For simplicity, we consider the growth of a single-layer film. In the developed (Kumaev et al. 2002, 2005; Alekseev et al. 2010c) model of the growth of a single-layer film, at the first stage, the iron flux "remaining" in the wall layer is determined as the difference between the fluxes through the oxide film and the flux that has gone into the coolant. The "oncoming" oxygen flow from the coolant is determined according to the stoichiometry of the resulting magnetite.

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 ⇔ Fe₃O₄ + 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 steel-coolant boundary, we distinguish four subregions: steel (conventionally Fe), oxide, an intermediate diffusion layer, and HLMC (Fig. 1).

Intermediate layer is a virtual layer where the above-described surface reaction is carried out (5). For flows of components and sources of impurities in the media presented in Fig. 1, we use the notation: J ^β_O – O (PbO) oxygen flow in the coolant; J ^β_Fe – Fe oxygen flow in the coolant; J ^δ_O – oxygen flow in oxide film; J ^δ_Fe – iron flow in oxide film; $J_O^{\gamma }~\frac{\partial v_O^{\gamma }}{\partial t}$ – oxygen flow (source) in the intermediate layer; $J_{Fe}^{\gamma }~\frac{\partial v_{Fe}^{\gamma }}{\partial t}$ – iron flow (source) in the intermediate layer; v ^γ_O,Fe – the amount of oxygen and iron in the intermediate layer, for impurity concentrations , where n = O, PbO, Fe, Fe₃O₄ and activity $a_n=\frac{C_n}{C_n^S}$ , C ^S_n – saturation concentration.

The chemical equilibrium of the oxygen-lead system with the formation of the [PbO] complex is described in the standard way:

$k_{\beta }^{ox}=\frac{a_{PbO}}{a_{Pb}{a}_O^{\beta }}=\exp \left(-\frac{\Delta {\mathrm{G}}_{\mathrm{Pb}\mathrm{O}}}{RT}\right)$ (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 (Nuclear Energy Agency 2015) for the Gibbs energy parameters for the formation of lead oxide and relation (6), it is easy to estimate that a_PbO / a_O ~10⁸, so for oxygen activity it is enough to understand the activity of the complex [PbO].

For HLMC medium with impurities (Pb, PbO, Fe, Fe₃O₄) we introduce the notation β. In the presence of iron in the coolant, three reactions are formally possible:

Pb + O = PbO

3Fe + 4O = Fe₃O₄ (7)

3Fe + 4PbO = Fe₃O₄ + 4Pb

Chemical equilibrium constant for the second reaction:

$k_{\beta }^O=\frac{1}{{\left(a_{Fe}^{\beta }\right)}^3{\left(a_0^{\beta }\right)}^4}=\exp \left(-\frac{\Delta {\mathrm{G}}_{{\mathrm{Fe}}_3}{\mathrm{O}}_4}{RT}\right)$ (8)

The chemical equilibrium constant for the third reaction, provided that the activity of magnetite is equal to unity:

$k_{\beta }^m={\left(a_{Fe}^{\beta }\right)}^3{\left(a_0^{\beta }\right)}^4=\exp \left(-\frac{4\Delta {\mathrm{G}}_{\mathrm{PbO}}-\Delta {\mathrm{G}}_{{\mathrm{Fe}}_3{O}_4}}{RT}\right)$ (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 lead-iron-oxygen 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 film-coolant (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:

Fe₃O₄ (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:

$k_{\gamma }={\left(a_{Fe}^{\gamma }\right)}^3{\left(a_O^{\gamma }\right)}^4=\exp \left(-\frac{4\Delta {\mathrm{G}}_{\mathrm{PbO}}-\Delta {\mathrm{G}}_{{\mathrm{Fe}}_3{O}_4}}{RT}\right)$ (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 "one-step" 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 oxygen-coolant 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 Fe₃O₄ (δ medium). In general, film growth also occurs on the metal-magnetite 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 metal-magnetite 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:

