Korosi-erosi Multiphase

Wear 255 (2003) 237–245

Case study

Evaluation of erosion–corrosion in multiphase flow via CFD and experimental analysis Benedetto Bozzini a , Marco E. Ricotti b,∗ , Marco Boniardi c , Claudio Mele a a

INFM, Dipartimento di Ingegneria dell’Innovazione, Università di Lecce, v. Monteroni, 73100 Lecce, Italy b Dipartimento di Ingegneria Nucleare, Politecnico di Milano, Via Ponzio 34/3, 20132 Milano, Italy c Dipartimento di Meccanica, Politecnico di Milano, v. Bonardi 9, 20133 Milano, Italy

Abstract A numerical simulation is proposed of erosion–corrosion phenomena in four-phase flows comprising two immiscible liquids, gas and particulate solid. The simulation geometry is a pipe bend and the evaluated quantity is the wall erosion–corrosion brought about by the flow of a fluid mixture of two liquid phases, one of which is corrosive, plus a gas phase flow and a solid phase. A computational fluid dynamic tool has been adopted for the simulation of the flow field inside the piping and for the simulation of the particle trajectories and impact rates. As far as corrosion is concerned, a passivating and an actively corroding metallic material have been considered. Erosion model parameters have been derived from experiments correlating particle impact angle and erosion rate. Corrosion model parameters have been obtained from electrochemical measurements. The effects of the key operating parameters (fluid flow velocity, particulate content and gas volume fraction) have been evaluated by a two-level design of experiments approach. The single most important effects on synergistic damaging and on the ratio of corrosive to overall damaging have been identified. Erosion-enhanced and erosion-limited effects of flow conditions have been highlighted for the passivating and for the actively corroding alloys, respectively. © 2003 Elsevier Science B.V. All rights reserved. Keywords: Erosion; Corrosion; Computational fluid dynamics; Multiphase flow

1. Introduction Investigations into the field of erosion–corrosion are typically daunted by the huge amount of experimental parameters which may have an effect on this synergistic damage mechanism, including: flow conditions, composition of the structural material, chemistry of the flowing system and temperature. Multiphase flows exhibit the additional challenge of requiring further phase composition parameters with the respective physical and modelling descriptors. Furthermore, experiments in realistic conditions are extremely hard to manage. An overall lack of predictivity is therefore characteristic for this kind of studies. The contribution of computational fluid dynamics (CFD) in simulating realistic flow patterns and in data reduction is proving fruitful. Even if several outstanding papers, among which [1–8], have been published, plenty of factors still await investigation. The computational approach to erosion–corrosion thus far has been dominated by the analysis of single-phase flow with a particulate solid, typically ∗ Corresponding author. Tel.: +39-02-2399-6325; fax: +39-02-2399-6309. E-mail address: [email protected] (M.E. Ricotti).

in two-dimensional geometry. The turbulence is typically simulated with the low Reynolds number k − ε model in non-disturbed and disturbed flows. Reynolds stress transport models have also been considered [6]. The motion of the dispersed particulate phase in the turbulent flow field has been modelled by Lagrangian, Eulerian or stochastic approaches. Slurry flows have been shown to differ very little from water flows, the predicted turbulence profiles are basically unchanged and Newtonian fluids have invariably been regarded as an acceptable approximation. CFD is suitable for providing local evaluation of the mechanical action of the fluid flow and mechanical parameters of the impacting solids suspended in the flow. This information ought to be linked with electrochemical information accounting for the corrosive contribution to erosion–corrosion. The synergy between erosion and corrosion phenomena mainly stems from alterations of the corrosion process brought about by the mechanical actions. These effects are widely diverse in nature and depend on the material–environment coupling. This variety of mechano-chemical phenomena can give rise to both enhancing and limiting effects of the corrosive component of the damage mechanism. Although the former generally dominates, the latter has been documented [9]. Enhancing effects are typical of passivating

0043-1648/03/$ – see front matter © 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0043-1648(03)00181-9

