# MHD R&D Activities for Liquid Metal Blankets

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Description

**B**. They describe the conservation of momentum and mass:

**v**denotes the velocity, p is the deviation of pressure from isothermal hydrostatic conditions at the reference temperature T

_{0}, $\mathit{g}$ stands for gravitational acceleration, and $\mathit{j}\times \mathit{B}$ is the electromagnetic Lorentz force induced by the interaction of imposed magnetic field

**B**and electric current density

**j**. The latter is determined via Ohm’s law $\mathit{j}=\sigma \left(-\nabla \varphi +\mathit{v}\times \mathit{B}\right)$ that, combined with the condition for charge conservation, $\nabla \cdot \mathit{j}=0$, results in a Poisson equation for the electric potential $\varphi $:

_{0}.

_{p}is the specific heat of the fluid at constant pressure, k the thermal conductivity and Q a volumetric thermal source.

_{0}are a typical size and mean velocity in the considered geometry. A characteristic temperature difference ΔT can be defined for instance through the volumetric heat as ΔT = QL

^{2}/k.

## 3. Theoretical Description of MHD Flows

#### 3.1. Numerical Simulations

#### 3.2. Asymptotic Analysis

#### 3.3. System Codes

- Multiphysics: to include all relevant phenomena encountered in a nuclear reactor from two-phase thermal-hydraulics to neutron kinetics diffusion equations and fuel thermomechanics.
- Best-estimate: to provide an accurate prediction of selected figures of merit and to be as close as possible to the physical reality that one aims to simulate.
- Safety: suitable to demonstrate the reactor safety for licensing purposes, namely developed according to a clearly defined quality assurance (QA) methodology, accompanied by uncertainty quantification (UQ), extensively verified and validated to prove the code scalability.
- Industrial: i.e., robust and computationally inexpensive to allow simulations on a wide test matrix for sensitivity analysis and UQ in a reasonable timeframe.

#### 3.3.1. MHD Pressure Loss, Electromagnetic Coupling and Heat Transfer

#### 3.3.2. State-of-the-Art of MHD Modelling in SYS-TH Codes

#### 3.3.3. A Validation Exercise for SYS-TH Code MHD Modelling: HCLL TBM Mock-Up

_{0}= σu

_{0}LB

^{2}. The reference pressure value is imposed at location A. The code shows an excellent agreement with the experimental data in terms of total pressure loss and local values (−1% < ε < +14%), even without a model to represent electromagnetic coupling. The latter does not affect the overall pressure loss estimate due to the low contribution from BUs. On the contrary, the system code is not able to reproduce the flow distribution in the BUs, since it is significantly influenced by coupling [49]. Further model development is required to provide accurate predictions of flow distribution in Bus, in particular when heat transfer has to be quantified.

#### 3.4. Requirements for Theoretical Analysis under Fusion Conditions

#### 3.4.1. Grid Generation

#### 3.4.2. Numerical Schemes

**j**is very small. This is because the two terms $\nabla \varphi $ and $\mathit{v}\times \mathit{B}$ are of nearly the same magnitude but with opposite signs. The computation of gradients of electric potential requires appropriate numerical schemes, such as Least-Squares (LS) or skew-corrected Green–Gauss (GG

_{corr}) [53].

_{uncorr}), the blue curve by Green–Gauss combined with a linear interpolation for the correction (GG

_{corr}), and the green profile by a least square (LS) scheme. For insulating walls, the determination of the electric potential gradient by means of GG

_{uncorr}introduces an error whose magnitude depends on the mesh topology, and this scheme is very sensitive to grid irregularities. Applying either corrected GG

_{corr}or LS for the electric potential gradient leads to a significant improvement of the solution. Skewness-related perturbations are minimized so that errors are caused mainly by the grid cell size and not by the mesh type. However, for very large Ha (fusion conditions) the most accurate results for MHD flow in circular pipe have been obtained by using a grid with a structured core, a layer of hexahedral elements to resolve the boundary layers, and an unstructured mesh to connect both of them, together with LS scheme for electric potential gradient discretization [56].