$k_O^{\delta }=\frac{1}{{\left(a_O^{\delta }\right)}^4}=\exp \left(-\frac{\Delta {\mathrm{G}}_{{\mathrm{Fe}}_3{O}_4}}{R{T}^{\delta }}\right)$ , (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)⁴C_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 steel-coolant 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 (Alekseev et al. 2010b, c) is a special case of the proposed model. According to (Alekseev et al. 2010b, c), iron fluxes at the interface can be calculated as follows:

${\left.J_{Fe}^{\delta }\right|}_{\delta }=D_{Fe}^{\delta }\frac{C_{Fe}^{\delta }-C_{Fe}^{\gamma }}{\delta }$ (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 (Alekseev et al. 2010b, c) and β respectively:

${\left.J_O^{\delta }\right|}_{\delta }=2D_O^{\delta }\frac{C_O^{\delta }-C_O^{\gamma }}{\delta }$ (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 $\theta =\frac{4}{3}\left(\frac{4M_O}{3M_{Fe}}\right)$ – for molar (mass) flows, respectively, or

$\frac{4M_O}{3M_{Fe}}\left(D_{Fe}^{\delta }\frac{C_{Fe}^{\delta }-C_{Fe}^{\gamma }}{\delta }-{\chi }_{Fe}\left(C_{Fe}^{\gamma }-C_{Fe}^{\beta }\right)\right)={\chi }_O\left(C_O^{\beta }-C_O^{\gamma }\right)-2D_O^{\delta }\frac{C_O^{\delta }-C_O^{\gamma }}{\delta }$ , (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):

$J_{Fe}^{\delta }=-\frac{C_{Fe}^{\delta }{D}_{Fe}}{RT}\frac{d{\mu }_{Fe}}{dx}=\frac{1}{2}\alpha D_{Fe}{C}_{Fe}^{\delta }\frac{d\ln P_{O_2}}{dx}=\frac{2}{3}{D}_{Fe}{C}_{Fe}^{\delta }\frac{d\ln P_{O_2}}{dx}$ , (23)

where μ_Fe – chemical potential of iron, P_O2 – partial pressure of oxygen, x – coordinate, α=⁴/₃ for Fe₃O_4.

The fulfillment of relation (23) gives a constant flow of iron [10]:

$J_{Fe}^{\delta }\delta =\frac{2}{3}{C}_{Fe}^{\delta }{\int }_{a_{O_2}\left(0\right)}^{a_{O_2}\left(\delta \right)}{D}_{Fe}d\ln a_{O_2}$ (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., $C_{Fe}^{\delta }=\frac{3M_{Fe}{\rho }_{ox}}{M_O}$ [kg/m³] ‒ 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, $a_{O_2}\left(0\right)=\frac{3M_{Fe}{\rho }_{ox}}{M_{ox}{C}_{O_2}^S\left(0\right)}$ , 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 (Topfer et al. 1995; Zhang 2005; Martinelli et al. 2008a, 2008b; Mikityuk 2010):

$D_{Fe}=D_V^0\exp \left(-E_V/RT\right)P_{O_2}^{2/3}+D_I^0\exp \left(-E_I/RT\right)P_{O_2}^{-2/3}$ , (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)

$A=D_V^0\exp \left(-E_V/RT\right){\left(P_{O_2}^0\right)}^{-2/3}$ ,

$B=D_I^0\exp \left(-E_I/RT\right){\left(P_{O_2}^0\right)}^{2/3}$ ,

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:

$\frac{4M_O}{3M_{Fe}}\left(J_{Fe}^{\delta }-{\chi }_{Fe}{C}_{Fe}^{\gamma }\right)\approx {\chi }_0\left(C_0^{\beta }-C_0^{\gamma }\right)$ (27)

Using the relations (26, 27) and k_γ = (a ^γ_Fe)³(a ^γ₀)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 (Maruyama et al. 2004) and Zhang (Zhang and Li 2005).

