Eurofer corrosion by the flow of the eutectic alloy Pb–Li in the presence of a strong magnetic field

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Abstract

A new investigation of the Eurofer 97 corrosion by the MHD flow of the liquid eutectic alloy Pb–17Li is presented. The experimental data previously obtained in Riga are confirmed and an attempt to model this phenomenon is presented. The model is based on a thermodynamic analysis of the dissolution and electro-dissolution mechanisms, leading to a relevant boundary condition at the liquid–solid interface. Then, analyzing the MHD flow, guiding ideas and scaling laws are derived for the dissolution rate of the Hartmann wall. The results obtained in the regime, where the solid wall is assumed to remain planar, allow determining a plausible value for an important non-dimensional number, the dissolution number Di. A linear analysis leads to predictions on the mechanism responsible for the formation of streaks imbedded within the Hartmann layer and associated with the wall shape disturbance, as well as for the selection of the unstable mode. It is found that this mechanism is related to an additional contribution due to the electric current, based on an electro-dissolution number Ed.

Introduction

Since the beginning of the research on the liquid metal concepts for the blankets of fusion reactors, such as ITER and DEMO, a number of experiments [1], [2], [3], [4] have demonstrated that the duct wall, made of the martensitic Eurofer 97 steel selected for the ITER blanket, can be significantly corroded by the liquid metal used to breed the tritium and to extract the heat. Similar results have been obtained for austenitic steel walls [5]. In particular, it has been shown that the necessarily present high magnetic field is responsible for an increase of the corrosion rate. Recently, two new experiments performed in Riga [6], [7], [8], in conditions representative of the fusion blankets, have given a quantitative basis to this corrosion phenomenon. They were made in rectangular ducts, the working fluid being the eutectic alloy Pb–17Li at a temperature of the order of 550 °C, as in the European HCLL (Helium Cooled Lead Lithium) concept, or in the US DCLL (Dual Coolant Lead Lithium) concept. Although the applied magnetic field is not as strong as in the actual blankets, it is large enough to provide the typical Hartmann flow whose properties are well established [9], [10]. Following the first postmortem analyses reported in [6], [7], a corroded sample of the Hartmann wall used in the first experiment has been analyzed with our own facilities. This investigation, summarized in Section 2, confirms the observation first made in Riga on the other samples, namely that the corroded interface, instead of remaining flat, exhibits quite spectacular grooves aligned with the fluid velocity and possessing a clear but not perfect periodicity in the spanwise direction.

It is the purpose of this paper to investigate the dissolution properties associated with the MHD flow present in the near-wall region. Our general strategy is to introduce in the model equations the necessary ingredients to explain the formation of the grooves observed in the Riga experiment [6], [7], to simplify all other details (keeping however consistent their physical influence), and to put the emphasis on what we consider as the generic mechanism responsible for the development of this quasi-sinusoidal disturbance. To achieve this goal, on a number of effects judged as secondary, we use classic simplifications to allow for analytical solutions. Among the key ideas, we establish and use a correct boundary condition for the Fe-concentration at the interface, and we escape from the assumption that the near wall flow is parallel to the duct axis (Antimirov and Chaddad [11]). Indeed, we introduce and express its weak three-dimensionality, which consists in streaks whose order of magnitude is the same as the amplitude of the wall periodic disturbance, and whose contribution to the wall shape disturbance is found to be crucial.

Our contribution to this effort is based, first, on a consistent thermodynamic expression for the dissolution and electro-dissolution processes and then, on the coupling between this physicochemical analysis and a 3D MHD model for the disturbance of the Hartmann flow. This thermodynamic attempt, described in Section 3, results in a consistent expression for the Fe-flux at the interface, from which a boundary condition is derived for the Fe-concentration in the melt. Unlike in previous attempts [12], where this boundary condition was reduced to the assumption that the Fe-concentration at the wall is given and equal to the solubility limit, we express and derive the difference between this concentration at the equilibrium ceq, and its local value in the melt at the interface ci. Beside pure dissolution, in the presence of the magnetic field, we show that an additional transport due to the induced electric current, that we call electro-dissolution, can significantly contribute to the whole corrosion process. In short, this electric current acts as a sort of electrons wind, which may extract Fe-atoms from the first atomic layers in the solid and carry them into the fluid. In the liquid metal, diffusion and convection are able to achieve the transport of the dissolved Fe-atoms and keep the difference ceq  ci large enough to avoid any saturation. To characterize these transport phenomena, three non-dimensional numbers are introduced, a dissolution number Di, an electro-migration number Em, and an electro-dissolution number Ed, whose definitions and orders of magnitude are discussed in Sections 4 Stable planar wall dissolution in the Hartmann flow, 5 Slow instability of the dissolving interface. They combine with the classic non-dimensional parameters in liquid metal MHD: the diffusion Péclet number PD for mass transport, the Hartmann number Ha and the conductance ratio C, which have the following orders of magnitude in the Riga experiment: PD  3.8 × 104, Ha  227 and C  0.39.

