Introduction

Study of magnetohydrodynamic (MHD) flow in capillary porous systems is recently of interest related to possible high temperature heat exchangers for various energy applications. Tungsten and liquid tin capillary porous system is considered as possible solution for fusion reactor divertor plates, which have to resist heat loads with nominal power densities in the range of 5–20 MW/m2 and transient events that will cyclically increase those levels to the range of GW/m2 during millisecond-range temporal scales [1].

Several different models of MHD flows in porous media are considered by a number of authors at microscopic and macroscopic level. CPS consisting of 100 μm grids is simulated in [2]. A simplified model geometry where the mesh is presented as overlapped wire groups is considered. The results show that liquid metal MHD flows in porous media exhibit a variety of different flow patterns depending on the electric properties of the porous material and orientation and strength of the magnetic field.

A macroscopic analytical model for magnetohydrodynamic flow in a porous medium comprised of a packed bed of uniform spheres is developed in [3, 4].

Regular SC sphere packing is one of the simplest porous media models. Modification of it by increasing spheres radius at fixed centre leads to the overlapping SC models which can mimic properties of natural porous media like sandstone - porosity and permeability, it can even describe percolation properties with reasonable accuracy. The vast literature about SC [5, 6] and other regular sphere packings and their hydrodynamic, thermal and electrical properties [7] position SC model as a good starting point for MHD topics.

In the present work we investigate the hydraulic properties of simple cubic (SC) periodic array of overlapping spheres both numerically and experimentally. The test section is 3D printed from electrically non-conducting PLA filament and consists of spheres with diameter 6 mm and distance between spheres centres 5.6 mm. The orientation of the SC unit cell to the main flow and magnetic field direction and used coordinate system is shown in Fig. 1. The experiments are carried out at room temperature, the flow of up to 50 mL/s of In-Ga-Sn in magnetic field of superconducting magnet up to 5T is investigated.

Fig. 1
figure 1

Left - The test section porous media model. Main flow is oriented in z direction, the orientation of magnetic field is\({B}_{x}\) and \({B}_{xy}\) (equal \({B}_{x}\) and \({B}_{y}\) components) and \({B}_{z}\). Right - cross-section at the throat plane. S1 = 1.888 mm2, S2 = 1.667mm2, S3 = 3.000mm2, Ssolid=1040.430mm2, Sfull=1158.117mm2, Spores = Sfull - Ssolid =117.687 mm2

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Equations and Non-dimensional Numbers

MHD flow in the pore space is described by the incompressible Navier-Stokes equation with magnetic body forces included. Assuming the steady state of the flow, velocity distribution is described by Navier-Stokes and continuity equations:

$$\left(\overrightarrow{v}\nabla \right)\overrightarrow{v}=-\nabla p+\mu {\nabla }^{2}\overrightarrow{v}+\overrightarrow{j}\times \overrightarrow{B},$$
(1)
$$\nabla \overrightarrow{v}=0.$$
(2)

The first equation represents the conservation of momentum, the second – the conservation of mass. The term on the left side of Eq. (1) is associated with acceleration, whereas the terms on the right side – with normal, tangential and body forces correspondingly.

The electric current distribution \(\overrightarrow{j}\) satisfies the Ohm’s law for conducting media and should be divergence-free due to charge conservation:

$$\overrightarrow{j}=\sigma \left(-\nabla \phi +\overrightarrow{v}\times \overrightarrow{B}\right),$$
(3)
$$\nabla \overrightarrow{j}=0.$$
(4)

Instead of dimensional form of momentum conservation Eq. (1) it is more convenient to rewrite it for non-dimensional variables and to introduce relevant non-dimensional numbers. That simplifies interpretation of experimental and numerical results by analyzing the values of non-dimensional criteria in each specific case.

