crevice mouth), is given by 1 @ @ j @ 2j r 0 r @r @r @x2
A1
The boundary conditions for Eq. (A1) have the form j0 ÿk
at r41
@j i0 @x
at x 0, ri 0, r ri @r
A5
where j is taken relative to a reference electrode (for example, SHE) which is situated away from the crevice mouth, k is the conductivity of the bulk solution, i0 is the average current density at the crevice mouth, isupp and itube(js) are the polarization curves for the support plate/electrolyte and tube/electrolyte surfaces, respectively, and js is the electrostatic potential at the electrode surface. Of course, the functions isupp(js) and itube(js) depend on the potential of the electrode, Ecorr, temperature, and the composition of the electrolyte outside the cavity. The general method for determining the non-linear boundary conditions for Laplace's equation in such systems is described in Reference [6]. The least resistance to transfer of charge in the external environment occurs when the electrochemical reactions on the external surface are so fast that we can neglect the surface overpotential relative to the ohmic potential drop in the solution (parameter G then reaches minimum value Gmin). Our numerical calculations show that, in this case, the reactive zone on the external surface does not exceed 3w, and since the width of the crevice is much smaller than the radius of the tube (w < ro ; and at x > 0, r ri
A10
must be used in place of Eqs. (A4) and (A6) for solving Laplace's equation. Importantly, Gmax is a function only of the ratio w/ri. As a speci®c example, according to our numerical calculations, Gmax=2.6 for w = 0.031 cm and ri=0.64 cm (w/r = 0.048) [14].
References [1] P.J. Millett, J.M. Fenton, A detailed model of localized concentration processes in porous deposits of SG's, in: Environmental Degradation of Materials. Nuclear Power Systems±Water Reactors, Monterey, CA, Aug. 25±29, 1991, NACE, Publ, 1992. [2] P. Cohen, Chemical thermohydraulics of steam generating system, in: Proceedings of Conference on Water Chemistry and Corrosion in the Steam-Water Loops of Nuclear Power Stations, Seillac, March, 1980. [3] A.J. Baum, Restricted Geometries and Deposits. in: P. Cohen (Eds.), The ASME Handbook on Water Technology for Thermal Power Systems, ch. Chapter 6, ASME, 1989. [4] P.J. Millett, J.M. Fenton, A review of PWR steam generator crevice impurity concentration mechanisms, in: Proceedings of the Steam Generators and Heat Exchanger Conference, Toronto, Canada, May, 1990. [5] P.J. Millett, Theoretical and experimental investigation in local concentration processes in PWR steam generators. PhD Dissertation, University of Connecticut, 1991. [6] D.D. Macdonald, M. Urquidi-Macdonald, Corros. Sci 32 (1991) 51. [7] D.D. Macdonald, M. Urquidi-Macdonald, An advanced coupled environment fracture model for prediction crack growth rate, in: S.M. Bruemmer, et al. (Eds.), Parkins Symposium on Fundamental Aspects of Stress Corrosion Cracking, Cincinnati, Ohio, 1991, pp. 443±455. [8] C.M. Marle, Multiphase Flow in Porous Media, Gulf Publishing Company, Houston, 1981. [9] G. Engelhardt, D.D. Macdonald, P. Millett, Transport Processes in Steam Generator Crevices. II. A Simpli®ed method for estimating Impurities Accumulating Rates, Corrosion Science (1998) submitted. [10] D.D. Macdonald, Corrosion 48 (1992) 194. [11] S.V. Patancar, Numerical Heat Transfer, Hemisphere, Washington, DC, 1980. [12] Y. Jaluria, K.E. Torrence, Computational Heat Transfer, Hemisphere, Washington, DC, 1986. [13] J.S. Newman, Electrochemical Systems, Prentice Hall, New Jersey, 1992. [14] D.D. Macdonald, G. Engelhardt, I. Balashov, J. Abella, Corrosion Potential Measurements and Modeling of Simulated PWR Steam Generator Secondary Environments, Research Project RPS520-13, Final Report, The Pennsylvania State University, Center for Advanced Materials, 1997. [15] S.M. Sharland, P.W. Tasker, Corros. Sci 28 (1988) 603. [16] M. Psaila-Dombrowski, A. Turnbull, R.G. Ballinger, Implications of crevice chemistry for cracking of BWR recirculation inlet safe-ends, in: Life Prediction of Corrodible Structures. Proceedings, Cambridge, UK, September, 1991, pp. 914±943. [17] A. Turnbull, Resolution of Key Issues in Deterministic Models of Environment Assisted Cracking, in: Proc. Res. Top. Sympos., CORROSION/96, NACE International, Houston, Texas, 1996, pp. 210±216. [18] P. Cohen (Ed.), The ASME Handbook on Water Technology for Thermal Power System, ASME, New York, 1989. [19] G. Engelhardt, M. Urquidi-Macdonald, D.D. Macdonald, Development of Fast Algorithm for Estimating Stress Corrosion crack Growth Rate In Sensitized Stainless Steels in Boiling Water Reactor, in: CORROSION/99, NACE International, San Antonio, Texas, April, 1999 submitted.
