Enhancement of EM Pump Performance Through Modeling and Testing

Project No. 16-10226

Enhancement of EM Pump Performance through Modeling and Testing

Reactor Concepts Research Development and Demonstration (RCRD&D)

Mark Anderson University of Wisconsin, Madison

Collaborators Fort Lewis College

Melissa Bates, Federal POC Yoichi Momozaki, Technical POC Nuclear Energy University Programs (NEUP)

Final Report on

Enhancement of EM Pump Performance through Modeling and Testing

Project Number: 16-10226

Authors: Jordan Reina, Mike Hvastab, William Nolletc, and Mark Andersona

aUniveristy of Wisconsin – Madison, Madison, WI 53706, USA bAlkali Consulting LLC, Lawrenceville, NJ 08648, USA cFort Lewis College, Durango, CO 81301, USA i

Executive Summary Liquid sodium has seen development as a nuclear reactor coolant in the United States for over 50 years. While sodium has many desirable characteristics as a heat transfer fluid, its reactivity poses numerous engineering challenges for even the most basic plant components like pumps. Despite the challenges, mechanical pumps have seen success in sodium service albeit at the expense of simplicity and flexibility. In fact, significant failures of these pumps have been recorded which can be attributed to their complexity. Notwithstanding the complexity, mechanical pumps are still widely proposed for the main circuit pump of sodium cooled reactors.

However, an alternative to mechanical pumps are Electromagnetic Induction Pumps (EMIPs) which use the principle of the Lorentz Force. EMIPs avoid mechanically moving parts in contact with sodium which simplifies the design by avoiding mechanical wear and seals. Therefore, these pumps are generally safer and more reliable than their mechanical counterparts. That being said, EMIPs suffer from relatively low efficiencies that range from 5% to 45%. Some of the inefficiencies are inherent to the EMIPs pumps; namely, inefficiencies arise in the form of resistive heating losses in the fluid itself. However, other inefficiencies arise from Edge Effects which include Finite Length and Finite Width effects.

Due to the significant safety advantages EMIPs have compared to mechanical pumps, there is a strong motivation to integrate these technologies into the design of Generation IV liquid-metal-cooled nuclear power plants. However, EMIPs have low efficiencies partly attributed to Edge Effects. Much of the literature focusing on Edge Effects is dated or simply does not exist in the public domain. More recently, numerical models of EMIPs have shown promising results. However, these models are often custom solvers or use complex software suits which limit their application.

This report describes the work completed to understand Edge Effects in two EMIPs designs. These designs are the Annular Linear Induction Pump (ALIP) and Permanent Magnet Induction Pump (PMIP) also known as a Moving Magnet Pump (MMP). The methodology developed during this project aim at measuring key performance parameters (pressure-flowrate and efficiency-slip performance) using high temperature liquid-sodium. Baseline experimental measurements were then used to verify the accuracy of numerical models developed in the software suit ANSYS/FLUENT.

On the experimental side, instrumentation was developed to characterize the pressure-flowrate and efficiency-slip performance of both the ALIP and PMIP in high-temperature liquid-sodium. Additionally, instrumentation was developed to measure potential output pressure pulsations and the magnetic field distributions. Moreover, a specialized ALIP was constructed to study the impacts magnetic field tapering, also known as coil grading, on the Finite Length Effect. Lastly, experimental performance curves of the PMIP were collected at 300 oC and 600 oC and compared to analytical theory which accounted for Finite Width Effects. These results showed that the Finite Width Effect and non-uniform magnetic field in a PMIP impact the performance.

On the numerical side, models were developed using the software suit FLUENT/ANSYS. These methods were initially compared against a simple MHD flow in a Permanent Magnet Electromagnet Flowmeter (PMEMFM). The results were significant in that they showed agreeance between the numerical model and demonstrated that PMEMFM’s could be calibrated numerically. These modeling methods were then applied to a prototypic PMIP. Results showed that this simple model provided reasonably accurate results of the PMIP p-Q performance. ii

Table of Contents

Executive Summary ...... i List of Figures ...... iv List of Tables ...... x 1 Introduction ...... 1 1.1 Project Objectives ...... 2 1.2 Methodology and Technical Approach ...... 2 2 Literature Review ...... 4 2.1 Permanent Magnet Electromagnet Flowmeters ...... 4 2.2 Electromagnetic Induction Pumps ...... 6 2.3 Annular Linear Induction Pumps ...... 15 2.3.1 Fundamental Components ...... 15 2.3.2 Analytical Solution of a Finite Length Pump ...... 15 2.3.3 Effects of the Finite Length ...... 16 2.3.4 Conclusions ...... 21 2.4 Permanent Magnet Induction Pumps ...... 21 2.4.1 Fundamental Components ...... 21 2.4.2 Analytical Solution of a Finite Width Pump ...... 22 2.4.3 Effects of the Finite Width ...... 27 2.4.4 Conclusions ...... 30 2.5 Literature Review Summary ...... 30 3 Numerical Calibration of a PMEMFM ...... 32 3.1 Model Development ...... 32 3.2 Experimental Setup ...... 37 3.3 Results ...... 41 3.4 Conclusions ...... 43 4 Experimental Measurements of ALIP Performance and the Finite Length Effect ...... 44 4.1 Technical Approach ...... 44 4.2 ALIP Specifications ...... 47 4.3 Experimental Facility ...... 49 4.4 Instrumentation ...... 51 4.4.1 Permanent Magnet Electromagnet Flowmeter ...... 51 4.4.2 Venturimeter ...... 52 4.4.3 Magnetic Field ...... 61 iii

