# Physical Background, Computations and Practical Issues of the Magnetohydrodynamic Pressure Drop in a Fusion Liquid Metal Blanket

## Abstract

**:**

## 1. Introduction

^{1}W/m-K) and low viscosity (~10

^{−7}m

^{2}/s) that make them very favorable for heat removal. PbLi is considered by many researchers as a more attractive breeder/coolant option than pure Li due to its lower chemical reactivity with water, air and concrete, but PbLi is more corrosive and has higher density and more undesirable activation products.

^{6}S/m) in a strong magnetic field induces electric currents, which, in turn, interact with the magnetic field resulting in strong flow opposing electromagnetic Lorentz forces, which can be several orders of magnitude higher than viscous and inertial forces in ordinary flows. The associated MHD pressure drop is typically high (up to several MPa), and can be unacceptable, because of several reasons:

- -
- Mechanical stresses in the structural walls are above the materials limit;
- -
- High pumping power that diminishes the overall blanket efficiency;
- -
- Unavailability of high capacity LM pumps.

## 2. Examples of LM Breeding Blankets and Their Pressure Drop

## 3. Mathematical Formulation of the Problem

#### 3.1. Governing Equations of LM MHD Flows

**g**= (g, 0, 0). The flow can change its direction and the duct can experience variations of the cross-sectional area along its axis as shown in the figure. Such a flow induces its own magnetic field ${\mathit{B}}^{\prime}$, so that the total magnetic field is a superposition of the applied and induced one: $\mathit{B}={\mathit{B}}^{0}+{\mathit{B}}^{\prime}$. The associated induced electric current

**j**$=({j}_{x,}{j}_{y},{j}_{z}$) interacts with the magnetic field resulting in the electromagnetic Lorentz force ${\mathit{F}}_{L}=\mathit{j}\times \mathit{B}$ opposing the flow. The flow in a blanket can also be affected by buoyancy forces that arise in the liquid due to density variations caused by neutron volumetric heating. A mathematical model for such a flow can be formulated in several ways, including a few choices for main variables, various approximations and coordinate systems (see, e.g., discussion in [34]).

**V =**($U$, V, W), pressure P, and temperature T as the main variables. Two basic approximations are employed. One is the so-called inductionless approximation (also known as the low magnetic Reynolds number approximation), where induced magnetic field is neglected compared to the applied one such that the Lorentz force term can be written using only applied magnetic field: ${\mathit{F}}_{L}=\mathit{j}\times {\mathit{B}}^{0}=\left({j}_{y}{B}_{z}^{0}-{j}_{z}{B}_{y}^{0},{j}_{z}{B}_{x}^{0}-{j}_{x}{B}_{z}^{0},{j}_{x}{B}_{y}^{0}-{j}_{y}{B}_{x}^{0}\right)$. The second one is the Boussinesq approximation, in which fluid density differences are ignored except where they appear in the buoyancy force term.

_{0}: ρ is the density, ν is the kinematic viscosity, $\beta $ is the thermal expansion coefficient, ${C}_{p}$ is the specific heat, σ is the electrical conductivity and k is the thermal conductivity. Equations (1)–(5) can be used for both laminar and turbulent flows. It should be noted that the equations neglect some molecular non-equilibrium effects on the temperature and velocity filed, such as magnetization, polarization or Peltier, Thompson, Seebeck and Doufour effects, which under the blanket conditions are weak compared to others [10].