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materials, while actively corroding ones can be beneficially affected by the presence of a plastic deformation process, resulting in a lower overall damage rate when the mechanical action is superimposed on the merely electrochemical one. A strong demand for the analysis of multiphase flow problems is felt in the field of process engineering. Specific stimulus comes from the oil and gas production industry. Only pioneering work is available in the literature on four-phase flows [10], which is the typical operating condition for the off-shore oil extraction industry implying pipe flow of two immiscible liquids (oil and sea water), one gas (hydrocarbon mixture) and one dispersed solid (sand). The scope of this particular research is the simulation of the erosion–corrosion of pipe walls, due to the internal flow of gas–liquid four-phase mixtures carrying an inert particulate solid dispersion. Reference is made to both passivating and actively corroding metals. A three-dimensional analysis of a 90◦ elbow of a circular cross-section pipe is carried out. The operating variables and the phase composition were investigated in a wide range and their effects were evaluated on the basis of a two-level design of experiments approach. The parameters relating to the individual effects of erosion and corrosion were evaluated experimentally.

2. The CFD approach A computational fluid dynamic tool has been selected for the simulation of the flow field inside the piping and for the simulation of the particle trajectories and their impact on the bend walls. CFD is currently one of the more sophisticated and promising approaches for the analysis and solution of a wide class of problems involving flow domains and in a wide set of research and industrial application fields. CFD codes are capable of solving the full set of fluid dynamic balance equations, usually in Navier–Stokes formulation for momentum balance. Turbulence can be approximated by different models. In particular, the FLUENT code [11] adopted for this study solves the balance equation set via domain discretisation, using a control volume approach to convert the balance partial differential equations (PDEs) into algebraic equations solved numerically. The FLUENT code has been used in the investigation of solid particle erosion in gas flow in components of complex geometry [12]. The solution procedure integrates the balance equations over each control volume, thus obtaining discrete equations that conserve primary quantities on a control volume basis. The numerical solution defines the flow field quantities, possibly used by routines implementing models for further flow-related quantities than, e.g. phases transported by a given fluid phase. One of the more important features of this class of fluid dynamic codes is the ability to simulate complex fluid flows and geometric domains, both in two- and three-dimensions, also accounting for turbulence. A set of models are usually

made available to the user, differing mainly in the scale of turbulence they can evaluate. The present case study has been performed by adopting a three-dimensional unstructured mesh for the pipe, an implicit method for the numerical solution of mass and momentum equations and a k − ε model for the turbulence. The mixture composition and phase velocities are defined at the inlet boundary. The system pressure is fixed at the outlet boundary.

3. Multiphase flow model The fluid flow under analysis is a ternary one, with two liquid and one gas immiscible phases. The fourth phase to be modelled, i.e. the solid dispersion, was dealt with separately, as described here. Specialised models enable the code to handle different multiphase flow domains, even if current computational fluid dynamics is still far from the possibility of simulating the details of multiphase flow, mainly owing to the complexity of flow regimes and related phenomena such as phase transition or interphase heat transfer. One of the most stringent assumptions necessary for our analysis is the perfect homogeneity among the phases. This hypothesis implies that the phase fraction composition of the flow mixture is homogeneous throughout the mesh. The simulation of the actual flow regime in the piping is therefore disregarded in this approach. This approximation is expected to be physically unsound only in the case of large void fractions, leading to large intermittent plugs, annular dispersed flow or free surface regimes, as might occur in at low fluid velocities. Among the available models for the simulation of fluid phases, the volume of fluid (VOF) option has been selected. Even if this model allows the simulation of the position of the interface between the fluids, this feature has been deactivated according to homogenisation hypothesis discussed above. A single set of momentum balance equations for the mixture is solved and the volume fraction of each of the phases in each computational cell is evaluated throughout the domain. Both the liquid and gas phases are treated as incompressible. The main physical properties for the fluid phases are reported in Table 1.

4. Discrete phase model A specialized model is also available within FLUENT for the simulation of particles transported in the continuous Table 1 Main physical properties for the fluid phases

Density, ρ (kg m−3 ) Viscosity, µ (kg m−1 s−1 )