4.4.4 Differential Pressure and Pressure Pulsations ...... 63 4.4.5 Frequency, Power, and Efficiency ...... 66 4.5 Results ...... 67 4.5.1 Duct Pressure Losses ...... 67 4.5.2 Magnetic Field Measurements ...... 68 4.6 Conclusions and Future Work ...... 72 5 Experimental Measurements of PMIP Performance and the Finite Width Effect ...... 74 5.1 Technical Approach ...... 74 5.2 PMIP Specifications ...... 74 5.3 Experimental Facility ...... 76 5.4 Instrumentation ...... 78 5.4.1 Frequency, Power, and Efficiency ...... 78 5.5 Results ...... 79 5.5.1 Duct Pressure Losses ...... 79 5.5.2 Magnetic Field Measurements ...... 81 5.5.3 Performance Curves ...... 83 5.6 Conclusions and Future Work ...... 85 6 Numerical Modeling of PMIP Performance ...... 87 6.1 Model Development ...... 87 6.2 Results ...... 92 6.3 Conclusions and Future Work ...... 95 7 Numerical Modeling of ALIP Performance ...... 96 7.1 Model Development ...... 96 7.2 Results ...... 98 7.3 Conclusions and Future Work ...... 99 8 Summary of Conclusions ...... 100 9 References ...... 103

List of Figures Figure 1: Schematic of a PMEMFM used to measure liquid metal flowrates...... 4

Figure 2: K2 values for PMEMFMs using rectangular magnetics [14] [15] [16] [17]...... 6 Figure 3: Simplified geometry for an infinite length and width EMIP...... 10 Figure 4: Normalized pressure output of an ideal EMIP as a function of . Note that as passes unity, the developed pressure decreases which is a common observation in EMIPs [25] [27] [28]...... 𝑅𝑅𝑅𝑅𝑅𝑅 ∙ 𝑠𝑠𝑠𝑠...... 𝑅𝑅𝑅𝑅𝑅𝑅....∙ 𝑠𝑠 14𝑠𝑠 Figure 5: On the left is a cross-section of the pump. A cut-away of a practical ALIP [6]...... 15 Figure 6: Simplified geometry for a finite length, infinite width, EMIP [25]...... 16 Figure 7: Numerical model of the force density along the length of an ALIP. In this model, large negative force pulsations developed at the entrance and exit of the pump, with the exit braking force have a larger magnitude than the entrance braking force [32]...... 17 Figure 8: Numerical model of the time averaged force density along the length of an ALIP. This model also predicts large negative braking forces at the inlet and exit of the pump [28]...... 18 Figure 9: Numerical model of pressure pulsations in a finite length ALIP. These pressure pulsations were found to occur at twice the supply frequency of the pump [33]...... 18 Figure 10: Numerical and experimental comparison of DSF pulsation magnitude as a function of slip [32]...... 19 Figure 11: Experimental measurements of DSF pulsations in a finite length ALIP [35]...... 19 Figure 12: Experimental measurement of DSF pulsation magnitude as a function of slip and supply frequency [35]...... 19 Figure 13: The reduction in DSF pulsation magnitude as a function of coil grading at three pump frequencies [35]...... 20 Figure 14: The reduction of electrical input power and resulting increase of efficiency as a function of coil grading [35]...... 21 Figure 15: A two-dimensional depiction of two types of PMIPs. On the left is a Drum-Type PMIP. On the right is a Disc-Type PMIP. In the middle is a Halbach array arrangement which can be used in the Drum- Type PMIP configuration [38] [39]...... 22 Figure 16: Simplified geometry of a Permanent Magnet Induction Pump...... 22 Figure 17: Normalized radial magnetic field distribution of a double-array Disc-Type PMIP. Here the mean channel radius to magnet width ration is 2.375. Γ values of 3, 7, 31, and 125 correspond to 12, 6, 4, and 2 magnets respectively [38]...... 23 Figure 18: Radial magnetic field distribution of a double-array Disc-Type PMIP with a mean channel radius to magnet width ratio of 2 [42]...... 23

Figure 19: The normalized attenuation coefficient Kat,1 as a function of normalized duct width = 0...... 25 𝑅𝑅𝑅𝑅𝑅𝑅 ∙ 𝑠𝑠Figure𝑠𝑠 20: The normalized attenuation coefficient Kat,1 as a function of normalized duct width plotted at various numbers. As approaches 1.7, Kat,1 rapidly approaches units...... 26

𝑅𝑅𝑅𝑅𝑅𝑅 ∙ 𝑠𝑠𝑠𝑠 𝑅𝑅𝑅𝑅𝑅𝑅 ∙ 𝑠𝑠𝑠𝑠 v

Figure 21: The normalized attenuation coefficient Kat,2 as a function of shape factor Γ...... 26