_{m}= ${U}_{*}$L/ν

_{m}has to remain much smaller than unity. This is typically the case in almost all liquid metal flows in a blanket due to the relatively low breeder velocities ${U}_{*}$, small characteristic blanket dimension L, and high magnetic viscosity ν

_{m}. One possible exception is the plasma disruption scenario [35], where the transients in the magnetic field may result in very high LM velocities of the order of ~10

^{2}m/s [36], such that Re

_{m}> 1. In such abnormal scenarios, the induced magnetic field cannot be neglected anymore, therefore the full magnetic induction formulation has to be used. The obvious advantage of the $\phi $-formulation, especially in the case of numerical computations, is that only one elliptic electromagnetic equation Equation (8) is required. The utilization of the magnetic field-based formulation requires three elliptic equations to be solved in a larger computational domain that includes not only the blanket but also some surrounding space (“vacuum”), where the induced magnetic field vanishes. This implies a significant computational effort compared to the $\phi $-formulation as solving elliptic equations is the most time-consuming part of the computations. When using the B-formulation, the electric current can be computed with the help of Ampèr’s law:

_{m}<< 1, Re

_{m}~ 1 and Re

_{m}>> 1.

**A**, as well as the current vector potential

**T**or some combinations of the above quantities, for instance

**A**− $\phi $ [38]. Attempts on implementation of a formulation making use of the induced electric current

**j**as the main electromagnetic variable are relatively recent. The equations for the induced electric current and thin wall boundary conditions were derived in [34]. The suggested formulation (denominated “j-formulation”) was then applied in the finite-difference computations of three common types of MHD wall-bounded flows: (i) high Hartmann number fully developed flows in a rectangular duct with conducting walls; (ii) quasi-two-dimensional duct flow in the entry into a magnet; and (iii) flow past a magnetic obstacle.

#### 3.2. Boundary Conditions

**V**= 0. In rare cases, for example at the interface between the PbLi breeder and SiC ceramics, a “slip” boundary condition may serve as a more appropriate condition:

**V**=

**V**

_{s}, where

**V**

_{s}is the “slip” velocity [39]. If there is no contact resistance between the liquid and the wall, the electrical contact is perfect. This implies the wall-normal component of the electric current and the electric potential to be continuous across the interface: ${{j}_{n}|}_{l}={{j}_{n}|}_{w}$ and ${\phi |}_{l}={\phi |}_{w}$. The tangential component of the current experiences a discontinuity at the interface: ${{j}_{\tau}/\sigma |}_{l}={{j}_{\tau}/\sigma |}_{w}$ because the electrical conductivity of the wall ${\sigma}_{w}$ is in general different from that of liquid ${\sigma}_{l}$. When the electrically conducting wall is “thin”, i.e., the wall thickness t

_{w}is much smaller than the characteristic duct dimension L, t

_{w}<< L, the so-called thin-wall boundary condition can be used [40]. Physically, the thin-wall boundary condition means that the electric current enters the thin wall from the liquid and then flows there in the tangential direction so that the electric potential does not vary across the wall to the leading order of approximation. This condition allows reducing the solution of Equation (6) to the fluid region. In the case of a “thick” conducting wall, the boundary conditions on the electric potential have to be imposed at the outer surface of the wall to enforce the wall-normal current density component to be zero. The temperature boundary conditions can be one of the three types, Dirichlet, Neumann or Robin, and can be imposed either at the liquid-solid interface or at the outer wall depending on the problem.

#### 3.3. Dimensionless Form of of Governing Equations and Basic Dimensionless Numbers

^{2}/Re is the ratio of electromagnetic to inertia forces. The Froude number $Fr=\frac{{U}_{*}}{\sqrt{gL}}$ is the ratio of inertia to gravity. The Grashof number $Gr=\frac{g\beta {L}^{3}\Delta T}{{\nu}^{2}}$ is the ratio between the buoyancy and viscous forces. The characteristic temperature difference in Gr is typically defined using the so-called neutron wall load (NWL), which is the integral of ${q}^{\u2034}$ over the radial depth of the blanket. For different LM blanket concept and designs, the NWL changes in the range from 0.7 MW/m

^{2}to 2.5 MW/m

^{2}. One more dimensionless parameter, Pr = ν${\rho}_{l}$C

_{p}/${k}_{l}$, is the Prandtl number, which is the ratio of momentum diffusivity to thermal diffusivity. Its small value in liquid metals of ~0.01 indicates that the heat diffusion in the liquid breeder dominates over the convection transport. For MHD flows in thin-walled ducts, another dimensionless parameter called “wall conductance ratio” can also be constructed that characterizes the electrical conductance of the wall in comparison with the conductance of the liquid: ${c}_{w}=\left({t}_{w}{\sigma}_{w}\right)/\left(L{\sigma}_{l}\right).$ This parameter, which in blanket applications is typically much smaller than unity, strongly influences the MHD pressure drop in all LM blanket concepts unless the walls are electrically insulated. Typical values of the key dimensionless parameters for the blanket concepts introduced in Section 2 of this paper are shown in Table 1.

