Corresponding author: Alexander I. Orlov ( oai@proryv2020.ru ) Academic editor: Boris Balakin
© 2020 Alexandr V. Avdeenkov, Oleg I. Achakovsky, Vladimir V. Ketlerov, Vladimir Ya. Kumaev, Alexander I. Orlov.
This is an open access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Citation:
Avdeenkov AV, Achakovsky OI, Ketlerov VV, Kumaev VYa, Orlov AI (2020) Basic models and approximation for the engineering description of the kinetics of the oxide layer of steel in a flow of heavy liquid metal coolant under various oxygen conditions. Nuclear Energy and Technology 6(3): 215234. https://doi.org/10.3897/nucet.6.59068

The article presents the results of corrosion processes, kinetics and changes in the oxide layer modeling using MASKALM software complex. The complex is intended for a numerical simulation of threedimensional nonstationary processes of mass transfer and interaction of impurity components in a heavy liquid metal coolant (HLMC: lead, leadbismuth). The software complex is based on the numerical solution of coupled threedimensional equations of hydrodynamics, heat transfer, formation and convectivediffusive transport of chemically interacting components of impurities.
Examples of calculations of mass transfer processes and interaction of impurity components in HLMC, formation of protective oxide films on the surfaces of steels are given to justify the coolant technology.
coolant, corrosion, films, HLMC, heavy liquid metal coolant, impurities, lead, leadbismuth, mass transfer
With the advent of liquid metal coolants necessary for nuclear power plants with fast neutron reactors, the key elements of the safety justification were attributed to the problems associated with maintaining the quality of the coolant. Maintaining the necessary physicochemical regime of the coolant (or coolant technology) has turned from an important but additional operation into one of the main components in substantiating the design and operating parameters of the reactor. There are two main tasks of the heavy liquid metal coolant (lead or leadbismuth) technology:
It should be noted that the main processes of the leadbismuth and lead coolant technology have a common physicochemical nature. In the leadbismuth eutectic, lead exhibits the dominant chemical activity. Considering also that the construction materials for both coolants are identical, the methods and means of the coolant technology are also similar in the principle of operation.
Therefore understanding the process of generation of the oxide layer of steel in a flow of heavy liquid metal coolant is highly important for justification of circuit integrity and reactor safety.
Kinetics of corrosion processes of steel components in HLMC is determined by the associated processes of hydrodynamics and the interaction of dissolved components, primarily such as oxygen and iron. The justification and appropriate modeling of these processes is an important component for substantiating heavy liquid metal coolant technology (Robertson et al. 1988;
The MASKALM software complex implements basic models and approximations with the following features:
The use of commercial CFD codes in this area has not yet been widely applied because of the need to build a complex apparatus of user functions for coordinated calculations of thermal hydraulics and processes of interaction and formation of impurities. For example, the works (
A system of threedimensional equations of an incompressible multicomponent medium in a Cartesian coordinate system was used to describe the processes of formation, transformation, and transfer of impurities in the primary circuit of a reactor with HLMC (
The medium is assumed to be incompressible with temperaturedependent thermophysical properties. The presence of dissolved and dispersed particulate impurities does not affect the hydrodynamics, heat transfer, and thermophysical properties of lead. Concentration distributions of dissolved impurities and particles are affected by molecular, Brownian, and turbulent diffusion. On the surfaces of structural materials, oxide films are formed that impede the diffusion exchange of oxygen and iron between the coolant and structural materials.
The system of equations describing the nonstationary thermohydrodynamics of the components of a multicomponent medium is presented in the form of equations of motion, conservation of mass and thermal energy of an inhomogeneous continuous medium (
The equations of motion for the characteristics of a multicomponent medium averaged over components that are not impurities were obtained in (
$\frac{\partial {U}_{i}}{\partial \tau}+\frac{\partial {U}_{i}{U}_{k}}{\partial {x}_{k}}=\frac{1}{\rho}\frac{\partial p}{\partial {x}_{i}}+\frac{1}{\rho}\left(\frac{\partial}{\partial {x}_{k}}\mu \frac{\partial {U}_{i}}{\partial {x}_{k}}\right)+{g}_{i}\frac{\stackrel{~}{\rho}\left(t\right)}{\rho}{\Lambda}_{ik}{U}_{k}$ , (1)
$\frac{\partial {U}_{i}}{\partial {x}_{i}}=0$
where: U_{i} – projections (components) of the medium velocity vector in the Cartesian coordinate system; p – pressure; ρ – density of medium averaged over temperature and components; μ – dynamic effective viscosity coefficient; ῀ρ (t) – averaged only over components and temperaturedependent density of the medium; g_{i} – projections of the gravitational acceleration vector on the coordinate axes; Λ_{ik} – tensor of drag coefficients.
The drag coefficients of the medium are also used to simulate the solid and liquid states of the medium. Zero values of the resistance coefficients correspond to the liquid state of the components; large values (tending to infinity) correspond to the solid state of the components.
Environmental properties are obtained by averaging the initial properties of the components and are determined by the following dependencies:
$\rho =\sum _{n}{\rho}_{0}^{n}{\epsilon}^{n}$ ; where ρ^{n}_{0} – density of n component;
$\mu =\sum _{n}{\mu}_{0}^{n}{\epsilon}^{n}$ ; where μ^{n}_{0} – n component dynamic viscosity coefficient;
${U}_{k}=\sum _{n}{u}_{k}^{n}{\epsilon}^{n}$ – projections of the velocity vector of the averaged motion of the medium;
ε^{n} – volume fraction of component with number n;
${u}_{k}^{n}={U}_{k}+\Delta {u}_{k}^{n}$ – projections of the velocity vector of the medium component with number n.
The drag coefficients of the medium Λ_{ik} are used to simulate the solid and liquid states of the medium. For components in the liquid state, Λ_{ik} is set equal to zero, for components in the solid state it is set to a larger value, which ensures that the velocity of the medium is equal to zero.
The equations of transport of nonimpurity components of the medium, taking into account stratification and taking into account the sources of components that can occur during chemical reactions, have the form:
$\frac{\partial {\epsilon}^{n}}{\partial \tau}+\frac{\partial}{\partial {x}_{k}}\left({\epsilon}^{n}{U}_{k}\right)=\frac{\partial}{\partial {x}_{k}}\left({\epsilon}^{fn}\Delta {u}_{k}^{n}\right)+\sum _{p=1}^{N}{J}^{np}$ , (2)
For components that are impurities, diffusion processes are additionally taken into account in the transport equations:
$\frac{\partial {C}^{n}}{\partial \tau}+\frac{\partial}{\partial {x}_{k}}\left({C}^{n}{U}_{k}\right)=\frac{\partial}{\partial {x}_{k}}\left({C}^{fn}\Delta {u}_{k}^{n}\right)+\frac{\partial}{\partial {x}_{k}}{D}^{n}\frac{\partial}{\partial {x}_{k}}{C}^{n}+\sum _{p=1}^{N}{G}^{np}$ , (3)
where Δu_{k}^{n} – components of the velocity vectors of the relative motion of the components of the medium (stratification velocity);
ε ^{fn} – volume fraction of component n, that is in a liquid state;
J^{np} – volume source of component n due to loss of component р;
῀N ^{fn} – mass concentration of the impurity component n, that is in the liquid;
G^{np} – mass source of the impurity component n due to a decrease in the impurity component р;
D^{n} – component n diffusion coefficient.
The lefthand sides of equations (2.3) describe the transfer of medium components due to averaged motion. The righthand sides of equations (2) describe the process of separation of a mixture of components in a liquid state, sources of components due to chemical reactions or other processes, as well as diffusion processes of component transfer. Solid components do not delaminate. The initial and boundary conditions that describe the flows of impurities onto solid surfaces, as well as the values of the concentrations of impurities at the input, are added to the transport equation.