By introducing the characteristic values as: \(L=\frac{1}{2}\sqrt{S3, }{U}_{0}=\frac{Q}{{S}_{pores}}\) (see Fig. 1. for S3 and Spores definitions), \({p}_{0}=\rho {U}_{0}^{2}\), the non-dimensional Navier-Stokes equations are rewritten in the form:

$$\left(\overrightarrow{v}\nabla \right)\overrightarrow{v}=-\nabla p+\frac{1}{Re}{\nabla }^{2}\overrightarrow{v}+\frac{{Ha}^{2}}{Re}\overrightarrow{j}\times \overrightarrow{B}$$
(5)

or

$$\nabla p=-\left(\overrightarrow{v}\nabla \right)\overrightarrow{v}+\frac{1}{Re}{\nabla }^{2}\overrightarrow{v}+\frac{{Ha}^{2}}{Re}\overrightarrow{j}\times \overrightarrow{B},$$
(6)

where:

\(\mu [Pa\cdot s]\) – dynamic viscosity of the liquid,

\(Re=\frac{Inertial ~forces}{Viscous~ forces}=\frac{{U}_{0}L}{\upsilon }\) - Reynolds number,

\(Ha={\left(\frac{Electromagnetic~ forces}{Viscous ~forces}\right)}^{1/2}={B}_{0}L\sqrt{\frac{\sigma }{\mu }}\) - Hartmann number,

\(N=\frac{Electromagnetic ~forces}{Inertia ~forces}=\frac{{Ha}^{2}}{Re}=\frac{\sigma {B}_{0}^{2}L}{\rho {U}_{0}}\) - Stuart number (interaction parameter).

The values of non-dimensional numbers for flowrates up to 50 mL/s, realized in the experiments, are shown in Table 1.

Table 1 Non-dimensional numbers depending on flowrate and magnetic field
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Numerical Model

For numerical simulations one cell of regular packing is used for computations. It is assumed that there is no macroscopic current across the cells related to the potential redistribution over the whole porous region or associated with external coupling with conductors. ANSYS Fluent software is used with MHD module in potential formulation. Two cases of solid matrix conducting properties are considered – ideally electrically conducting and non-conducting spheres.

In the numerical approach velocity is calculated in an element of the porous media. Assuming laminar flow with periodic velocity conditions, pressure difference can be summed over elements in the direction of the flow according to:

$${\Delta }p=\frac{{\Delta }{p}_{cell}L}{{L}_{cell}},$$
(7)

where \({{\Delta }p}_{cell}\left[Pa\right]\) – pressure difference over a single cell,

\({L}_{cell}\left[m\right]\) – length of a cell in the direction of the flow,

\({\Delta }p\left[Pa\right]\) – pressure difference,

\(L\left[m\right]\) – length of the porous media in the direction of the flow.

Similarly, the average velocity per cell (superficial velocity) is assumed to be equal everywhere in the porous media so that we have:

$$v=\frac{Q}{A}=\frac{{Q}_{cell}}{{A}_{cell}},$$
(8)

where \(v[m/s]\) – superficial velocity,

\({Q}_{cell}[{m}^{3}/s]\) – flow rate for a single cell,

\({A}_{cell}\left[{m}^{2}\right]\) – area of a cell boundary normal to the flow,

\(Q[{m}^{3}/s]\) – flow rate,

\(A\left[{m}^{2}\right]\) – area of the porous media normal to the direction of the flow.

The pressure gradient is applied in the negative z-axis direction, thus, the flow main flow is oriented in z-axis direction. A specified pressure boundary condition is applied to the inlet and outlet of the computational cell.

The no-slip condition \(\overrightarrow{v}=0\) is applied on the pore boundaries. The normal component of the electric current and the tangential component of the electric field must be continuous on the pore boundaries. In the limiting case of electrically non-conducting matrix this results in boundary condition \({\nabla }_{n}\phi =0\) on the pore boundaries. Conversely, φ = 0 applies in case of perfectly conducting matrix.