## 4. Special Classes of MHD Flows in a LM Blanket

#### 4.1. Fully Developed MHD Flows

_{x}= 0, j

_{y}= j

_{y}(y,z) and j

_{z}= j

_{z}(y,z). Such a 2D electric current distribution in the cross-sectional plane suggests existence of only one non-zero component of the induced magnetic field ${B}_{x}^{\prime}$. The two electric current components can be computed from it using Ampèr’s law: ${j}_{y}=\frac{1}{\mu}\frac{\partial {B}_{x}^{\prime}}{\partial z}$ and ${j}_{x}=-\frac{1}{\mu}\frac{\partial {B}_{x}^{\prime}}{\partial y}$. Outside the duct, in the surrounding vacuum domain, the induced magnetic field is zero. Using these intrinsic features of the fully developed flow, and also assuming that the conditions of the inductionless approximation are met (i.e., Re

_{m}<< 1), the governing equations can be written as follows:

_{w}= 0 at the liquid-wall interface corresponds to a non-conducting duct where all currents are closed in the liquid domain. Other asymptotic case of $\frac{\partial {\tilde{B}}_{x}^{\prime}}{\partial \tilde{n}}=0$ at ${c}_{w}\to \infty $ corresponds to a duct with perfectly conducting walls, such that the electric currents induced in the liquid domain close completely through the electrically conducting wall. Equation (19) was first introduced in [41] by Shercliff. It can be considered as a particular form of the more general thin wall boundary condition proposed later by Walker in [40] for 3D MHD flows:

_{w}= 0), for which analytical solution was obtained independently in [43]. All solutions were derived in the form of infinite series.

_{w}= 0, c

_{w}= 0.05 and c

_{w}= 0.1 at Ha = 200. In this figure, the velocity is scaled by the mean velocity ${U}_{m}$, and the induced magnetic field by $R{e}_{m}{B}_{z}^{0}.$ In the fully developed MHD flows, the induced axial magnetic field serves as a stream function of the induced electric current, such that the magnetic field isolines plotted in the figure also depict the electric current distribution. As seen from the figure, the flow has thin boundary layers at the duct walls and a core region. The electric currents generated in the core close their circuit through the boundary layers and/or the electrically conducting walls. The current density in the boundary layers and in the electrically conducting walls (in the case of high wall electrical conductivity) is significantly higher compared to the core, where the current is distributed uniformly. The greatest changes of the velocity profile occur in the boundary layers, while the core demonstrates almost uniform velocity distribution in both conducting and non-conducting wall cases. The two primary boundary layers at the duct walls perpendicular to the applied magnetic field are known as Hartmann layers. As shown by Hunt and Shercliff, their thickness is scaled as 1/Ha.

^{1/2}, i.e., they are significantly thicker than the Hartmann layers. The most noticeable feature of the Hunt flow in a duct with electrically conducting Hartmann walls is formation of the “M-shaped” velocity profile with high velocity jets at the side walls. These jets carry a significant portion of the flow rate, especially if the wall conductance ratio is high. Unlike the Hunt flow, the Shercliff velocity profile does not exhibit the jet pattern. In the context of the main topic of the present article, the effect of magnetic field and the wall electrical conductivity on the MHD pressure drop is of particular interest. This effect is demonstrated in Figure 8 where the MHD pressure drop was computed as a function of the applied magnetic field using the Hunt and Shercliff solutions for a PbLi flow in a 10 m long square duct. In the compliance with the earlier conclusions, the pressure drop in the Shercliff flow increases linearly with the magnetic field, while in the Hunt flow it changes as a square of the magnetic field. At the highest magnetic field of 10 T that corresponds to the IB blanket, the pressure drop in the flow in the electrically conducting duct is about 1.2 MPa. The pressure drop in the case of a non-conducting duct is almost three orders of magnitude lower.