Modeling of heat transfer and heat exchange processes in a multicomponent medium taking into account stratification is carried out by solving the energy equation in the following form:
$\frac{\partial \left(c\rho t\right)}{\partial \tau}+\frac{\partial \left(c\rho {U}_{k}t\right)}{\partial {x}_{k}}+\sum _{n}\left[\frac{\partial}{\partial {x}_{k}}\left({c}^{n}{\rho}^{n}\Delta {u}_{k}^{n}{\epsilon}^{fn}t\right)\right]=\frac{\partial}{\partial {x}_{k}}\lambda \frac{\partial t}{\partial {x}_{k}}+{q}_{v}$ , (4)
where:
с – the specific heat of the medium averaged over the components;
λ – thermal conductivity coefficient averaged over the components of the medium;
U_{k} – medium volume velocity;
Δu_{k}^{n} – projections of the component stratification velocity vector n;
ε ^{fn} – volume fraction of a component n in a liquid state.
The first two terms on the left side of equation (4) describe the convective transfer of thermal energy. The third term is the transfer of thermal energy due to the separation of components.
Mass transfer processes in nonisothermal circuits with HLMC are closely related to the interaction of the coolant with structural steels. Structural steels are the main sources of metallic impurities entering the coolant (iron, chromium, nickel, etc.). In turn, the coolant is a supplier of dissolved oxygen for structural steels, which is necessary for the formation of protective oxide films on steel surfaces that prevent the development of corrosion processes.
Dissolved oxygen in the coolant is both in free form, and in the form of various compounds with a coolant and metal impurities. Moreover, in quantitative terms, free oxygen in the solution is negligible relative to the bound forms of dissolved oxygen.
The main metal impurity, which should be taken into account when calculating the simulation of mass transfer processes in the circulation circuit with HLMC, is an iron impurity. In lead melt, iron can be both free and in oxygenbound form. The relationship between these forms of existence of dissolved iron in the coolant depends on the oxygen content and can vary widely.
In the model of interaction of dissolved impurities in the coolant, only those forms of existence of oxide compounds that play a decisive role should be taken into account: PbO, Fe_{3}O_{4}.
The most important in modeling physicochemical interaction of impurities are the processes of formation and growth of protective oxide films on the surfaces of structural steels. The composition of oxide film, as a rule, varies from chrome spinel near the boundary with the coolant to chromium spinel at the border with steel. In this work, only the process of magnetite formation at the "coolant – oxide film" interface was taken into account, and the formation of chromium spinel is not modeled separately, but is completely determined by the model of magnetite film formation. This approach can be considered as simplified, but it gives quite adequate description of the evolution of the oxide film.
Oxidative kinetics leading to the parabolic law of growth of an oxide film is usually used when the effective diffusion of reagents is independent of time and size of the film. The oxidation mechanism of ferriticmartensitic steels is determined by the diffusion of Fe ions in the direction of the coolant and variations in the oxygen potential. In this approach, oxidation of the surface of steels is described by the parabolic velocity equation with the rate constant predicted by Wagner’s theory if the resulting oxide film has an "ideal" crystal lattice, and ion diffusion is dominant.
For simplicity, we consider the growth of a singlelayer film. In the developed (
In this approximation, it is unambiguously assumed that the growth of the film (magnetite) occurs due to the reaction on the surface of the oxide layer:
3Fe + 4PbO ⇔ Fe_{3}O_{4} + 4Pb, (5)
and at the same time, all four components of the reaction are in chemical equilibrium. Since the rate of chemical reactions is much higher than the characteristic rates of all other physical processes, the chemical equilibrium can be considered steady in the volume of the coolant at each moment of time and the distribution of components substantially depends on thermohydraulic processes.
Let us consider a model assuming the presence of a diffusion "reaction" layer at the interface between the oxide film and the coolant. The main components interacting with oxygen are iron and lead. We consider the oxide film as a single layer and consisting of magnetite. At the steelcoolant boundary, we distinguish four subregions: steel (conventionally Fe), oxide, an intermediate diffusion layer, and HLMC (Fig.
Intermediate layer is a virtual layer where the abovedescribed surface reaction is carried out (5). For flows of components and sources of impurities in the media presented in Fig.
The chemical equilibrium of the oxygenlead system with the formation of the [PbO] complex is described in the standard way:
${k}_{\beta}^{ox}=\frac{{a}_{PbO}}{{a}_{Pb}{a}_{O}^{\beta}}=\mathrm{exp}\left(\frac{\Delta {\mathrm{G}}_{\mathrm{Pb}\mathrm{O}}}{RT}\right)$ (6)
Lead activity is equal to one; therefore, this ratio shows the ratio of oxygen concentrations and the dynamic complex of PbO. For the liquid phase, the existence of a separate oxygen phase [O] does not have deep physical meaning, and its concentration is much lower than the concentration of [PbO]. The concentration of [O] can be considered as the concentration of "instantly free" oxygen.
Using the data (
For HLMC medium with impurities (Pb, PbO, Fe, Fe_{3}O_{4}) we introduce the notation β. In the presence of iron in the coolant, three reactions are formally possible:
Pb + O = PbO
3Fe + 4O = Fe_{3}O_{4} (7)
3Fe + 4PbO = Fe_{3}O_{4} + 4Pb
Chemical equilibrium constant for the second reaction:
${k}_{\beta}^{O}=\frac{1}{{\left({a}_{Fe}^{\beta}\right)}^{3}{\left({a}_{0}^{\beta}\right)}^{4}}=\mathrm{exp}\left(\frac{\Delta {\mathrm{G}}_{{\mathrm{Fe}}_{3}}{\mathrm{O}}_{4}}{RT}\right)$ (8)
The chemical equilibrium constant for the third reaction, provided that the activity of magnetite is equal to unity:
${k}_{\beta}^{m}={\left({a}_{Fe}^{\beta}\right)}^{3}{\left({a}_{0}^{\beta}\right)}^{4}=\mathrm{exp}\left(\frac{4\Delta {\mathrm{G}}_{\mathrm{PbO}}\Delta {\mathrm{G}}_{{\mathrm{Fe}}_{3}{O}_{4}}}{RT}\right)$ (9)
where ${a}_{n}^{\beta}=\frac{{C}_{n}^{\beta}}{{C}_{n}^{S}}$ – activities corresponding to the coolant component.
The equilibrium constants (6, 8, 9) in the dynamic and chemical equilibrium of all components are obviously related:
${k}_{\beta}^{m}={\left({k}_{\beta}^{ox}\right)}^{4}/{k}_{\beta}^{O}$ (10)
Obviously, in conditions of dynamic equilibrium, it is sufficient to consider any two of the three equations. From the point of view of the thermodynamics of the leadironoxygen system, it is more convenient to consider the first two equations. And the area of existence of iron oxides of various compositions in terms of oxygen potentials overlaps with the region of existence of oxygen solutions in the lead melt.
Let’s consider the surface layer at the media interface of oxide filmcoolant (medium γ). As we discussed above, if we formally approach this region as well as the coolant region, then there will be no differences between them and, moreover, there will not even be a need to introduce this region for consideration. Since a possible film growth is assumed in this medium, that is, a kind of phase transition:
Fe_{3}O_{4} (solid phase) + Pb (liquid phase) →
dissolved (Fe, PbO) in Pb (liquid phase), (11)
then in this environment we assume the possibility of only the third reaction from (11).