Figure 22: The normalized attenuation coefficient Kat,2 as a function of normalized duct width b. Here, Γ is set so that the magnetic field magnitude at ± /2 is zero. As the duct width decreases Kat,2 approaches Kat,1...... 26 𝑏𝑏 Figure 23: Qualitative distribution of currents in a finite width duct with no electrically conducting walls...... 27 Figure 24: Qualitative distribution of currents in a finite width duct with electrically conducting walls. .. 27 Figure 25: Qualitative current distribution in a rectangular under the influence of a stationary and uniform magnetic field [48]. On the left is the case where the side walls have a finite conductivity. On the right is the case where the side walls have zero conductivity...... 28 Figure 26: Geometry used in a FLIP numerical model. While the results do not directly represent a PMIP, the channel geometry is similar and the results do provide insight into the finite width effects of a PMIP [49]...... 28 Figure 27: Geometry used in a FLIP numerical model. While the results do not directly represent a PMIP, the channel geometry is similar and the results do provide insight into the finite width effects of a PMIP [49]...... 29 Figure 28: Experimental setup of a Drum-Type PMIP used to pump GaInSn. Four configurations were used to study the impact of outer ferrous yokes and copper side bars [50]...... 29 Figure 29: Experimental pressure output of the Drum-Type PMIP with, 1 no ferromagnetic yoke or copper side bars, 2 with a ferromagnetic yoke and no copper side bars, 3 no ferromagnetic yoke and with copper side bars, and 4 with a ferromagnetic yoke and copper side bars [50]...... 30 Figure 30: Three-dimensional depiction of the PMEMFM geometry. The flowmeter conduit was modeled as 12.7 mm outer diameter and 9.4 mm inner diameter 316-stainless steel tube centered between a pair of cube magnets. The magnets used are a pair of grade N42 NdFeB cubes. Magnet sizes of 2.5 cm and 5 cm were investigated in this work...... 32 Figure 31: Calculated magnetic Reynolds number in the experimental conditions of the PMEMFM. Note that for the measured conditions, Rmf is always less than unity...... 33 Figure 32: Two different mesh types and several inflation layer thicknesses were investigated. Note that CC is a Cut-Cell mesh type and TET is a Tetrahedral mesh type...... 33 Figure 33: Example of the steady-state voltage distribution on the outer surface of the PMEMFM flow conduit. These results were obtained at a sodium temperature of 250 oC, mass averaged velocity of 1.5 m/s, cube magnets of 5 cm in length, whose faces were separated by 10 cm...... 34 Figure 34: Cross-section of the PMEMFM flow conduit at maximum induced axial voltage. Again, these results were obtained at a sodium temperature of 250 oC, mass averaged velocity of 1.5 m/s, cube magnets of 5 cm in length, whose faces were separated by 10 cm...... 34 Figure 35: Bottom-to-tope voltage profile in the PMEFM conduit from Figure 33. The shaded region above indicates the wall of the conduit...... 35 Figure 36: The results of the mesh sensitivity study using a TET mesh and assuming 250 oC sodium at a velocity of 1 m/s...... 35 vi

Figure 37: A comparison of the PMEMFM simulation results using the CC and TET mesh types. These mesh types yielded similar results and the maximum difference in predicted slope was approximately 1.1%...... 36 Figure 38: The PMEMFM output solved by FLUENT’s Electrical Potential MHD solver deviates from linear theory at high flowrates. This deviation begins when the magnetic Reynolds number Rmf is approximately 0.2...... 37 Figure 39: Two-Dimensional schematic of the University of Wisconsin – Madison’s sodium test facility and test section used for PMEMFM testing...... 37 Figure 40: Calibration curves of the theoretical reference PMEMFM output versus the measured Micromotion F025A Coriolis Flowmeter flowrate [53]...... 38 Figure 41: Experimental flowmeter setup. The configuration pictured has a pair of 2.5 cm magnet cubes with their faces separated by 10 cm...... 39 Figure 42: Measurement lead positioning reference to the centerline of the conduit...... 40 Figure 43: Measurement lead positioning reference to the centerline of the magnets...... 40 Figure 44: Distances to flowmeter grounding points and a sectional diagram...... 40 Figure 45: Experimentally measured and theoretically calculated centerline magnetic field measurements as a function of distance from the face of the magnet...... 41 Figure 46: Experimental and Numerical results of PMEMFM calibration...... 42 Figure 47: Custom designed and built CMI-Novacast model LA125 ALIP used in experimental testing. 47 Figure 48: Qualitative sketch of coil shifting configurations of the experimental ALIP...... 48 Figure 49: Sketch of the tapped coils installed in the first three and last three coil locations. Each of the six coils have taps at the 1st, 20th, 31st, or 80th turn...... 49 Figure 50: Cut away of the experimental ALIP showing each of the 12 polyphase coils in the pump. Note that the first three coils and last three coils are tapped...... 49 Figure 51: The Medium-Scale-Component Sodium Test Facility constructed to measure the performance of the experimental ALIP. Note that the reservoir in the lower left of the frame will be placed in its own catch-pan...... 50 Figure 52: Two-dimensional sketch of the Medium-Scale-Component Sodium Test Facility constructed for ALIP performance testing...... 51 Figure 53: PMEMFM used to measure sodium flowrates in a 27 mm inner diameter conduit...... 51 Figure 54: As-fabricated dimensions of the Sodium Pump Loop (Loop 4) Venturimeter with entrance and exit regions. All dimensions were made in accordance to the ISO 5167-4 2003(3) specifications [57]. ... 53 Figure 55: Finished two-piece Venturimeter produced by Electron Discharge Machining (EDM). On the left is the convergent section. Note the pressure tapping at the bottom of the piece. On the right is the divergent section. Both pieces received boss-and-receiver features which were referenced off the internal features. These features guaranteed concentric alignment of both pieces to a common axis of rotation. .. 54 Figure 56: Two-dimensional drawing of the throat-insert and two positions. On top is the throat-insert itself. In the middle is the 'fully closed' position of the throat-insert while on the bottom is the 'fully open' position...... 55 vii