#### 4.2. Quasi-Two-Dimensional Turbulent MHD Flows

^{1/2}. As applied to the side layer, the ratio Re/Ha

^{1/2}represents the Reynolds number based on the boundary layer thickness. In the sub-region between the two threshold lines in the Ha-Re diagram, the Hartmann layers are laminar and stable while the side layers are expected to be unstable. The primary instability mechanism is the inflectional Kelvin-Helmholtz instability associated with the inflection points in the velocity profile. The flow eventually turns to turbulence but it appears in a special form of quasi-two-dimensional (Q2D) turbulence.

#### 4.3. MHD Flows with Buoyancy Effects

^{3}, which is typical to the DCLL blanket, $\gamma $ was computed as 36.7 K/m and $\Delta P$ = 61 × 10

^{3}Pa. This $\Delta P$ is significantly lower than the maximum allowable blanket pressure drop of 2 MPa. Compared to the pressure drop data plotted in Figure 8 for a single duct, the $\Delta P$ of 61 × 10

^{3}Pa is much higher than the friction pressure loss in Figure 8b for a non-conducting duct but lower than the total pressure drop in Figure 8a for a conducting duct. Therefore, it can be concluded that the MHD pressure drop in a blanket caused by buoyancy forces should be taken into account, especially in the case of electrically insulated walls.

## 5. Origins of the MHD Pressure Drop in a Blanket

^{4}times higher compared to the Poiseuille flow at the same Re number. In the above formulas for the resistance factor, both Ha and Re are constructed using the channel half width b.

_{y}and j

_{z}and the transverse component of the applied magnetic field: ${F}_{Lx}={j}_{y}{B}_{z}^{0}-{j}_{z}{B}_{y}^{0}$. The axial current component j

_{x}does not have a direct effect on $\Delta {P}_{III}$. In a fully developed flow in a conduit with non-conducting walls, $\Delta {P}_{III}$ is zero because the electric currents close their circuit in the cross-sectional plane inside the liquid. In ducts with electrically conducting walls, the induced current is closed through the wall, such that the result of the integration of the Lorentz force term is non-zero. The associated pressure drop depends on the magnetic field strength and the wall electrical conductivity and thickness. Typically, this component of the MHD pressure drop in ducts with electrically conducting wall is very high compared to others. For example, in the Hartmann flow between two electrically conducting plates, the resistance factor associated with the pressure opposing Lorentz force (see, e.g., [6]) is:

^{3}. An equation showing conditions when the pressure loss related to the Lorentz forces becomes dominating over the viscous friction loss, i.e., ${\lambda}_{em}\gg {\lambda}_{f}$, can be obtained from Equations (41) and (42), resulting in:

## 6. 2D and 3D MHD Pressure Drop

_{w}) and the magnetic field strength (Ha). This coefficient can be obtained from analytical solutions of evaluated from the experimental or numerical data.

## 7. MHD Pressure Drop in Electrically Coupled Blanket Components

## 8. Approaches to Calculation of the MHD Pressure Drop in a Blanket

- R
_{1}is associated with the poloidal flow in the “cold” feeding duct; - R
_{2}—radial flow from the cold duct to a module; - R
_{3}—flow in the expansion at the entry to a module; - R
_{4}—poloidal (upward) flow in the front duct facing the plasma; - R
_{5}—flow in the U-turn at the top of the module; - R
_{6}—poloidal (downward) flow in the return duct; - R
_{7}—flow in the contraction at the exit from the module; - R
_{8}—radial flow from the exit of a module to the collecting “hot” duct; - R
_{9}—poloidal flow in the “hot” collecting duct.

