Acta Polytechnica https://doi.org/10.14311/AP.2021.61.0324 Acta Polytechnica 61(2):324–335, 2021 © 2021 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague A CONCEPTUAL STUDY OF A LIQUID METAL ALLOY IN A DISK-SHAPED MAGNETOHYDRODYNAMICS CONVERSION SYSTEM Ayokunle O. Ayeleso∗, Atanda K. Raji Cape Peninsula University of Technology, Electrical, Electronic & Computer Engineering, Bellville 7535, PO Box 1906, Cape Town, South Africa ∗ corresponding author: kulama2005@gmail.com Abstract. The use of solar-heated liquid metal in a magnetohydrodynamics (MHD) generator provides an alternative and direct conversion method for electric power generation. This prompted the present study to conduct a three-dimensional numerical analysis for a liquid Ga68In20Sn12 flow exposed to several uniform magnetic field intensities (Bo of 0.5T, 1T and, 1.41T) within a disk channel geometric boundary. The aim is to study the influence of the external magnetic fields on the generator performance and the fluid flow stability at a high Reynolds number (Re) and Hartmann number (Ha) using the Ansys Fluent software. The simulation results show that at Re of ≈ 2.44e6, the fluid velocity decreases inside the generator regardless of Bo. When Bo of 1T and 1.41T are applied, the velocity magnitude decreases and spreads within the disk channel and walls due to high Ha values (5874 and 8282). The fluid pressure increases from the nozzle pipe inlet to the disk channel and decreases towards the outlet. The induced current density in the radial direction, jx, increases within the disk channel and near the inner electrode edge as Bo increases. A significant observation is that the current densities obtained for Bo of 1T and 1.41T cases are higher than in other cases. The numerical analysis obtained in this study showed that the Bo of either 1T or 1.41T is needed to achieve the required flow stability, current density, and output powers. Keywords: Solar, liquid metal, disk MHD generator, magnetic field, current density. 1. Introduction In the past decades, electrically conducting fluid such as liquid metal under externally applied magnetic field has become the subject of various engineering and industrial applications. These applications in- clude nuclear reactor cooling, liquid metal flow con- trol, nanofluid flow in thermal and energy systems, high-temperature plasma, mini and micro magnetohy- drodynamics (MHD) pumps, and MHD power gener- ators [1–9]. Mebarek-Oudina et al. [1] investigated the stability of natural convection in an inclined cylindrical annulus ring containing molten potassium under the influence of a radial magnetism and a small number of Prandtl liquids (Pr = 0.072). They found that the best stabi- lization of the natural oscillatory convection occurred with the strongest magnetic field, the high radii ratio, and the inclination of the annulus for γ = 30°. The an- gle of inclination and radii ratio of the annulus have a significant effect on magneto-convective flux stabiliza- tion. Teimouri et al. [10] also investigated the natural convection of molten potassium in a long horizontal ring under the influence of radial magnetism. Their results showed that increasing the radii ratio reduces the magnetic field influence on the natural convection. In another study, Afrand et al. [11] examined the flow in an inclined cylindrical ring containing molten potassium under the influence of magnetism. Their results showed that the average Nusselt number de- creased with increasing Hartmann number, Ha, when the magnetic field was perpendicular to the ring axis. Yadav et al. [12] investigated the influence of the Ha and Brinkman number on the MHD convection flow of a viscous fluid between two horizontal concentric cylinders. They found that with increasing Ha values, there is a reduction in fluid velocity and irreversibility