In the steady state LBE pipe flow without protective oxide layer at wall under isothermal condition, the steel corrosion depends on the mass transfer rates and dissolution/reaction rates of steel constituents at wall. If the mass transfer rate is greater than dissolution rate, the corrosion is dominated by the mass transfer, which is called mass transfer controlled corrosion. In the other case, it is called activation controlled or dissolution controlled corrosion (Zhang et al., 2010). The dissolved iron at wall is firstly transferred through the diffusion boundary layer by molecular diffusion, then the transfer rate of iron becomes higher in the viscous sublayer. The diffusion boundary layer could be very thin as the Schmidt number is considerably high (order of 10³).

The mass flux of iron J at wall obeys the Fick’s law and could be expressed as follow: J = D m ∂ C ∂ n = D m y 0 ( C w − C 0 ) (1) where D _m is the molecular diffusion coefficient of iron, C and n are, respectively the concentration and normal distance to wall in Fick’s law, respectively, C _w is the concentration of iron at wall, y ₀ is the distance from wall and C ₀ is the concentration of iron at distance y ₀. In general, the distance y ₀ represents the thickness of the first inflation mesh layer in CFD simulation. Meanwhile, the mass flux from dissolution location to bulk fluid can be expressed by the following equation, where C _w denotes concentration at wall, C _b denotes concentration in bulk fluid and K is mass transfer coefficient of iron: J = K ( C w − C b ) (2)

Since the mass transfer rate in diffusion boundary layer is very low, it dominates the mass transfer from wall to bulk fluid. Thus, the iron mass flux is the same in Eqs 1, 2, and the mass transfer coefficient can be deduced by K = D m y 0 ( C w − C 0 ) ( C w − C b ) (3)

For the mass transfer controlled corrosion, the corrosion rate (CR) is proportional to the mass flux and therefore can be directly calculated as follow (Davis and Frawley, 2009): C R = J M ρ F e = K ( C w − C b ) M ρ F e (4) where ρ _Fe and M denote the density and molar mass of iron, respectively. In addition, many studies have found that local wall-to-liquid mass transfer behavior and corrosion rate in practical geometries have strong dependence on the near-wall turbulent level (Poulson, 1999; Nešić, 2006; Ikarashi et al., 2017), the near-wall concentration distributions of solutes are considered to be influenced by the near-wall turbulent kinetic energy (Wan and Saito, 2018).

For the mass transfer in a steady-state or fully developed flow, correlations between Sherwood number Sh, ( S h = K l / D m with l the characteristic length), Reynolds number Re and Schmidt number Sc [ S c = μ / ( ρ D m ) ] were proposed as follow (Dawson and Trass, 1972; Grifoll et al., 1986; Postlethwaite and Lotz, 1988; Poulson, 1990): S h = a 1 R e a 2 S c a 3 (5) where the coefficients a ₁ depends on the geometry, a ₂ generally ranges from 0.5 to 1, while a ₃ is close to 1/3. As the near-wall mass transfer is affected by the wall roughness, the rough-to-smooth ratio of mass transfer coefficient K _r /K _s has been characterized by the similarity function (Dawson and Trass, 1972; Tantirige and Trass, 1984; Postlethwaite and Lotz, 1988; Mobarak et al., 1997; Zhao and Trass, 1997; Lolja, 2005). The mass transfer similarity function contains several parameters, such as the dimensionless roughness number, Schmidt number and Reynolds number: K r K s = f ( e + ,   S c ,   R e ,   . . . ) (6)

The dimensionless roughness number (or roughness Reynolds number) e ⁺ is defined by roughness height e and friction velocity u*: e + = ρ u ∗ e μ ,   u ∗ = U f 8 (7) The dimensionless friction velocity u∗ is determined by the stream flow velocity U and friction factor f, while the friction factor f could be determined by the measured pressure drop for pipe flow with roughened wall: f = Δ P ρ U 2 2 D L (8)

The resolution of the near-wall iron concentration is critical since the near-wall concentration dominates the mass transfer behavior in LBE corrosion. To achieve a precise simulation of the near-wall flow behavior, LRN models with damping functions are used. LRN models have been proposed by Jones and Launder (1972), Launder and Sharma (1974), Lam and Bremhorst (1981), and Chien (1982). In this study, the Launder-Sharma LRN model is adopted for calculation, the transport equations of the turbulent kinetic energy and its dissipation rate are given as follows: ∂ ρ k ∂ t + ∂ ∂ x j [ ρ k u j − ( μ + μ t σ k ) ∂ k ∂ x j ] = P k − ρ ε − 2 μ ( ∂ k ∂ y ) 2 (9) ∂ ρ ε ∗ ∂ t + ∂ ∂ x j [ ρ ε ∗ u j − ( μ + μ t σ ε ) ∂ ε ∗ ∂ x j ] = ( C 1 f 1 P k − C 2 f 2 ρ ε ∗ ) ε ∗ k + 2 μ μ t ρ ( ∂ 2 u ∂ y 2 ) 2 (10) μ t = ρ C μ f μ k 2 ε ∗ (11) ε ∗ = D ε + ε (12)