#### 8.1. Exact Analytical Solutions

#### 8.2. Asymptotic Solutions

_{w}, and the duct aspect ratio $\beta =a/b$. In these formulas, the Hartmann and Reynolds numbers are built using the dimension b, the half width of the duct in the direction of the applied magnetic field. In the case of the pipe flow, Ha and Re are built using the inner pipe radius R.

#### 8.2.1. Rectangular Duct with Non-Conducting Walls in a Transverse Magnetic Field

#### 8.2.2. Rectangular Duct with Non-Conducting Walls in an Inclined Magnetic Field

#### 8.2.3. Rectangular Duct with Non-Conducting Hartmann Walls and Ideally Conducting Side Walls in a Transverse Magnetic Field

#### 8.2.4. Rectangular Duct with Ideally Conducting Side and Hartmann Walls in a Transverse Magnetic Field

#### 8.2.5. Rectangular Duct with Thin Electrically Conducting Walls in a Transverse Magnetic Field

#### 8.2.6. Circular Pipe with Thin Electrically Conducting Walls in a Transverse Magnetic Field

#### 8.3. Asymptotic Numerical Techniques. Core Flow Approximation

#### 8.4. Full Numerical Computations

^{2}. A diagram summarizing progress in high Ha computations is shown in Figure 16. MHD computations for duct flows were pioneered in the 1970s but at that time were limited to Hartmann numbers of a few tens [88]. The computations progressed quickly over the next decades reaching Hartmann numbers on the order of hundreds in the late 1980s (e.g., [89]) and a few thousands around 2010 [90]. Significant acceleration in MHD computations can be seen at around 2005 due to development of a new consistent and conservative scheme [91]. However, the progress has been different between simple geometry flows (e.g., in a straight rectangular duct) and more complex flows in blanket-relevant geometries as also shown in Figure 16. At present, computations for simple geometry flows can be performed at any blanket relevant Hartmann numbers. Computations for complex 3D blanket components are less advanced but are rapidly progressing to the target Ha number.

^{4}in the poloidal flows and Ha = 3.3 × 10

^{4}in the manifold) were performed for a full blanket geometry. The most interesting conclusions from this study on the part of the MHD pressure drop are summarized below in Section 9.1.

#### 8.5. Experiments

_{p}(in Equation (44)) or k

_{p}in Equation (45) can be tabulated as a function of the duct and flow parameters. As a matter of fact, the experiments and analytical studies guided the development of LM magnetohydrodynamics in the earlier years. The significant increase in the drag and turbulence suppression in channel flows in the presence of a wall-normal magnetic field were first investigated in the experiments by Hartmann in Lazarus in 1937 [136]. Further experiments also demonstrated suppression of flow instabilities and turbulence as the magnetic field increases for various liquid metals, flow configurations and wall materials [7]. Later, experimental observations of high-amplitude low-frequency fluctuations in duct flows in a strong magnetic field [13] led to the development of the concept of Q2D MHD turbulence. All correlations for MHD pressure drop shown in Section 8.2 have been verified in the experimental studies. Here, we would like to refer to the experiments and analysis by Miyazaki et al. in [137] and [138] where good accuracy of simple correlations for the MHD pressure drop in rectangular duct and circular pipe flows were confirmed with the experimental data. Even now, when the numerical computations take a leading role in the blanket design and analysis, the importance of experiments as a primary validation tool is very high.

## 9. Examples of 3D Numerical Computations of the MHD Pressure Drop

#### 9.1. MHD Flow Computations for a PbLi Blanket Prototype at Ha ~ 10^{4}

^{4}in the manifold region and 1.1 × 10

^{4}in the poloidal channels on a very fine mesh of ~ 320 × 10

^{6}cells to accurately capture all flow features using a finite-volume code “MHD-UCAS” [94]. As shown in Figure 17, four cases have been simulated, including: Case 1 for electrically conducting walls, Case 4 for non-conducting walls, and Cases 2 and 3 with partial electrical insulation of the blanket conduits using SiC flow channel inserts (SiC FCIs) placed at selected locations. In these four cases, the computed MHD flows were analyzed for: (1) MHD pressure drop, (2) flow distribution, (3) flows in particular blanket sub-components, (4) 3D and flow development effects, and (5) unsteady flows.