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 (Martinelli et al. 2008a). In this case

$\frac{4M_O}{3M_{Fe}}{J}_{Fe}^{\delta }=J_O^{\beta }$ (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:

δ² = 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 (Zhang and Li 2005):

$J_O^{\beta }=\frac{4M_O}{3M_{Fe}}\left(J_{Fe}^{\delta }-J_{Fe}^{\beta }\right)\equiv \frac{4M_O}{3M_{Fe}}\left(J_{Fe}^{\delta }-{\chi }_{Fe}\left(C_{Fe}^{\gamma }-C_{Fe}^{\beta }\right)\right)$ (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 (Martinelli et al. 2008b). If the parameters of the equation are considered as constants, then the equation has an analytical solution:

$t=-\frac{\delta -{\delta }_0}{{\chi }_{Fe}^{\text{'}}}-\frac{K_p}{2{\chi }_{Fe}^{\text{'}2}}\left[\ln \left|1-\frac{2{\chi }_{Fe}^{\text{'}}}{K_p}\delta \right|-\ln \left|1-\frac{2{\chi }_{Fe}^{\text{'}}}{K_p}{\delta }_0\right|\right]$ (35)

where δ₀ – 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 δ₀ << 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 (Topfer et al. 1995), where, when modeling the diffusion of iron cations, the assumption was made that the concentration of vacancies and interstitial ion concentrations are independent of porosity and grain size of magnetite, that is, ion diffusion within the framework of the vacancy model for ideal "magnetite" (Hallstrom et al. 2011). Such models are quite fundamental in nature and allow one to calculate the diffusion coefficient of iron ions, which is necessary for calculating the film growth rate constant in the framework of the Wagner model. The main result of the model (Topfer et al. 1995) and similar ones (Diekmann 1998) is that in the region of low partial oxygen pressures $K_p~P_{O_2}^{2/3}~{\left(d_O^{\gamma }\right)}^{4/3}$ . Experimental studies of film growth on various alloys show slightly different dependences of the film growth constant on the partial pressure of oxygen than was obtained for pure magnetite using models of the type (Topfer 2005; Hallstrom et al. 2011).

Since in reality the film for the overwhelming majority of materials under consideration is not pure magnetite, but is at least two-layer, the models developed in (Topfer 2005; Hallstrom et al. 2011) it’s not necessarily to describe the dependence of the film growth rate constant (conditional name) on the partial oxygen pressure like $K_p~P_{O_2}^{2/3}$ . Since in the works (Surman 1973; Smith 1982; Saito. et al. 1985; Sato et al. 2002) the dependencies $P_{O_2}^{0.145}$, $P_{O_2}^{0.135}$, $P_{O_2}^{0.141}$ and $P_{O_2}^{0.279;0.313}$ have been experimentally found for different types of steel correspondingly. These empirical estimates imply the fulfillment of the parabolic law of film growth (31), i.e., without taking into account possible mass transfer processes leading to its dissolution (corrosion). The latter was taken into account in (33) and for large times provides a nonparabolic change in the growth of the oxide film. Such a different dependence of the film velocity constant is determined by both the composition of the metal so that by the properties of the grains, which leads to the failure of the theoretical dependence $K_p~P_{O_2}^{2/3}$ , characteristic to the "ideal" magnetite (Diekmann 1998). In (Sato et al. 2002), it was assumed that the film growth rate constant is generally determined by the diffusion of chromium ions in chromium oxide, the theoretical dependence of which $K_p~P_{O_2}^{3/16}$, which, in principle, is somewhat closer to the experimental values.

Note that in (Surman 1973; Smith 1982; Saito et al. 1988; Diekmann 1998; Sato et al. 2002) the oxidation of steel surface was carried out at low partial oxygen pressures typical for partial pressures in liquid lead, but in a gaseous medium. Therefore, the processes of film dissolution and erosion, in principle, could not play a significant role. The results of experiments in a stationary coolant (Steiner 2009, Askhadullin et al. 2011) led to an empirical dependence $P_{O_2}^{0.125}$ for steels EP823-(Sh) and E302-(Sh). For T91 steel, the available experimental data shows a dependence of the type $P_{O_2}^{0.11}$ .

The parabolic dependence of the film growth (31), in principle, is quite well confirmed experimentally (Aiello et al. 2004), if corrosion-erosion processes have low intensity (for example, in a gaseous medium), and the Wagner approach is currently the only reasonably substantiated approximation. Deviations from the parabolic law can be explained within the framework of the approach (Zhang and Li 2005; Steiner et al. 2008), in which the film growth is described by equation (34).