Water–liquid phase

Gasoline– liquid phase

Gasoline– gas phase

998.2 0.001003

830 0.00332

9.4 7E−6

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flow field. The discrete phase model (DPM) option solves the equation of motion for a discrete phase dispersed in the continuous phase, by adopting a Lagrangian frame of coordinates and leading to the computation of the particle trajectories. The force balance equation on the particle is solved using the local continuous phase conditions: dvp (ρp − ρf ) (1) = FD (vf − vp ) + g + Fx dt ρp where vp and vf are the particle and fluid velocities, ρp and ρf are the particle and fluid densities, respectively, g is the gravitational acceleration, Fx is a term accounting for additional forces, FD (vf − vp ) is the drag force per unit particle mass, FD is dimensionally an inverse of time and reads:  
 1 18µ CD Re FD = α= (2) τa 24 ρp Dp2 where τ a is the aerodynamic response time for the particle, Re is the Reynolds number for the particle referred to the relative velocity and CD is the drag coefficient: ρf |vf − vp |Dp Re = (3) µf a2 a3 (4) + 2 CD = a1 + Re Re with coefficients a derived in [13] for two-phase flows. Eq. (1) needs to be coupled with the trajectory equation in the Lagrangian frame, ds/dt = vp , where s is the trajectory abscissa, in order to close the solving equation set and to give velocity and position for the particle. Moreover, in turbulent flows the effect of turbulence on the particle dispersion can play a significant role. A stochastic model, the discrete random walk (DRW) or “Eddy lifetime” model, predicts the turbulent dispersion of particles by integrating the trajectory equations for each particle by adopting the instantaneous fluid velocity along the particle path. The fluid velocity in that space position where the particle is located, is v = v¯ + v , where v¯ is the mean fluid phase velocity and v is the turbulent part of the fluid velocity. The turbulent contribution is given by a Gaussian distributed random fluctuation, which is taken as constant over the solution advancement time step t that cover the Eddy time scale τ e or “Eddy lifetime”: k (5) τe = 0.30 ε

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where k and ε are the turbulent kinetic energy and the turbulent dissipation rate, respectively. The velocity and trajectory calculations for the particles can be treated both as uncoupled or as coupled with the continuous fluid field solution. Obviously, the second option requires iterations between each set of balance equations. This option is useful in the case the discrete phase injection is such that a feedback on the fluid field quantities (e.g. velocity, pressure) is expected. Otherwise, the segregation between the two solutions is acceptable. The coupled method has been adopted in order to directly evaluate the erosion mass flux. The steady-state analysis is carried out through the following steps: (i) solving of the continuous phase flow; (ii) injection of the discrete phase; (iii) solving of the coupled flow and particle trajectories; (iv) tracking of the discrete phase. 4.1. Boundary conditions Particles are assumed to be spherical and monodispersed in diameter. They are injected in a single point at the inlet boundary, in the centre of the pipe cross-section area, and released into the flow as from a spray cone dispenser at the same inlet multiphase flow velocity. The particle stream impact on the pipe wall boundaries is treated as an an elastic reflection, with fixed momentum restitution coefficients in both normal and tangent directions with respect to the hitting direction on the wall. The main assumptions for the DPM analysis are summarized in Table 2. A total inert particle mass flow rate is defined for each injection. Thus, the total solid phase flow rate is proportionally split in a number of stream flow rates equalling the number of streams defining the whole injection. The total number of particle streams or stochastic histories analysed (S) is given by the product of the number of particle injections (Nj ) and the number of particle streams per injection (Ms ). In our case, being a single point of injection chosen, S = Nj × Ms = 1 × 10, 000 = 1E+4. The number of histories gives sufficient confidence that the simulation results are statistically meaningful. A sensitivity analysis was carried out in order to ensure statistical reliability, by increasing the number of particle streams or the injection points while keeping constant the total injected mass flow rate of inert phase. An increase of one order of magnitude in the stream number did not lead to significant improvement in the simulation.

Table 2 Main assumptions for the discrete phase model Particle type

Density (kg m−3 )

Mean diameter (spherical particles) (␮m)

Number of injections/type

Number of particle streams per injection

Reflection at wall normal coefficient

Reflection at wall tangent coefficient

Sand

2800

300

1 injection/cone (20◦ semiangle)

10000

0.8

0.8

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5. Erosion

impact angle is reported in Fig. 1. The fit curve has been implemented in the CFD code for the erosion analysis.