ratio. In other research, Zhang et al. [13] experimen- tally investigated the liquid Galinstan (Ga 68%, In 20%, and Sn 12%) based mini channel cooling for high heat flux thermal devices. Their results showed that liquid Galinstan driven by a high-efficiency direct current electromagnetic pump (DC-EMP) dissipate heat with a heat flux of 300W/cm2, heat power of 1500W, and a pressure of 100 kPa. Taheri et al. [14] use a numerical finite volume method (FVM) and artificial neural network (ANN) to compute the flow at the entrance length of a laminar MHD rectangular channel at different values of the Reynolds numbers, Re, (600 < Re < 1100) and Ha (4 < Ha < 10). They obtained results for various physical parameters and found that Ha increases the Lorentz force while de- creasing the velocity profile and the MHD entrance length. To our knowledge, there is still no published study that has investigated the influence of axial magnetism on the flow of liquid Galinstan in a disk channel at a high Re and Ha. Therefore, the present study aims to 324 https://doi.org/10.14311/AP.2021.61.0324 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en vol. 61 no. 2/2021 A conceptual study of a liquid metal alloy. . . Figure 1. The liquid Ga68In20Sn12 flow cycle. conduct a computational analysis for the flow of liquid Ga68In20Sn12 in a disk channel under the influence of several external magnetic fields, Bo: 0T, 0.5T, 1T, and 1.41T. The study also investigates the effects of high Re and Ha on the flow stability and generator performance in the low Prandtl number fluid (Pr = 0.028) and steady-state conditions using the Ansys Fluent software. 2. Theoretical background 2.1. Liquid Galinstan flow cycle The fluid considered in this study is liquid Galinstan (Ga68In20Sn12). It is a non-toxic, non-radioactive, low melting point (−19 °C), high thermal conductivity, and chemically stable fluid [15–18]. This fluid does not stick to superlyophobic structured surfaces and behaves like a liquid in an environment with oxygen levels below 1 part per million (ppm) [16, 17]. Further- more, liquid Ga68In20Sn12 does not have wettability issues in channels and pipes developed with materials like Plexiglas, Teflon, Organic materials, Tungsten, Silicon tubes, Nickel, PVC, and Glass [18]. Fig. 1 de- picts the schematic diagram of the liquid Ga68In20Sn12 flow cycle. In this figure, the fluid flows out of the reservoir through a valve and is pre-heated by a solar heater before entering the disk MHD generator. Sub- sequently, the fluid exiting the generator is recycled back through an electromagnetic pump at a pressure head of 100 kPa for the next round of circulation. 2.2. Disk MHD generator geometry and material properties Fig. 2 depicts the geometry of the disk MHD gener- ator considered in the present study. In this figure, the fluid flows through the nozzle pipe inlet into the disk channel in the axial direction (z-axis) and then towards the outlet in the radial direction (x-axis). The diameter of the nozzle pipe inlet is 66.03mm. The disk channel diameter is 152.12mm, and the height of the outlet is 37.9mm. Furthermore, there are two circu- lar tungsten copper electrodes in the y-axis direction perpendicular to the direction of Bo. Figure 2. The disk MHD generator. 