Mechanical energy harvesting using a liquid metal vortex magnetohydrodynamic generator

Supplementary Material

Materials

Since the operation of the MHD is based on the flow of a liquid metal though a magnetic field to produce electrical current, the first task was to evaluate possible candidates for the liquid metal.

There are only two choices for liquid metals, which remain liquid at room temperature: Hg and

Ga/In/Sn alloys, of which Galinstan [1] (Ga 68.5% / In 21.5% / Sn 10%) is a commercially available product. Mercury is toxic [2] enough to cause concerns for worker’s health during product manufacturing, waste disposal, and environmental contamination. In recent years,

Galinstan has been investigated and implemented as a replacement for Hg used in mechanical switches, MEMS sensors and switches, and thermometers [3] [4]. According to MDS listings for

Galinstan, there are no know harmful effects or health issues [5] . The risk of intoxication from inhalation is negligible due to the extremely low vapor pressure and from ingestion also is negligible as the material is not retained in the body. Table S1 compares the physical properties of Hg and Galinstan. In every case, the physical properties of Galinstan are equal to or better than Hg for the MHD application, particularly having lower density and higher electrical conductivity. Galinstan does have one issue, which has been reported more recently, in that its melting point may be as high as 10ºC, but it often acts as a super-cooled liquid eventually crystallizing at -19ºC.

Property Galinstan Hg

Boiling point (oC) >1300 357

Melting point (oC) -19 -39

Vapor pressure (Torr) <10-8 (500oC) 16.3 x 10-6 (20oC)

Mass density (g cm-3) 6.44 13.546

Viscosity (Pa s) 0.0024 0.0015

Electrical conductivity (S/m) 3.46 x 106 1.02 x 106

Thermal conductivity (W m-1 K-1) 16.5 8.65

Surface tension (N/m) 0.718 0.487

TABLE S1 Physical properties of Galinstan and Hg at ambient conditions

Another major problem in using Galinstan as the liquid metal is that its surface rapidly oxidizes when exposed to air or moisture. The thin solid oxide film, composed mainly of Ga2O3 [6], changes the surface properties of the liquid metal affecting its electrical conductivity and flow properties. However, the oxide can be easily removed by rinsing the Galinstan in dilute hydrochloric or other acids.

A major concern was to achieve the lowest possible contact resistance between the liquid metal and the electrodes. Since the MHD device operates at low voltages and high currents, any increase in internal resistances within the device would greatly affect the power performance.

We found that the lowest contact resistance was obtained when the liquid metal completely wet

(0º contact angle) the surface of the metallic electrode. Most metals exposed to air, even immediately after cleaning, will have enough oxide to prevent complete wetting by Hg or

Galinstan. A simple solution to solve this problem is to submerge both the electrode and liquid metal in an acidic solution. After the surface oxide of the electrode (or on the Galinstan) is removed by the acid, the liquid metal will immediately wet [7] the surface. Attempts were made to measure the contact resistance of samples prepared in this manner, but the resistances were lower than the sensitivity of our measuring equipment. We estimate that the upper limit on the contact resistance that we observed is several micro-ohms.

Many metals readily amalgam with both Hg and Ga, especially after the surface oxides have been removed. The initial surface reaction or amalgamation produces the low contact resistance interface, but for certain metals the reaction will not stop, and as more and more electrode material is dissolved into the liquid metal, its prosperities will eventually change. Several different ways were investigated to solve this problem. Metals, which will not amalgamate with

Hg and Ga alloys, generally will not allow complete wetting, but many alloys containing one component, which will amalgamate with Hg and Ga alloys, will be completely wetted. After the initial reaction, the surface composition of the alloy changes becoming enriched with the non- amalgamating component, and dissolution of the electrode material is greatly slowed. One example of such alloy, which was successfully tested, was Ni 400 (Monel) [8]. Although the surface wetting was excellent with no long-term degradation, the bulk resistivity of Monel is significantly higher than Cu adding to the internal resistance of the MHD generator. Another alloy Cu 182 (Copper alloyed with trace amounts of Chromium) with conductivity 85% IACS was also tested. Results indicated that the small amount of Cr was enough to stop the Cu dissolution and thus the interface degradation with time while showing excellent wetting for both

Hg and Galinstan. For this study, Cu 182 was used for the electrode material for the MHD generator fabrication. Electro co-plating [9] from a Ni and Cu solution to deposit a thin Ni-Cu alloy layer on the Cu electrode surface was also investigated showing excellent surface wetting and interface stability.