The model available in the FLUENT code for the calculation of the erosion flux is a simplified model taking into account the mass flow rate of the impacting stream (Γ s ), the surface area of the impacted wall boundary cell (A) and an impact angle function f(θ). Other terms in the erosion equation could be defined (a function of the particle diameter C(Ds ) and a function of the particle stream relative velocity b(v)), but were not considered in the present case for the sake of simplicity: E=

N stream i=1

b(v)

Γs,i f(θi )C(Ds,i )vi Ai

(6)

The angular dependence f(θ i ) to be used in Eq. (6) was evaluated experimentally by performing sand-blasting erosion tests on disk-shaped 2205 Duplex stainless steel specimens (disk diameter: 2 cm, disk thickness: 0.5 cm, impact velocity 4 m s−1 , sand diameter ca. 400 ␮m). Duplex stainless steel (DSS, composition 22Cr–5.5Ni–3Mo–1.5Mn–17N + Si, P, C, S; hardness 248 HV; yield stress 516 MPa; ultimate tensile stress 790 MPa) have been successfully used in process industry, including food and biomedical, due to their mechanical and corrosion resistance properties which are better than conventional austenitic or ferritic grades. The high chromium and molybdenum contents allow the use of DSSs under conditions of pitting, crevice and above all stress corrosion cracking that would be critical for the traditional stainless steels grades. The erosion rate was estimated gravimetrically. The experimental data could be approximated with the Eq. (7), f(θi ) = B sin θi

(7)

where B = 8.5 mg cm−2 s−1 . A graph showing experimental data of erosion rate and their fitting as a function of

6. Corrosion As previously mentioned, the synergistic action typical of erosion–corrosion processes is obtained by modelling the merely electrochemical corrosion mechanism allowing for the mechanical effects. Simple approximate models are proposed for both passivating and actively corroding materials. The electrochemical component of corrosion has been taken into account by the “recovering target” concept. The particles are modelled with rigid monodisperse spheres of radius R. The particle impact process is assumed to be Poissonian with parameter λ impacts m−2 s−1 . Each independent impact is assumed to give rise to an alteration of the corrosion mechanism whose effect is a transient local variation of the corrosion current density for a typical time. The effective corrosion current density icorr (nA cm−2 ) at a given electrode potential (typically the corrosion potential) can be related to the mechanically affected corrosion component of the synergistic damage through a coefficient fa (such that 0 ≤ fa ≤ 1) expressing the fraction of the corroding surface which is affected by the erosive action of impinging particles, by Eq. (9), icorr = fa ia + (1 − fa )iu

where the pedices a and u stand for “affected” and “unaffected”, respectively. The current densities ia and iu are characteristic for the corroding material in the absence and in the presence of the relevant mechanical action and in principle can be measured separately by means of suitable experiments. In general the coefficient fa can be defined as Eq. (10): 
 damaged area no. of impacts control area impact ×recovery time = λAa τ


fa =

Fig. 1. Experimental erosion rate data (dry angle-dependent sand blasting) for 2205 DSS.

(9)

(10)

The numerical value of fa depends on the fluid dynamics through λ and on the material properties through Aa , and τ. Aa of course depends on the specific mechanism through which the chemical behaviour is affected by the mechanical action. The recovery time τ, instead, depends merely on materials properties and, in particular, on the metal–environment coupling. In the present investigation, the material-related parameters Aa have been related to the impinging particle radius by simple modelling. As far as the estimation of Aa values for passivating and actively corroding metals is concerned, we followed a simple mechanical modelling approach. The need for somewhat crude approximations at this stage of the research is due to the fact that a velocity spectrum for the impinging particles has not been considered, since it is not possible to extract

