Numerical Simulation of Core-Free Design of a Large Electromagnetic Pump with Double Stator V

MAGNETOHYDRODYNAMICS Vol. 52 (2016), No. 3, pp. 287–301

NUMERICAL SIMULATION OF CORE-FREE DESIGN OF A LARGE ELECTROMAGNETIC PUMP WITH DOUBLE STATOR V. Geˇza 1,2, B. Nacke 2 1 Laboratory for Mathematical Modelling of Environmental and Technological Processes, University of Latvia, Latvia 2 Institute of Electrotechnology, Leibniz University of Hannover, Germany

Induction pumps are most promising electromagnetic pumps for use in power plants with sodium as a coolant liquid. These pumps usually contain an iron core in stators. This paper aims to investigate an electromagnetic pump with core-free double stator design. Numerical results show that such pump configuration is of particular interest, it exhibits little efficiency decrease if compared to standard electromagnetic pumps, elimi- nates magnetic field limitation due to saturation and decreases design complexity. The velocity field in the core-free pump is smoother than in the standard pumps. Introduction. Historical development of electromagnetic pumps. The idea of moving conducting material by the interaction of magnetic field and current is not new. One of the first applications of this principle is the Faraday’s motor or its reversed device – disk dynamo, which is known from the 19th century. Although the de- vices which utilize such principle had been reported long before, the first machine with this principle designed for pumping purposes was patented in 1920 by Danish physicist Hartmann [1]. The induction pumps were developed later, when in 1927 Spencer patented the fluid conductor motor [2]. The principle is the same as for the induction pumps nowadays, i.e. the alternating magnetic field from two coils with a phase shift induces the current in the conducting liquid and the Lorentz force drives the liquid along the channel. However, this pump was designed for non-conducting liquids. Mercury (mostly used conducting liquid at that time) was pumped from one tank into another; when the tank was full, the pumping direction was reversed. The fluid to be pumped was located above the mercury, the latter played the role of pistons. Nowadays, the invention of the induction pump is attributed to Albert Ein- stein and Leo Szilard [3] which they designed for refrigerators. The pump had no moving parts, but the moving force was created using the phase-shifted AC current in the coils. The Einstein–Szilard pump was the basis for the induction pumps used nowadays. The AEG company had developed the pump technology, but at that time the pump was very noisy due to cavitation. In the late 1940–50’s the demand for EM pumps without seals and moving parts increased, first from the aluminum industry [4], later from the nuclear industry. Argonne National Laboratory had carried out a major work on DC conduction pumps. During 1947–1958, about 30 different pump designs were employed for the flow rates 5–10000 gpm (1–2270 m3/h) [5]. The largest of those was the 10000 gpm pump which was supplied by a specially designed homopolar generator with a nominal current of 250,000 A and a voltage of three DC volts. The conduction pumps have a simple design and can provide high pressures, but develop small flow rates (if compared to the induction pumps) and require special power supply 287 V. Geˇza,B. Nacke with high currents, which is especially challenging for the DC pumps. Watt was the first to formulate the single-phase electromagnetic pump [6]. This pump is generally a transformer, consisting of a primary and a secondary iron core. The primary magnetic circuit, energized by the primary coil, induces the current in liquid metal which acts as a secondary winding. It magnetizes the second iron circuit which is shaped in a special way to create a Lorentz force driving the liquid along the pipe. The single-phase pumps have not been not intensively developed further because of their very low efficiency. In 1953, a patent was issued to Crever for the centrifugal conduction pump [7]. The active pumping zone was a disk with the current passing in the radial and a magnetic field in the axial direction. That resulted in acceleration of the fluid flow while it was rotating about the disk axis and moving closer to the outer rim due to the centrifugal forces. The outlet was placed at the periphery of the disk. However, Crever was not the first one to use the disk concept for the EM pumps – the disk with the spiral duct was used almost a decade before. In 1956, Donelian invented the centrifugal induction pump [8]. The liquid disk had a shape similar to the one in the conduction centrifugal pump. The centrifugal pumps have not been developed further either. The 1950–60’s was the period of the most rapid electromagnetic (EM) pump evolution. Large interest was coming from NASA due to the nuclear space reactor development. SNAP-10A was the first space nuclear reactor to use the electromagnetic pump. NaK was used as a coolant