Equilibrium constant in this environment:
${k}_{\gamma}={\left({a}_{Fe}^{\gamma}\right)}^{3}{\left({a}_{O}^{\gamma}\right)}^{4}=\mathrm{exp}\left(\frac{4\Delta {\mathrm{G}}_{\mathrm{PbO}}\Delta {\mathrm{G}}_{{\mathrm{Fe}}_{3}{O}_{4}}}{RT}\right)$ (12)
Once again, we note that in general case, knowledge of the equilibrium constant is not enough for this medium, since we will assume that the existence of equilibrium is not a prerequisite. In this case, it is necessary to know the parameters of forward and reverse reactions (7). In addition, the third reaction (7), which occurs on the surface of magnetite, is in reality not "onestep" as it is written, but contains steps for adsorption and desorption of the elements involved. Therefore, the reaction constants do not have to be identical with the constants in a homogeneous medium. The oxygencoolant system is assumed in chemical equilibrium, similarly to formula (6). It should also be noted that the heterogeneous reaction rate is directly proportional to the area of contacting reagents. Therefore, "rough" surface, obviously, will provide a more intensive reaction process. Therefore, oxygen consumption by the steel surface will depend not only on its type, but also on the method of its manufacture. The latter is almost impossible to take into account when modeling, which can significantly reduce the physical justification for the direct application of the Wagner model (Robertson et al. 1988).
Let’s consider the layer of the film itself, which at this stage is assumed to be composed only of magnetite Fe_{3}O_{4} (δ medium). In general, film growth also occurs on the metalmagnetite interface, but since the oxygen diffusion rate is much lower than the rate of the chemical reaction of incorporation of oxygen into the magnetite lattice, it is sufficient to know the concentration at the metalmagnetite interface to calculate the oxygen flux.
The constant of chemical equilibrium in this medium, taking into account the fact that the activity of magnetite and iron are equal to unity:
${k}_{O}^{\delta}=\frac{1}{{\left({a}_{O}^{\delta}\right)}^{4}}=\mathrm{exp}\left(\frac{\Delta {\mathrm{G}}_{{\mathrm{Fe}}_{3}{O}_{4}}}{R{T}^{\delta}}\right)$ , (13)
where a ^{α}_{O} – oxygen activity in the medium δ, Τ ^{δ} – film temperature.
The condition of chemical equilibrium in the intermediate layer during the formation of magnetite:
$\frac{{\nu}_{O}^{\gamma}}{{\nu}_{Fe}^{\gamma}}=\frac{\partial {\nu}_{O}^{\gamma}}{\partial {\nu}_{Fe}^{\gamma}}=\frac{4}{3}=\frac{{J}_{O}^{\gamma}}{{J}_{Fe}^{\gamma}}$ (14)
The formulas for the sources of oxygen and iron in the surface layer γ, J ^{γ}_{O} and J ^{γ}_{Fe}, are proportional to the growth rate (or degradation) of this "reaction" layer, which in turn is proportional to the amount of oxygen and iron substance in it. Since the system in question is assumed in chemical equilibrium, the velocities of direct, R^{→} = K^{→} (C ^{γ}_{Fe})^{3} (C ^{γ}_{O=PbO})^{4} and reverse, R^{←} = K^{←} (C_{Pb})^{4}C_{O=Fe3O4} reactions are the same and from the chemical equilibrium condition (12) it is possible to determine only the ratio of concentrations and "fluxes" of iron and oxygen. That is, the condition of chemical equilibrium and the assumption of the instantaneous establishment of this chemical equilibrium in comparison with the slow diffusion process does not make it possible to determine the slow growth of the film and additional assumptions about the mechanism of film growth are needed, namely, that the molar fluxes of oxygen and iron are treated as 4/3, in accordance with the kinetics of reaction (7), with instant subsequent formation of magnetite on the surface.
The source of oxygen in the system is the coolant (oxygen content is regulated through external sources). Therefore, "diffusion" of oxygen from the coolant to the steelcoolant surface section and the formation of a conventional oxide film (magnetite in our model) are initially assumed. That is, oxygen is "consumed" on the growth of the inner and outer parts of the film, respectively:
${\left.{J}_{0}^{\beta}\right}_{x=\delta +\gamma}={J}_{0}^{\gamma}+{\left.{J}_{0}^{\delta}\right}_{\delta}$ (15)
Similarly for iron that diffuses from steel:
${\left.{J}_{Fe}^{\delta}\right}_{\delta}={J}_{Fe}^{\gamma}+{\left.{J}_{Fe}^{\beta}\right}_{x=\delta +\gamma}$ (16)
Note that the model considered in (
${\left.{J}_{Fe}^{\delta}\right}_{\delta}={D}_{Fe}^{\delta}\frac{{C}_{Fe}^{\delta}{C}_{Fe}^{\gamma}}{\delta}$ (17)
${\left.{J}_{Fe}^{\beta}\right}_{x=\delta +\gamma}={\chi}_{Fe}\left({C}_{Fe}^{\gamma}{C}_{Fe}^{\beta}\right)$ , (18)
where χ_{Fe} – mass transfer coefficient of iron at the border.
Similarly, oxygen flows in media can be represented by δ (
${\left.{J}_{O}^{\delta}\right}_{\delta}=2{D}_{O}^{\delta}\frac{{C}_{O}^{\delta}{C}_{O}^{\gamma}}{\delta}$ (19)
${\left.{J}_{O}^{\beta}\right}_{x=\delta +\gamma}={\chi}_{O}\left({C}_{O}^{\beta}{C}_{O}^{\gamma}\right)$ , (20)
where χ_{O} – oxygen mass transfer coefficient.
Using ratios (16) and (17) we get:
$\theta \left({J}_{Fe}^{\delta}{J}_{Fe}^{\beta}\right)={J}_{O}^{\delta}{J}_{O}^{\beta}$ , (21)
where $\theta =\frac{4}{3}\left(\frac{4{M}_{O}}{3{M}_{Fe}}\right)$ – for molar (mass) flows, respectively, or
$\frac{4{M}_{O}}{3{M}_{Fe}}\left({D}_{Fe}^{\delta}\frac{{C}_{Fe}^{\delta}{C}_{Fe}^{\gamma}}{\delta}{\chi}_{Fe}\left({C}_{Fe}^{\gamma}{C}_{Fe}^{\beta}\right)\right)={\chi}_{O}\left({C}_{O}^{\beta}{C}_{O}^{\gamma}\right)2{D}_{O}^{\delta}\frac{{C}_{O}^{\delta}{C}_{O}^{\gamma}}{\delta}$ , (22)
where ${J}_{O,Fe}^{\beta ,\delta}$ ‒ flows of oxygen and iron in the coolant stream and the film, respectively, C ^{δ}_{Fe} – iron concentration in the layer δ, C ^{β}_{Fe} – iron concentration in the coolant flow, C ^{γ}_{Fe} – iron concentration in the layer γ, C ^{β}_{O} – oxygen concentration in the coolant flow, C ^{δ}_{O} – oxygen concentration in the layer δ.
Using relations (17–20) and the relationship between the activities of oxygen and iron in the wall layer γ, we can obtain a nonlinear equation for the concentration (or activity) of oxygen in the wall layer in which the concentrations of iron and oxygen, C ^{β}_{Fe} and C ^{β}_{O}, are calculated from equations (3–5). Concentrations C ^{δ}_{Fe} and C ^{δ}_{O} are calculated based on the properties of magnetite, namely, the activity of iron in magnetite is assumed equal to one. The activity of oxygen in magnetite is found from the condition of chemical equilibrium (13). Thus, the equation obtained from relation (20) will contain one unknown parameter, namely, the film thickness δ. Note that the obtained equation for the concentration C ^{γ}_{O} (or the activity corresponding to it) parametrically depends only on the concentrations in other regions (β and δ), which in turn depend on time. The thickness of the film cannot be expressed only through these parameters (concentration or fluxes), and the film growth will directly depend on time.
As was determined above, the diffusion processes of iron and oxygen in the oxide film are slow, and therefore the determining processes of its growth. Therefore, formulas (17) and (19) are a rather crude approximation.