In the above equations, μ _t and P _k denote turbulent viscosity and production rate of turbulence respectively. σ _k , σ _ɛ , C ₁, C ₂, and C _μ are constants which have the same values as those in the standard k-ε model (i.e. 1.0, 1.3, 1.44, 1.92, and 0.09, respectively). The modified turbulent dissipation rate ε* is related to the origin turbulent dissipation rate ε with a damping term D _ε whose value is significant near wall but can be negligible away from the wall. With Eq. 12, equation of ε* could be solved by applying a zero wall boundary condition, which greatly simplifies the computation. f ₁, f ₂, and f _μ are damping functions accounting for near-wall effects: f 1 = 1.0 ,   f 2 = 1 − 0.3 ⁡ exp ( − R e t 2 ) ,   f μ = exp [ − 3.4 ( 1 + 0.02 R e t ) 2 ] (13) wherein the turbulent Reynolds number Re _t is defined as: R e t = ρ k 2 μ ε ∗ (14)

The direct resolution of the viscous sublayer in LRN model requires very fine grids at walls, the first node (or cell centroid) should be located at y ⁺ ≤ 1.

The rough wall geometry consists of periodic square cavities, Figure 1 shows the 2D geometry of computational domain. An axis-symmetry model is adopted, with a radius of the cylinder pipe R = 4.9 mm. λ is the periodic distance between two adjacent cavities, while w and d denotes the width and depth of the cavity, respectively. The width-to-depth ratio w/d is one for square cavities, hence a d-type roughness is represented. The d-type roughness is considered as a good representative for a natural roughness and allows for comparison of the flow field with CFD and electrochemical experiments (Lari et al., 2013). Three different sizes of cavities w = 500, 250, 50 μm are adopted and the ratio λ/d = 2 and 4 are considered, respectively. The total length of computational domain is 80 mm, which is more than eight times of the inlet diameter, thus the pipe flow could be considered as fully developed. The mass transfer of smooth wall with the same radius R and length is also studied for comparison. Table 1 gives the models with different geometrical dimensions in this study.

Unstructured triangle grid was applied to the LBE bulk, with coupled mesh inflation that consists of fine layers near the wall. The thickness of the first inflation layer was fixed approximately at y ⁺ = 0.1 or below for LRN model, which is considered as the thickness of the diffusion boundary layer (Nešić et al., 1992). Hence, in the above geometrical models, y ⁺ was set as approximately 0.1 in the whole computational domain. For the LBE flow in condition Re = 5 × 10⁴, the thickness of the first layer is 3.4 × 10⁻⁷ m. With extension ratio of 1.2 in normal direction to the wall, sixteen layers were defined, which could cover the viscous sublayer region. In the region away from the wall, coarse triangle meshes were applied. The maximal total number of cells does not exceed two million. A typical mesh structure is given in Figure 2.

The steady-state simulation of LBE flow and iron transfer was carried out by the CFD code Ansys FLUENT. Table 2 summarizes the main boundary conditions applied in the computation. In each model, the inlet bulk fluid velocity ranges from 0.07 to 1.4 m/s, correspondingly the Reynolds number ranges from 5 × 10³ to 10⁵. With an assumption that the oxygen concentration in LBE is sufficiently low, Fe₃O₄-based oxide film does not form at the metallic wall surface, so that LBE has direct contact with wall and the dissolution equilibrium of iron is established at wall. Thus, a constant iron concentration of 8.978 × 10⁻⁵ wt% is imposed at wall and the iron concentration in LBE bulk fluid is set as 9.576 × 10⁻⁶ wt% (Li, 2002). The viscosity of LBE adopted in calculation is 0.001402 kg/m.s, the density of LBE in simulation is 10130.2 kg/m³ at 450°C, the molecular diffusion coefficient of iron in LBE at 450°C is 3.16 × 10⁻¹⁰ m²/s, which is deduced through an extrapolation from Robertson’s law (Abella et al., 2011; OECD, 2015). A mesh independent check was preliminarily performed in the smooth pipe geometry. The Figure 3 shows the radial distribution of iron concentration at pipe outlet under Re = 5 × 10⁴ for different meshes number. The convergence of the resolution is reached when the relative residuals are all below 10⁻⁷.