In the general case, the film growth constant can be written as:

$K_p=A\exp \left(-\frac{Q}{RT}\right)\left({\left(P_{O_2}^{\gamma }\right)}^n-{\left(P_{O_2}^{\delta }\right)}^n\right)$ (37а)

P ^γ_O2, P ^δ_O2 – partial oxygen pressure at the interface between the coolant-oxide and steel-oxide media, respectively, while unknown parameters are determined empirically.

The partial pressure at the coolant-oxide interface, using relation (10), is defined as:

$P_{O_2}^{\gamma }={\left(\frac{C_O}{C_O^S}\right)}^2\exp \left(\frac{2\Delta G_{PbO}}{RT}\right)$ (37b)

The value of the partial pressure at the oxide-steel 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:

$\frac{4M_O{C}_{Fe}^{\delta }}{3M_{Fe}}\left(\frac{K_p}{2\delta }-\frac{{\chi }_{Fe}\left(C_{Fe}^{\gamma }-C_{Fe}^{\beta }\right)}{C_{Fe}^{\delta }}\right)\approx {\chi }_0\left(C_0^{\beta }-C_0^{\gamma }\right)$ (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:

$\frac{4M_O}{3M_{Fe}}\left(\frac{{\delta }_c}{\delta }-1\right)\approx \frac{{\chi }_0\left(C_0^{\beta }-C_0^{\gamma }\right)}{{\chi }_{Fe}\left(C_{Fe}^{\gamma }-C_{Fe}^{\beta }\right)}$ (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:

${\delta }_c=\frac{K_p}{2{\chi }_{Fe}{C}_{Fe}^s{a}_{Fe}^{\gamma }\rho }\frac{3M_{Fe}}{M_{ox}}{\rho }_{ox}\approx \frac{0.2K_p}{{\chi }_{Fe}{C}_{Fe}^s{a}_{Fe}^{\gamma }}$ (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 δ₀, ${\delta }_c=\frac{0.2K_p}{{\chi }_{Fe}{C}_{Fe}^S{a}_{Fe}^{\gamma }}>{\delta }_0$ , 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 (Nuclear Energy Agency 2015), we determine the expected order of the film thickness in the region of several tens of microns, for example, 50 μm, and suppose that the film grew parabolic to this value. Then the maximum film thickness is about 150 microns. Further, the expected order of the mass transfer member can be estimated:

${\chi }_{Fe}^{\text{'}}=\frac{{\chi }_{Fe}\left(C_{Fe}^{\gamma }-C_{Fe}^{\beta }\right)}{C_{Fe}^{\delta }}\approx {\chi }_{Fe}{c}_{Fe}^s{a}_{fe}^{\gamma }\rho \frac{M_{ox}}{3M_{Fe}{\rho }_{ox}}\approx {10}^{-10}{a}_{fe}^{\gamma }$ , [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 m²/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; Martinelli et al. 2008a; Alekseev et al. 2010a; Nuclear Energy Agency 2015). The inner layer is chromium-rich spinel, in which the concentration of chromium is approximately equal to its concentration in the metal, while the outer layer is a porous iron-rich oxide. A thin transition layer is observed in which the iron concentration gradually changes from the concentration in the oxide to the concentration in the alloy, the chromium concentration is practically unchanged, and the oxygen concentration gradually changes from the concentration in the oxide to zero.

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; Martinelli et al. 2008a). The model assumes that the spinel layer Fe-Cr grows into an accessible space created by vacancies formed in steel due to diffusion of iron, while the mobility of iron ions is several orders of magnitude higher than the mobility of chromium ions. Vacancies can accumulate upto the formation of nanocavities and nanochannels, which facilitates the access of oxygen and provides the formation of spinel. But the rate of vacancy formation is completely determined by the diffusion of iron cations, and it is this process that determines the rate of both magnetite growth and the spinel formation rate. That is why the ratio of the thicknesses of magnetite and spinel Fe₁₃Cr_0.6O₄ is approximately 1.16 (Robertson and Manning 1988; Martinelli et al. 2008a). In turn, the degree of diffusion of iron cations depends on the partial pressure at the interface between the media and determines the rate constant of the (parabolic) film growth К_р (40).