2.3. Governing equations The present study considered the turbulence flow of a viscous, incompressible, and electrically conducting fluid in magnetic fields. The governing equations are mass conservative (continuity), Navier–Stokes momen- tum, and Ohm’s law [19], ∇× v = 0 (1) ρ Å ∂v ∂t + (v · ∇)v ã = ρ+ η∇2v −∇P + j × B (2) where ρ is the density, P is the pressure, v is the velocity, η is the viscosity, B = B0 + b is magnetic flux density, B0 is the external magnetic field, b is the induced magnetic field produced by the motion of the fluid through B0. The current density, j, expression is given by [14], j = σ(E + v × B) (3) In a generalized form, the Ohm’s law equation in Equation 3 can be re-written as [9], j + β |B|j × B = σ(E + v × B) (4) where E is the electric field, σ is the electrical con- ductivity, β = eB/meveh is the hall parameter, e is the electron charge, me is the electron mass and veh 325 Ayokunle O. Ayeleso, Atanda K. Raji Acta Polytechnica General setup Solver Type Density-Based 3D Space Planar Time Steady Velocity formulation Absolute. Models Energy equation On Viscous model Standard k-epsilon model Standard Wall Function. Models Fluid Ga68In20Sn12 Melting point 10.8 °C Density 6363.2 kg/m3 Dynamic viscosity 0.00222 kg/(m·s) Thermal conductivity 26.72 W/(m·K) Specific heat 331 J/(kg·K) Magnetic permeability 8e-7 h/m Electrical conductivity 3.31e6 s/m Kinematic viscosity, ν 3.49e-7 m2/s Thermal diffusivity, α 1.27e-5 Wm2/J Cell boundary zone Fluid Domain Boundary conditions Inlet Velocity-Inlet Velocity 5.606311 m/s Outlet Pressure-Outlet Gauge Total Pressure 1e5 Pa Inlet Temperature 323 K = 49.85 °C Outlet Temperature 300 K Table 1. Setup and boundary conditions. is the average momentum transfer collision frequency for an electron (e) with a heavy particle (h). Taking the curl of E in Equation 3 gives, ∇× E = −∇× Å j σ ã +∇× (v × B) (5) By neglecting the displacement current from Maxwell’s equation (∇ × H = j + ∂D/∂t), the re- maining expression gives ∇×H = j. Substituting the Maxwell’s induction field (H = B/µm) into∇×H = j, the induced current density becomes [20], j = ∇× B µm (6) Inserting Equation 6 into Equation 5, and using the following properties, ∇ × B, ∇ × E = ∂B/∂t ∇ × (∇ × B) = ∇(∇ · B) − B(∇ · ∇), the magnetic induction equation is derived as [21–23], ∂B ∂t + (v · ∇)B = 1 µmσ ∇2B + (∇ · B)v (7) where µm is the magnetic permeability. 3. The three-dimensional modelling of the disk MHD generator The physical and thermodynamic properties consid- ered in the simulation of the liquid Ga68In20Sn12 are mainly the density, electrical conductivity, viscosity, and magnetic permeability. Table 1 presents the setup and boundary conditions considered for this study. The boundary conditions in Table 1 are chosen based on the physical and thermal properties of liquid Ga68In20Sn12. The disk channel boundary condition is a no-slip wall boundary, which means that the ra- dial velocity, vr, and tangential velocity, vq, are equal to the wall velocity. A pressure-based solver is used to solve these boundary conditions. Furthermore, the pressure-velocity couplings are solved using the coupled scheme solution method. The pressure cal- culations are solved using the second-order upwind discretization. The first order upwind is chosen for the momentum, energy, and magnetic induction cal- culations. The external magnetic field is added as a volume force once the initial hydrodynamic solution is converged and completed. Grid independence test The disk MHD generator geometry consists of non- uniform grids around the electrodes and walls. Thus, to reduce numerical errors in the simulation results, several grid sizes are investigated for a grid indepen- dence study. For this purpose, the grid systems of 84 867 × 75 176, 107 354 × 79 613, 159 518 × 89 321, 212 249× 99 106, 260 378× 107 846, 303 372× 115 539, 358 068×125 355, 402 795×133 596, 441 995×140 739, and 509 521 × 152 988 are investigated for liquid Ga68In20Sn12 flow at B0 = 0 and Pr = 0.028 (Ta- 326 vol. 61 no. 2/2021 A conceptual study of a liquid metal alloy. . . Case Element Node Velocity (m/s) 1 84 867 75 176 3.97 2 107 354 79 613 3.95 3 159 518 89 321 4.35 4 212 249 99 106 4.50 5 260 378 107 846 4.56 6 303 372 115 539 4.61 7 358 068 125 355 4.66 8 402 795 133 596 4.67 9 441 995 140 739 4.67 10 509 521 152 988 4.68 Table 2. Maximum velocity at the disk outlet for different