The radial distributions of velocity at different positions near outlet are compared. In Figure 4, at distance of 70, 75, and 79 mm of the pipe, high consistency of the velocity profiles consist is observed respectively for smooth pipe and roughened pipe at Re = 10⁵, which could verify the developed state of flow near outlet.

Since the mass transfer coefficient K is locally determined, a profile of K is obtained along the wall. The mass transfer coefficient of smooth wall K _s is the averaged value of K near the outlet. The mass transfer coefficient of rough wall K _r is the averaged value of K along the cavity walls and crest horizontal wall in a periodic length (see Figure 1). In addition, the mass transfer coefficient of the crest K _cr is the averaged value of K along the crest horizontal wall. The last periods of cavities in each roughness geometrical models are chosen in order to neglect the entrance effect of flow.

The results of mass transfer data Sherwood number Sh determined by predicted mass transfer coefficient are shown in Figure 5. Sh increased with a higher flow velocity, this is consistent with the fact that the diffusion boundary layer and viscous sublayer are thinner at high Re, therefore a higher near-wall concentration difference was formed and the mass transfer rate was enhanced. A higher corrosion rate could be observed for the flow at higher velocity. Berger and Hau (1977) have proposed a correlation for fully developed flow in smooth pipe where Re ranges from 8 × 10³ to 2 × 10⁵ and Sc varying from 1 × 10³ and 6 × 10³: S h = 0.165 R e 0.86 S c 0.33 (15)

In previous mass transfer study, a relation between thickness of viscous sublayer δ _v and thickness of diffusion boundary layer δ _d was proposed (Levich, 1962; Nešić and Postlewaite, 1991): δ d = δ v / Sc 0.33 (16)

The diffusion boundary layer is much thinner than viscous sublayer. In this diffusion boundary layer, the mass transfer is dominated by molecular diffusivity, while out of this layer the mass transfer is dominated by turbulence in viscous region. In the correlation between Sh, Re and Sc, Re affects the viscous layer of flow, while the relation between δ _v and δ _d depends on Sc. For this reason, the variation of mass transfer is mainly impacted by Re. The factor Sc ^0.33 in Eq. 16 is also verified by Wan and Saito (2018) in LBE pipe flow. Thus, we assume the same exponent value on Sc, which is the same approach as in Dawson and Trass (1972), Berger et al. (1979), and Postlethwaite and Lotz (1988), then the following correlation is obtained by regression: S h = 0.425 R e 0.5 S c 0.33 (17)

The corrosion rate of smooth wall is also estimated and compared with experimental data which were reported by Wan and Saito (2018). The averaged corrosion depth of a 9.8 mm-diameter straight pipe under LBE flow at a velocity of 0.7 m/s ranges from 0.06–0.16 mm after 3,000 h operation. With the same flow condition, the predicted corrosion depth calculated by Eqs 3, 4 is about 0.22 mm, which is near 1.4 to 3.6 times of the measured value. This discrepancy may be caused by the varying surface condition of the pipe during operation and the assumption of saturated solute concentration at solid-fluid interface.

Iron mass transfer from roughened walls is different compared to smooth wall, since the near-wall velocity field can be affected by the rough wall. The results show that stagnant flow is formed in cavities. Figure 6 gives the flow velocity fields in cavities under different roughness at Re = 5 × 10⁴ and Figure 7 gives the corresponding iron concentration distributions. Given the same inlet condition, the velocity magnitude in cavity decreased as the size of cavity decreased, resulting in the increased near-wall concentration, which led to weaker transfer of iron from cavity wall to LBE bulk fluid. Similar results were observed at different Reynolds numbers as well.

The size of the crest can impose influence on the iron transfer at crest wall since it is depended by the distribution of K along the crest wall. The averaged coefficient K _cr of each rough geometry are given in case of Re from 5 × 10³ to 10⁵ in Figure 8, and are compared with K _s of smooth wall. It is shown that, as the flow velocity increased, K _cr increased obviously and the values are much greater than K _s . At the same ratio of λ/d, value of K _cr is larger as the cavity has a smaller size. For the same cavity size, value of K _cr is smaller with higher ratio of λ/d, which represents a longer crest.