Based on the foregoing and following the ideas developed in (Robertson and Manning 1988; Martinelli et al. 2008a), it does not make much sense to complicate the engineering approach for calculating the changes not only in the magnetite film, but also in its spinel part, since the latter is completely determined by the diffusion (departure) of iron cations from the metal into magnetite. Let’s consider, in general terms, an equation similar to equation (28) for film growth, taking into account spinel formation and magnetite growth:

$\frac{1}{S}\frac{\partial {\nu }_{ox}}{\partial t}\equiv \frac{1}{S}\frac{\partial {\nu }_{Sp}}{\partial t}+\frac{1}{S}\frac{\partial {\nu }_{mag}}{\partial t}=\frac{{\rho }_{mag}}{M_{mag}}\frac{\partial {\delta }_{mag}}{\partial t}+\frac{{\rho }_{Sp}}{M_{Sp}}\frac{\partial {\delta }_{Sp}}{\partial t}=\frac{J_o^{\beta }}{4M_o}$ ,

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:

$\frac{{\rho }_{ox}}{M_{ox}}\frac{\partial \delta }{\partial t}\equiv \frac{{\rho }_{mag}}{M_{mag}}\left(\frac{1+\alpha \frac{{\rho }_{Sp}{M}_{mag}}{{\rho }_{mag}{M}_{Sp}}}{1+\alpha }\right)\frac{\partial \delta }{\partial t}=\frac{J_o^{\beta }}{4M_o}$ ,

or

$\left(\frac{1+\alpha \frac{{\rho }_{Sp}{M}_{mag}}{{\rho }_{mag}{M}_{Sp}}}{1+\alpha }\right)\frac{\partial \delta }{\partial t}=\frac{J_o^{\beta }}{4M_o}\frac{M_{mag}}{{\rho }_{mag}}$ , (43)

where ρ_mag,Sp, M_mag,Sp – densities and molar masses of magnetite and spinel Fe_3-xCr_xO₄, where x reflects the stoichiometry of the compound, respectively. Note that the densities of spinel and magnetite are fairly close to each other (Zhang 2013) and since $\frac{{\rho }_{Sp}{M}_{mag}}{{\rho }_{mag}{M}_{Sp}}\approx 1$ with accuracy of less than 3%, then $\left(\frac{1+\alpha \frac{{\rho }_{Sp}{M}_{mag}}{{\rho }_{mag}{M}_{Sp}}}{1+\alpha }\right)~1$ and equation (43) takes the form of (31) with high accuracy. Thus, within the framework of this model, a separate calculation of the growth of magnetite and spinel films does not make much sense, but only significantly complicates the calculations.

The interface between the outer and inner layers with sufficient accuracy coincides with the original metal surface (Robertson and Manning 1988; Schroer et al. 2012; Tsisar et al. 2017). Based on the mass balance and on the assumption that iron does not enter the coolant, but remains in the oxide outer and inner layers, and that all oxidized chromium is in the inner layer, we can write the ratio of the thicknesses of the outer layer and the inner (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 /m³).

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)) (Zhang and Li 2007; Weisenburger et al. 2008; Schroer et al. 2009; Tsisar et al. 2017), the magnetite layer is most likely absent due to erosion or dissolution of iron oxide films in the coolant. Although in some cases the same steels (for example, T91) show the presence of a layer of magnetite in dynamic experiments (Alekseev et al. 2010a; Schroer et al. 2012).

In static experiments, as a rule, a sufficiently bright two-layer 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 near-wall 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 (Ivanov et al. 2005), the diffusion yield of metal components into the coolant is very intense, and its fraction can reach ~ 50% and the diffusion flux does not have to be proportional to the oxygen flux. According to the data of (Ivanov et al. 2005), for an oxidation duration of more than 400 hours, the proportion of steel components entering the coolant of the total amount of oxidized is ~ 1% and less, with an oxidation duration of 200 to 400 hours, this proportion is ~ 10%. Therefore, during long campaigns, at least more than several hundred hours, oxygen consumption occurs only on the surface and parabolic film growth in static experiments should be manifested to a greater extent than, for example, in dynamic ones. Thus, the empirical determination of the parabolic constant in the best way can be made by the thickness of the film after a sufficiently long oxidation (more than ~ 400 hours) and when using the static mode. Unfortunately, there are extremely few such experimental data.