#### 3.4.3. Suitable Closing Laws for SYS-TH Codes

## 4. Experiments

_{0}= 0.1 m/s and for PbLi properties as shown in Table 2, is characterized by a Hartmann number close to Ha = 9000. Experimental reproduction of such numbers is challenging for several reasons. In laboratory experiments, the strength of the magnetic field is limited to about B

_{exp}$\lesssim $ 2 T when normal conducting copper magnets are used. Moreover, the available space in the magnets reduces the typical length scale for mock-up experiments by at least a factor of two (see, e.g., [58]), i.e., L

_{exp}$\lesssim $ 0.05 m. If thermal insulation is required, L

_{exp}could become even smaller. Therefore, Hartmann numbers in PbLi mock-up experiments are limited to Ha < 1000–2000. Higher values of Ha are possible only in superconducting solenoids, where, however, the length of channels with transverse magnetic field is limited by the size of the magnet bore [59]. In addition to this, MHD experiments with PbLi have to be performed at high temperatures. This results in thermoelectric disturbances of signals of the measured induced electric potential and difficulties in operating the flow meters and pressure transducers.

^{3}/h, differential pressure between 30 pressure taps, and distribution of potential by up to 600 sensors on the wall or at the fluid–wall interface. The latter ones are of particular importance, since for $Ha\gg 1$, potential is constant along magnetic field lines and hence, the wall data give a good picture of the values inside the core of the fluid. Moreover, it is possible to calculate from Ohm’s law the components of velocity in the plane perpendicular to the magnetic field according to

_{1}; and T

_{2}> T

_{1}; to generate the horizontal temperature gradient driving the flow. To provide clear boundary conditions, the pipes were made of copper to ensure that their temperature is as uniform and constant as possible. Their outer surface is coated with a very thin electrically insulating layer to prohibit induced currents from closing into their walls and to avoid parasitic thermoelectric effects that could occur in contact with the model fluid GaInSn. The box is made of PEEK plastic and thermally insulated to provide adiabatic conditions before being inserted in the MEKKA magnet that produces a vertical magnetic field (Figure 6a). Temperature and electric potential were recorded with high-accuracy instrumentation at the most pertinent locations identified by preliminary simulations [72]. As an example of the data collected, nondimensional temperature profiles measured at the center of the cavity for hydrodynamic flows (Ha = 0) and various Gr are presented in Figure 6c. A convection cell forms in the center of the cavity, between the two pipes, and the buoyant flow results in a thermal stratification with the hot fluid staying on the top and the cold fluid on the bottom. Experiments have been performed for a variety of temperature differences (Gr) and magnetic field strengths up to Ha = 3000.

## 5. MHD Phenomena and Coupling with Heat and Mass Transfer

#### 5.1. Electromagnetic Flow Coupling

#### 5.1.1. Flow Distribution in Electromagnetically Coupled Parallel Channels

- Channels are stacked along magnetic field direction, α = 0° (Hartmann wall coupling);
- Channels are stacked transverse to the magnetic field, α = 90° (sidewall coupling).

#### 5.1.2. Coupled Flow in Manifolds of LM Blankets

_{in}) decreases, moving towards the top of the blanket, while the internal duct flow rate (Γ

_{out}) increases. Local flow reversals caused by electromagnetic coupling can be observed in Figure 11. In general, it is advisable to tailor the manifold configuration to ensure that all the channels have a similar mean velocity along their length in order to avoid coupling-mediated flow reversals.

#### 5.2. Turbulence and Heat Transfer

^{4}typical for blanket conditions implies that 3D turbulence cannot occur. This does not mean that the flow necessarily becomes laminar and steady-state. From the general physical perspective, it is evident that even a very strong magnetic field cannot prevent the growth of hydrodynamically unstable perturbations, if the perturbations are quasi-two-dimensional (with nearly zero velocity gradients along the magnetic field outside the boundary layers) and, therefore, do not generate significant Joule dissipation. The physical nature of such instability may vary. For example, growth of perturbations has been demonstrated for shear-flow instability of the Kelvin–Helmholtz type, thermal convection, or a combination of the two (see, e.g., [95,96,97,98,99,100,101,102] or the review [5] for further examples).

#### 5.3. Tritium Transport

^{−3}) at a certain interface is linked to tritium partial pressure ${p}_{T}$ through Sieverts’ law [104]

^{−3}Pa

^{−0.5}) is the Sieverts’ constant. At the interface between liquid metal and steel, the continuity of ${p}_{T}$ is assumed,

_{2}O

_{3}, ZrN, Al-Cr-O, Al

_{2}O

_{3}. These coatings may guarantee PRF higher than 1000 in the range 400–700 °C. Alumina Al

_{2}O

_{3}serves also to mitigate corrosion of Eurofer steel and represents the reference coating for DEMO [107]. Additional research is necessary in order to ensure high adhesive strength of coatings, compatibility with complex shapes, and good performance in a radiation environment.