Experimental Setup

Experimental setup consists of the liquid metal loop with porous media test section, electromagnetic pump, flowrate and pressure sensors. As liquid metal In-Ga-Sn eutectic alloy is used. For flowrate measurements the conduction type electromagnetic flowmeters are used [8].

The loop and its stand are built of non-magnetic materials not to interact with magnetic field. The stand is positioned on wheels allowing to move test section into 5T magnets opening. The overall view of experimental setup is shown in Fig. 2. The pressure is measured by 2 pressure sensors of type 154 N with the pressure measurement range up to 0.3 bar. The sensors have been tested in strong magnetic field up to 5T and their signals are shown to be independent from magnetic field. The experiments were performed at room temperature, the dimensions and pressure sensor positions in test section are shown in Fig. 3.

Fig. 2
figure 2

Drawing of experimental setup and test section position for \({B}_{x}\) and \({B}_{xy}\) orientation of magnetic field. 1- superconducting magnet, LM pump – liquid metal electromagnetic induction pump with permanent magnets, FM1, FM2 – electromagnetic flowmeters

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Fig. 3
figure 3

Test section drawing, dimensions and pressure sensor positions

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As can be seen from cross-section at the throat plane in Fig. 1. right, the pores are connected with channels which cross-section shape is close to rectangular. For the magnetic field orientation perpendicular to the main flow direction two field orientations are considered – one with x component and another rotated by 45 degrees - with equal x and y components, what corresponds to the rectangular channel flow with magnetic field oriented diagonally and perpendicularly to the channel walls respectively.

Results

The experimental results for three magnetic field orientations are presented in Figs. 4, 5. All measurements exhibit good reproducibility for the case without magnetic field, as measurements are done at different days after rebuilding the loop for different orientations of the test section.

Fig. 4
figure 4

Experimental P-Q curves for \({B}_{x}\) and \({B}_{xy}\) orientation of magnetic field for electrically non-conducting spheres

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Fig. 5
figure 5

Experimental P-Q curves for \({B}_{z}\) orientation of magnetic field for electrically non-conducting spheres

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As expected, perpendicular orientation of magnetic field (Fig. 4.) leads to an increased pressure drop across the test section with the increase of magnetic field. The \({B}_{xy}\) orientation of magnetic field exhibits higher interaction with magnetic field than \({B}_{x}\) orientation. For both orientations for high values of magnetic field the relation between pressure drop and flowrate is linear, for lower tends to quadratic.

For \({B}_{z}\) orientation of the magnetic field (Fig. 5.) the influence of field strength on the pressure drop is almost non-existent. This result is consistent with numerical calculations which show, that in this case flow is forming quasi one-dimensional jet-like structures colinear with magnetic field direction (a form of “channeling”, see Fig. 6.), thus the effective interaction with magnetic field is weak.

Fig. 6
figure 6

Calculated velocity vectors at SC cell inlet, outlet and midplane for electrically non-conducting spheres at different magnetic field orientations. Upper left figure corresponds to the case with no magnetic field

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The pore space resembles a deformed rectangular channel with expansion; therefore, it might be useful to compare the results with well-known channel flow in magnetic field. If only the z-axis velocity component is present as in classical channel flow [9], there is no influence of magnetic field on the flow, but in the considered geometry there are all three velocity components, thus it is a more complex interaction of velocity and magnetic field. Nevertheless, at high values of magnetic field, the flow tends to be essentially one-dimensional and colinear with magnetic field.

The analogy with classical channel flow is further supported by numerical simulation results shown in Fig. 6. This analogy is determined by the fact that highest velocity and, consequently, induced currents and electromagnetic forces values are located in the throat region with geometry close to rectangular, as seen in Fig. 1. right.