and was pumped by the thermoelectric (TE) pump [9]. The magnetic field there was created by permanent magnets and the currents by the thermoelectric effect. The development of the thermoelectric pumps in the 1960’s was intensively driven by the North American Aviation, Inc. [10, 11]. The main advantage of this type of pumps is the self-generation of currents, so there is no need in a homopolar generator or in a three-phase AC generator. This fact makes the TE pump very lightweight, and the mass of parts is an important factor in space. The drawbacks of this type of pump are the relatively small developed pressure and flow rate. Approximately at the same time an idea appeared that the rotating magnetic field could be created by rotating permanent magnets [12] or by winding poles [13] instead of generating it by the three-phase AC generator. Findlay [12] stated that his invention was “simple in construction, economical to manufacture, and will pump liquid metals over a wide range of temperatures, pressures and flow rates and with greater efficiency than presently available in electromagnetic pumps”. In the invention of Findlay, two rotating magnet systems were used, one on each side of the rectangular channel. The invention of Baker [13] was in principle similar, but it included a helical channel which allowed obtaining a larger pressure. Another significant difference from the Findlay invention was the need in brushes and slip rings for feeding the rotating windings. The pumps with rotating permanent magnets were evolved in the last decade when strong magnets became available. They show good performance and efficiency at small to medium flow rates, but up to now they have been manufactured mostly for lead and lead alloys [14]. These pumps are promising at relatively small flow rates; the pumps with up to 5 bar pressure and 50 m3/h have been built. From all available electromagnetic pumps, only the induction pumps can be applied for very high flow rates (1 m3/s and more). Moreover, this pump type had been thoroughly discussed in many research papers published in the 1980– 90’s. Different effects which can reduce the efficiency of the induction pumps were investigated, in particular, the side effects and end effects [15]. The side effects which are present in flat pumps can significantly decrease their efficiency, therefore, 288 Numerical simulation of core-free design of a large electromagnetic pump ... different approaches were proposed to enhance the pump performance [16]. A summary of theoretical aspects of induction machines can be found in the book by Voldek [17]. The induction pumps of different sizes and for different purposes were built, but the largest of these pumps were developed for the use in fast breeder reactors. Fast breeder reactor era. In the 1960’s, the rapid development of liquid metal fast breeder reactors (LMFBR) had led to the increased demand for pumping power. The most common choice for the primary loops of LMFBR was mechanical centrifugal pumps; usually 2 to 4 pumps were used in parallel. For the EBR-II, the mechanical pumps with the 5.86 bar pressure, the 1250 m3/h flow rate and 78% efficiency were used for the primary loop, and the electomagnetic induction pump (EMP) with the 3.65 bar pressure, the 1476 m3/h flow rate and 45% efficiency for the secondary loop [18]. Although the EMPs were not used for the primary loop, it was obvious that their characteristics were close to the requirements for the LMFBR at that time. However, in the next decades, the powers of the LMFBR and mechanical pumps increased faster than the development of EMP. In Superphenix built in France in 1985, the 4 MW mechanical pump developed the 6.25 bar pressure with the 18000 m3/h flow rate [19]. At the same time, in Russia, the EMP for the LMFBR secondary loop was developed with the 3.0 bar pressure and 3600 m3/h flow rate [20]. That was the largest built EMP at that time, although several larger pumps had been designed but not built [21]. The number of the built reactors had dramatically decreased in the 1990’s as well as the development of EMPs. Several new reactors are planned in the 2020’s; this arise interest to the EMPs. Recently, the EMP has been widely discussed as a candidate for the LMFBR primary loop [22]. Recent activities in Japan have resulted in an EMP with the highest flow rate ever built up to now (2.5 bar pressure and 9600 m3/h flow rate) [23]. Tables 1 and 2 summarize the characteristics of the EMP and mechanical pumps; the most notable sodium pumps are collected there. Fig. 1 summarizes the working points of the pumps from Tables 1 and 2. In the tables it can be seen that the EMP efficiency does not exceed 45%; on the other hand, the mechanical pump efficiency varies from 65% to 83% in these examples. Despite the concerns about electromagnetic pump efficiency, its influence on the overall plant performance is small. It is determined by the fact that part of

Table 1. EPM data for sodium in the literature.