Such experiments were carried out at JSC "SSC RF-IPPE" (Askhadullin et al. 2011), where a technique was developed that made it possible to estimate the flows of oxygen consumed for the oxidation of the steel surface under variations in temperature and oxygen conditions of the coolant. There is a matrix of various experimental values of the flow of oxygen consumed for steel oxidation, and the temperature and partial pressure of oxygen in the coolant (Askhadullin et al. 2011; Ivanov et al. 2017).

Based on the available experimental data (Askhadullin et al. 2011; Ivanov et al. 2017), the following dependence was proposed for EP-832 steel:

$K_p=\exp \left(39.5\right)p_{O_2}^{0.125}\exp \left(-\frac{30000}{T}\right),[\mu m^2/h],$ (50)

Similar experimental studies, by definition, were also carried out for EP-302 steel. The following dependence is proposed for a parabolic constant:

$K_p=\exp \left(31\right)p_{O_2}^{0.125}\exp \left(-\frac{20000}{T}\right),[\mu m^2/h]$ (51)

For comparison, we present the dependences of parabolic constants for the basic oxygen regime, obtained in (Hwang and Lim 2010), based on an analysis of the available experimental data (Fazio et al. 2001; Barbier and Rusanov 2001; Aiello et al. 2004; Balbaud-Celerier et.al. 2004; Zhang et al. 2005; Lim et al. 2010; Martinelli et al. 2008a; Hwang and Lim 2010; Nuclear Energy Agency 2015).

For steels of ferritic-martensitic 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:

$K_p=1.156\times {10}^{-3}\exp \left(-\frac{135693}{RT}\right),\left[{\mathrm{cm}}^2/\mathrm{s}\right]$ (53)

In (Zhang 2013), the following dependences were also obtained for HT9 and T91 steels that take into account the partial pressure at the interface between media, namely, steel Т91:

$K_p=2.096\times {10}^{-5}{c}_o^{0.27}\exp \left(-\frac{157802}{RT}\right),\left[{\mathrm{m}}^2/\mathrm{s}\right],$ (54)

And for steel HT9:

$K_p=9.735\times {10}^{-8}{c}_o^{0.27}\exp \left(-\frac{126802}{RT}\right),\left[{\mathrm{m}}^2/\mathrm{s}\right],$ (55)

where с₀(ppm, millionth share) ‒ mass fraction of oxygen.

An analysis of the empirical dependences of parabolic constants (Fig. 2) shows that among ferritic-martensitic steels, oxygen consumption will be the smallest for EP-823 steel, which, under similar other conditions, will provide the smallest film growth. Austenitic steels (for example, the AISI 316L shown here) generally have lower oxygen consumption. Note that in (Barbier and Rusanov 2001; Balbaud-Celerier et al. 2004) the geometric data of the circulation loops are not presented in sufficient detail, on the basis of which the temperature dependences for the parabolic constant are obtained, which introduces additional calculated errors.

It was also noted in (Zhang and Li 2007; Steiner 2009; Alekseev et al. 2010c) that the parabolic dependence does not necessarily describe the available experimental data. Further, when taking into account corrosion processes, we will adhere to the general classification of corrosion oxidation regimes adopted in (Zhang and Li 2007), namely, a comparison of K_p,c ~ K_p / δ (K_p ‒ parabolic constant) and Q_c,o ‒ corrosion rates during corrosion oxidation:

– 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 (Zhang and Li 2007):

Q_c,0 ~ Q_c = χ (c_s – c_b) ~ χ_o (ν,D,d)Vⁿ 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 (Alekseev et al. 2010c).