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velocity information for each impinging particle from the available CFD code. In the case of passivating materials, the alteration of the corrosion mechanism caused by particle impact is depassivation due to the mechanical removal of the protective layer. In the present treatment we assume that the passivating film behaves like a brittle thin ceramic layer and that the area experiencing cracking and spalling after application of the mechanical action by the impinging particle can be identified with the activated area Aa . The area damaged by a spherical particle of radius R pressed against the metal surface with a force typical for the case at hand can be estimated to be of the order of π/4R2 [14]. This quantity can thus be identified with Aa . In the case of actively corroding metals, the specific mechano-chemical effect is related to the plasticisation of the material, affecting its activity through variations of metal density and grain defectivity. We take into account a model system consisting of an initially annealed material, which is locally deformation-hardened by the action of particle impacts. An effective impact of a particle of radius R can be though of as producing a hemispherical plasticised volume. For the material and mechanical actions at hand, the radius of the plasticised hemisphere can be estimated to be ∼0.3R, giving rise to a typical value for activated area Aa of ∼8 × 10−3 R2 [15]. Values for τ, iu and ia have been evaluated by suitable experiments. In the case of a passivating metal, provided the surface is initially in the passive state, iu corresponds to the passivity current density and ia to the current density which can be measured after removal of the passive layer (e.g. by scratching under controlled conditions) at the electrode potential of interest. τ is the time the abraded surface takes for reforming the passivation layer under the relevant environmental and electrochemical conditions. As a typical passivating material of interest for the petrochemical industry, we tested 2205 DSS polarised potentiostatically in NaCl 35 g l−1 in the passive range at +0.6 V versus immersion potential and scratched. The thereby following current density transient was recorded and interpolated with a single exponential decay, whose time constant is used as an estimate of τ. The relevant current density contribution was evaluated after measuring the scratched area under an optical microscope. Electrode scratching was carried out on a horizontal upward-facing surface. The scratching tip—a standard Vickers pyramidal diamond indenter—was loaded with 25 g and slid across the surface. The resulting scratching exhibited an asymptotic width of ca. 35 ␮m. The experimentally derived quantities were: τ = 2.5 s, ia = 96 nA cm−2 , iu = 8 nA cm−2 . In the case of an actively corroding metal, the small deformed undersurface volume which displays the corrosion behaviour typical for the mechanically affected material, is dissolved by the ongoing corrosion process in a time τ. The annealed, mechanically unaffected areas and the deformed ones exhibit corrosion rates which are proportional to iu

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and ia , respectively. The hemispherical volume plasticised by the particle impact can be approximated by a cylinder with a plasticisation depth hp = 0.2ap , where ap is the radius of the above-mentioned plasticised hemisphere. Since the corrosion rate of the deformed material N (␮m s−1 ) is ∼5 × 10−5 ia /ρ for a bivalent metal of density ρ (g cm−3 ), the recovery time τ can be estimated as the time required to dissolve the plasticised volume τ ∼ hp /N. In this research we considered an electrochemically very simple and instructive system of admittedly limited industrial value. We studied a 0.1% carbon steel in the annealed (120 HV) and laminated (180 HV) states to represent the pristine and hardened conditions, respectively. The employed electrolyte was NaCl 35 g l−1 acidified to pH 2 by addition of H2 SO4 . This metal–electrolyte system provides a very straightforward generalised corrosion behaviour in which the cathode reaction is hydrogen reduction. Galvanic coupling effects between undeformed and deformed areas have been disregarded. Potentiodynamic polarisation measurements (scan rate: 5 mV s−1 ) have been performed for the two materials (Fig. 2) and the corrosion current densities have been estimated by the polarisation resistance and Tafel extrapolation methods, yielding pretty consistent values: ia = 5 × 104 nA cm−2 , iu = 6×105 nA cm−2 . This result implies that deformation results in a beneficial effect on the corrosion behaviour of this material. A typical value for the recovery time in these conditions is τ ∼7 × 10−3 R (t expressed in s and R in ␮m). As commented above in Section 1, this effect—which has a bearing on the surface activity of the alloy, in a Nernstian sense—of course does not apply in general. In actual facts, most of the occurrences of surface plastic deformation are reported to give rise to a decrease of the material stability against corrosion. Nevertheless, in some instances of high plastic strain and plastic flow localisation, such as sand impingement in a corrosive environment, the reverse behaviour has been observed beyond any reasonable doubt [9]. The experiments carried out in this research are a case

Fig. 2. Corrosion current densities for laminated and annealed materials, via potentiodynamic polarisation.

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of such an unexpected behaviour, which can however be easily rationalised in terms of surface compaction and/or variation of quality of surface crystalline defects or chemical stability of the thereby ensuing oxidation products. The dramatic difference in the potentiodynamic curves measured in this research, between the deformed and undeformed specimens ensures that, in this specific case, the results are not an artefact. The above-estimated numerical values of the corrosion parameters are inserted in Eqs. (9) and (10) for the evaluation of the effective corrosion current density. If the numerical value of fa computed during the simulation procedure exceeds unity, it is forced to be equal to one, in order to account for saturation of the mechanical effect on the corrosion behaviour corresponding to the whole of the surface being affected by the erosive action.