grid resolutions. Figure 3. Profiles of the local outlet velocities with different mesh grid sizes. ble 2). The maximum velocity at the disk outlet (probing point) serves as the control parameter. For the different grid sizes investigated, the flow field vari- ables reach a fully developed state and then converge. From Table 2, the maximum differences between 402 795 × 133 596, 441 995 × 140 739, and 509 521 × 152 988 grid systems are within 0.01%. The velocity results also showed that a change of less than 1% is observed in computed values when varying the grid sizes. Furthermore, to determine the exact turning point and optimum grid point at which the velocity results variation is no longer significant, a quadratic regression model is performed using the polynomial degree order of greater than one. This method creates the best-fitting curve alongside the initial velocity graph, as shown in Fig. 3. As seen from Fig. 3, the polynomial of degrees 3, with a coefficient of determination (r-square) of 98%, produces the best fitting curve. Also, the polynomial of degree 2, with an r-square of 96%, clearly shows that the turning point occurs at 402 795 × 133 596 grid size. At this point, the numerical solution does not change significantly (i.e., independent) with varying mesh sizes. Therefore, the adopted mesh grid for all numerical simulations is a polyhedral dominant-shaped with 402 795 elements and 133 596 nodes (Fig. 4). The selected grid optimizes the CPU (2.3GHz) time and the cost of computations. Based on the selected mesh grid size, the conver- gence criteria at a given time-step are declared when the iteration level of the residuals of x-, y- and z- velocities, continuity, and k-epsilon fall below 10−3. Furthermore, the mass-weighted average of the veloc- 327 Ayokunle O. Ayeleso, Atanda K. Raji Acta Polytechnica Figure 4. Mesh display of the disk MHD generator. ity magnitude at the disk outlet under different B0 cases in Fig. 5 gives a further insight into the solu- tion convergence. The result of the first simulation without magnetic fields (B0 = 0T) shows that the ve- locity solution converges at a steady value of 3.13m/s. Subsequently, by applying the B0 values of 0.5T, 1T, and 1.41T to the initial solution, the velocity magni- tude reaches a steady state with convergence values of 1.64m/s, 1.43m/s, and 1.32m/s, respectively. 4. Disk MHD generator modelling results This section discusses the effect of the Re, Ha, and interaction parameter, N , on the fluid velocity, pres- sure gradient, and current density for different values of B0. 4.1. Magnetic flux density Fig. 6 presents the plot of the magnetic flux den- sity along the nozzle pipe and disk channel. The flux density in Fig. 6a increases from the nozzle pipe inlet (z = 0.044m) and reaches optimum at about (z = −0.024m) and then slightly decreases at the disk channel centre. From the disk centre (x = 0.0007m, y = 0, z = −0.032m), the flux decreases towards the outlet, as shown in Fig. 6b. The maximum magnetic flux density of 0.034T, 0.065T, and 0.088T are ob- tained along the disk centre when the B0 values of 0.5T, 1T, and 1.41T are applied (Fig. 6b). The above results show that the B0 value of either 1T or 1.41T is sufficient to decelerate the fluid flow inside the disk channel geometry considered in this study. 