At low coolant speeds, the corrosion rate of a metal surface is fully or partially controlled by the mass transfer effect (Chen et al. 1992; Balbaud-Célérier and Barbier 2001). The corrosion rate increases with increasing coolant velocity. At higher coolant velocities, the reactions of diffusion transfer of metal ions become equal to the diffusion rates of metal particles in the coolant or even become lower than them. As a result, the overall corrosion rate begins to be controlled by the reaction rates in the metal (activated controlled), and the corrosion rate ceases to depend on the speed of the coolant. At even higher speeds, mechanical erosion begins.

Thus, using the approach of (Zhang and Li 2007) for a pure metal, the corrosion rate is defined as Q_c ≈ χ_FeC ^s_Fe, while $Q_{c,o}\approx {\chi }_{Fe}{C}_{Fe}^{\gamma }\frac{M_{ox}}{3M_{Fe}{\rho }_{ox}}$ , that is

$\frac{Q_{c,o}}{Q_c}=\frac{M_{ox}{\rho }_{metall}}{3M_{Fe}{\rho }_{ox}}{a}_{Fe}^{\gamma }\approx a_{Fe}^{\gamma }\equiv a_{Fe}^{\gamma }\left(d,T,C_0\right)$ (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 2, 8) obtained on the basis of experimental data (Harriott and Hamilton 1965; Berger and Hau 1977; Silverman 1984):

k_b₋h = 0.0165υ⁻^0.53D^0.67V^0.86d⁻^0.14

k_s = 0.0177υ⁻^0.579D^0.704V^0.875d⁻^0.125 (58)

k_h₋h = 0.0096υ⁻^0.567D^0.654V^0.913d⁻^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_b-h. Fig. 3 shows typical values of the rates of dissolution (corrosion) of the film taking into account this mass transfer coefficient (58) depending on temperature.

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. 4.

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 δ² ~ 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 (Askhadullin et al. 2011; Ivanov et al. 2017; Salaev et al. 2018) make it possible to relatively accurately determine only the parameters of the dependence on temperature and partial pressure (37b). Therefore, the absolute values of the parabolic constant can be determined only by reference experiments. In our case, these are experiments (Barbier and Rusanov 2001; Balbaud-Celerier et.al. 2004), since the experiments were performed for different types of steels on the same installation. Based on the data in Figs 8–10 (Balbaud-Celerier and Terlain 2004), we accept ε_Kp ≈ 0.6. Since the accepted empirical dependence (37a) is determined by three parameters, A, Q, and n, the relative uncertainty of the parabolic constant is determined by the relative errors for these parameters. The dependence on the uncertainty of the degree of partial pressure can be neglected due to its smallness, then the uncertainty of the parameters A and Q, ε_A, ε_Q contribute to the uncertainty of the parabolic constant as

${\epsilon }_{K_p}=\sqrt{{\left({\epsilon }_A\right)}^2+{\left(Q/T{\epsilon }_q\right)}^2}$ (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

$\epsilon =\frac{1}{N}\sum _{i=1}^N\frac{abs\left(x_i^{calc}-x_i^{\exp }\right)}{x_i^{\exp }}$ (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 MASKA-LM and STAR-CCM + 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 well-established 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. 11).

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 EP-823 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 5–7 show the typical model dependences for the film saturation thickness on the oxygen velocity and concentration, as well as the characteristic time of transition to parabolic growth. The simulation parameters are taken for lead coolant and steel EP-823.

The growth of the oxide film upon saturation, depending on the coolant speed, is shown in Fig. 5. Obviously, at low speeds, a substantial, almost exponential, increase in the thickness of the saturation film is observed. That is, the saturation thickness substantially depends on the velocity of the coolant.

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. 6).

The calculations shown in Fig. 7 demonstrate that under the basic oxygen regime (oxygen concentration in the region of 10^-8 kg/kg) and at standard coolant velocities (1–2 m/s), a very small film growth is expected (up to 1 μm). If you deviate, in the direction of increasing the oxygen content, from the basic mode, you can expect film growth to several tens of micrometers. At a low oxygen content (dash dotted line in Fig. 7), the formation of an oxide film is practically unrealistic, which can lead to corrosion.