#### 9.2. Computations and Analysis of MHD Pressure Drop in the Inlet and Outlet Manifolds of the DCLL Blanket

_{exp}< 12 accurate correlations for the 3D MHD pressure drop were established. Here, the Hartmann and Reynolds number are based on the dimension and the velocity of the large duct and r

_{exp}is the expansion ratio. In doing so, two flow regimes (originally postulated by Hunt and Leibovich [142]) were observed: (1) an inertial-electromagnetic (IE) regime characterized by the balance of inertia and Lorentz forces inside the internal boundary layer at the sudden change in duct aspect ratio, and (2) a viscous-electromagnetic (VE) regime characterized by the balance of viscous and Lorentz forces. As determined in [62], flows with Re⁄√Ha > 3 are in the IE regime while those with Re⁄√Ha < 3 are in the VE regime. Most of the blanket flows are expected to be in the IE regime where inertial forces are especially important.

_{ve}, d

_{ve}, k

_{ie}, and d

_{ie}are functions of the expansion ratio, r

_{exp}:

## 10. Practical Approaches to Mitigate the MHD Pressure Drop

#### 10.1. Electrical Insulation

^{2}due to higher currents closing through the electrically conducting walls. The walls often have higher electrical conductivity than the LM breeder itself resulting in high electric current densities in the core of the flow. For example, RAFM steel has electrical conductivity about two times higher than PbLi. Electrical decoupling between the liquid breeder and the electrically conducting walls can significantly reduce the electric current density, and as a consequence, the MHD pressure drop. Taking into account high Ha in blanket applications, the reduction of the MHD pressure drop in long poloidal breeding ducts or radial supply pipes by electrical insulation can achieve, in theory, three orders of magnitude compared to the ducts without insulation. Strictly speaking, using electrical insulation in self-cooled blankets is mandatory. Otherwise, at high breeder velocities of 0.5–1 m/s in self-cooled blankets, the MHD pressure drop in the breeding zone can easily reach 10 MPa or higher. The electrical insulation is usually not considered for low breeder velocity HCLL and WCLL blankets within the breeding zone, but it might be needed inside the pipes of the supply system where the breeder velocity is significantly higher. In the DCLL blanket, the electrical insulations is mandatory for all IB blankets where the magnetic field is ~10 T. The magnetic field of OB blankets is two times lower ~5 T, such that the requirements on the electrical insulation may be mitigated, but insulating FCIs can still be needed for thermal insulation to reduce the heat leakage from the PbLi into the cooling He streams.

#### 10.1.1. Flow Channel Inserts

#### 10.1.2. Electroinsulating Coatings

_{2}O

_{3}, CaO, MgO and Y

_{2}O

_{3}or nitrides, such as A1N and Si

_{3}N

_{4}were considered in the past as candidate materials. They have sufficiently low electrical conductivity such that the thickness < 0.2 mm can provide required MHD pressure drop reduction as demonstrated, for example, in [156]. However, there are many practical issues associated with the fabrication of the coatings and their stability in the harsh blanket environment. Both oxides and nitrides need to be applied onto the structural material with sufficient bonding. When doing this, it is problematic to keep the coating intact in a long run under thermal cycling conditions. The coatings are likely to suffer from the radiation effects and will experience mechanical stability issues. Their insulating properties can also degrade in time due to radiation effects. Development of cracks in the coating is one of the most serious feasibility issues as the electric current can leak through even tiny cracks of a few microns. The influence of cracks in perfectly insulating coatings on the MHD flow in a circular pipe was studied in [156] and that for a rectangular duct in [60,157]. The worst case occurs when two small line cracks are formed at the walls of the conduit parallel to the applied magnetic field. In the computed cases, the flow pattern near the crack shows a reversed flow and a more than 10 times increase of the MHD pressure drop compared to the coating with no cracks. These examples suggest that the number of cracks should not exceed a certain limit. In practice, it is hard to control the crack formation and their development, so that in spite of significant promise, utilization of thin insulating coatings in blanket applications looks questionable.