Q, [m3/h ∆p, [bar] P , [kW] η, [%] Year Comment 273 2.8 70 36 1953 [33] 1885 5.2 600 45 1955 Intended for EBR-II reactor [34] 170 4.0 98 19.3 2011 [35] 9600 2.8 1680 45 1998 Development for ASTRID [23] 174 4.4 41 CLIP-150 [36] 12 1.3 1.9 23 1999 [37] 3600 3.0 1000 30 1989 CLIP-3/3500 [20] 3300 12.6 2500 46 1979 Pump was not tested [21] 2385 8.3 1300 42 1987 PRISM primary loop project [38] 1476 3.7 372 45 1963 EBR-II secondary loop pump [18] 426 1.5 71 24 2012 BN-800 secondary loop [39] 26,1 4.0 13 22 2012 BN-800 auxillary pump [39] 3600 3.0 1000 30 1986 BN-350 secondary loop pump [? ]

289 V. Geˇza,B. Nacke

Table 2. Data on the mechanical pumps for sodium in the literature.

Q, [m3/h ∆p, [bar] P , [kW] η, [%] Year Comment 1250 5.9 260 78 1963 EBR-II primary [18] 1635 4.8 260 83 1963 Hallam Nuclear Station [40] 2680 9.3 790 87 1963 Enrico Fermi Nuclear Station [40] 5040 8.9 1975 PFR [41] 16200 2.4 80 1985 SPX secondary pump [42] 4250 6.3 1200 62 1977 Pheonix primary loop [19] 3200 5.4 700 69 1977 Pheonix secondary loop [19] 18000 6.3 4000 78 1985 Superpheonix primary loop [19] 14000 2.7 1200 86 1985 Superpheonix secondary loop [19] 6000 7.7 2000 64 1992 MONJU Primary [43] 8050 6.7 1985 FTBR, 4 pumps used [44] 2830 10.8 1200 70 1973 Primary loop pump in BN 350 [45] 3100 6.7 800 72 1973 Secondary loop pump in BN 350 [45] 16000 8.7 5100 76 1980 Primary loop pump in BN 600 [45] 15000 5.7 2900 80 1980 Secondary loop pump in BN 600 [45] 12300 9.5 5000 65 2015 Primary pump in BN 800 [46]

15.0

Induction pumps Mechanical pumps 12.5

Region of interest 10.0

7.5 , [bar] p

5.0

2.5

0 1 2 3 4 5 6 Q, [m3/s]

Fig. 1. Characteristics of EMP and mechanical pumps. the energy which dissipates in the liquid metal as heat can be regained in the turbine for power generation. In [24], it is estimated that the differences between the efficiency of the mechanical pump and that of the EMP can influence the total plant efficiency only by 0.4%. The pump developed pressure rarely exceeds 7 bar, especially at high flow rates. Recent research shows some prospects in using liquid metals as coolants in concentrated solar power (CSP) plants [25]. For the CSP, high developed pressures (10 bar and more) are required [26], [27], whereas the flow rates are within the same 290 Numerical simulation of core-free design of a large electromagnetic pump ... range as for the LMFBR. It can be seen that the EMP are missing in the circle in Fig. 1 which represents the region of interest for the plants with sodium as the heat transfer medium (both LMFBR and CSP). In [28], the rule of thumb is suggested for sodium at 260◦ C that the induction pump can develop a pressure gradient of 7 kPa per inch, or 2.76 kPa/cm. For the 4 m long pump, this results in the 11 bar pressure which is twice than that in [23]. This relatively low pressure gradient in the referenced pump is attributed to the iron core saturation in the high magnetic field. All mentioned pumps are large, heavy and require a large amount of materials for production. The most massive part of the pump is the ferromagnetic yoke. The yoke-free design would allow to reduce the production and material costs and could also be used in an environment, where the mass plays an important role, e.g., in space applications. In this paper, the possibility to remove the iron core from the double stator electromagnetic pump design is considered. The performance of the standard pump and of the core-free pump is compared. For the core-free pump design a simulation model has been developed. In this paper, an example of the core-pump is described. 1. Simulation model. Any electromagnetic process is described by the Maxwell equations which are simplified for the conditions used in technological applications. For numerical simulation purposes, the A–ϕ formulation is used (A is the magnetic vector potential, ϕ is the electric scalar potential): ∂A E = −∇ϕ − , (1) ∂t B = rot A. (2) Here E is the electric field intensity, B is the magnetic flux density. For the solution of AC current systems, simplifications are used further. Since the harmonic source current can be written as I = I0 exp(iωt), the magnetic field (and the vector potential) can be expressed in the same way B = B0 exp(iωt). ω = 2πf is the angular frequency, where f is the field oscillation frequency. The induction equation written for the vector potential becomes 1 iωA = u × rot A + ∆A. (3) µ0σ In this approximation the process is treated as quasi-stationary and the change of variables in time is neglected. In EM simulations, the sodium flow was assumed to be uniform across the channel (moving like a solid body). The pump developed pressure (electromagnetic pressure) was obtained by integrating the EM force in the channel. The motion of incompressible Newtonian fluid is described by the Navier– Stokes equation with the continuity equation ∂u 1 1 + (u grad) u = − grad p + ν∆u + f(r), (4) ∂t ρ ρ div u = 0. (5) The oscillations of the EM force are neglected in this model, because it is known that liquid metal does not react to force oscillations below 4 Hz [29]. Simulations were performed in the ANSYS software package; ANSYS Emag was used for the EM field simulations, FLUENT was used to calculate the velocity profile in the duct. The calculation algorithm for coupled approach can be described as follows: 291 V. Geˇza,B. Nacke