According to Wagner’s theory, the flow of iron through the oxide layer as a function of coordinates can be represented (Zhang 2005):
${J}_{Fe}^{\delta}=\frac{{C}_{Fe}^{\delta}{D}_{Fe}}{RT}\frac{d{\mu}_{Fe}}{dx}=\frac{1}{2}\alpha {D}_{Fe}{C}_{Fe}^{\delta}\frac{d\mathrm{ln}{P}_{{O}_{2}}}{dx}=\frac{2}{3}{D}_{Fe}{C}_{Fe}^{\delta}\frac{d\mathrm{ln}{P}_{{O}_{2}}}{dx}$ , (23)
where μ_{Fe} – chemical potential of iron, P_{O2} – partial pressure of oxygen, x – coordinate, α=⁴/₃ for Fe_{3}O_{4.}
The fulfillment of relation (23) gives a constant flow of iron [10]:
${J}_{Fe}^{\delta}\delta =\frac{2}{3}{C}_{Fe}^{\delta}{\int}_{{a}_{{O}_{2}}\left(0\right)}^{{a}_{{O}_{2}}\left(\delta \right)}{D}_{Fe}d\mathrm{ln}{a}_{{O}_{2}}$ (24)
where a_{O2} – oxygen activity in oxide at a point with a coordinate x, determined as ${a}_{{O}_{2}}=\frac{{P}_{{O}_{2}}}{{P}_{{O}_{2}}^{0}}$ , where P ^{0}_{O2} = 1 atm., ${C}_{Fe}^{\delta}=\frac{3{M}_{Fe}{\rho}_{ox}}{{M}_{O}}$ [kg/m^{3}] ‒ the average density of iron in magnetite, where M_{Ox} = 3M_{Fe} + M_{O}, M_{Fe} and M_{O} – molar masses of iron and oxygen, respectively, ρ_{ox} ‒ magnetite density, ${a}_{{O}_{2}}\left(0\right)=\frac{3{M}_{Fe}{\rho}_{ox}}{{M}_{ox}{C}_{{O}_{2}}^{S}\left(0\right)}$ , C ^{S}_{O2} (0) – oxygen saturation concentration in magnetite.
Using relation (24) and the dependence of the diffusion coefficient of iron cations, which takes into account the transport of iron ions over vacancies and along the grain boundaries of the metal in the framework of the theory of point defects (
${D}_{Fe}={D}_{V}^{0}\mathrm{exp}\left({E}_{V}/RT\right){P}_{{O}_{2}}^{2/3}+{D}_{I}^{0}\mathrm{exp}\left({E}_{I}/RT\right){P}_{{O}_{2}}^{2/3}$ , (25)
D ^{0}_{V} and D ^{0}_{I} – constants characterizing the transport of vacancies and between grains, respectively), we can obtain the value of the iron flux from the layer δ as a function of oxygen activity at the layer boundary (x = δ):
${J}_{Fe}^{\delta}=\frac{{C}_{Fe}^{\delta}}{\delta}\left(A\left[{\left({a}_{{O}_{2}}\left(\delta \right)\right)}^{2/3}{\left({a}_{{O}_{2}}\left(0\right)\right)}^{2/3}\right]B\left[{\left({a}_{{O}_{2}}\left(\delta \right)\right)}^{2/3}{\left({a}_{{O}_{2}}\left(0\right)\right)}^{2/3}\right]\right)$ , (26)
$A={D}_{V}^{0}\mathrm{exp}\left({E}_{V}/RT\right){\left({P}_{{O}_{2}}^{0}\right)}^{2/3}$ ,
$B={D}_{I}^{0}\mathrm{exp}\left({E}_{I}/RT\right){\left({P}_{{O}_{2}}^{0}\right)}^{2/3}$ ,
In the accepted notation a_{O2} (δ) = a ^{γ}_{O2}, a_{O2} (0) = a ^{δ}_{O2}.
Assuming that in the coolant flow C ^{β}_{Fe} = 0 and neglecting the last term in equation (22) we obtain that:
$\frac{4{M}_{O}}{3{M}_{Fe}}\left({J}_{Fe}^{\delta}{\chi}_{Fe}{C}_{Fe}^{\gamma}\right)\approx {\chi}_{0}\left({C}_{0}^{\beta}{C}_{0}^{\gamma}\right)$ (27)
Using the relations (26, 27) and k_{γ} = (a ^{γ}_{Fe})^{3}(a ^{γ}_{0})^{4}, we can obtain the equation for C ^{γ}_{O} (which does not have a simple analytical solution, but can be solved numerically (the only solution)).
The rate of change of the film is determined by the flow of oxygen J ^{β}_{O}:
$\frac{d\delta}{dt}=\frac{{J}_{o}^{\beta}}{{\rho}_{ox}}\left(1+\frac{3}{4}\frac{{M}_{Fe}}{{M}_{o}}\right)$ (28)
From the general formulas (22) and (28), one can derive particular approximations of Martinelli (
Neglecting both the oxygen flow in the film and the iron flow in the coolant flow, we can obtain the approximation of film growth according to Martinelli (
$\frac{4{M}_{O}}{3{M}_{Fe}}{J}_{Fe}^{\delta}={J}_{O}^{\beta}$ (29)
Then the film growth equation (28) takes the form:
${C}_{Fe}^{\delta}\frac{d\delta}{dt}={J}_{Fe}^{\delta}$ (30)
Taking into account expression (26), we obtain an expression for the dependence of film growth on time:
δ^{2} = t . K_{p}, (31)
${K}_{p}=2\left(A\left[{\left({a}_{O}^{\gamma}\right)}^{4/3}{\left({a}_{O}^{\delta}\right)}^{4/3}\right]B\left[{\left({a}_{O}^{\gamma}\right)}^{4/3}{\left({a}_{O}^{\delta}\right)}^{4/3}\right]\right)$
Neglecting the oxygen flux in the film, we can obtain the equation of film growth in the Zhang approximation (
${J}_{O}^{\beta}=\frac{4{M}_{O}}{3{M}_{Fe}}\left({J}_{Fe}^{\delta}{J}_{Fe}^{\beta}\right)\equiv \frac{4{M}_{O}}{3{M}_{Fe}}\left({J}_{Fe}^{\delta}{\chi}_{Fe}\left({C}_{Fe}^{\gamma}{C}_{Fe}^{\beta}\right)\right)$ (32)
Then the film growth equation takes the following form:
$\frac{d\delta}{dt}=\frac{{K}_{p}}{2\delta}\frac{{\chi}_{Fe}\left({C}_{Fe}^{\gamma}{C}_{Fe}^{\beta}\right)}{{C}_{Fe}^{\delta}}$ (33)
Denote the value ${\chi}_{Fe}^{\text{'}}=\frac{{\chi}_{Fe}\left({C}_{Fe}^{\gamma}{C}_{Fe}^{\beta}\right)}{{C}_{Fe}^{\delta}}$ , which reflects the dissolution rate (corrosion) of the film. Then the equation for changing the film takes an even simpler form:
$\frac{d\delta}{dt}=\frac{{K}_{p}}{2\delta}{\chi}_{Fe}^{\text{'}}$ (34)
This equation corresponds to the Tedmon model (
$t=\frac{\delta {\delta}_{0}}{{\chi}_{Fe}^{\text{'}}}\frac{{K}_{p}}{2{\chi}_{Fe}^{\text{'}2}}\left[\mathrm{ln}\left1\frac{2{\chi}_{Fe}^{\text{'}}}{{K}_{p}}\delta \right\mathrm{ln}\left1\frac{2{\chi}_{Fe}^{\text{'}}}{{K}_{p}}{\delta}_{0}\right\right]$ (35)
where δ_{0} – initial film thickness.
Expression (35) shows that the film thickness at t → ∞ reaches its limit ${\delta}_{c}=\frac{{K}_{p}}{2{\chi}_{Fe}^{\text{'}}}$ .
At δ_{0} << δ_{c} and $\frac{2{\left({\chi}_{Fe}^{\text{'}}\right)}^{2}}{{K}_{p}}\ll 1$ expression (35) can be approximately written as
$\delta {\delta}_{0}={\left({K}_{p}t\right)}^{1/2}\frac{2}{3}{\chi}_{Fe}^{\text{'}}t={\left({K}_{p}t\right)}^{1/2}\left(1\frac{\sqrt{2}}{3}\sqrt{\left.\frac{2{\chi}_{Fe}^{\text{'}2}}{{K}_{p}}t\right)}\right.$ (36)
In this case, we can assume that the parabolic film growth is quite well performed and the film corrosion can be neglected at times $t\ll \frac{{K}_{p}}{2{\chi}_{Fe}^{\text{'}}}$ .