Figure 57: Finished throat-insert...... 55 Figure 58: Water loop used for calibration of the Sodium Pump Loop (Loop 4) Venturimeter. This loop can achieve a maximum volumetric flowrate of 5.9 m3/hr at 30 oC...... 56 Figure 59: Discharge Coefficient 'C' versus Reynolds number 'Re'. Note that the Reynolds numbers are lower than the specified values in the ISO standard and therefore 'C' does not take on the predicted constant value...... 57 Figure 60: Discharge Coefficient 'C' versus Re-0.4. The change in behavior as a function of Reynolds number can be distinctly observed. An arbitrary cutoff of Re = 60,000 was chosen. Using the two fits, a conservative flowrate error estimate of 3% was calculated...... 58 Figure 61: Pressure recovery coefficient ξ as a function of mass flowrate...... 59 Figure 62: Flow Coefficient 'Cv' as a function of mass flowrate. Over most mass flowrates, the flow coefficient takes on a value of roughly 15.5 GPM/psi0.5...... 60 Figure 63: Flow Coefficient 'Cv' with the throat-insert installed. These data are plotted against the throat- inset retraction which is given in 'Number of Turns'...... 61 Figure 64: Air-cored pick-up coil used to measure the radial magnetic field distribution in the ALIP channel...... 62 Figure 65: Calibration curve for the pick-up coil in Figure 52 used to measure the radial magnetic field component in the ALIP channel...... 63 Figure 66: Photo of PCB Model 112A05 pressure transducer...... 64 Figure 67: Sketch of thermal stand-offs for the PCB high frequency pressure sensors. Note that the sensor is recessed from the flow roughly 8.6 cm. Also shown are the manufactured mounts for the PCB sensors. These mounts were machined into a Cone-and-Thread fitting plug...... 64 Figure 68: Initial calibration of two PCB model 112A05 pressure transducers in water. Note that a new calibration will need to be performed at lower pressures since the expected ALIP pressure pulsation magnitude is on the order of 0.15 bar...... 65 Figure 69: Test vessel used to evaluate the seal ring compatibility in sodium. On the left is the mount for the pressure sensor while on the right is the stand after heating and insulation...... 66 Figure 70: Entrance length and relative positioning of the pressure taps on the experimental ALIP. Note that the pressure taps are in the same location as the experimental ALIP test facility...... 67 Figure 71: Friction pressure losses in ALIP duct as a function mass flowrate...... 68 Figure 72: Radial magnetic field distribution as a function of ALIP length with no coil grading...... 70 Figure 73: Radial magnetic field measurements of Configuration 1. Note that Configuration 1-1 and Configuration 1-2 only apply the coil grading to the pump inlet and outlet respectively...... 70 Figure 74: Radial magnetic field measurements of Configuration 2. Note that Configuration 2-1 and Configuration 2-2 only apply the coil grading to the pump inlet and outlet respectively...... 71 Figure 75: Radial magnetic field measurements of the Entrance Shifted Configuration compared to measurements in the centered configuration...... 71 Figure 76: Radial magnetic field measurements of the Exit Shift Configuration compared to measurements in the centered configuration...... 72 viii

Figure 77: Photo of UW Disc-Type PMIP installed in UW-Madison’s Small-Scale Component Sodium Test Facility...... 75 Figure 78: Blow apart of the magnet array’s used in UW-Madison’s Disc-Type PMIP [38]...... 75 Figure 79: A two-dimensional sketch of the Small-Scale Component Sodium Test Facility used during PMIP testing...... 76 Figure 80: Proposed drive-train to accommodate an in-line torque transducer for power measurements. . 77 Figure 81: Photo of PMIP drive-train used in power measurements...... 77 Figure 82: Picture of the sample holder used to throttle the Small-Scale-Component Sodium Test Facility...... 77 Figure 83: Two-dimensional schematic of PMIP drive-train that will be used to measure pump power. .. 79 Figure 84: PMIP duct configuration used during frictional pressure loss measurements...... 80 Figure 85: Two-dimensional sketch of PMIP with some relevant dimensions...... 80 Figure 86: Experimental frictional pressure loss measurements in water...... 81 Figure 87: Sketch of magnetic field measurement locations in the experimental PMIP...... 81 Figure 88: Axial magnetic field magnitude as a function of axial distance between the PMIP discs. Note that in the conduit region of ±3.2 mm the distribution is roughly uniform...... 82 Figure 89: Axial magnetic field magnitude as a function of radial distance between the PMIP discs. Note that in the region of the conduit region of ±25 mm the distribution is roughly symmetric...... 82 Figure 90: Initial p-Q performance curves of the experimental ...... 83 Figure 91: Corrected experimental PMIP pressure output at 300 oC compared to theoretical calculations...... 84 Figure 92: Corrected experimental PMIP pressure output at 600 oC compared to theoretical calculations...... 84 Figure 93: A schematic showing the sodium flow-path between the experimental pressure transducers. These transducers are located at the inlet and outlet of this geometry. Note that all units are presented in inches...... 88 Figure 94: The mid-plane of the ‘active channel’ modeled using the FLUENT MHD module...... 88 Figure 95: Detailed look at the cut cell used in Model 1...... 89 Figure 96: Detailed look at the symmetric tetrahedral mesh used in Model 2...... 89

Figure 97: Contours of the Bz magnetic field in Tesla. Note that the z-axis is directed out of the page. The magnets on the PMIPs were arranged to produce an alternating polarity magnetic field...... 90 Figure 98: The magnetic Reynolds number of sodium traveling through the PMIP duct as a function of velocity...... 91 Figure 99: The calculated hydraulic losses across the total system in Figure 92 and the active channel in Figure 93...... 92 Figure 100: A comparison of the calculated differential pressures and the experimental measured values. The x-error bars corresponds to ±3.6% while the y-error bars correspond to ±0.1254 psid...... 93 ix