#### 5.3.1. Tritium Analysis Methods, Transport Modeling and Coupling with MHD

#### 5.3.2. Tritium Transport under Fusion-Relevant Conditions

**WCLL DEMO blanket.**A number of studies have been performed to address the effect of MHD on tritium transport for the WCLL blanket concept for DEMO reactor [123,124]. In particular, flow in a portion of the breeding unit, as shown in Figure 12 on the top, has been simulated adopting a novel coupling strategy for the physics involved, and differences between pure hydrodynamic (Gr = 4.78 × 10

^{10}, Ha = 0) and magnetohydrodynamic (Gr = 4.78 × 10

^{10}, B = 4 T, Ha ≈ 11,000) conditions have been highlighted. In both models, the system has been assumed to be operated at steady-state conditions. Buoyancy effects have been introduced using the Boussinesq approximation. The radial profile of the volumetric nuclear heating on the equatorial midplane has been determined by means of the MCNP Monte Carlo code.

^{3}. By applying a toroidal magnetic field, the concentration is much more evenly distributed between the first wall and the edge of the stiffening plate, and the maximum value is smaller than $0.4$ mol/m

^{3}. Nevertheless, the presence of recirculation zones increments the tritium mean permanence time in the breeder unit, and the tritium concentration decreases rapidly with the radial coordinate.

**DCLL breeding blanket**. The DCLL blanket concept is characterized by relatively high PbLi velocities compared to the WCLL blanket. The MHD pressure drop is reduced by electrically decoupling the PbLi flow from the metallic structure by using insulating Flow Channel Inserts (FCI) [127]. In the case of the European DCLL blanket design [128], a sandwich-type insert is proposed, which consists of a thin alumina layer protected by two Eurofer sheets. The FCI divides the flow into two regions: the core flow and the gap flow between insert and wall.

#### 5.4. Corrosion

#### 5.4.1. Modelling of Steel Corrosion in PbLi

Material | Loop | T_{hot}°C | T_{cold}°C | Velocity m/s | Corrosion µm/year | Reference |
---|---|---|---|---|---|---|

IN-RAFM | Indian | 465 | 400 | 0.10 | 31–44 | [136] |

Eurofer | LIFUS 2 | 480 | 400 | 0.01 | 40 | [143] |

Manet I, F82H-mod, Optifer IVa, Eurofer | PICOLO | 480 | 350–400 | 0.22 | 90 | [138] |

CLAM, Eurofer, ODS-Eurofer | PICOLO | 550 | 400 | 0.10 | 200–220 | [141,142] |

Eurofer | PICOLO | 550 | 400 | 0.22 | 400 | [137] |

#### 5.4.2. Corrosion with Magnetic Field

**B**, and a larger mass loss occurs at the sidewalls (2–3 times stronger corrosion rates if compared to the Hartmann wall). These studies highlight the need for further experimental campaigns to increase the amount and the accuracy of corrosion data in MHD flows. Moreover, the method used to achieve improved correlations for the properties needed for corrosion modelling by matching calculated and experimental data, represents a valid procedure to obtain reliable predictions of corrosion rates in MHD flows.

## 6. Conclusions and Future R&D

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**MHD flow in a circular pipe in a fringing magnetic field $B\left(x\right)$ as used in [12]. In the figure, $B\left(x\right)$ is normalized by the value ${B}_{0}$ in the center of the magnet. Axial pressure gradients $\partial p/\partial x$ are calculated at the top and side of the circular pipe for $Ha=6600$. Results displayed for various numerical resolutions n

_{x}and n

_{z}in x and z directions highlight the rapid convergence of the asymptotic method.

**Figure 3.**(

**a**) Poloidal–radial sketch of the TBM geometry. The flow path (A–I) is depicted with colored lines; (

**b**) Numerical results (stars) and experimental data (dots) for nondimensional pressure drop vs. normalized length of the flow path (Ha = 3000, Re = 3360).