Figs 8–10 show the calculations for steels EP-823, T91 and 316L for which both static and dynamic experiments were performed (Barbier and Rusanov 2001; Balbaud-Celerier and Terlain 2004), which allows a consistent analysis of these data, that is, the same parameterization of the parabolic constant is used for both types of tests. The coolant speed in the dynamic test is 1.9 m/s. Below is an analysis for T = 600 °C. Taking into account the numerical errors from the variation of the parameters, we obtain fairly good agreement with the experimental data. An interesting feature of the above calculations and experimental data is that although the steels EP-823 and T91 are quite close to each other in composition, the growth and thickness of the film for them is significantly different, which determines the adequacy of our engineering approach.

The authors of (Zhang and Li 2007; Weisenburger et al. 2011) analyzed the obtained experimental information for T91 steel. The experiments were carried out on the same installation of JSC "SSC RF-IPPE" as in (Balbaud-Celerier and Terlain 2004; Zhang et al. 2005), but with an average coolant speed of 1 m/s and under the basic oxygen regime. As discussed earlier, it is not possible to reconstruct the exact geometric parameters. Therefore, a certain spread in the values of the hydraulic diameter in the range of 0.005–0.1 m, which gives no more than 20% error in the value of the film thickness, is accepted. As an example, a calculation with an average coolant velocity of 0.3 m/s is presented to demonstrate sensitivity to this parameter. This speed was chosen intentionally to place most of the experimental data between the two calculated curves (Fig. 11). The calculated estimate demonstrates that having a spread of experimental data can be, inter alia, related to a "spread" in speed from 0.3 m/s to 1 m/s.

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 MASKA-LM code of own design.

The STAR CCM+ code interface allows you to use user-defined functions to model certain physical and chemical processes. The system of equations (33), (38), functions dependent on them, and semi-empirical 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 10⁵; 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/m²∙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 MASKA-LM 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 MASKA-LM codes are shown in Figs 12, 13 (for coolant temperature of 400 °C and 500 °C, respectively, the differences are minimal). Calculations are given for various turbulence models (StarCCM+) and for various turbulent diffusion coefficients used in the MASKA-LM code. Changing the coefficient of turbulent diffusion by an order of magnitude gives the maximum error in calculating the oxygen concentration of less than 15%. It is worth noting that at the moment, the methodology for determining the measurement error of the oxygen activity sensor has not been developed, and according to qualitative estimates it can be at least 50%. In this case, the choice of a turbulence model or a simplified description of turbulent motion will not affect the description of available experimental results at all.

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 EP-823 steel pipe and a beam of rods at various flow rates of HLMC and temperatures.

Figs 14–16 show the calculations for different temperatures, oxygen concentrations and average speeds in the pipe using the MASKA-LM and STAR CCM+ codes.

Fig. 14 shows film growth versus time. All models, in accordance with Zhang’s approach, demonstrate the approximation of the film thickness to the maximum possible value. The calculation shown by the black solid line is made taking into account only the parabolic constant (there is no dependence on the coolant speed through taking into account mass transfer). As can be seen, for calculations at a speed of 1.8 m/s, the film growth rate close to the parabolic behavior is observed only from about 1 to 50 hours, while at a speed of 0.1 m/s it is observed up to several thousand hours. Although it is more correct to say that film growth is closer to the paralinear law in these areas. The Tedmon’s approach fairly well describes only areas close to saturation in film growth.

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. 15). All models, including the simple 0D model, show very similar results. Iron concentration distributions also show fairly similar behavior, and the main difference, again, is observed near the surface.

Fig. 16 shows the calculations for an increased oxygen concentration; in general, the concentration of the basic oxygen regime is more than an order of magnitude higher. In this case, for a rather long time, a pronounced parabolic growth of the film is observed. Comparison of calculations using the codes in question was carried out only for temperatures Т = 500 °С and Т = 650 °С. The maximum difference in the thickness of the oxide film is not more than 20%.

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 × 10⁶ 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~10⁵), turbulence model K-ε when using STAR CCM+ code.

Fig. 17 shows the calculation results for the thickness of the oxide film at various temperatures. As in the previous case, the difference in the obtained calculations does not exceed 20%.

Note that in the above examples only isothermal calculations were considered.