7. Criteria for the definition of the case matrix Three fluid dynamic characteristic parameters have been selected as key points for the case matrix definition, namely: (i) fluid flow inlet velocity, (ii) inlet volumetric flow ratio for the gas–liquid phase; (iii) mass flow rate of inert particles injected. A further fourth parameter was initially selected, namely the inlet volumetric flow ratio for the water (liquid) phase, but the model adopted in the multiphase mixture treatment (VOF) together with the simplifying assumptions and the scope of the work, suggested to eliminate the investigation of this parameter. No significant differences are expected to be encountered by changing the volumetric ratio between water and gasoline, keeping fixed the gas phase volume: only a mean density change in the mixture would result, with minor effects on the inertia forces and the particle trajectories. Therefore, the further possible parameter quantifying the inlet volumetric flow ratio for the water (liquid) phase was disregarded in the parametric analysis. In addition, the way the multiphase mixture is treated in the code (VOF) limits the physical insight which can be gleaned by considering this particular phase ratio. Two values for each parameter have been selected to compose the eight cases set, representing typical high and low levels of the relevant quantities, in a design of experiments frame of mind. The values assigned to the parameters define a range sufficiently wide to cover a representative domain for the phenomena. The resulting case matrix is reported in Table 3.

8. Results 8.1. Erosion The space domain for the CFD analysis refers to a 90◦ bend, 3.5 in. i.d. pipe. A three-dimensional mesh has been set up, by adding further volumes both at the inlet and the outlet of the bend. The former, corresponding to several tens of diameters in length with respect to the Reynolds number of the flow, in order to reach both a steady, fully developed flow and a sufficient dispersion of the particles injected in the stream, prior to reaching the bend zone. The latter in order to avoid possible recirculation flow paths at the outlet surface of the domain, thus leading to numerical convergence errors or unphysical results. A scheme of the regular, hexahedral mesh made up by 115,000 volumes is depicted in Fig. 3. A preliminary sensitivity study on the mesh size led to an acceptable compromise between accuracy in flow field simulation and computational time required by the code runs. As far as the numerical and the turbulence models are concerned, a segregated, implicit, steady-in-time solver has been adopted, together with a standard k−ε model for turbulence. Standard wall functions for the near-wall zone treatment have been selected. According to the case matrix in Table 3, eight simulations have been performed. The complete computational procedure consisted of the following successive steps: (i) convergence in the flow field domain, (ii) calculation of the particle steam trajectories, (iii) computation of the erosion rate at the bend surface, (iv) evaluation of the impact-induced corrosion effect, (v) evaluation of the synergic erosion–corrosion damage. The adoption of a three-dimensional geometry is a mandatory choice, in order to take into account possible effects of secondary flow paths on the particle stream trajectories and on the corresponding erosion effect. These secondary flow paths develop in transverse planes with respect to the main flow direction and are well known fluid dynamic characteristic of flow in bends, arising from centrifugal effects on the fluid due to the curvature of the domain. The corresponding, typical velocity field in a cross-section close

Table 3 Case matrix for the CFD analysis Values

Particle mass flow rate (kg s−1 )

Fluid velocity (m s−1 )

Phase volume ratio (gas–water–gasoline) (%)

High Low

0.1 0.01

10 1

40–30–30 5–30–65

Fig. 3. Hexahedral mesh adopted in the CFD analysis: bend zone and tube outlet zone.

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Fig. 4. Secondary recirculation path in the tube cross-section area at bend outlet: velocity vector plot (up) and velocity magnitude contour plot.

to the outlet of the simulated bend, a sort of recirculation path, is depicted in Fig. 4, obtained from one of the analysis listed in the case matrix. This feature of the flow field will lead to some peculiar results in the simulation, as will