FIG. S1. 3D model views of the experimental chamber apparatus. An exploded view (a) shows (from bottom to top) a disc magnet, the chamber with copper contacts, an O-ring seal, the chamber top, and a disc magnet. A top view of the chamber (b) shows electrodes with labels, while an isometric view (c) shows the pattern of fluid flow from the inlet and around the chamber. Cutting the load bridge in (b) allows for a four-point resistance measurement of the internal resistance of the system using the four contacts shown.

Experimental setup design

An experimental apparatus which was used in this study can be seen in Fig. S1. Fig. S1(a) shows an exploded view of the apparatus containing the magnets discussed above as well a copper electrode piece, two clear acrylic pieces to form the chamber, and an O-ring to provide a seal between those two pieces. Fig. S1(b) shows the locations of the central and peripheral electrodes as well as the load bridge in between them. The dimensions of this load bridge can be varied to provide different load resistances which can be verified using a four-point resistance measurement on the two contacts for each electrode. These contacts allow for an accurate determination of the internal resistance via a four-point resistance measurement when the system is full of liquid metal as well as voltage measurements over the load that can allow for the calculation of current and power. Another version of this experiment can be performed to determine the open circuit voltage by severing the load bridge. It is expected that the results of experiments with thinner load bridges providing more resistance should approach the results of the open circuit experiment. This apparatus includes a hole for filling the chamber that can be seen at the bottom of Fig. S1(b). Pressure measurements can be taken using a pressure tap coming off the inlet at the top of the image.

In order to operate the MHD, it is necessary to flow the liquid metal at relatively high velocities though the MHD chamber. To achieve this, the MHD generator was attached to an external fluidic circuit to create the flow. Two experimental flow circuits were used due to the different properties of Galinstan and mercury. Both of the experiments used a setup that allowed the conductive liquid metal to flow at high rate between the two reservoirs.

For the case of mercury, the setup shown in Fig.2 was constructed to allow the flow of mercury between two reservoirs – one connected to the MHD exhaust outlet and one connected to the chamber inlet. The mercury, collected in the second reservoir, could be returned to the first reservoir flowing through the MHD chamber to setup the next run. The flow rate could be regulated by changing the air pressure applied to the mercury reservoir. The flow rate was measured from the change of weight of the mercury per run time and by using a custom-built flowmeter.

FIG. S2. Schematic of experimental setup for MHD generator experiments with mercury

The flowmeter was connected to the inlet line of the MHD generator as shown in Fig. S2. The flowmeter was a simple device, illustrated in Fig. S3, conceptually similar to a duct MHD device. The flow of the mercury through a known cross section in a magnetic field produces a voltage. By knowing the distance between the electrodes and the strength of the magnetic field in the flow area, one can calculate the flow rate. The flow rates calculated from the flowmeter voltages agreed well with those obtained by the mercury weight change per run time.

FIG. S3. Schematic representation of the flowmeter

As Galinstan is more expensive than Hg, glass syringes were used to contain and inject Galinstan in order to minimize the amount of material required. In that case the pressure application was intermittent. The setup for Galinstan is shown in Fig. S4. Galinstan has the additional problem of forming surface oxides when exposed to air. These oxides can be easily removed by rinsing material with dilute hydrochloric acid. The MHD chamber was first filled with dilute hydrochloric acid and then Galinstan was added to replace the acid. This procedure ensured that the surface of the electrodes was wetted by the Galinstan and free of any oxide formed during the transfer process. After the Galinstan was enclosed in the MHD system, no further oxidation occurred. The design and construction of the MHD chamber was the same as previously discussed.

FIG. S4. Schematic of the MHD generator setup used for Galinstan experiments

The flow rate of Galinstan was determined by analyzing the video recording of the exhaust syringe plunger motion as Galinstan was injected by applying pressure to the inlet syringe.

Additional results

Results for a mercury experiment with the external loads of 65 µΩ and 316 µΩ (compared to an internal resistance of roughly 222 µΩ) and with velocity measured using the MHD flowmeter can be seen in Fig. S5 and S6 respectively. These figures show good agreement between theoretical and experimental results for all experiments except for those with a magnetic field of

0.19 T and external resistance of 65 µΩ. Additionally, voltage and power density results for mercury were plotted with respect to load resistance in Fig. S7. The results shown were for the highest inlet velocity experiments (consistently measured at 1.6 m/s). This figure shows that our experimental results were obtained both above and below peak power production with respect to load resistance.

FIG. S5 Experimental (dots) voltage results plotted against fluid velocity from mercury experiments with external load 65 µΩ for different magnetic field strengths. Theory (lines) provides reliable predictions for experimental resultss.

FIG. S6 Experimental (dots) voltage results plotted against fluid velocity from mercury experiments with external load 316 µΩ for different magnetic field strengths. Theory (lines) provides reliable predictions for experimental results.