The derivation of the film growth rate constant of the parabolic law (31) is based on the model (
Since in reality the film for the overwhelming majority of materials under consideration is not pure magnetite, but is at least twolayer, the models developed in (Topfer 2005;
Note that in (
The parabolic dependence of the film growth (31), in principle, is quite well confirmed experimentally (
In the general case, the film growth constant can be written as:
${K}_{p}=A\mathrm{exp}\left(\frac{Q}{RT}\right)\left({\left({P}_{{O}_{2}}^{\gamma}\right)}^{n}{\left({P}_{{O}_{2}}^{\delta}\right)}^{n}\right)$ (37а)
P ^{γ}_{O2}, P ^{δ}_{O2} – partial oxygen pressure at the interface between the coolantoxide and steeloxide media, respectively, while unknown parameters are determined empirically.
The partial pressure at the coolantoxide interface, using relation (10), is defined as:
${P}_{{O}_{2}}^{\gamma}={\left(\frac{{C}_{O}}{{C}_{O}^{S}}\right)}^{2}\mathrm{exp}\left(\frac{2\Delta {G}_{PbO}}{RT}\right)$ (37b)
The value of the partial pressure at the oxidesteel interface is much smaller than the value of the partial pressure of oxygen at the surface and can be neglected (Robertson and Manning 1988). In the case of lead – bismuth coolant, it can be estimated that since the binding energy of bismuth oxide is much lower than the binding energy of lead oxide, the formation of the former does not occur until a significant amount of lead oxide is formed. Therefore, in all calculations, the formation of bismuth oxide is neglected for its smallness.
When describing the film growth constant using formula (37), the ratio of fluxes at the boundary (22) takes the following form:
$\frac{4{M}_{O}{C}_{Fe}^{\delta}}{3{M}_{Fe}}\left(\frac{{K}_{p}}{2\delta}\frac{{\chi}_{Fe}\left({C}_{Fe}^{\gamma}{C}_{Fe}^{\beta}\right)}{{C}_{Fe}^{\delta}}\right)\approx {\chi}_{0}\left({C}_{0}^{\beta}{C}_{0}^{\gamma}\right)$ (38)
Given the discussion of relation (31), this expression means that $\left(\frac{{\delta}_{c}}{\delta}1\right)~{\chi}_{0}\left({C}_{0}^{\beta}{C}_{0}^{\gamma}\right)$, where ${\delta}_{c}=\frac{{K}_{p}}{2\frac{{\chi}_{Fe}\left({C}_{Fe}^{\gamma}{C}_{Fe}^{\beta}\right)}{{C}_{Fe}^{\delta}}}$ .
That is, upon reaching the maximum value of the film δ_{c}, the oxygen concentrations in the film and in the volume are equal. That is, the global isoconcentration regime is established.
Based on the foregoing, in the previous section, we rewrite equation (38) in the following form:
$\frac{4{M}_{O}}{3{M}_{Fe}}\left(\frac{{\delta}_{c}}{\delta}1\right)\approx \frac{{\chi}_{0}\left({C}_{0}^{\beta}{C}_{0}^{\gamma}\right)}{{\chi}_{Fe}\left({C}_{Fe}^{\gamma}{C}_{Fe}^{\beta}\right)}$ (39)
The expression for the maximum value of the film, provided that there is no iron in the coolant flow and the saturation concentration of iron is reached at its surface, takes the form:
${\delta}_{c}=\frac{{K}_{p}}{2{\chi}_{Fe}{C}_{Fe}^{s}{a}_{Fe}^{\gamma}\rho}\frac{3{M}_{Fe}}{{M}_{ox}}{\rho}_{ox}\approx \frac{0.2{K}_{p}}{{\chi}_{Fe}{C}_{Fe}^{s}{a}_{Fe}^{\gamma}}$ (40)
Here, the densities of lead and magnetite are taken for temperature 550 °С, C ^{S}_{Fe} – fractional concentration of iron saturation. Thus, the film growth condition from the initial value δ_{0}, ${\delta}_{c}=\frac{0.2{K}_{p}}{{\chi}_{Fe}{C}_{Fe}^{S}{a}_{Fe}^{\gamma}}>{\delta}_{0}$ , which is accompanied by simultaneous natural conditions (39) that the concentration of oxygen in the coolant is greater than at the wall and vice versa for iron. Note that the parameter δ_{c} ≡ δ_{c} (T,P ^{γ}_{O2},υ,d_{h}) will characterize either the growth or dissolution of the film with respect to the initial thickness to an equilibrium value δ_{c}.
From relations (22) and (38) it follows that the condition of the parabolic law of film growth corresponds to the condition
$\frac{{J}_{Fe}^{\delta}}{{J}_{Fe}^{\beta}}=\frac{{\delta}_{c}}{\delta}\gg 1$ (41)
Based on the smallness parameter for equation (39), parabolic behavior is expected for times $t\ll {t}_{k},{t}_{k}=\frac{{K}_{p}}{2{\left({\chi}_{Fe}^{\text{'}}\right)}^{2}}$ . The relation also holds ${\delta}_{c}^{2}=\frac{{K}_{p}}{2}{t}_{k}$ . Thus, if we conditionally consider the fulfillment of the parabolic law with time less than t ≈ t_{k} / 20, which is due to the smallness parameter of equation (36), then this is expected for film thickness no more than $\delta \approx {\delta}_{c}/\sqrt{10}$ . The latter meets the condition $\frac{{J}_{Fe}^{\delta}}{{J}_{Fe}^{\beta}}\approx \sqrt{10}$ .
Let us estimate the order of times at which parabolic growth of the oxide film is expected. Based on the available experimental information (
${\chi}_{Fe}^{\text{'}}=\frac{{\chi}_{Fe}\left({C}_{Fe}^{\gamma}{C}_{Fe}^{\beta}\right)}{{C}_{Fe}^{\delta}}\approx {\chi}_{Fe}{c}_{Fe}^{s}{a}_{fe}^{\gamma}\rho \frac{{M}_{ox}}{3{M}_{Fe}{\rho}_{ox}}\approx {10}^{10}{a}_{fe}^{\gamma}$ , [m/s] (42)
Let us take the activity of iron at the surface of the film of the order 10^{1}–10^{2}, and mass transfer constant χ_{Fe} ~ 10^{4}[m/s] at characteristic parameters of temperatures and coolant velocity. Therefore, ${t}_{k}=\frac{{\delta}_{c}}{{\chi}_{Fe}^{\text{'}}}$ ~ 300–3000 hours. The latter means that parabolic behavior is expected at a time of less than 15–150 hours. Thus, if the experimentally observed film thicknesses are relatively close to the saturation film, then a parabolic approximation to this value will take place over a relatively short period of time, up to several hundred hours.
In order for the parabolic film growth to last for several thousand hours, the saturation parameter of the film growth δ_{c} must be in the order of 10^{2} m.
This corresponds to the fact that the film growth constant should be of the order of 10^{11} m^{2}/s at Т = 600 °С and an order of magnitude or two less at temperatures up to 400 °С. Such a film growth constant is very large and gives a huge value of the film thickness, which does not correspond to the known experimental data.
When the activity of iron at the surface of the film is of the order 10^{3} parabolic film growth is provided for times from several thousand hours to several tens of thousands of hours depending on temperature.