From the Table 1 it follows that considered experimental setup allows the realization of flows with different balance of forces, because for all non-dimensional parameters except Reynolds number it is possible to obtain values lower or higher than 1. In theory it is possible to realize a flow with very low Reynolds number, but our measurement setup did not allow to measure flowrates smaller than 1 mL/s. Alternative solution would be to use smaller pore size, but in that case the magnetic interaction also would decrease linearly, because we are limited to the maximal field value of 5T. Thus the experimental setup is a compromise for existing measurement techniques and equipment.

As can be seen from non-dimensional Navier-Stokes equations, there are several flow regimes, depending on the values of Reynolds, Hartmann and Stuart numbers.

For small Reynolds (\(Re\ll 1\)) number the flow depends on the value of the Hartmann number. For low values of Hartmann number, the flow is viscosity dominated and the pressure drop \(\varDelta p \sim v\). For high Hartmann number values the flow is dominated by electromagnetic forces and again \(\varDelta p \sim v\), due to \(j \sim v\). For high Reynolds number (\(Re\gg 1\)) the value of Stuart number is important. For small values of Stuart number (\(Re\gg {Ha}^{2}\)), the inertia forces dominate and \(\varDelta p \sim {v}^{2}\). For high values of Stuart number \(Re\ll {Ha}^{2}\), electromagnetic forces dominate and \(\varDelta p \sim v\). All these regimes can be recovered in measurements, see Fig. 7.

Fig. 7
figure 7

P-Q curves for \({B}_{x}\) orientation of magnetic field in double logarithmic coordinates and power dependences for 0 and 5T. dP ~ Q1- viscosity or EM forces dominated flow regime, dP ~ Q2 – inertia dominated flow regime

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For flowrates higher about 10 mL/s the pressure difference dependence on the flowrate is quadratic due to increase of flow inertia and, consequently, convective momentum transfer domination. This is confirmed by the value of the Reynolds number which is more than 200 (see Table 1).

The increase of magnetic field leads to the increase of the braking force, which is proportional to the flowrate. Even at magnetic field values of 1T the viscous forces are small in comparison with electromagnetic, as confirmed by the Hartmann number value of 18 (Table 1). The pressure dependence on the flowrate, however, depends on the value of the interaction parameter. At low values of interaction parameter inertial forces dominate and pressure drop dependence on flowrate is quadratic. In case of high values of interaction parameter electromagnetic forces dominate and pressure drop is linear on the flowrate. As follows from Table 1., in the considered interval of flowrates this transition occurs at magnetic field values 1-2T.

At magnetic field values higher than 3T the electromagnetic forces dominate and the pressure dependence on the flowrate is linear. At high values of magnetic field this linear behaviour is present from zero flowrate.

The comparison of the \({B}_{x}\) and\({B}_{xy}\) cases of magnetic field orientation reveals, that at high flowrates flow inertia dominates till \(B=2T\) and the field orientation influence on the pressure difference is small. At higher magnetic field values \({B}_{xy}\) orientation leads to higher pressure differences than \({B}_{x}\) orientation. This fact is in agreement with numerical simulation results discussed later.

The CFD simulation results for the relative flowrate (ratio of flowrate at specific magnetic field to that without magnetic field) vs. Ha number for both cases of electrically ideally conducting and non-conducting spheres are presented in Fig. 8. As can be seen, the \({B}_{z}\) field orientation (colinear with main flow direction) has small influence on the relative flowrate, regardless of the conductive properties of the solid matrix.

For magnetic field orientation perpendicular to the main flow, the solid matrix conductive properties are essential. In contrast with the \({B}_{z}\) case, for the \({B}_{x}\) and \({B}_{xy}\) orientation magnetic field influence strongly depends on the solid matrix conducting properties, whereas the field orientation has much smaller influence if at all.

As shown in Fig. 8., at high magnetic field values for electrically non-conductive spheres the asymptotic behaviour of relative flowrate tends to 1/Ha, whereas for ideally electrically conducting spheres to 1/Ha2. That means that for high magnetic fields non-conductive solid matrix is preferable if convective transport is considered to be important, because the flowrate ratio between these two cases grows as Ha. This relates to both the heat transfer as to the replenishment of evaporated liquid metal.