1. EM simulations in ANSYS Emag with the ﬁxed frequency ω, voltage V and the uniform velocity u = Q/A (A is the duct cross-section area, Q is the ﬂow rate) are performed. The force distribution f(r) is obtained.

2. Flow simulations in FLUENT with the ﬂow rate Q and the force distribution f(r) obtained at step 1 are made. A new velocity distribution u∗(r) is obtained.

3. Step 1 is repeated with the new velocity distribution u∗(r). A new force distribution f ∗(r) is obtained.

4. Step 2 is repeated with the new force distribution f ∗(r). The velocity distribution is updated. Steps 3 and 4 are repeated until the pump developed head does not change by more than 1%.

The EM simulations are performed in a 2D axisymmetric formulation with PLANE53 elements; stranded coils are connected via CIRCUIT124 elements which are connected to a voltage supply [30]. Losses in the circuit elements are neglected. For the simulation of velocity effects, the real-constant ANSYS feature was used. To include the non-uniform distribution of velocity, a real-constant set for each element in the duct was created. The velocity distribution was mapped on an EM mesh using the the built-in *MOPER command. The flow simulations were performed using the k-ω-SST turbulence model [31]. The user defined function (UDF) feature was used to import the external force distribution. The fixed flow rate was used as an inlet boundary condition, the pressure was set to zero at the outlet. The resulting pressure at the inlet is the developed hydraulic pressure of the pump. 2. Results of pump characteristic simulations with an electromagnetic model. 2.1. Reference case. As a reference case, the largest pump ever built was chosen [23]. However, not all details of the pump geometry are specified in that paper. In the descibed simulations, the reference case pump has 4.5 m long stators with the 75 mm duct thickness. The pump contains 84 inner stator and 84 outer stator coils which create 14 poles. The schematic principle of the pump is illustrated in Fig. 2; only 14 coils of the pump are shown. In this model, each coil consists of 9 windings; all coils which are in one phase are connected in series to a voltage supply.

Fig. 2. Axisymmetric geometry of the investigated pump. 14 coils of the pump are shown.

292 Numerical simulation of core-free design of a large electromagnetic pump ...

3 (a) , [bar] p 2

1 25 Hz 30 Hz 5 Hz 10 Hz 15 Hz 20 Hz

0 1 2 3 4 5

40 (b) , [%] η

20 25 Hz 20 Hz

30 Hz 5 Hz 10 Hz 15 Hz 0

0 1 2 3 4 5 Q, [m3/s]

Fig. 3. (a) p-Q characteristics. (b) Pump characteristics and eﬃciency for the reference case.

The voltage to frequency ratio V/f is kept constant in the simulations to en- sure simple control over a wide flow rate range [32]. V/f = 70 is chosen. Fig. 3a shows the obtained p-Q characteristics and the pump efficiency in the EM simulations. Pressures up to 4 bar are reached with flow rates up to 4.5 m3/s. The obtained characteristics are similar to the simulations in [23], except for the pressure decrease with frequency. To check the validity of this effect, simulations were repeated for an empty pump (no sodium and duct). As expected, the empty pump test has led to a constant current over the whole frequency range (this is ensured by V/f = const). The pressure decrease at high frequencies is determined by the skin effect which changes the direction of the magnetic field. At low frequencies the magnetic field in the duct is mostly directed across the duct; at higher frequencies part of the magnetic field lines encloses the stator coils without penetrating the sodium. Such sensitivity of the magnetic field distribution with frequency is connected with the large air gap. Fig. 3b shows the efficiency increase from 34% at 5 Hz to 53% at higher frequencies. A similar tendency is observed for the experimental data in [23]. Ab- solute efficiency values differ by some constant offset because in the present sim- 293 V. Geˇza,B. Nacke ulations no other losses than losses in the windings and in the conducting parts (sodium and steel walls) are taken into account. In experiment, additional losses are present in the bus bars and iron cores. 2.2. Core free double-stator pump. Since the induction pumps are nothing but electrical motors, they are usually made with laminated steel cores which are used to decrease losses and increase efficiency. But as already mentioned, the pump efficiency has small effect on the plant performance. Therefore, sacrificing the pump efficiency to obtain higher pressures within the same pump length might be a promising idea in terms of the design complexity reduction. The reference case pump can reach only 5 bar before the iron core gets saturated. Removing the core in the reference design pump would decrease efficiency but will increase the allowed magnetic flux density and, as a result, the maximum pump pressure. Fig. 4 shows the p-Q characteristics and efficiency for the core-free pump. Lower efficiency is reached which is still above 40%. The efficiency ratio between the reference and the core-free pump at the 20 Hz frequency is ηref /ηcf = 1.16. The main difference between both concepts from the energetic point of view is the losses in the coils because the current has to be increased in the core-free