An oxide film, as a rule, consists of two different layers and a transition region between the inner oxide layer and steel (Robertson and Manning 1988;
Numerous experimental data show that the layers of magnetite and spinel are comparable in their thicknesses, which is explained in principle in the framework of the "accessible space" model (Robertson and Manning 1988;
Based on the foregoing and following the ideas developed in (Robertson and Manning 1988;
$\frac{1}{S}\frac{\partial {\nu}_{ox}}{\partial t}\equiv \frac{1}{S}\frac{\partial {\nu}_{Sp}}{\partial t}+\frac{1}{S}\frac{\partial {\nu}_{mag}}{\partial t}=\frac{{\rho}_{mag}}{{M}_{mag}}\frac{\partial {\delta}_{mag}}{\partial t}+\frac{{\rho}_{Sp}}{{M}_{Sp}}\frac{\partial {\delta}_{Sp}}{\partial t}=\frac{{J}_{o}^{\beta}}{4{M}_{o}}$ ,
where υ_{ox}, υ_{mag}, υ_{Sp} – the amount of the substance of the oxide, which is the sum of the substance of magnetite and spinel. Based on the above accepted model, spinel growth is proportional to magnetite growth, δ_{Sp} = αδ_{mag} and the film growth equation can be rewritten as:
$\frac{{\rho}_{ox}}{{M}_{ox}}\frac{\partial \delta}{\partial t}\equiv \frac{{\rho}_{mag}}{{M}_{mag}}\left(\frac{1+\alpha \frac{{\rho}_{Sp}{M}_{mag}}{{\rho}_{mag}{M}_{Sp}}}{1+\alpha}\right)\frac{\partial \delta}{\partial t}=\frac{{J}_{o}^{\beta}}{4{M}_{o}}$ ,
or
$\left(\frac{1+\alpha \frac{{\rho}_{Sp}{M}_{mag}}{{\rho}_{mag}{M}_{Sp}}}{1+\alpha}\right)\frac{\partial \delta}{\partial t}=\frac{{J}_{o}^{\beta}}{4{M}_{o}}\frac{{M}_{mag}}{{\rho}_{mag}}$ , (43)
where ρ_{mag,Sp}, M_{mag,Sp} – densities and molar masses of magnetite and spinel Fe_{3x}Cr_{x}O_{4}, where x reflects the stoichiometry of the compound, respectively. Note that the densities of spinel and magnetite are fairly close to each other (
The interface between the outer and inner layers with sufficient accuracy coincides with the original metal surface (Robertson and Manning 1988;
$\frac{{\delta}_{out}}{{\delta}_{in}}=\frac{x\left(1+{C}_{Fe}/{C}_{Cr}\right)}{3}1$ , (44)
where C_{Fe} and C_{Cr} determine the concentration of iron and chromium in the alloy (mol /m^{3}).
It should be noted that the steel studied in sufficient detail for the structure of the oxide film does not give an unambiguous answer to the question of the presence or absence of an external layer of magnetite. But, in general, it can be argued that under nominal and similar modes, and in the case of dynamic experiments (coolant velocity 1–3 m/s)) (
In static experiments, as a rule, a sufficiently bright twolayer structure of the film is observed and, as a rule, it is significantly thicker compared to dynamic experiments, in equal conditions.
For an individual unit volume of coolant, the activities of oxygen, iron and magnetite in a state of chemical equilibrium are connected by the relation (the coolant index β in this section is omitted):
$k=\frac{{a}_{F{e}_{3}{O}_{4}}^{\tau +\Delta \tau}}{{\left({a}_{O}^{\tau +\Delta \tau}\right)}^{4}{\left({a}_{Fe}^{\tau +\Delta \tau}\right)}^{3}}$ . (45)
Chemical equilibrium occurs due to the loss of oxygen and iron, the accumulation of magnetite:
${a}_{F{e}_{3}{O}_{4}}^{\tau +\Delta \tau}={a}_{F{e}_{3}{O}_{4}}^{\tau}+\Delta {a}_{F{e}_{3}{O}_{4}};$${a}_{Fe}^{\tau +\Delta \tau}={a}_{Fe}^{\tau}\Delta {a}_{Fe};$${a}_{O}^{\tau +\Delta \tau}={a}_{O}^{\tau}\Delta {a}_{O}.$ (46)
The accumulation of magnetite is associated with a decrease in oxygen and iron ratio:
$\Delta {a}_{Fe}=\alpha \Delta {a}_{F{e}_{3}{O}_{4}}\frac{{C}_{s}^{F{e}_{3}{O}_{4}}}{{C}_{s}^{Fe}};$$\Delta {a}_{O}=\beta \Delta {a}_{F{e}_{3}{O}_{4}}\frac{{C}_{s}^{F{e}_{3}{O}_{4}}}{{C}_{s}^{O}};$$\alpha +\beta =1.$ (47)
Considering the stoichiometric coefficients of the reaction of magnetite formation and the ratio of the atomic weights of oxygen and iron, we can obtain the coefficients α = 0.726 and β = 0.274.
Combining relations (45–47), we can obtain an equation for determining the profit of magnetite in the process of establishing equilibrium:
${a}_{Fe}^{\tau}\alpha \Delta {a}_{F{e}_{3}{O}_{4}}\frac{{C}_{s}^{F{e}_{3}{O}_{4}}}{{c}_{s}^{Fe}}=\frac{{\left({a}_{F{e}_{3}{o}_{4}}^{\tau}+\Delta {a}_{F{e}_{3}{O}_{4}}\right)}^{1/3}}{{\left(k{\left({a}_{O}^{\tau}\beta \Delta {a}_{F{e}_{3}{0}_{4}}\frac{{C}_{s}^{F{e}_{3}{O}_{4}}}{{C}_{s}^{O}}\right)}^{4}\right)}^{1/3}}.$ (48)
Equation (48) is solved at each time step by the tangent method. According to the found values of the magnetite profit, the values of the loss of oxygen and iron are found at each step of solving the transport equations. Thus, the volume sources of impurities in the coolant are calculated by the relations:
${G}_{F{e}_{3}{O}_{4}}=\Delta {a}_{F{e}_{3}{O}_{4}}{C}_{s}^{F{e}_{3}{O}_{4}}/\Delta \tau ;$ (49)
${G}_{Fe}=\alpha \Delta {a}_{F{e}_{3}{O}_{4}}{C}_{s}^{F{e}_{3}{O}_{4}}/\Delta \tau ;$
${G}_{O}=\beta \Delta {a}_{F{e}_{3}{O}_{4}}{C}_{s}^{F{e}_{3}{O}_{4}}/\Delta \tau $
For the nearwall cells of the computational grid, surface flows of oxygen and iron are added to volume sources.
As a rule, the parabolic film growth constant is determined experimentally in a gaseous medium or under the condition that the processes of corrosion (dissolution) of the film and erosion can be neglected. Obviously, these two effects will be quite small in a static experiment. Under the assumption that the yield of iron from steel to the coolant is small compared to its consumption by an oxide film, the Martinelli approximation (29) should work well, which ensures parabolic film growth when solving equation (28). In this case, the oxygen flux is inversely proportional to the thickness of the oxide film. Thus, to determine the parabolic film growth constant, it is necessary to measure the oxidation rate of the surface at various temperatures and oxygen concentrations.
