Neal A. Sondergaard

Liquid-Metal Flows in Current Collectors for Homopolar Machines: Fully Developed Solutions for the Primary Azimuthal Velocity

Liquid metals in the small radial gaps between the rotors and stators of homopolar machines represent low‐resistance electric current collectors. Design predictions for these liquid‐metal current collectors require a thorough knowledge of liquid‐metal flows in a narrow gap between a fixed and a moving surface, with a strong applied magnetic field and a free surface beyond each end of the gap. The radial and axial velocities in the secondary flow are reduced by a strong axial or radial magnetic field. For a sufficiently strong field, the azimuthal momentum transport by the secondary flow can be neglected. This assumption reduces the problem for the primary azimuthal velocity to a fully developed magnetohydrodynamic duct flow problem with a moving wall and two free surfaces. Asymptotic solutions for large Hartmann numbers are presented for skewed magnetic fields with both radial and axial components. Collectors without any electrical insulation or with insulation on the stator sides, or rotor sides, or both...

Hydrodynamic stability threshold of liquid metal current collectors

A mechanism derived from hydrodynamic theory to explain the ejection instability of liquid metal current collectors is presented. The ejection mechanism is shown to be caused by the onset of the Kelvin–Helmholtz instability resulting from the gradient of azimuthal (primary) flow at the interface between the liquid metal and cover gas. This new mechanism differs from the previous theory developed by Eriksson [in Electrical Engineering Series, no. 48, edited by M. Luukkala (The Finnish Academy of Technical Sciences, Helsinki, Finland, 1982), who analyzed the onset of the Kelvin–Helmhotz instability resulting from the gradient of meridional (secondary) flow at the interface. Considering the solution to the linearized Navier–Stokes equations at the liquid metal and gas interface, the azimuthally driven (primary flow) instability mechanism for the onset of ejection is much more prevalent than the meridional (secondary) flow driven mechansim. Furthermore, Eriksson’s theory requires an empirical multiplicative f...

Liquid‐metal flows in sliding electrical contacts with arbitrary magnetic‐field orientations

In certain situations, liquid‐metal sliding electrical contacts for high‐current and low‐voltage electrical machines may prove a viable alternative to solid metal brushes. Before it can be ascertained whether such an option is feasible, the problems inherent in a liquid‐metal flow through a narrow gap between a fixed and a moving surface with free surfaces beyond each gap end must be explored. The flow occurs in the presence of an arbitrarily oriented magnetic field. By assuming that the secondary flow is negligible, the problem reduces to a fully developed magnetohydrodynamic (MHD) duct flow problem. In the parameter range presented here, the liquid‐metal flow can be laminar or turbulent, requiring that both regimes be analyzed. The numerical results from the mathematical model presented herein for laminar flow with arbitrary Hartmann number M and with arbitrary magnetic‐field orientation indicate that, even with an O(1) Hartmann number, the flow is already beginning to evolve into the distinct regions p...

Liquid-metal flows and power losses in ducts with moving conducting wall and skewed magnetic field

Fully developed, laminar liquid‐metal flows, currents, and power losses in a rectangular channel in a uniform, skewed high external magnetic field were studied for high Hartmann numbers, high interaction numbers, low magnetic Reynolds numbers, and different aspect ratios. The channel has insulating side walls that are skewed to the external magnetic field. Both the perfectly conducting moving top wall with an external potential and the stationary perfectly conducting bottom wall at zero potential act as electrodes and are also skewed to the external magnetic field. A solution is obtained for high Hartmann numbers by dividing the flow into three core regions, connected by two free‐shear regions, and Hartmann layers along all the channel walls. Mathematical solutions are presented in each region in terms of singular perturbation expansions in negative powers of the Hartmann number. The free‐shear layers are treated rigorously and in detail with fundamental magnetohydrodynamic theory. Numerical calculations ...

Magnetohydrodynamic liquid‐metal flows and power losses in a rectangular channel with a moving conducting wall

Fully developed, viscous liquid-metal flows and power losses in a rectangular channel in a uniform, external magnetic field were studied for moderate Hartmann numbers and different channel aspect ratios. The channel was assumed to have insulating side walls parallel to the field, a perfectly conducting moving top wall, and a stationary bottom wall perpendicular to the field. Exact-series solutions and numerical calculations are presented for velocity profiles, induced magnetic-field distributions, current densities, voltage profiles, and viscous and joulean power losses. The joulean and viscous dissipation are computed from squared quantities involving infinite series with double summations for finite M. The diverging series derived for the viscous power losses is made convergent by slightly modifying the velocity profile of the conducting fluid at the moving interface. The effect of the applied field is to produce an eddy current that accelerates the bulk fluid to velocities greater than those without the field, but because of the presence of the side walls, the velocities are less than in one-dimensional couette flow. Except at small aspect ratios and Hartmann number, almost the entire power loss resulted from the component of the induced current parallel to the applied field.