**Figure 4.**Influence of discretization schemes on MHD flow in an insulating pipe at Ha = 1000 [54]. Axial velocity profile along the diagonal marked in red in the sketch in (

**a**), for two grid types, O-grid (

**a**) and polyhedral mesh (

**b**), and various discretization schemes for the electric potential gradient: Least-Squares (LS), skew-corrected Green–Gauss (GG

_{corr}), uncorrected Green–Gauss (GG

_{uncorr}). Vertical dashed lines indicate the boundary of the core grid. The black dashed curve shows the asymptotic solution according to Chang and Lundgren [55].

**Figure 5.**Pipe flow in an axially increasing magnetic field (Ha = 5485 and Re = 10,043). Colored contours of axial velocity on the pipe symmetry plane, displayed in the upper plot, have been obtained from potential measurements at the electrically conducting wall.

**Figure 6.**(

**a**) Magneto-convection test section in front of the magnet during installation. For thermal insulation the box is surrounded from all sides by Styrofoam. The top insulation has been removed to obtain a view of the box and of the instrumentation. (

**b**) Design; (

**c**) nondimensional temperature profiles measured at the center of the cavity for several Gr and Ha = 0.

**Figure 7.**Sketch of liquid metal flow paths in an experimental MHD mock-up of a HCLL TBM for ITER. Nondimensional results for measured electric potential ϕ (symbols) along the upper wall (dashed line) are compared with numerical simulations (solid line).

**Figure 9.**Qualitative representation of flow rate imbalance in parallel ducts due to electromagnetic coupling (Ha = 10,000). (

**a**) Coupling between channels with equal mean velocity through Hartmann wall ($\alpha =0\xb0$) and side walls ($\alpha =90\xb0$). (

**b**) Coupling between channels with large difference in mean velocity and α = 90°. Channel no. 1 collects and returns all the flow rate incoming from no. 2 to no. 5 and Γ

_{0}represents the mean flux in ducts 2–5 assuming uniform flow distribution.

**Figure 10.**MHD flow in three electrically coupled ducts exposed to an inclined magnetic field, α = 67.5°, for Ha = 2500 and conductance parameter c = σL/(σt

_{w}) = 0.038. Velocity is plotted along the central line of the duct array and contours on the horizontal midplane.

**Figure 11.**Coupling-mediated flow reversal in a concentric co-axial manifold. Numerical results for Ha = 2000 for half channel (symmetry with respect to x = 0). (

**a**) Sketch of flow rate distribution along poloidal coordinate (${\mathsf{\Gamma}}_{in}+{\mathsf{\Gamma}}_{out}={\mathsf{\Gamma}}_{tot}$) and qualitative representation of regions where the flow is expected to recirculate in inflow and outflow manifolds. Velocity distribution in the manifold channel (

**b**) at the first cell (${\mathsf{\Gamma}}_{out}=0,{\mathsf{\Gamma}}_{in}={\mathsf{\Gamma}}_{tot}$), (

**c**) at the equatorial plane (${\mathsf{\Gamma}}_{out}={\mathsf{\Gamma}}_{in}={\mathsf{\Gamma}}_{tot}/2$), and (

**d**) at the last cell (${\mathsf{\Gamma}}_{out}={\mathsf{\Gamma}}_{tot})$. Crossed circles ⊗ mark flow reversal regions.

**Figure 12.**On the top the investigated geometry and the used mesh are shown. Contours of PbLi velocity for (

**a**) hydrodynamic (Ha = 0, Gr = 4.78 × 10

^{10}) and (

**b**) magnetohydrodynamic (Ha = 10,830, Gr = 4.78 × 10

^{10}) cases [124,125] are displayed on a radial poloidal plane in the middle of the central submodule.

**Figure 13.**Tritium concentration in LiPb and Eurofer for (

**a**) hydrodynamic (Ha = 0, Gr = 4.78 × 10

^{10}) and (

**b**) MHD (Ha = 10830, Gr = 4.78 × 10

^{10}) cases.

**Figure 14.**Steady-state molar flux exiting from the BU in PbLi (Out) and permeated through the piping system into the water (permeation) for the two cases.

**Figure 15.**Left: fully developed velocity profile in the gap of a central outboard poloidal channel of a DCLL blanket for $Ha=7630$ and $\partial p/\partial x=1740\mathrm{Pa}/\mathrm{m}$. Right: tritium concentration contours in the middle section of the PbLi channel, including PbLi gap, Eurofer walls and external Eurofer layer of the FCI.