3 (a) , [bar] p 2

5 Hz 10 Hz 15 Hz 20 Hz 25 Hz 30 Hz

0 1 2 3 4 5

30 (b) , [%]

η 20

10 15 Hz 10 Hz 20 Hz 25 Hz 30 Hz 0 5 Hz

0 1 2 3 4 5 Q, [m3/s]

Fig. 4. Pump p-Q characteristics (a) and eﬃciency (b) for the case with no iron core.

294 Numerical simulation of core-free design of a large electromagnetic pump ... pump to reach a similar magnetic flux density in the channel as in the reference pump. Inductance also differs significantly, and for this reason the V/f ratio was 1.5 times decreased to keep the developed pressure within the same range. The current in the core-free design was 2.6 times higher. The reference case has a small ratio between the power in the coils and the dissipated heat in the melt Pcoils/Pmelt = 0.11 − 0.15 depending on the flow rate. The core-free pump has the ratio between 0.68 and 0.98. This leads to significantly increased losses for the regime when no flow is present (Q = 0). But the hydraulic power to coil power ratio Phydr/Pcoils at the working point is 1.6 and, therefore, the core-free pump has no dramatic efficiency loss if compared with the reference pump. The magnetic field distribution along the pump channel with the constant current is calculated for different slip parameter values. For the reference case I = 700 A, for the core-free pump I = 1820 A was set. The current was increased in the core-free pump to obtain the same magnetic flux density in the channel as in the reference pump. This leads to higher losses in the coils, whereas other losses remain at the same level. The shape of magnetic field distribution has a slight maximum in the bottom part of the pump (Fig. 5) in the reference design, especially with the slip values 0.2 – 0.4. For the highest slip values, the magnetic flux density values and the distribution do not differ in the reference and core-free pumps. The significant difference was the waveness of the magnetic flux density distribution observed in the reference design. This effect is connected with the ferromagnetic teeth which

0.25 s = 0

0.20

0.15 s = 0.13

, [T] (a)

B s = 0.22 0.10

s = 0.38 s = 0.53 0.05

s = 1.0 0 0 1 2 3 4 5

0.25

0.20

, [T] (b) 0.15 B s = 0 s = 0.13 0.10 s = 0.22 s = 0.38

0.05 s = 0.53 s = 1.0

0 1 2 3 4 5 z, [m]

Fig. 5. Magnetic ﬂux density distribution along the pump height for diﬀerent slip values, f = 20 Hz; (a) reference design; (b) core-free pump.