In Section 4, assuming that the wall remains planar and smooth during the corrosion process, the elementary solution for the Hartmann flow is used to derive expressions for the Fe-concentration in the melt and Fe-mass flux at the wall. Using simplifying approximations, such as the assumption that the inlet Fe-concentration is exactly zero, the continuous reduction of the wall thickness can then be derived and compared to the measurements. This leads to a direct estimation for the dissolution number (Di  0.6 × 10−10) in the Riga experiment.

Then, in Section 5, we show that, in the presence of the deformation of the corroded wall, the near wall flow does not remain exactly parallel to the axis and we model its quasi-steady 3D disturbance, essentially due to the small displacement of the fluid flow from its initial position. This flow analysis is based on the continuity equation and the classic vorticity equation including advection, stretching, as well as viscous and electromagnetic damping. Since the deviation from the plane shape remains very small, this analysis can be linear. Additional simplifications are made in assuming that the MHD damping can be expressed using a linear drag, expressed with a Hartmann damping time tem and in expressing the closure of the electric currents in the solid wall using the classic thin wall approximation. Thanks to these approximations, the equations for the velocity field and the electric currents are decoupled, the differential order of the problem remains low and the set of equations is analytically solvable. In turn this model allows determining the additional Fe-flux due to the disturbance. The boundary condition derived in Section 3 together with the equation for the Fe-transport within the liquid metal allow expressing the corrosion disturbance and estimating the most amplified wave length of the wall shape perturbation. Predictions are finally derived for the extremely small but non-zero amplification rate and propagation speed of the Eurofer wall wavy-shape.

In Section 6, concluding remarks and a few comments are proposed aiming at first guidelines before any extrapolation to the ITER or DEMO conditions can be safely carried out. Orientations are also suggested for next experiments and theoretical analyses in regimes closer to the ITER or DEMO conditions.

Section snippets

Morphological analysis of a corroded sample

The interaction between Eurofer and liquid lead–lithium is a complex process and the purpose of this paper is not to give an exhaustive description of the experimental features. Several authors have previously investigated [1], [2], [12] the overall dissolution rates, with and without an externally applied magnetic field. The purpose of the present section is to carry out a simple investigation in order to specify the minimum ingredients necessary for the model developed in the next sections.

Boundary conditions for electro- and chemical-dissolution

The corrosion of Eurofer by liquid lead is the response of the atoms in the solid to a driving force, which is both of chemical and electrical origin. For the sake of simplicity let us consider that Eurofer consists mainly of iron: since there is no evidence of a chemical gradient coming close to the sample surface, this indicates that all the atoms in Eurofer behave in a similar manner as far as their interaction with liquid lead is concerned. Therefore, in our attempt to model this

A reminder on the Hartmann flow main properties

Consider first the fully established flow of the eutectic alloy Pb–17Li at a uniform temperature above its melting point, whose density, kinematic viscosity and electrical conductivity are denoted ρ, ν, σ, in a rectangular duct made of Eurofer. The duct width is denoted 2h, the wall thickness and conductivity are denoted a (the initial value being a0) and σw. As indicated in [7], the values of these parameters can be taken as: ρν1.02×103Pas m1,σ0.73×106Ω1 m1,σw0.95×106Ω1 m1, h  5 

The instability mechanism

Fig. 10 illustrates how the wall deformation does influence the near wall flow. Let us denote ζ=δε(t)ei(kr) the distance between the actual position of the interface and its undisturbed plane position, and characterize the disturbed shape by the wave vector k=(m,n,0), or K=(M,N,0) with M = , N =  in non-dimensional notations (the distances are then measured as multiples of the Hartmann layer thickness), and by the still unknown time-dependent small amplitude ɛ(t). In the vicinity of a hump

Concluding remarks

This paper first confirms previous analyses on the challenging experiments on Eurofer corrosion by the eutectic alloy Pb–Li in the presence of a strong magnetic field performed in Riga [7], [8]. It namely shows that the key phenomenon is the Fe-dissolution into Pb–Li, without any Pb-redeposition. Then, a thermodynamic analysis of the dissolution and electro-dissolution mechanisms leads to a consistent boundary condition for the Fe-concentration at the liquid–solid interface. On this basis, an

Acknowledgements

The authors are indebted to their Riga colleagues for having attracted their attention to this challenging problem, and particularly to Andrej Shishko, who provided them with detailed information on the experiments [6] and with whom they had stimulating exchanges. They are also pleased to thank their colleagues Ahcene Bouabdallah, Gérard Cognet and Evgeny Glickmann for fruitful discussions.

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