Figure 101: A contour plot showing the magnitude of the induced electrical current density on the mid- plane in A/m2...... 93 Figure 102: A vector plot showing the induced electrical currents on the mid-plane in A/m2...... 94 Figure 103: A contour plot showing the magnitude of the Lorentz-force on the mid-plane in N/m3...... 94 Figure 104: A vector plot showing the Lorentz-force on the mid-plane in N/m3. Note that the vectors are directed against the flow since the magnetic field was assumed to be stationary...... 95 Figure 105: Model of the simple ALIP geometry...... 96 Figure 106: ALIP model with a constant 0.25 T magnetic field within the active region...... 97 Figure 107: ALIP model with a sawtooth grading profile. Here the peak field is 0.25 T...... 97 Figure 108: ALIP model with a sinusoidal grading profile. Here the peak field is 0.25 T...... 97 Figure 109: Contour plot of the initial current distribution results in a general ALIP with a constant, uniform, 0.25 T field...... 98 Figure 110: Logarithmic contour plot of the initial current distribution results in a general ALIP with a constant, uniform, 0.25 T field...... 98 Figure 111: Contour plot of the initial current distribution results in a general ALIP with a sinusoidal magnetic field with 0.25 T magnitude...... 99 Figure 112: Logarithmic contour plot of the initial current distribution results in a general ALIP with a sinusoidal magnetic field with 0.25 T magnitude...... 99

List of Tables Table 1: Duct width in centimeters required to achieve the specified normalized attenuation coefficient for two different values...... 25

Table 2: Theoretical𝑅𝑅𝑅𝑅𝑅𝑅 ∙ 𝑠𝑠calibration𝑠𝑠 constants at three temperatures...... 38 Table 3: Correction constants for each ...... 39 Table 4: Experimentally obtained slopes using a = model and Linear Regression analysis...... 42

Table 5: Comparison of experimental and numerical𝑦𝑦 slopes𝐴𝐴 ∙ 𝑥𝑥 to their theoretical values...... 43

Table 6: ALIP Testing Campaign 1 and 2. These campaigns will form the baseline p-Q and η-sm data that will be used to evaluate the effectiveness of the modified pump configurations...... 46 Table 7: Testing campaigns for coil grading Configurations 1, 1-1, and 1-2. More details on the coil grading can be found in Section 4.5.2. Measurements are performed at 200 oC so that pressure pulsations can be measured by the high frequency pressure instrumentation. Only a single supply frequency is studied since the frequency dependence will be deduced from Campaign 1...... 46 Table 8: Testing campaigns for coil grading Configurations 2, 2-1, and 2-2. More details on the coil grading can be found in Section 4.5.2. Measurements are performed at 200 oC so that pressure pulsations can be measured by the high frequency pressure instrumentation. Only a single supply frequency is studied since the frequency dependence will be deduced from Campaign 1...... 47 Table 9: Testing campaigns for Entrance and Exit coil shifting configurations. More details on the coil shifting can be found in Section 4.5.2. Measurements are performed at 200 oC so that pressure pulsations can be measured by the high frequency pressure instrumentation...... 47 Table 10: Rated conditions for CMI-Novacast’s LA125 ALIP...... 48 Table 11: Some relevant geometric properties for CMI-Novacast’s LA125 ALIP...... 48 Table 12: Relevant properties of PMEMFM used in experimental ALIP facility...... 52 Table 13: Relative contributions of the measured variable uncertainties in the calculated calibration curves...... 58 Table 14: Relative contributions for the calculated uncertainties in the pressure recovery curves...... 59 Table 15: Relevant dimensions of the air-cored pick-up coil in Figure 52...... 62 Table 16: Relevant properties of PCB Model 112A05 pressure transducer...... 64 Table 17: Relative contributions of each measured quantities uncertainty towards the calculated efficiency...... 67 Table 18: Relevant dimensions of the tapped coils in the ALIP...... 69 Table 19: Experimentally measured ALIP coil grading configurations...... 69 Table 20: Testing campaigns to evaluate PMIP performance and the impact of the Finite Width Effect. . 74 Table 21: Properties of UW-Madison ...... 76 Table 22: Relative contributions of each measured quantities uncertainty towards the calculated efficiency...... 79 xi

Table 23: An overview of the meshing and modeling methods used in this work...... 87 Table 24: Sample FLUENT data for 180 Hz...... 92 Table 25: Sample input to model the magnetic field in the active region of the ALIP...... 96

1 Introduction Beginning in the early days of the nuclear power industry, sodium-metal has seen development as a candidate reactor coolant. In the United States, one of sodium’s first nuclear applications was in the nuclear-powered submarine USS Seawolf (SSN-575) [1]. Constructed as a variant to the historic USS Nautilus (SSN-571), the Seawolf studied the feasibility of a sodium cooled reactor for marine use. At the same time, sodium coolant was tested in land-based reactors with the aptly named Sodium Reactor Experiment (SRE). Notably, the SRE produced the first electricity for a commercial grid by powering the nearby community of Moorland Park California [2]. Since the Seawolf and SRE, several more experimental reactors using sodium were built and operated such as the Experimental Breeder Rector-II and Enrico Fermi-I. Even as recently as 2019, sodium is the chosen coolant for reactors such as GE- Hitachi’s PRISM and TerraPower’s Traveling Wave Reactor.

Basic to nearly every nuclear reactor design are pumps. Typically, mechanical pumps such as Centrifugal Pumps have filled the role of driving forced convection for liquid metal systems [3]. However, there are limitations in their application to sodium. Firstly, sodium’s reactivity impacts the shaft-seal design. Normally, a Centrifugal Pump eliminates fluid leakage by using a seal in direct contact with the working fluid and the rotating shaft. However, it has been noted that dynamic seals in direct contact to sodium have failed due to excessive wear caused by reactions with the seal materials [4]. Therefore, shaft-seals are generally kept in the gas spaces [5]. This eliminates material compatibility issues but forces the pump to be oriented vertically.