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be described in the following paragraphs, and would not be observed in a two-dimensional domain. Moreover, gravitational effects like particle settling could be difficult to account for in two-dimensional geometry. By changing the configuration values for the flow velocity, the CFD analysis shows the qualitative results reported in Fig. 5, in terms of evaluated erosion rate. Two images of the bend and outlet pipe zone are shown, representing a lateral view from the outer side of the bend of the erosion rate distribution. The erosion rate distribution for a low velocity (2 m s−1 ) and a high velocity (10 m s−1 ) fluid test show the typical shift of the erosion peak, from the bottom to the side of the bend, as the fluid speed increases. The corresponding quantitative results are described in Figs. 6 and 7, where the erosion rate values are reported over the 360◦ angle of the bend outlet section. As far as the tests included in the case matrix are concerned, the low values configuration case (low flow velocity, low gas volume, low particulate content) demonstrates the low transport capability of the mixture on the solid phase, mainly due to the low inertia and low flow velocity, hence the reduced drag force on the particles. Thus, the particle streams tend to collect on the bottom of the straight pipe and the bend and to concentrate their erosion effect in the first half of the bend. Due to the secondary flow paths, the particles tend also to move towards the lower, intrados part of the bend. The maximum value reaches 8.46E−11 kg m−2 s−1 . This concentration effect is enhanced by the increase of the gas content in the mixture. The bend surface affected by the erosion is further reduced and the corresponding rate increased, with a maximum value at 3.91E−10 kg m−2 s−1 . To augment the solid phase content has the reasonable effect

Fig. 5. Different erosion rates and their distribution on the bend surface (test cases: fluid velocity of 10 and 2 m s−1 ).

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the bend, where the separation of the two transverse, recirculation path takes place, as already reported in Figs. 4 and 5. The peaks of erosion are at 2.59E−9 kg m−2 s−1 for the low value in solid phase flow rate. They increase up to 2.97E−8 kg m−2 s−1 for the high value. In this case, the weight of the impact angle (Fig. 1) leads to concentrate the erosion effect as the stream flow rate increases. The high value on the gas content is taken into account as well: when compared with the low void fraction configuration, the presence of a significant gas phase percentage decreases the drag and inertia capability of the mixture on the particles, thus leading to a more effective gravitational settling. The overall result is a lower peak in the top half of the wall bend, extrados side. As a further effect, an increase of the maximum erosion rate occurs in the bottom half, from 2.59E−9 to 3.66E−9 kg m−2 s−1 . Fig. 6. Erosion rate on the bend surface, outlet circumference, for fluid velocity equal to 10 m s−1 .

of increasing the erosion rate without significantly perturbing the flow field in the pipe and the bend. The qualitative results are the same as in the previous cases, but the maximum erosion rates of 8.38E−10 and 3.96E−9 kg m−2 s−1 , for a solid particle phase injected of 0.1 kg s−1 , with 5 and 40% gas void fraction, respectively. The case matrix high values configurations in term of flow velocity show the dramatic effect of the fluid dynamic field both on the erosion distribution on the bend surface and on the erosion rate. The main effect is to push the particle streams to impact on the extrados, side wall of the bend, towards the exit of the bend zone, due to the inertial and drag forces. In this case almost half of the bend surface is significantly interested by the erosion. It is also clearly recognisable from the results that the secondary flow paths effects on the erosion rate are that a double peak zone develops, one above and one under the horizontal symmetry plane of

8.2. Erosion–corrosion The synergic effects of erosion and corrosion give rise to an overall damage which can be related to the simulation conditions with a design of experiments approach. The single and joint effects of the simulation parameters have been evaluated as described, e.g. in [16] and are reported in Table 4. The intensity of the effects on a given quantity are normalised to 1, positive and negative values refer to correlation and anti-correlation between the parameter and the effect, respectively. The damage estimators we considered are: (i) the mean damage (MD), defined as the sum of the erosion and corrosion damage rates (expressed in kg m−2 s−1 ) and (ii) the relative mean corrosion damage (RMCD), defined as the ratio of the mean corrosion damage to the MD. The effects on the MD estimator have been evaluated from the decimal logarithms of the damage rates. From Table 4 it can be concluded that the synergic damage effects can be typically related to a single dominating factor. The flow velocity v is the leading factor driving synergistic damage for the stainless steel. The next single most important factor is the particle injection rate p. It is worth

Table 4 Normalised single and joint effects of operating parameters on erosion–corrosion damage estimators

Fig. 7. Erosion rate on the bend surface, outlet circumference, for fluid velocity equal to 2 m s−1 .