Kelvin–Helmholtz instability of Couette flow between vertical walls with a free surface

A novel type of Kelvin–Helmholtz instability model is developed from hydrodynamic theory. The classical Kelvin–Helmholtz instability involves a horizontal interface between two fluids with different parallel, uniform, horizontal velocities. If the upper fluid is a gas with a much smaller density than the lower fluid which is a liquid, then the phase velocity of the critical disturbance equals the liquid’s velocity, so that the liquid sees a standing interfacial wave. The inertial force driving the interfacial instability involves only the gas, no matter how small its density is. In a much more realistic flow model, the liquid velocity at the free surface is not uniform, but varies across the free surface. The disturbance phase velocity can only equal the liquid velocity at one point, while liquid on either side of this point moves faster or slower than the wave. The inertial forces in the liquid then dominate and the gas plays a negligible role. The concept is developed from a Couette flow hydrodynamic model where the fluid flows between two parallel vertical walls with a free surface. The importance of a nonuniform liquid velocity is demonstrated. This modified theory will be applied in future work to study the ejection instability at the interface of the liquid metal and inert cover gas in sliding electrical contacts.

Effects of magnetic field orientation on a liquid‐metal free surface in a sliding electrical contact

This paper treats a free surface between a liquid metal and an inert gas in the presence of a magnetic field with arbitrary orientation relative to the free surface. The free surface intersects a perfectly conducting surface at rest and an insulated surface rotating about an axis which is perpendicular to both surfaces and which is far from the liquid‐metal region. This problem models free surfaces in liquid‐metal sliding electric contacts for motors and generators. There is a primary azimuthal liquid‐metal velocity which is driven by the rotation of the insulated surface, and there is a secondary flow which involves radial and axial velocities and which is driven by the centrifugal force due to the primary velocity. The free‐surface positions, pressures, and velocities are presented as functions of the magnetic‐field orientation and strength.

Further studies of liquid‐metal flows and power losses in ducts with a moving conducting wall and a skewed magnetic field

In a previous paper the authors initiated studies of fully developed laminar liquid‐metal flows, currents, and power losses in a rectangular channel with a moving perfectly conducting wall and with a skewed homogeneous external magnetic field for high Hartmann numbers, high interaction parameters, low magnetic Reynolds numbers, and different aspect ratios. The channel had insulating side walls that were skewed to the external magnetic field, while the perfectly conducting moving top wall with an external potential and the stationary perfectly conducting bottom wall at zero potential acted as electrodes. These electrodes were also skewed to the external magnetic field. A mathematical solution was obtained for high Hartmann numbers by dividing the flow into three core regions, two free shear layers, and six Hartmann layers along the channel walls. The free shear layers were treated rigorously and in detail with fundamental magnetohydrodynamic theory. The previous work, however, left the solution for the vel...

A liquid‐metal free surface in a sliding electrical contact with a strong magnetic field

A low‐resistance electrical contact is provided by a liquid metal in a small gap between the perimeter of a rotating disk (rotor) and a static surrounding surface (stator). The liquid metal extends radially inward on both sides of the rotor to free surfaces with an inert cover gas, and there is a strong axial magnetic field. This paper presents results for the shape of the free surface and for the liquid‐metal velocity and pressure adjacent to the free surface. The results depend on the magnetic‐field strength, the surface tension, the wetting angle at the free‐surface–solid intersections, and the voltage difference between the stator and rotor. The copper stator and rotor are perfect conductors compared to the liquid metal. Two cases are considered, with and without electrically insulating coatings on the sides of the rotor.

Magnetohydrodynamic liquid‐metal flows in a rectangular channel with an axial magnetic field, a moving conducting wall and free surfaces

Fully developed, viscous liquid‐metal velocity profiles and induced magnetic field contours were studied for Hartmann numbers of M=2 and 10 and for different load currents for a particular rectangular channel configuration (two‐dimensional Couette flow). The rectangular channel was assumed to have a homogeneous external (axial) magnetic field parallel to the moving, perfectly conducting top wall and the stationary, perfectly conducting bottom wall. The two stationary side walls were also perfect conductors. The small gap between the moving wall and each side wall was an insulating, free surface. The method of weighted residuals was used to obtain truncated series solutions for the variables of interest. The heavy load currents across the channel were obtained by simulating an external potential to the conducting moving wall. The load currents in each case were opposed by the induced electric field. Since there is no pressure gradient, the flow along the channel is driven by the viscous effects of the movi...