Secondly, sodium’s reactivity impacts the bearing design. Normally, a Centrifugal Pump will use a bearing lubricated by the working fluid or some other lubricant. However, it has been noted that sodium has poor lubricating properties itself and can react with many common bearing lubricants [5]. Therefore, bearings are kept in the gas space away from directly contacting sodium [5]. Again, this eliminates the material compatibility issues but forces the pump to be oriented vertically. Additionally, this may also impose restrictions on the length of the pump shaft.

Lastly, Centrifugal Pumps inhibit natural circulation. Many reactor vendors are developing passive cooling driven by natural circulation in plant designs. However, the complex flow-path through a Centrifugal Pump’s impeller and vanes can prevent natural circulation from forming. Therefore, passive cooling methods may become more complex with the addition of Centrifugal Pumps.

While limitations exist, Centrifugal Pumps have been successfully used in sodium service [2] [3]. For example, the SRE successfully used mechanical pumps in the main coolant circulation loop. However, while operating the SRE did experience a significant failure of these pumps. The SRE pumps bearings and seals were lubricated by an organic fluid. And while precautions were taken to prevent the organic from contacting sodium, these systems failed, leaking the organic lubricant into the system. This leak contaminated the reactor coolant which clogged several reactor fuel channels, resulting in a partial meltdown [2].

Electromagnetic Induction Pumps (EMIPs) are an alternative to mechanical pumps for electrically conductive fluids. EMIPs have several advantages over mechanical pumps [3] [6] [7]. Firstly, EMIPs are hermetically sealed. Therefore, these pumps avoid complex seal designs and a potential failure mode. Secondly, EMIPs have no rotating parts in direct contact with sodium. Therefore, these pumps avoid complex bearing designs and can use standard lubricants if needed. Thirdly, EMIPs components are 2

mounted externally and avoid direct contact with sodium. Therefore, these pumps simplify material selections and repair procedures. Fourthly, EMIPs can operate in any orientation. These pumps do not have a free-surface limitation like many Centrifugal Pump designs which may simplify plant design. Lastly, EMIPs have no impellers or vanes which may impeded the formation of natural circulation. Therefore, these pumps may simplify passive cooling system and enhance plant safety.

While EMIPs have many notable advantages, their main disadvantage is efficiency. Efficiencies of EMIPs typically range from as high as 45% to as low as 5% [6] [8]. The low efficiencies are generally caused by resistive heating in the conduit walls [8], resistive heating in the electrical equipment [9], and edge effects from the finite dimensions of the pump [9]. Thus, EMIPs have not seen widespread deployment. Consequently, little publicly available research exists considering the inefficiencies caused by Edge Effects in EMIPs. However, some private reactor vendors such as GE-Hitachi are proposing EMIPs in the main coolant circuit. This suggests that these companies may have overcome these challenges.

1.1 Project Objectives Due to the significant safety advantages EMIPs have compared to mechanical pumps, there is a strong motivation to integrate these technologies into the design of Generation IV liquid-metal-cooled nuclear power plants. That being said, EMIPs suffer from low efficiencies with Edge Effects being identified as a contributor to this low efficiency. However, much of the literature focusing on Edge Effects is dated or simply does not exist in the public domain. More recently, numerical models of EMIPs have shown promising results. However, these models are often custom solvers or use complex software suits which limit their application. These challenges in EMIP development for liquid-metal nuclear systems have defined four project objectives that guided the course of this work.

1. Experimentally quantify the impact of Edge Effects on the pressure-flowrate (Δp-Q) and efficiency- slip (η-sm) performance of EMIPs.

2. Develop simple, yet accurate numerical models of EMIPs using a commercial software suit.

3. Verify the developed numerical models using experimental Δp-Q data from prototypic EMIPs.

4. Publish experimental Edge Effect quantification and simple numerical EMIP models to expand the publically available knowledge.

1.2 Methodology and Technical Approach The project objectives were achieved using the methodology described below.

1. Identify important Edge Effects in Permanent Magnet Induction Pumps (PMIPs) and Annular Linear Induction Pumps (ALIPs) through a comprehensive literature review. The review will focus on identifying literature which quantifies the impact of Edge Effects on the Δp-Q and η-sm performance of PMIPs and ALIPs. Additionally, this review will be keenly concerned with identifying additional experimental measurements that can practically quantify or characterize Edge Effects and approaches to reduce the impact of Edge Effects on the Δp-Q and η-sm performance of these EMIPs.

2. Measure experimental Δp-Q and η-sm performance of prototypic PMIPs and ALIPs. Additionally, this work will also perform the experimental measurements identified in literature which characterize 3

Edge Effects in these EMIPs. Lastly, the effectiveness each approach on reducing the impact of Edge Effect will be evaluated through experimental measurements in the prototypic PMIPs and ALIPs.

3. Develop numerical PMIP and ALIP models using the commercial software suit FLUENT. The numerical solver in FLUNET will be benchmarked against simple MHD flows in a Permanent Magnet Electromagnet Flowmeter (PMEMFM). After successfully benchmarking against a simple chase, models of the prototypic pumps will be developed and benchmarked against experimental p-Q performance data.

2 Literature Review 2.1 Permanent Magnet Electromagnet Flowmeters Permanent Magnet Electromagnet Flowmeters (PMEMFMs) are a type of flowmeter used to measure flowrates in liquid metal systems. These flowmeters use permanent magnets to induce a voltage that is proportional to the liquid metal flowrate. Figure 1 shows a schematic of a PMEMFM [10].

Figure 1: Schematic of a PMEMFM used to measure liquid metal flowrates.