The distribution of concentration at the first node, C ₀, along the crest wall depends not only on the flow velocity, but also on the crest length. Figure 9 gives the distributions of (C _w −C ₀) along the crest walls for d = 500 μm and for Re = 5 × 10³, 2.5 × 10⁴ and 5 × 10⁴. At the same Re, compared with the relatively uniform distribution in the case of the smooth pipe, C ₀ has much lower values at the front of the crest wall, while it increases along the crest, then it stays very close to C _w until the end of the crest wall. This distribution is similar to that of the entrance part of the smooth pipe wall. Hence, higher local mass transfer coefficients K are obtained at the front of crest wall. However, when λ/d = 4, which represents the horizontal length of crest is three times of that of λ/d = 2, K remains at a low level at the end of crest wall, so the average coefficient K _cr owns a lower value. The distributions of turbulent kinetic energy at the first node along the crest are given in Figure 10, for d = 500 μm and for Re = 5 × 10³, 2.5 × 10⁴, and 5 × 10⁴. It is shown that, higher values of turbulent kinetic energy are obtained at higher flow velocities, the turbulent kinetic energy decreases from the front part of crest wall and then keeps at a stable level until the end of crest wall. It could be observed that the turbulent kinetic profiles are similar to those of (C _w −C ₀), thus it is reasonable to relate the near-wall mass transfer rate to the near-wall turbulence level.

By calculating the mass transfer coefficients along the cavity walls and crest wall, the averaged coefficient K _r on a periodic length λ is determined for each rough geometry. The obtained Sherwood number for roughened walls Sh _r are correlated with Sc and Re by Eq. 5, which is shown in Figure 11. Assuming that the correlation expression for Sh _r has the same exponent values on Re and Sc as in Eq. 16, Table 3 gives the correlative coefficient a for different cases. The value of a ranges from 1.810 to 2.357 for the correlations, which is thought to be a geometry-dependent coefficient for mass transfer in this study.

The mass transfer enhancement phenomenon caused by the roughen walls was described by the rough-to-smooth ratio K _r /K _s . The values of K _r /K _s for iron mass transfer in LBE are plotted against the dimensionless roughness number e ⁺ in Figure 12. At Sc = 438, for each rough geometry, K _r /K _s increased from near two to approximately six as e ⁺ increased. It should be noticed that for the same rough geometry, the increase of e ⁺ is due to the rise of flow velocity. Hence, K _r /K _s has low dependence on e ⁺ but has great dependence on the flow velocity. In previous reported researches, the mass transfer similarity functions have been developed and K _r /K _s was characterized under different roughness regimes depending on e ⁺ (Dawson and Trass, 1972; Postlethwaite and Lotz, 1988). However, the same approach cannot be adopted to analyze the results in this study, more parameters are required to characterize K _r /K _s . The difference between the results in previous studies and in this study can be explained from several aspects. First, the properties of fluid are involved in the definition of e ⁺, which could induce a great different range of e ⁺. Secondly, the Schmidt number characterizes the mass transfer properties of the species in fluid, the value is relatively much lower in this study comparing with Sc = 2,000–5,000 in previous reported researches. Moreover, in this study, K _r are determined by averaging the values along the cavity wall and crest wall in a roughness period, which is quite different from the electrochemical method.

In previous sections, the dimension and geometry of the crest and cavity are found to have considerable influence on C ₀. The value of K _r is actually dominated by the value of K _cr . For this reason, a correlation of K _r /K _s is proposed with the following parameters: dimensionless roughness number e ⁺, Reynolds number Re and the geometrical ratio (λ-d)/λ: K r K s = b 1 + b 2 ( e + ) b 3 R e b 4 ( λ − d λ ) b 5 (18)

In the correlation expression, the value of the first constant b ₁ is one since it is the asymptotic value of K _r /K _s as e ⁺ and Re tend to zero, which means that the mass transfer behavior on rough walls is similar to that of smooth wall. The values of K _r /K _s are plotted by using Eq. 18 in Figure 13, fitting values of correlative constants in Eq. 18 are obtained with a deviation within 4%. Thus, the mass transfer enhancement is well correlated for Re ranging from 5 × 10³ to 10⁵: K r K s = 1 + 0.0117 ( e + ) 0.1 R e 0.5 ( λ − d λ ) 0.1 (19)

The exponent values on e ⁺ and Re are, respectively 0.1 and 0.5, which is consistent with the results that e ⁺ has less contribution while the flow velocity has higher contribution on K _r /K _s . To a certain degree, the mass transfer enhancement is also affected by the geometrical ratio of the crest. The effect of Sc is not included in the correlation since all predicted results are obtained at the same Sc = 438.