Fig. 8
figure 8

Calculated relative flowrate (ratio of flowrate at specific magnetic field and that without magnetic field) vs. Hartman number. CS – ideally conducting spheres, NC – non-conducting spheres

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The comparison of experimental results with simulations for relative flowrate (Fig. 9.) shows a reasonable agreement both qualitatively and quantitatively. For the \({B}_{z}\) field orientation experimental results confirm small magnetic field influence on the relative flowrate, whereas for perpendicular orientation the influence is essential. As simulations were carried out al low Reynolds numbers Re < < 1, but in experiment these values were not achieved, experimental results are extrapolated to low Reynolds number values for comparison.

Fig. 9
figure 9

Experiment vs. calculations for relative flowrate (ratio of flowrate at specific magnetic field and that without magnetic field) in case of electrically non-conducting spheres. Each point in graph represents one calculation or measurement. CFD – simulations, Exp – experimental data

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For strong magnetic field values (Ha > 20), the relative flowrate drops faster than theoretically predicted 1/Ha dependence. One possible explanation might be related to the assumption that there is no macroscopic current across the cells related to the potential redistribution over the whole porous region or associated with external coupling with conductors. Taking these effects into account require much more sophisticated and computationally demanding models and are subject of further studies.

Conclusions

SC sphere model is one of the simplest models for porous media but extending it to overlapping spheres model offers new possibilities, namely, the possibility of creation of porous media with predictable properties in volume defined by only one parameter.

For magnetic field orientation collinear with main flow, regardless of boundary conditions (ideally conducting or non-conducting spheres), the increase of magnetic field (Hartmann number) leads to the flow orientation in magnetic field direction and the flow resembles a simple rectangular channel flow without magnetic field. The magnetic field strength influence on the flowrate is weak.

For magnetic field orientation perpendicular to the main flow, the boundary conditions are essential. Electrically non-conducting spheres tend to have asymptotic behaviour as 1/Ha, but ideally conducting as 1/Ha2, what is in accordance with theory for infinite and rectangular channel flows. That means that for high magnetic fields nonconductive solid matrix is preferable if convective transport is to be considered important, because the flowrate ratio between these two cases grows as Ha.

While the flow has several similarities with the channel flow in strong magnetic field [9], convective transport in porous media is always present due to the curved streamlines instead of a straight ones in a channel. Nevertheless, in strong magnetic field this convective transport influence on the pressure drop is not of high importance. For the perpendicular magnetic field orientation to the main flow the main contribution to pressure drop comes from the main flow direction velocity component interaction with magnetic field, thus curvature of the streamlines and contribution of other velocity components interaction with magnetic field is of a small importance. In the case of magnetic field orientation colinear with main flow, flow tends to form jet-like flow pattern in the main flow direction, as shown by numerical calculations, thus other velocity components interaction with magnetic field is reduced.

Concerning to the applicability of present results to the CPS as PFC, there are several aspects which have to be taken into account.

Firstly, it is strongly recommended to use electrically non-conducting or weakly conducting solid matrix. That allows significantly decrease hydraulic resistance of porous media in magnetic field.

Secondly, it is imperative that the pores geometry allow the main flow direction to be aligned with the magnetic field direction. In this context channel-like solid matrix shapes are preferable to chaotic ones.

Thirdly, care should be taken by choosing the pore size. While at the surface one is limited to the small characteristic size for capillary forces to be strong enough, deeper in volume the situation is not so obvious. Small pore size leads to strong hydraulic resistance (but weak interaction with magnetic field), whereas big pore size – strong interaction with magnetic field and, consequently, strong hydraulic resistance. Apparently, for each case there is an optimal pore size for smallest hydraulic resistance of porous media.