In principle, this voltage is induced by the Hall Effect. The Hall Effect, electrons flowing in a stationary conductor are deflected by an applied magnetic field. This deflection force obeys the Lorentz Force Law shown in Equation 1 where F is the force exerted on the electrons, E is the electric field, u is the velocity of the electrons, and B is the magnetic field. The Lorentz Force is not specific to electrons flowing in a stationary conductor. In general a Lorentz Force will occur given the presence of an electric field and any relative motion between a magnetic field and a charged particle. For example, in a PMEMFM the Lorentz Force will be exerted on the electrons in a liquid metal moving through a stationary magnetic field.

= q( + × ) Equation 1 𝐅𝐅 𝐄𝐄 𝐮𝐮 𝐁𝐁 At steady state, F in Equation 1 will be zero. Therefore, an electric field must be established that follows Equation 2.

= × Equation 2 𝐄𝐄 −𝐮𝐮 𝐁𝐁 By assuming the magnetic field does not vary with time, Faraday's Law of Induction states that × = 0. Thus, the electric field is conservative and can be calculated by the gradient of a scalar potential field φ shown in Equation 3. ∇ 𝐄𝐄

= Equation 3 𝐄𝐄 −∇φ

Substituting this definition of E into Equation 2 results in Equation 4 which states that the gradient of the potential field φ is proportional to the particle velocity cross product with the magnetic field.

= × Equation 4 ∇φ 𝐮𝐮 𝐁𝐁 Equation 4 can be simplified in an idealized case. In this case, the magnetic field B is infinite in the and directions and only has a single component of constant magnitude in the direction. This case also assumes that the particle velocities u are symmetric about the and directions and only have a single𝐱𝐱� 𝐲𝐲component� in the direction. With these assumptions, Equation 4 reduces to𝐳𝐳� Equation 5. 𝐲𝐲� 𝐳𝐳� 𝐱𝐱� = u B y ∂φ Equation− x 5 z ∂ Equation 5 is then averaged over the direction and finally integrated with respect to y. By noting that Bz is uniform, Equation 5 can be simplified to Equation 6 where V is the voltage at two points separated by 𝐲𝐲� distance d, u is the average particle velocities across distance d, and Bz is the magnitude of the applied magnetic field. Δ x �� 1 V = d u dy B d = u B d d �2 Equationx 6 z x z ∆ � �−d� � �� 2 In practice, the measured voltage VM could deviate considerably from the ideal output in Equation 6. Three factors, K1, K2, and, K3, and an offset V0 attempt to account for these non-ideal effects. Thus, the actual measured potential will followΔ Equation 7. Δ V = K K K V + V Equation 7 ∆ M 1 2 3∆ ∆ 0 Correction factor K1 corrects for wall-shunting effects. Practical flowmeters often use an electrically conducting conduit which promotes electron-ion recombination at the walls. This Wall-Shunting effect occurs during steady state operation when the electric field in Equation 3 causes negative charges to drift towards one wall and positive metal atoms to drift towards the other wall. Once the electrons reach the stationary wall, they are no longer under the influence of a Lorentz Force and can freely travel through the conduit walls to recombine with positive atoms on the other side. This has the effect of reducing the measured voltage. To account for this, a correction factor K1 in Equation 8 was derived for annular geometries, where d is the inner conduit diameter, D is the outer conduit diameter, ρf is the electrical resistivity of the fluid, and ρw is the electrical resistivity of the wall [11] [12] [13].

d d (T) d K (T) = 2 1 + + 1 D D 2 (T) D 2 −1 ρf 1 � � Equation� 8 � − � � �� ρw Correction factor K2 accounts for charge recombination near the inlet and outlet of the flowmeter. The magnetic field produced by an actual PMEMFM is neither infinite in extent nor perfectly uniform. At the 6

entrance and exit of the flowmeter, the weak magnetic field cannot keep electrical charges separated. To account for this shunting or recombination effect, an entrance and exit correction factor K2 must be used to calibrate experimental systems. As shown in Figure 1, several semi-empirical K2 correction factors L exist. Equation 9 can be used to approximate the factor over the range of 1 d where L is the length of the flowmeter magnet parallel to the flow and d is the inner diameter of the circular pipe or tube. ≤ �

K ~1 0.18812 L Equation 9 d 2 −

Figure 2: K2 values for PMEMFMs using rectangular magnetics [14] [15] [16] [17].

Correction factor K3 accounts for demagnetization at high temperatures. Generally, the strength of a permanent magnet is not constant. Permanent magnets can be quite sensitive to their absolute temperature. This is particularly true for the rare-earth variety of magnets like NdFeB type which will lose their magnetization at temperatures greater than 80 oC [18]. To account for this, a magnet temperature factor K3 was experimentally measured for NdFeB type magnets. K3 was found to follow o o o Equation 10 where Tm is the temperature of the magnet in C of the temperature range of 25 C to 45 C [19].

K (T ) = e . × ( ) −3 Equation−�1 1 10 10 � Tm−20℃ 3 m Lastly, V0 accounts for system voltage offsets. These offsets can be caused by the thermoelectric effect [20]. In this effect, a voltage is induced by either a temperature difference between the junction of two dissimilarΔ metals [21] or simply by the temperature difference between the junction of the same metal [22].

2.2 Electromagnetic Induction Pumps Electromagnetic Induction Pumps (EMIPs) are a type of pump used for electrically conducting fluids. In principle, EM pumps produce a Lorentz Force which can be described by Equation 11. Note that j is the current density, B is the magnetic field, and FV is the volume force density, also known as the pressure gradient [6].

= × Equation 11 𝐅𝐅V 𝐣𝐣 𝐁𝐁 By assuming the magnetic field and current density only has a single, constant component in the y- direction and x-direction respectively, Equation 11 can be simplified greatly. Thus, for a duct of height a, width b, and length L, the ideal developed EM pressure is given in Equation 12.

j B p = L xaby EMEquation 12

Of course, frictional pressure losses will exist. These losses are proportional to the square of the mean fluid velocity and take the form of Equation 13 where K is a loss coefficient and U0 is the characteristic velocity [23].