FIG. S7 Experimental (dots) voltage (a) and power density (b) results plotted against load resistance for mercury experiments with an inlet velocity of 1.6 m/s and different magnetic field strengths. Theory (lines) show reasonable quantitative agreement for the voltage curve and qualitative agreement for the power density curve.

Theoretical model

Here we consider a simple order-of-magnitude estimate of the maximum power density that can be achieved by the proposed vortex flow MHD approach. Total electrical power 푊푒푙̇ that can be generated by a cylindrical control volume similar to the one shown in Fig. 1 of the article can be expressed as

푉2 푊̇ 푒푙 = (S1) 푅푡 where 푅푡 is the total electrical circuit resistance and 푉 is the voltage generated in the radial direction between the center and the periphery the control volume. For the control volume shown in Fig. 1 of the article the voltage 푉 can be expressed as

푉 = 퐵푎 〈푣〉 (S2) where 퐵 is the magnetic flux density, 〈푣〉 is the average flow velocity, 푎 = 푟2 − 푟1 , and 푟1 and

푟2 are the exhaust tube radius and the peripheral control volume radius respectively. Similarly, the control volume resistance 푅 can be expressed as

휌푒푙 푟2 푅 = ln (S3) 2휋ℎ 푟1 where 휌푒푙 is the fluid resistivity, and h is the control volume height. Substituting Eq. (S2, S3) in

Eq (S1) and assuming the impedance matching load so that 푅푡 = 2 푅 we obtain the following estimate for the generated power 푊푒푙̇

휋ℎ퐵2 ̇ 2 2 (S4) 푊푒푙 = 2 푎 푣 훼휆 휌푒푙 where 〈푣〉 = 푣/휆 and 훼 = ln 푟2⁄푟1 , where 훼 and 휆 are constants on the order of one for the typical experimental conditions explored in this work. The amount of kinetic energy 퐸̇푘 entering the control volume in unit time can be expressed as

1 퐸̇ = 푚̇ 푣2 (5) 푘 2 where 푚̇ is the mass flow rate of the entering fluid and 푣 is its velocity. The mass flow rate 푚̇ is equal to

푚̇ = 푆휌푚푣 (S6) where 휌푚 is the density of the flowing fluid, and S is the inlet cross-sectional area. Substituting

Eq. (S6) into Eq. (S5) we obtain the following expression for 퐸̇푘

1 퐸̇ = 푆휌 푣3 (S7) 푘 2 푚

By equating Eq. (S7) and (S4) we can define the characteristic velocity 푣∗ which represents the upper limit on the inlet flow velocity at which full conversion of the incoming kinetic energy into electrical energy within the control volume is still possible

2휋퐵2푎2ℎ ∗ (S8) 푣 = 2 훼휆 푆휌푚휌푒푙

The resulting estimate for the characteristic volumetric electrical power density 푤̇ ∗ takes the form

푊̇ 퐵2 ∗ 푒푙 ∗2 (S9) 푤̇ = 2 = 2 푣 2휋ℎ푎 2훼휆 휌푒푙

Substituting typical values for our experimental system and assuming Galinstan as a working fluid

kg 휌 = 7 ∙ 103 , 휌 = 2.8 ∙ 10−7Ω ∙ m, 퐵 = 0.8 T, 푆 = 8 mm2, 푎 = 8.8 mm, 푚 m3 푒푙

h = 3.5mm, λ = 2, α = 2 we obtain 푤̇ ∗ = 27 W∙cm-3 . For a stronger field with 퐵 = 1.3 T we obtain 푤̇ ∗ = 191 W∙cm-3 which indicates that very high power densities on the order of 102 W∙cm-3 are potentially possible using the proposed approach.

Here we utilize the angular momentum equation to establish a connection between the fluid velocities 푣2 and 푣1 expressed by Eq. (1) of the article and the torques generated by the Lorentz's force and the forces acting on the conductive fluid from the chamber walls. The angular momentum equation takes the form

푚̇ (푟2푣2 − 푟1푣1) = 푇푎 + 푇표 (S10) where 푚̇ is the mass flow rate of the fluid and 푇푎 and 푇표 are the torques generated by the

Lorentz's force and the chamber walls respectively. Let us first consider the open circuit case where the Lorentz's force is equal to zero. In this case Eq. (S10) takes the form

푇표 = 푚̇ (푟2푣2 − 푟1푣1) = 푚̇ 푣2(푟2 − 푟1훾) (S11) 푣 where γ = 1 . The experimental data are consistent with the assumption that 훾 ≈ 1. 푣2

We now turn our attention to the calculation of the torque developed by Lorentz's force. The