In the empirical determination of the parameters of the parabolic constant, the assumption of a parabolic growth of the oxide film in the static mode is fundamental. This assumption works fine provided that the diffusion yield of steel components in the coolant is small. In passivation mode in the initial period of formation of oxide films (
Such experiments were carried out at JSC "SSC RFIPPE" (
Based on the available experimental data (
${K}_{p}=\mathrm{exp}\left(39.5\right){p}_{{O}_{2}}^{0.125}\mathrm{exp}\left(\frac{30000}{T}\right),[\mu {m}^{2}/h],$ (50)
Similar experimental studies, by definition, were also carried out for EP302 steel. The following dependence is proposed for a parabolic constant:
${K}_{p}=\mathrm{exp}\left(31\right){p}_{{O}_{2}}^{0.125}\mathrm{exp}\left(\frac{20000}{T}\right),[\mu {m}^{2}/h]$ (51)
For comparison, we present the dependences of parabolic constants for the basic oxygen regime, obtained in (
For steels of ferriticmartensitic class (9–12Cr) НТ9, Т91 the following dependence is obtained:
${K}_{p}=1.156\times {10}^{3}exp\left(\frac{135693}{RT}\right),\left[{\mathrm{cm}}^{2}/\mathrm{s}\right]$ (52)
For austenitic steel AISI 316L:
${K}_{p}=1.156\times {10}^{3}\mathrm{exp}\left(\frac{135693}{RT}\right),\left[{\mathrm{cm}}^{2}/\mathrm{s}\right]$ (53)
In (
${K}_{p}=2.096\times {10}^{5}{c}_{o}^{0.27}\mathrm{exp}\left(\frac{157802}{RT}\right),\left[{\mathrm{m}}^{2}/\mathrm{s}\right],$ (54)
And for steel HT9:
${K}_{p}=9.735\times {10}^{8}{c}_{o}^{0.27}\mathrm{exp}\left(\frac{126802}{RT}\right),\left[{\mathrm{m}}^{2}/\mathrm{s}\right],$ (55)
where с_{0}(ppm, millionth share) ‒ mass fraction of oxygen.
An analysis of the empirical dependences of parabolic constants (Fig.
It was also noted in (
– K_{p,с} ~ 0, metal corrosion without oxide formation;
– K_{p,с} < Q_{c,o}, oxidation affects corrosion;
– K_{p,с} > Q_{c,o}, corrosion oxidation process;
– Q_{c,o} ~ 0, oxide corrosion, slight erosion;
– K_{p,с} = Q_{c,o}, no new oxide formations.
The general concept of corrosion oxidation due to mass transfer effects is the assumption that the rate of corrosion oxidation is proportional to the corrosion of the metal itself (
Q_{c,0} ~ Q_{c} = χ (c_{s} – c_{b}) ~ χ_{o} (ν,D,d)V^{n} exp(q (ν,D,d)/RT), (56)
where Q_{c} ‒ corrosion rate of the actual metal surface, V ‒ coolant speed, χ_{o} and q ‒ constants depending on the properties of the coolant and flow geometry, n – determined by type of flow (
At low coolant speeds, the corrosion rate of a metal surface is fully or partially controlled by the mass transfer effect (
Thus, using the approach of (
$\frac{{Q}_{c,o}}{{Q}_{c}}=\frac{{M}_{ox}{\rho}_{metall}}{3{M}_{Fe}{\rho}_{ox}}{a}_{Fe}^{\gamma}\approx {a}_{Fe}^{\gamma}\equiv {a}_{Fe}^{\gamma}\left(d,T,{C}_{0}\right)$ (57)
Thus, the activity of iron at the surface of the oxide film gives an understanding of how much the corrosion process on the film is slower compared to corrosion of the metal surface. As a rule, this value is about 100–1000. Based on the chemical equilibrium on the wall, the greater the oxygen activity in the coolant, the lower is the activity of iron and the less corrosion of the surface. Of course, it must be kept in mind that a large (close to saturation) oxygen activity can lead to excessive and undesirably large film growth, that is, the yield of iron can be minimized by an increase in oxygen in the coolant, but thereby its output will increase, in fact, in the oxide film, which also is not always desirable.
There are several expressions of mass transfer coefficients (Figs
k_{b₋h} = 0.0165υ⁻^{0.53}D^{0.67}V^{0.86}d⁻^{0.14}
k_{s} = 0.0177υ⁻^{0.579}D^{0.704}V^{0.875}d⁻^{0.125} (58)
k_{h₋h} = 0.0096υ⁻^{0.567}D^{0.654}V^{0.913}d⁻^{0.087}
where υ ‒ viscosity, D ‒ diffusion coefficient, V ‒ speed and d ‒ hydraulic diameter.
Using formulas (58), numerical values are obtained for the mass transfer coefficients, but these differences do not exceed 20% at a speed of 2 m/s and less at lower speeds. In further calculations, we use the expression k_{bh}. Fig.
When using experimental data to verify the calculation models, it is not always possible to extract data on the geometry of the working sections with sufficient accuracy, but from general considerations, the limits of the hydraulic diameters of these sections can be estimated. As a rule, hydraulic diameters are within the range of 10–100 mm. Corrosion rate parameter change χ_{Fe} (34) when changing the hydraulic diameter from 10 to 100 mm lies within 20%, as can be seen in Fig.
The main goal of a computer program is to calculate the thickness of an oxide film and the associated concentrations of oxygen and iron. Thermohydraulic calculations are a necessary basis, but the accuracy of the calculated thermohydraulic characteristics (in our case, this is the field of temperatures and velocities) does not have to exceed the accuracy of the calculation of physicochemical processes (in our case, this is the thickness of the oxide film and the concentrations accompanying it). The main errors in the calculation are introduced by the uncertainty of empirical parameters.
The film growth in the static case is parabolic in nature. It is static experiments that serve as the basis for determining the parameters of the parabolic constant (37a), which is determined by the oxygen consumption of the film. Based on the fact that in the static case there is a parabolic dependence of the film thickness growth δ^{2} ~ K_{p}∙t, then the relation between the relative errors of the film thickness and the uncertainty of the parabolic constant is written as ε_{Kp} = 2ε_{δ}.
The experiments on oxygen consumption (
${\epsilon}_{{K}_{p}}=\sqrt{{\left({\epsilon}_{A}\right)}^{2}+{\left(Q/T{\epsilon}_{q}\right)}^{2}}$ (60)
Since, based on the available data, it is impossible to unambiguously indicate which of the uncertainties makes the largest contribution, we accept this contribution to be the same, i.e.
${\epsilon}_{A}=Q/T{\epsilon}_{Q}=0.6/\sqrt{2}\approx 0.43$ (61)
That is, uncertainty ε_{A} may be high enough while uncertainty ε_{Q} should be small since Q/T >> 1.
Thus, during variational calculations of the film thickness and impurity concentrations, the boundaries of the varied parameters are estimated А and Q.
The errors of partial pressure and average flow rate, temperatures are taken from the available experimental data.
In dynamic experiments, the flow rate and hydraulic diameter were additionally varied. The speed varied based on the data of the error of the flow meter, this value, as a rule, lies within ±10%. The hydraulic diameter ranged from 10–100 mm (see previous section).
Based on the selected intervals, the average deviation of the calculated value from the experimental ones was calculated to get the relative error
$\epsilon =\frac{1}{N}\sum _{i=1}^{N}\frac{abs\left({x}_{i}^{calc}{x}_{i}^{\mathrm{exp}}\right)}{{x}_{i}^{\mathrm{exp}}}$ (62)
where x_{i}^{calc}, x_{i}^{exp} calculated and experimental values of the simulated value, respectively.
Subsequent numerical calculations showed that the errors in the calculation of dissolved oxygen concentrations are not more than 15% when comparing the results obtained using the MASKALM and STARCCM + codes, and the oxide film thickness is not more than 30% for the static case (stationary coolant) and not more than 55% for the dynamic case (coolant with average speeds of 0.1–3 m/s) when comparing the calculated and experimental data.
When considering experimental errors for oxygen concentration, it should be noted that at the moment in the literature there is no described wellestablished measurement procedure with a proven error in a wide range of oxygen concentrations and it, in principle, can reach significant values. When considering experimental errors for oxide film thicknesses, we note that experimental results are very often presented with no errors at all, and those data presented with errors should be considered as a spread of experimental data without evaluating the experimental errors themselves (see, for example, Fig.
The joint solution of equations (34) and (38) within the framework of the point model, that is, without taking into account the exact geometry (Tadmon’s approach), allows a reasonable accuracy to estimate the film growth rate depending on temperature and coolant velocity. For this, we used the film growth parameters selected for EP823 steel, since for this grade of steel there is a fairly large amount of experimental data.