Code | 2D Loss | 3D Loss | Coupling | Heat Transfer |
---|---|---|---|---|

RELAP5-3D [40] | Yes | Only for fringing magnetic field | No | No |

RELAP5/MOD3.3 [23] | Yes | Yes | No | No |

MARS-FR [45] | Yes | No | No | No |

MELCOR 1.8.5/1.8.6 [46] | Yes | No | No | No |

MHD-SYS [43] | Yes | Estimated through coupling | Based on analytical relations | Based on analytical relations |

GETTHEM [47] | Yes | Yes | No | No |

**Table 2.**Thermophysical properties of PbLi at 400 °C and examples of possible model fluids that allow experimentation at room temperature. Nondimensional parameters have been evaluated for experimental conditions with B

_{exp}= 2 T, L

_{exp}= 0.05 m and u

_{0}= 0.1 m/s. For the Grashof number Gr a wall heat flux of 10 W/cm

^{2}is assumed.

ρ kg/m ^{3} | ν × 10^{6}m ^{2}/s | σ × 10^{−6}1/Ω/m | k W/m/K | β × 10^{3}1/K | Re | Ha | Gr | |
---|---|---|---|---|---|---|---|---|

PbLi _{400 °C} | 9719 | 0.161 | 0.849 | 22.4 | 0.122 | 29,585 | 2273 | 1.2 × 10^{9} |

Hg _{20 °C} | 13,546 | 0.115 | 1.04 | 8.72 | 0.181 | 43,478 | 2584 | 9.6 × 10^{9} |

GaInSn _{20 °C} | 6353 | 0.340 | 3.32 | 24 | 0.122 | 14,706 | 3920 | 2.7 × 10^{8} |

NaK _{20 °C} | 868 | 1.06 | 2.87 | 21.8 | 0.29 | 4717 | 5585 | 7.3 × 10^{7} |

Material | Loop | T_{hot}°C | T_{cold}°C | Velocity m/s | B T | Exposure h | Corrosion µm/year | Ref. |
---|---|---|---|---|---|---|---|---|

P-91 steel | IPUL/Riga | 550 | 370–430 | 0.15–0.3 | 1.7 | 1000–2700 | 320–360 | [152,153] |

0.15–0.3 | Entrance | 1000–2700 | 200–226 | [152,153] | ||||

0.15–0.3 | Exit | 1000–2700 | 100–218 | [152,153] | ||||

Eurofer | 0.05 | 1.7 | 1000 | [156] | ||||

0.05 | 1.7 | 2000 | +50 ÷ 100% | [154] | ||||

550 | 350 | 0.05 | 1.8 | 1000 | +80 ÷140% | [155] | ||

515 | 350 | 0.05 | 1.8 | 2500 | +80 ÷ 180% | [155] | ||

316 L 1.4914 | CELIMENE ALCESTE | thermal gradient DT = 40K, convection | 0.001 and 0 | 1.4 | ÷50% +30% | [151] | ||

316 L | CELIMENE annular geometry | thermal gradient | Natural convection movements | 1.4 | Corrosion rate in quasi- stagnant conditions is larger on average than with B = 0 Magnetic field results in dissymmetry between dissolution and deposition rates in directions ⊥ or θ to B | [150] |

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## Share and Cite

**MDPI and ACS Style**

Mistrangelo, C.; Bühler, L.; Alberghi, C.; Bassini, S.; Candido, L.; Courtessole, C.; Tassone, A.; Urgorri, F.R.; Zikanov, O.
MHD R&D Activities for Liquid Metal Blankets. *Energies* **2021**, *14*, 6640.
https://doi.org/10.3390/en14206640

**AMA Style**

Mistrangelo C, Bühler L, Alberghi C, Bassini S, Candido L, Courtessole C, Tassone A, Urgorri FR, Zikanov O.
MHD R&D Activities for Liquid Metal Blankets. *Energies*. 2021; 14(20):6640.
https://doi.org/10.3390/en14206640

**Chicago/Turabian Style**

Mistrangelo, Chiara, Leo Bühler, Ciro Alberghi, Serena Bassini, Luigi Candido, Cyril Courtessole, Alessandro Tassone, Fernando R. Urgorri, and Oleg Zikanov.
2021. "MHD R&D Activities for Liquid Metal Blankets" *Energies* 14, no. 20: 6640.
https://doi.org/10.3390/en14206640