#### 10.2. Slotted Channel Geometry

#### 10.3. Poloidal, Toroidal and Radial Flows

**V**×

**B**

^{0}= 0. The radial and poloidal field components can influence the flow but they are much smaller compared to the toroidal field and cannot be responsible for strong Lorentz forces. Therefore, a toroidal blanket where most of the breeder flows occur along the magnetic field lines may exhibit lesser pressure drop compared to poloidal or radial blankets. However, a purely toroidal blanker is an idealization. In practice, there are always blanket components where the flow is radial or poloidal. Moreover, strong MHD pressure losses can be expected when the flow changes its direction from radial to toroidal as it happens when the LM is redistributed at the entrance to the blanket to feed the cooling channels at the FW. This occurs in the inlet manifold, where the flow is essentially 3D resulting in high 3D MHD pressure drop. High pressure losses can also be expected in the flow in the outlet manifold where the liquid is collected before leaving the blanket.

#### 10.4. Geometrical Discontinuities

## 11. Concluding Remarks

^{4}, a result that was not foreseen just a few years ago. However, such computations are still unique as they require effective, well-validated codes, mesoscale computers, and fine spatial resolution of large multi-material domains that need computational meshes of ~10

^{8}cells. Further code developments to possibly enable parametric computations of the MHD flows in the entire blanket and integration of the MHD codes into the multi-purpose computational suites [159] are underway such that significant progress in blanket design and analyses is immanent.

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**The US self-cooled lithium/vanadium blanket [3].

**Figure 3.**Schematics of the DCLL blanket with poloidal channels and SiC flow channels inserts [30].

**Figure 4.**Two possible designs of the DCLL blanket. (

**a**) full segment “banana” blanket; and (

**b**) modular blanket.

**Figure 5.**Schematics of the WCLL blanket. Shown is the CEA WCLL design for 2012 EU DEMO [31].

**Figure 6.**Sketch of a pressure driven LM flow in a complex geometry thin-walled duct with applied magnetic field and volumetric heating in the presence of gravity forces.

**Figure 7.**Velocity profiles (

**top**) and induced magnetic field distributions (

**bottom**) in the MHD flow in a square duct with electrically conducting Hartmann walls and non-conducting side walls (Hunt flow) at Ha = 200: (

**a**) c

_{w}= 0, (

**b**) c

_{w}= 0.05, and (

**c**) c

_{w}= 0.1. The applied magnetic field is along the z axis.

**Figure 8.**Effect of the magnetic field on the MHD pressure drop in the PbLi flow in a square duct at b = 0.1 m, l = 10 m, and ${\mathrm{U}}_{\mathrm{m}}$ = 0.1 m/s: thin steel Hartmann walls with t

_{w}= 5 mm (Hunt solution 2), and all walls are non-conducting (Shercliff flow).

**Figure 10.**Computed MHD fully developed upward (buoyancy-assisted) mixed convection flow of PbLi in a non-conducting square duct with volumetric heating at Ha = 200, Re/Gr = 10

^{3}and m = 1: (

**a**) velocity distribution, (

**b**) induced magnetic field, and (

**c**) temperature.

**Figure 11.**Sketch of a fully developed flow of electrically conducting viscous fluid between two infinitely long parallel plates. Without a magnetic field the velocity profile is parabolic (Poiseuille flow). With the magnetic field, the velocity profile has a uniform core and thin boundary layers with the thickness scaled as 1/Ha (Hartmann flow).