The growth of an oxide film as a function of time is considered in detail in a subsequent chapter. The most indicative characteristics parameters of film growth are its maximum value (see equations (34–36)) and the characteristic film growth time according to the parabolic law $t\ll {t}_{k},{t}_{k}=\frac{{K}_{p}}{2{\left({\chi}_{Fe}^{\text{'}}\right)}^{2}}$ . Based on (40), the saturation yield corresponds to the equality of the fluxes of iron through the film and in the coolant, J ^{δ}_{Fe} = J ^{β}_{Fe}, which corresponds to a zero flow of oxygen. This, of course, is performed provided that a constant concentration is maintained at the selected oxygen regime.
Figs
The growth of the oxide film upon saturation, depending on the coolant speed, is shown in Fig.
Also, calculations showed that at temperatures of 450–650 °C, parabolic film growth is expected for times from less than an hour to several hundred hours (Fig.
The calculations shown in Fig.
Figs
The authors of (
This section provides calculations of model oxygen distributions at various temperatures. Test calculations of the elementary model of HLMC flow in a pipe were carried out using the commercial StarCCM+ code and MASKALM code of own design.
The STAR CCM+ code interface allows you to use userdefined functions to model certain physical and chemical processes. The system of equations (33), (38), functions dependent on them, and semiempirical dependencies of type (2, 20–25) were incorporated into the code for joint solution with the STAR CCM+ thermohydraulic models.
For numerical analysis, a simple model was chosen for the flow of lead coolant of a given temperature in the pipe: the length of the computational domain is 1 m; diameter 50 mm; design grid 1 million cells; Reynolds number is 10^{5}; input mass concentration of oxygen 2∙10^{8} kg/kg. To simulate oxygen corrosion, the following value was selected for the drain (consumption) of oxygen to the wall 4∙10^{8} kg/m^{2}∙s. The selected amount of oxygen consumption is very significant compared to that usually found in real calculations and was chosen only to demonstrate the distribution of oxygen.
When using the MASKALM code, a rectangular computational grid with constant spatial steps in the Cartesian coordinate system, consisting of 275,000 cells (50 × 50 × 110) (along the directions of the coordinate axes) was entered into the computational domain. That is, the number of cells and the quality of the grid is noticeably inferior to the StarCCM+ code.
The results of calculations using the StarCCM+ and MASKALM codes are shown in Figs
Thus, a turbulent diffusion coefficient of the order of (1–3)*10^{5} gives a completely adequate description of the oxygen distribution with an error of no more than 15% at the surface. With smaller oxygen sinks, the scatter of values is even smaller. The scatter in the values of the coefficient of turbulent diffusion gives a very small error in the thickness of the oxide film.
This section presents model calculations of the growth of an oxide film in an EP823 steel pipe and a beam of rods at various flow rates of HLMC and temperatures.
Figs
Fig.
A similar behavior of the increase in the thickness of the oxide film is also observed at a temperature of T = 500 °C. The main difference from the previous case is that the approximation to the asymptotic value of the film thickness occurs much later (Fig.
Fig.
For the following geometry, a bunch of rods was chosen with a diameter of 10 mm, a length of 0.97 m, in a triangular package, with a step of 14 mm; the simulated area is cut off along the planes of symmetry, and the grid contains 3.1 × 10^{6} cells when using the STAR CCM+ code; mass oxygen content of 10^{8} kg/kg; temperature 500 °С, 650 °С; coolant velocity 1.7 m/s (Re~10^{5}), turbulence model Kε when using STAR CCM+ code.
Fig.
Note that in the above examples only isothermal calculations were considered.
An engineering model is presented for a selfconsistent calculation of the growth of an oxide film in circulation loops with a heavy liquid metal coolant and concentrations of impurities (oxygen, iron, magnetite). The modeling of thermohydraulic and physicochemical processes is based on solving the associated threedimensional equations of hydrodynamics, heat transfer, convectivediffusive transport, and the formation of chemically interacting impurity components in the coolant volume and on the surface of steels.
The change in the thickness of the oxide film on the metal surface and the corresponding exit of steel components into the coolant substantially depends on the steel grade. For a more adequate justification of the evolution of the oxide film, a semiempirical model is proposed for using the empirical parameterization of the parabolic constant not only in the equation for changing the thickness of the oxide film, but also in the mass balance equation associated with it. The parabolic constant, which is determined by the degree of oxygen consumption by a steel, obviously significantly depends on the type of this steel and on the method of its preparation. Therefore, the direct application of the Wagner approach is unlikely to adequately describe such differences in steels, while the experimental parametric dependences on oxygen consumption by steel depending on temperature and oxygen partial pressure are obviously unique for each steel.
For an adequate experimental determination of the parametric dependences for a parabolic constant, at least two conditions are necessary: the experiment should be carried out for a static case or the average coolant velocity should be low in order to exclude the influence of corrosionerosion processes; steel should have a sufficient initial oxide film to exclude the influence of the initial large flow of iron, and possibly corrosion of steel, since with a large yield of iron the film growth does not have to obey the parabolic law even in the static case.
The developed approach regarding accounting for the main physical and chemical processes is used not only in our MASKALM code, but is also implemented as userdefined functions in the STAR CCM+ code. The crossverification calculations of the two test models showed a fairly good agreement between the results in terms of describing the increase in the thickness of the oxide film.
In the future, the developed approach will be applied to more complex systems in terms of geometry, expanded for nonisothermal loops, and applied to simultaneously account for the formation of deposits in loops.
Other Necessary Closing Relations
As the closing relations for the system of equations of motion and equations for the transport of impurities, we used dependences to estimate the values of the coefficients of turbulent thermal conductivity, turbulent viscosity, and turbulent diffusion (
To calculate the coefficient of turbulent viscosity, we use the relations of the threelayer Karman model (
$\left\{\begin{array}{c}\frac{{v}_{T}}{v}=0\text{for}{\mathrm{y}}_{+}<5\\ \frac{{v}_{T}}{v}=\frac{{y}_{+}}{5}1\text{for}5\le {y}_{+}\le 30\\ \frac{{v}_{T}}{v}=\frac{{y}_{+}}{2,5}1\text{for}{\mathrm{y}}_{+}>30\end{array}\right.$
where ν_{Т} – turbulent viscosity,
${y}^{+}=\frac{u*y}{\nu}$ – dimensionless coordinate in the boundary layer, ν – kinematic viscosity,
${u}_{*}=\sqrt{\frac{\tau}{\rho}}$ – dynamic speed, τ – wall shear stress, ρ – density. For pipes with a circular crosssection and the average shear stress along the perimeter of the bar in the assembly of smooth bars, the shear stress is calculated by the formula $\tau =\frac{\xi \rho {u}^{2}}{8}$ (
For pipes with circular crosssection at Re = 4 × 10^{3} ₋ 10^{8}:
ξ = 1/(1.821∙lgRe₋1.64)^{2}
For assembling smooth rods in triangular grid with Re = 6 × 10^{3} ₋ 10^{8}:
ξ = ξ_{0} (1+(h/d₋1)^{0.32}), ξ_{0} = 0.21/Re^{0.25}, where h – grid period, d – rod diameter.
In (
According to the Prandtl model, the average turbulent viscosity at y ≥ δ (that is, in the flow core) is calculated by the formula (
ν_{T} = 0.4u*δ , where the thickness of the boundary layer is calculated in the formulation of the threelayer Karman model (derivation of the relation in [51]):
$\delta =\frac{v}{{u}_{*}}30\mathrm{exp}\left[0.4\left(\frac{\overline{u}}{{u}_{*}}13.96\right)\right]$
Thus, the turbulent viscosity in the flow is defined as:
$\frac{{\nu}_{T}}{\nu}=12\mathrm{exp}\left[0.4\left(\frac{\overline{u}}{{u}_{*}}13.96\right)\right]$
The formulas used to determine the turbulent viscosity are not the only possible ones for use in calculations. In (
Estimates of the coefficient of turbulent diffusion of an admixture are based on the analogy of transport processes, the turbulent Schmidt number Sc_{T} = ν_{T} / D_{T} ≈ 1.