Lorentz's force 푑퐹푎 acting on the elementary volume of the fluid 푑푉 can be expressed as

푑퐹푎 = 푗 ∙ 퐵 ∙ 푑푉 (S12) where j is the electrical current density at the point where the elementary volume is located, and

B is the magnetic flux density. For the circular chamber geometry shown in Fig. 1 of the article the values of 푗 and 푑푉 can be expressed as

퐼 푗 = (S13) 2휋푟ℎ

푑푉 = ℎ ∙ 푟푑휃푑푟 (S14) where h is the chamber height and I is the total electrical current flowing radially through the chamber. The torque created by the force 푑퐹푎 can be expressed as

퐼퐵 푑푇 = 푟푑퐹 = 푟푑푟푑휃 (S15) 푎 푎 2휋 and the total torque generated by the Lorentz's forces takes the form

1 푇 = ∫ 푑푇 = 퐼퐵(푟 2 − 푟 2) (S16) 푎 푎 2 2 1 We now assume that for our experimental conditions the forces acting from the chamber walls do not appreciably depend on electrical current and act essentially independently from the

Lorentz's force. Thus, we can use Eq. (S11) with γ = 1 to estimate the torque 푇표 not only for the open circuit case but for the non-zero electrical load as well. Thus, substituting Eq. (S11) and Eq.

(S16) into Eq. (S10) we obtain

1 푚̇ (푟 푣 − 푟 푣 ) = 퐼퐵(푟 2 − 푟 2) + 푚̇ 푣 (푟 − 푟 ) (S17) 2 2 1 1 2 2 1 2 2 1 The mass flow rate 푚̇ can be expressed as

푚̇ = 휌 ∙ 푆 ∙ 푣2 (S18) where 휌 is the fluid density and S is the inlet cross sectional area. Note that 푣1 ≠ 푣2 in this case due to the presence for the Lorentz’s force. The current I can be expressed as

푉 퐼 = (S19) 푅푡 where Rt is the total electrical resistance of the circuit. By substituting Eq. (S18) and Eq. (S19) into Eq. (S17) and solving for v1 we obtain

훽 1 − 푣 푣 = 푣 2 (S20) 1 2 훽 1 + 푣2 where the characteristic velocity 훽 takes the form

푎2퐵2 1 푟 훽 = ∙ (1 + 2) (S21) 2휌푆푅푡 2 푟1 It is important to notice that Eq. (S20) and all the subsequent equations based on it are valid only if the inlet velocity v2 is much larger than 훽. It is also expected that the experimental results should progressively deviate from the analytical solution as v2 approaches 훽. Substituting Eq.

(S20) into Eq. (S2) we obtain the expression for the electromotive force V as 1 푉 = 퐵푎푣 ∙ 2 훽 (S22) 1 + 푣2 The voltage drop 푉푙 on the load takes the form

푅푙 푉푙 = 푉 ∙ (S23) 푅푡 where 푅푙 is the load resistance and 푅푡 is the total circuit resistance.

Once we have this voltage formula we can calculate the total power generated and the power dissipated in the load. We can start by squaring the voltage and grouping terms for convenience in forming non-dimensional groups.

퐵2푎 푎2퐵2 1 1 1 푉2 = 푣2 = 2휌푆푅 푣2 = 휌푣24푆푅 훼 2 훽 푡 2 2 2 푡 2 (S24) 1 + 2휌푆푅푡 훽 2 훽 ( 푣 ) (1 + ) (1 + ) 2 푣2 푣2 This leads to the following formula for power in the load

푣2 푉2 1 2훽푆 2 푅 ̇ 푙 2 훽 푙 푊푙 = = 휌훽 푟 ( 푣 ) (S25) 푅푙 2 (1 + 2) 1 + 2 푅푡 푟1 훽 Similarly, the total power generated is

푣2 1 2훽푆 2 ̇ 2 훽 푊푡 = 휌훽 푟 ( 푣 ) (S26) 2 (1 + 2) 1 + 2 푟1 훽 Assembly example

FIG. S8 Cross-section view of a simplified columnar generator assembly. An axial force applied to the assembly would push liquid metal from the top bellows through the spiral inlet into the vortex MHD chamber. The metal exits the chamber passing through the ring-shaped magnet and central electrode before entering the top electrode. Restoring force could be provided by a spring (not shown). Fig. S8 shows a simplified assembly of the MHD generator chamber as well as the system for handling the inlet and exhaust for the liquid metal. This columnar shape could be easily integrated into a wide variety of devices including the below-the-knee section of a leg prosthetic.

Such a generator would harvest energy from the wearer’s strides and that energy could be used to operate the prosthetic.

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