**Figure 12.**Basic 2D [(

**a**) and (

**b**)] and 3D [(

**c**)–(

**h**)] hydraulic elements of LM blankets: (

**a**) circular pipe, (

**b**) rectangular duct, (

**c**) single-channel bend, (

**d**) multi-channel bend, (

**e**) sudden change of cross-section, (

**f**) expansions and contractions, (

**g**) manifold, (

**h**) U-turn. All 2D and 3D elements can include FCIs to reduce the MHD pressure drop.

**Figure 13.**Example of induced 3D electric currents in the flow with sudden expansion computed in [62] at Ha = 1465 and Re = 50. The liquid moves from left to right. The magnetic field is in the transverse direction along the longest duct wall. Near the inlet inside the small duct, and also at the exit inside the larger duct, the currents are 2D. The 3D current circuits are clearly seen right upstream and downstream of the expansion.

**Figure 14.**Schematics of modular IB DCLL blanket of three modules and an equivalent hydraulic network.

**Figure 15.**Asymptotic structure of a dully developed MHD flow in a rectangular duct with either conducting or non-conducting walls in a transverse magnetic field.

**Figure 17.**The PbLi blanket prototype with radial supply ducts, inlet and outlet manifolds, poloidal channels and a U-turn section at the top [67]. Four cases of electrical insulation 1–4 have been considered.

**Figure 18.**Pressure drop along the module centerline (

**left**) and the summary of the MHD pressure drop for the four insulation cases (

**right**).

**Figure 19.**The simulation geometry is a duct with a sudden expansion or contraction depending on the flow direction. The flow in the inlet manifold features expansion (the flow direction is from left to right), while the flow in the outlet manifold (the flow direction is from right to left) experiences contraction.

**Figure 20.**Dimensionless 3D MHD pressure drop P* scaled by the dynamic head in a LM duct flow with sudden expansion or contraction. (

**a**) Effect of the Reynolds number. (

**b**) Effect of the Hartmann number.

**Table 1.**Examples of the dimensionless parameters in several blanket concepts for DEMO, ITER TBM and FNSF, including inboard (IB) and outboard (OB) blankets. Dimensional parameters ${B}_{*}$, ${U}_{*}$, and L are also shown.

Parameter | Li/V Self-Cooled, DEMO IB [3] | HCLL, ITER TBM OB [27] | DCLL, FNSF IB/OB [21] | WCLL, DEMO OB [18] |
---|---|---|---|---|

Ha | 4.5 × 10^{4} | 1.1 × 10^{4} | 3.7 × 10^{4}/1.5 × 10^{4} | 9.8 × 10^{3} |

Re | 3.2 × 10^{4} | 670 | 7.5 × 10^{4}/1.7 × 10^{5} | 120 |

Gr | 6.0 × 10^{8} | 1.0 × 10^{9} | 6.6 × 10^{11}/1.0 × 10^{12} | 5.4 × 10^{11} |

N | 6.0 × 10^{4} | 1.8 × 10^{5} | 1.8 × 10^{4}/1.3 × 10^{3} | 8.0 × 10^{5} |

${\mathit{B}}_{\mathbf{*}}$, T | 10 | 4 | 10/5.5 | 4 |

${\mathit{U}}_{\mathbf{*}}$, m/s | 0.5 | 0.001 | 0.087/0.203 | 0.0002 |

L, m | 0.05 | 0.07 | 0.152/0.109 | 0.117 |

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**MDPI and ACS Style**

Smolentsev, S.
Physical Background, Computations and Practical Issues of the Magnetohydrodynamic Pressure Drop in a Fusion Liquid Metal Blanket. *Fluids* **2021**, *6*, 110.
https://doi.org/10.3390/fluids6030110

**AMA Style**

Smolentsev S.
Physical Background, Computations and Practical Issues of the Magnetohydrodynamic Pressure Drop in a Fusion Liquid Metal Blanket. *Fluids*. 2021; 6(3):110.
https://doi.org/10.3390/fluids6030110

**Chicago/Turabian Style**

Smolentsev, Sergey.
2021. "Physical Background, Computations and Practical Issues of the Magnetohydrodynamic Pressure Drop in a Fusion Liquid Metal Blanket" *Fluids* 6, no. 3: 110.
https://doi.org/10.3390/fluids6030110