4.2. Velocity distribution Figs. 7 and 8 show the contour (x− z plane, y = 0) and plot of velocity magnitude along the nozzle pipe and disk channel. The velocity distribution changes as the B0 value gradually increases from 0 to 1.41T. In the case where there is no magnetic field (equiv- alent to Ha = 0), the fluid moves from the nozzle pipe and gradually decelerates as it reaches the disk channel (Fig. 8a). The velocity of 5.606m/s at the nozzle pipe inlet (z = 0.044m) decelerates to about 2.717m/s at the disk centre. From the disk centre (x = 0.0007m, y = 0, z = −0.032m), the velocity decreases towards the outlet and forms an M-shaped profile in the radial direction (Fig. 8b). When the B0 values of 0T, 0.5T, 1T, and 1.41T (i.e., Ha of 2 937, 5 874, and 8 282) are applied, the maximum velocities obtained are 4.943m/s (x = −0.036m), 4.331m/s (x = 0.038m), 3.283m/s (x = −0.038m), and 3.166m/s (x = −0.033m), respectively. More- over, we can see in Fig. 7 that for B0 values of 1T and 1.41T (Ha = 5874 and Ha = 8282), the decelerated fluid spreads rapidly within the disk and the velocity near the conducting walls increases. In general, the velocity at the disk centre is higher than at the side- walls. These observations fully agree with the findings reported by [14, 24, 25]. 4.3. Pressure gradient The plot in Fig. 9a presents the pressure gradient along the nozzle pipe. In this figure, the pressure behaviour showed that the magnetic force has almost no effect on the fluid velocity in the axial direction (z-axis). In addition to this, an elevated pressure is noticeable from the nozzle pipe inlet (z = 0.044m) to the disk centre (x = 0.0007m, y = 0, z = −0.032m). Hence, we can deduce that the imposed disk inlet and outlet geometries are the essential parameters responsible for the pressure elevation. Conversely, from the centre of the disk, the pressure decreases towards the outlet, as shown in Fig. 9b. Similar outcomes have been reported by [26] using a rectangular channel. In the outlet of the disk channel (x = −0.076m, x = 0.076m), the pressure obtained is approximately 100 kPa. This pressure exits the disk channel outlet and activates the electromagnetic pump in Fig. 1 for the next round of circulation. 4.4. Current density The plot and contour in Figs. 10 and 11 present the induced current density along the inner and outer electrode regions. As shown previously in Fig. 2, the magnetic force is in the axial (z-axis) direction and the conductive fluid flow traversing the magnetic force is in the radial (x-axis) direction. The effect of the retard- ing Lorentz force on the current density in the radial direction, jx, can be seen in Fig. 11 for the different B0 andHa cases. When B0 values of 0.5T, 1T, and 1.41T are applied, the maximum jx values obtained near the inner electrode (x = −0.04072m, x = 0.04072m) are 1.03e6A/m2, 2.89e6A/m2, and 3.79e6A/m2, respec- tively. Whereas the maximum jx values obtained near the outer electrode (x = −0.07145m, x = 0.07145m) are 8.79e5A/m2, 5.49e5A/m2, and 1.15e6A/m2, re- spectively. Moreover, we can see in Fig. 11 that as B0 andHa increase, the current density increases near the inner electrode edge and gradually decreases toward the disk outlet. For the cases where the B0 values 328 vol. 61 no. 2/2021 A conceptual study of a liquid metal alloy. . . (a). (b). (c). Figure 5. The mass-weighted average of velocity magnitude at the disk outlet under different magnetic fields: a. 0.5T, b. 1T, c. 1.41T. 329 Ayokunle O. Ayeleso, Atanda K. Raji Acta Polytechnica (a). (b). Figure 6. Plot of magnetic flux density along the nozzle pipe and disk channel: 6a. (0, 0, z), 6b. (x, 0,−0.032m). Figure 7. Contour of velocity magnitude along the nozzle pipe and disk channel (x− z plane, y = 0). 330 vol. 61 no. 2/2021 A conceptual study of a liquid metal alloy. . . (a). (b). Figure 8. Plot of velocity distribution along the nozzle pipe and disk channel: 8a. (0, 0, z), 8b. (x, 0,−0.032m). (a). (b). Figure 9. Plot of pressure distribution along the nozzle pipe and disk channel: 9a. (0, 0, z), 9b. (x, 0,−0.032m). 331 Ayokunle O. Ayeleso, Atanda K. Raji Acta Polytechnica Figure 10. Plot showing the current density distribution along the inner and outer electrode regions: (x, 0,−0.0224). Figure 11. Contour showing the current density distribution along the inner and outer electrode regions (x − y plane, z = 0). 