295 V. Geˇza,B. Nacke

0.20

reference core-free

0.15

0.10 , [T] B

0.05

0 0.2 0.4 0.6 0.8 1.0 s

Fig. 6. Average B in the channel, dependence on the slip. Averaging is done over the channel middle line r = (rout + rin)/2; the ends are excluded (0.5 m from both ends); f = 20 Hz. have the maximum magnetic field on their tips. Such behaviour is proven experi- mentally in [23]. The core-free pump has a smoother magnetic field distribution. Both designs have shown a significant difference in magnetic flux density value with the smallest slip values (s = 0.22 and s = 0.13). To quantify this effect, the magnetic field was averaged across the channel along its middle line. This simulation considered also end effects, and to exclude them, the averaging was performed between the coordinates z = 1 m and z = 4 m. The averaged magnetic flux density dependence on the slip is plotted in Fig. 6. It can be seen that both pumps have the same magnetic field for large slip values, but this difference becomes crucial below s = 0.22. The pump efficiency should not be significantly affected, because the EMPs have the efficiency maximum between 0.2 and 0.4 (the present pump at s = 0.19). 3. Simulations with EM field – velocity coupling. The coupled model (EM field + velocity field) demonstrated a stable performance at small slip values in the way described. Not more than 15 iterations were necessary for the solution to converge. However, at larger slip values, the convergence was very slow, and the relaxation factor for force density update (step 4 in the calculation algorithm) had to be introduced. The factor 0.5 resolved the convergence problems, and again not more than 15 iterations were required. The p-Q characteristics obtained with the coupled model are similar to the results of the electromagnetic model. However, they differ slightly due to the additional fluid friction losses and redistribution of flow across the channel. With this model, lower pressure and efficiency values (Fig. 7) were obtained. The sodium flow simulations for both cases show a slight difference in velocity profile across the duct. Fig. 8 compares the velocity profiles with the 25 Hz frequency for the reference case and for the core-free pump. The velocity is nor- malized with respect to the channel mean velocity uave = Q/A (A is the channel cross-section area). A tendency is observed that at smaller slip values the velocity profile has peaks near both walls; the higher peak is located at the outer wall. It is arrtibuted to the magnetic field redistribution due to the skin effect. Part of the magnetic field lines encloses the stator coils without penetrating the sodium and, therefore, 296 Numerical simulation of core-free design of a large electromagnetic pump ...

1.50 1.0 p, EM+HD K, EM+HD p, EM K, EM 1.25 0.8

1.00 0.6

0.75 , [%] , [bar]

p 0.4 η 0.50

0.2 0.25

0 0 1 2 3 4 Q, [m3/s]

Fig. 7. Comparison of the results obtained by two diﬀerent approaches: electromagnetic calculations and electromagnetic + hydrodynamic calculations. Pressure (left axis) and eﬃciency (right axis) are compared.

2.0

1.5

ave 1.0 u/u (a) 0.5 s = 0.85 s = 0.25 s = 0.70 s = 0.17 s = 0.58

0 0.500.51 0.52 0.53 0.54 0.55 0.56 0.57 2.0

1.5

ave 1.0 (b) u/u

0.5 s = 0.85 s = 0.25 s = 0.70 s = 0.17 s = 0.58

0 0.500.51 0.52 0.53 0.54 0.55 0.56 0.57 r, [m]

Fig. 8. Radial velocity profile for different slip values, f = 25 Hz; (a) reference design; (b) core-free pump. the highest forces are located near the walls. This effect is less pronounced in the core-free case. 297 V. Geˇza,B. Nacke

1.030

1.025

1.020

ave 1.015 u/u 1.010

1.005 reference core-free

1.0 1.5 2.0 2.5 3.0 z, [m]

Fig. 9. Velocity distribution along the pump length in the middle of the channel; f = 20 Hz, Q = 3.0 m3/s.

At smaller slip values the velocity profile is almost flat in the middle part of the channel. Below s = 0.25 the differences between the profiles are very small. The reference design has a slightly higher (∼2%) velocity value near the inner wall. Although the radial profiles look similar, the velocity distribution along the channel length exhibits a significant difference. As shown in Fig. 6, the magnetic field density distribution along the channel length has a more pronounced waveness than the core-free pump. This leads also to a less uniform force density distribution and velocity field. Fig. 9 illustrates the velocity distribution along the pump channel length for the core-free pump; the velocity is slightly higher (by approximately 1%) because of the larger dynamic boundary layer thickness, which is barely notable in Fig. 8. The reference case velocity exhibits small spatial oscillations (around 0.5% which increase near the wall up to 2%), whereas the core-free pump has a smoother distribution. Such waviness can generate turbulence and, in critical cases, can lead to flow instabilities in the pump. The influence of this phenomenon on the pump performance has to be further investigated. 4. Conclusions. Simulations for the core-free electromagnetic pump have shown that this design slightly differs from the reference case (the pump with the iron core) in terms of the hydraulic performance. The core-free pump has a smoother magnetic field and velocity distribution along the pump channel which could positively affect the pump stability. The core-free pump has smaller efficiency. However, it differs from the reference pump only 1.16 times. Moreover, with this value, the losses in the core are negligible. For the core-free pump, only the losses in the coil are higher if compared with the reference pump. Other losses are at the same level. Acknowledgements. The authors wish to acknowledge the support of the Helmholtz Association in the framework of the Helmholtz Alliance LIMTECH (Liquid Metal Technology), Subtopic B2: Liquid Metals for Solar Power Systems.

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