Operating parameters

MD DSS

MD carbon steel

RMCD Duplex

RMCD carbon steel

Flow velocity, v Gas fraction, g Particle injection rate, p v⊗g v⊗p g⊗p v⊗g⊗p

1.000 −0.02777 0.5092

−0.2841 −0.2291 −1.000

−1.000 −0.006327 −0.2239

−1.000 −0.1551 −0.7219

0.1245 0.1842 −0.02600 0.06879

0.02065 −0.1845 −0.1238 0.05558

−0.02502 0.006773 −0.01647 −0.01015

−0.1644 −0.6940 −0.1368 −0.1445

MD: mean damage (mean erosion + mean corrosion). RMCD: relative mean corrosion damage (mean corrosion/MD).

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noting that the introduction of a corrosion model allowing for damaging saturation brings about a reversal of the relative importance of v and p with respect to the linear model reported in [16]. According to our simulation, the gas volume fraction g and all the joint effects play a minor role in the erosion–corrosion damaging of the DSS. v exhibits the single most important effect on the ratio of erosion versus overall damage. A limited enhancing effect of the relative corrosion damage is due to the combination of v and p. As far as the carbon steel is concerned, the overall damage is negatively correlated with p, v and g, p being the single most important factor. Joint effects seem to play a role comparable to that of secondary single effects. A slight enhancing effect is attributed to the triple joint effect. The relative corrosion damage is mainly negatively affected by vp and the joint effect of v and p are the next most important factors tending to enhance the relative erosion damage. The differences in the coupling of the erosive and corrosive actions for the two different materials are evident from the above discussion. v and p tend to enhance the overall damage of the passivating alloy and to depress that of the actively corroding one. If these quantities are increased, the erosion fraction of the overall damage tends to increase for both kinds of material. v is the single most important factor in damage enhancement for the stainless steel, while p is the single most important factor in damage reduction by plasticisation for the carbon steel. As far as the stainless steel is concerned, the factor by far dominating the relative enhancement of the erosion to overall damage is v, while for the carbon steel even though v is still the key factor, p and joint actions of v and p play quantitatively similar roles.

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• gas volumetric void fraction effects: the main effects of the mixture composition on the solid phase—in terms of drag force or momentum exchange, inertial and gravitational forces—have been pointed out; the bottom half of the bend is more damaged than the top half when a high gas content is present; • fluid velocity effects: the gravitational settling is the main effect at low velocity values, while the drag force is more important at high values; this moves the zone of main erosion from the bottom, entrance zone of the bend to the extrados, side wall, outlet zone of the bend, as the fluid velocity increases; also the damaged area increases with fluid speed, as the particle streams are more spread in trajectory; • solid phase content effects: increasing the solid content injected into the mixture leads to a concentration of the erosion damage, due to a corresponding increase of importance of the streams impinging with high impact angle on the bend walls; • other general, qualitative effects arose from the CFD analysis, namely (i) secondary flow paths develop in the bend region, that decrease the erosion damage on the bend middle line in the horizontal plane and (ii) erosion damage tends to concentrate towards the outlet of the bend zone. • The flow velocity is the single most important variable affecting the erosion–corrosion behaviour of both passivating and actively corroding metals. The joint effects of phase composition and flow parameters seem to have a limited effect on the nature and amount of damage. • Corrosion-enhancing and corrosion-limiting mechanochemical effects have been identified for the passivating and actively corroding alloys, respectively.

9. Conclusions References The following conclusive remarks ensue from our analysis. As far as the fluid dynamic analysis is concerned: • the fluid dynamics of four-phase mixtures has been studied by means of a CFD tool. The treatment of the mixture as an homogeneous mixture does not allow a thorough evaluation of the different flow regimes which could occur in the pipe and the bend (e.g. slug, intermittent, bubbly), hence the particular effects they could have on the solid phase transport have been disregarded in the present approach. The evaluation of the effects of these regimes on erosion–corrosion is a challenging topic, especially in horizontal multiphase flows. Our essentially homogeneous treatment of the mixture has been chosen also in order to limit the computational effort when compared with that required by the main goal of the study, i.e. the investigation of erosion–corrosion effects; • the use of a three-dimensional geometry proved to be vital for similar investigations, because secondary flow paths develop owing to turbulence and affect the particle dynamics and ensuing erosion and erosion–corrosion rates;

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