332 vol. 61 no. 2/2021 A conceptual study of a liquid metal alloy. . . Disk inlet area Magnetic fields Flow rate Reynolds number Hartmann number Interaction parameter Current density (m2) B0 (T) Q (m3/sec) Re Ha N jx (A/m2) vA vρDh)/µ B0Dh √ σ/µ (Ha)2/Re Max Literature 0.6540 0.3 0.18e-3 ≈ 4 000 140 ≈ 5 ≈ 1e5 Present study 0.0185 0 0.1038 ≈ 2.44e6 0 ≈ 0 0 0.0185 0.5 0.1038 ≈ 2.44e6 2 937 ≈ 4 ≈ 1.03e6 0.0185 1 0.1038 ≈ 2.44e6 5 874 ≈ 14 ≈ 2.89e6 0.0185 1.41 0.1038 ≈ 2.44e6 8 282 ≈ 28 ≈ 3.79e6 Note: Dh is the characteristic diameter or length of the channel. Table 3. Summary of current densities from past and present studies. of 1T and 1.41T are applied, the current densities observed at the inner electrode are significantly higher than when B0 values of 0T and 0.5T are applied. Based on these observations, we can deduce that the higher the fluid velocity near the electrode region, the stronger the jx and the retarding force are. Similar deductions have been reported by [27]. For validation purposes, the current densities from the past and present studies on liquid Ga68In20Sn12 are presented in Table 3 [28–31]. In this table, the jx values are almost of the same order of magnitude. The slight difference may be due to an increase in Ha values. We can also see that at high Re and Ha values, N and jx values increase. 5. Conclusion This study has investigated the flow of liquid Ga68In20Sn12 in a disk-shaped MHD channel under different cases of B0. The effects of high Re and Ha on the flow stability in the low Prandtl number fluid (Pr = 0.028) have been discussed in detail, with the main conclusions as follows. (1.) At high Re, the fluid velocity decreases along the nozzle pipe and disk channel regardless of B0. (2.) When B0 values of 1T and 1.41T are applied, the fluid velocity decreases and spreads within the disk channel and walls due to high Ha values (5 874 and 8 282). (3.) The pressure increases from the nozzle pipe inlet to the disk channel and decreases to the outlet for different B0 cases. (4.) The induced current density in the radial direc- tion, jx increases within the disk channel and near the inner electrode edge as B0 increases. Moreover, the jx obtained for B0 values of 1T and 1.41T cases are significantly higher than in other cases. (5.) The liquid Ga68In20Sn12 does not stick to chan- nels coated with anti-stiction layers (Plexiglas and Teflon) and superlyophobic structured surfaces. (6.) The numerical analysis obtained in this study showed that the B0 value of either 1T or 1.41T is needed to achieve the required flow stability, current density, and output powers. The study also lays the groundwork for a future research on liquid metals and MHD systems. Acknowledgements The presented research was financially supported by Cape Peninsula University of Technology. List of symbols Re Reynolds number [–] Ha Hartmann number [–] B Magnetic flux density [T] B0 External magnetic field [T] b Induced magnetic field [T] P Pressure [Pa] E Electric field [N/C] vr Radial velocity [m/s] vq Tangential velocity [m/s] Pr Prandtl number [–] Rh Dimensionless factor [–] Q Flow rate [m3/sec] N Interaction parameter [–] Dh Characteristic diameter or length of the channel [mm] v Fluid velocity [m/s] j Current density [A/m2] e Electron charge [C] me Electron mass [kg] veh Average momentum transfer collision frequency for an electron (e) with a heavy particle (h) µm Magnetic permeability [h/m] H Maxwell’s induction field [A/m] k Thermal conductivity [W/(m K)] cp Specific heat [J/(kg K)] Greek symbols α Thermal diffusivity of the fluid [W m2/J] ν Kinematic viscosity of the fluid [m2/s] ρ Density of the fluid [kg/m3] σ Electric conductivity [S/m] η Dynamic viscosity [kg/(m s)] β Hall parameter [–] 333 Ayokunle O. Ayeleso, Atanda K. 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