1 p = U K 2 2 lossEquation 13 0 ρ Now, the actual developed pressure can be estimated using Equation 14.

p = p p Equation 14 ∆ EM EM − loss 8

While useful conceptually, Equation 12 is impractical for EMIPs since the current density is rarely known a priori. Therefore, EMIP problems are solved using the Magnetohydrodynamic (MHD) Equations. These equations are formed by the set of Maxwell Equations for the electromagnetic field and the Navier-Stokes Equations for the fluid field. The Maxwell Equations, neglecting displacement currents are given in Equation 15 to Equation 20. The Navier-Stokes Equations, assuming an incompressible fluid, are given in Equation 21 and Equation 22.

× = t Equation 15∂ 𝐁𝐁 ∇ 𝐄𝐄 − ∂ q = Equation 16 ∇ ∙ 𝐄𝐄 ϵ × = Equation 17 ∇ 𝐁𝐁 μ𝐣𝐣 = 0 Equation 18 ∇ ∙ 𝐁𝐁 = ( + × ) Equation 19 𝐣𝐣 σ 𝐄𝐄 𝐮𝐮 𝐁𝐁 = 0 Equation 20 ∇ ∙ 𝐣𝐣 + ( ) + p = t ∂𝐮𝐮 Equation 21 2 ρ � 𝐮𝐮 ∙ ∇ 𝐮𝐮� ∇ − μ∇ 𝐮𝐮 𝐅𝐅 ∂ = 0 Equation 22 ∇ ∙ 𝐮𝐮 Equation 10 through Equation 22 can be used to derive the Induction Equation [9] [24] [25]. The results of this derivation are shown in three useful forms. Firstly, Equation 23 is the Induction Equation in terms of the total magnetic field.

= × ( × ) t 2 ∂𝐁𝐁 ∇ 𝐁𝐁 μσ �Equation− ∇ 23 𝐮𝐮 𝐁𝐁 � ∂ Equation 23 can be expanded if the total magnetic field can be assumed to be a superposition of the applied external field Be and the induced fluid field Bi as shown in Equation 24 [25].

= + Equation 24 𝐁𝐁 𝐁𝐁e 𝐁𝐁i

Thus, Equation 23 can be written as Equation 25.

( ) + ( ) = ( ) + ( ) ti te 2 ∂𝐁𝐁 Equation 25 ∂𝐁𝐁 ∇ 𝐁𝐁i − μσ � − 𝐁𝐁i ∙ ∇ 𝐮𝐮 𝐮𝐮 ∙ ∇ 𝐁𝐁i� −μσ � − 𝐁𝐁e ∙ ∇ 𝐮𝐮 𝐮𝐮 ∙ ∇ 𝐁𝐁e� ∂ ∂ When solving 2D MHD problems, it is often useful to write Equation 25 in terms of a vector potential A defined in Equation 26 and Equation 27.

× = Equation 26 ∇ 𝐀𝐀 𝐁𝐁 = 0 Equation 27 ∇ ∙ 𝐀𝐀 Thus, substituting Equation 26 into Equation 23 results in another form of the Induction Equation in Equation 28 which is the vector potential form [24].

1 = + × × t ∂𝐀𝐀 2 Equation∇ 𝐀𝐀 28𝐮𝐮 𝛁𝛁 𝐀𝐀 ∂ μσ Closer inspection of Equation 28 provides valuable qualitative insight into the behavior of magnetic induction. Consider the nondimensionalized version of Equation 28 shown in Equation 29 where and are the nondimensional forms of A and u [9]. Note that the time rate of change of is caused by diffusive � effects, given by the Laplacian term, and advective effects, given by the velocity term. The relative𝐀𝐀 𝐮𝐮� � scaling of these two effects is governed by the Magnetic Reynolds Number Rm. 𝐀𝐀

1 = + × × t Rm ∂𝐀𝐀� 2 Equation∇ 𝐀𝐀� 29𝐮𝐮� ∇ 𝐀𝐀� ∂ f The Magnetic Reynolds Number for the fluid is defined Equation 30. This nondimensional parameter describes the relative effects of magnetic induction to magnetic diffusion. When Rmf is small, diffusion dominates, and the solution is like a solid body solution. However, when Rmf is large, convection dominates, and a considerable entrance length may be required for the magnetic field to fully diffuse into the fluid.

U Rm = f kf B Equationf μ σ 30 0 Note that k0 is the fundamental wave number, defined in Equation 31 where τ is the pole pitch, or half- wavelength, of the magnetic wave.

k =

Equation0 π 31 τ 10

Also note that UB is the velocity of the magnetic wave. This velocity, called the synchronous velocity, is defined in Equation 32 where f is the frequency.

U = f Equation 32 B 2τ Now, the set of differential equations used to solve MHD induction problems are given in Equation 33 and Equation 34.

1 = + × × t ∂𝐀𝐀 2 Equation∇ 𝐀𝐀 33𝐮𝐮 ∇ 𝐀𝐀 ∂ μσ + ( ) + p = × ( × ) t ∂𝐮𝐮 Equation 34 2 ρ � 𝐮𝐮 ∙ ∇ 𝐮𝐮� ∇ − μ∇ 𝐮𝐮 𝐣𝐣 ∇ 𝐀𝐀 ∂ In practice, a general analytic solution to Equation 33 and Equation 34 is impossible. However, in specific special cases where the velocity profile is known, the differential equations can be decoupled and solved independently [7] [9]. One specific case for which an analytical solution can be obtained is the case where the fluid velocity is constant with time and uniform as in Equation 35.

= U Equation 35 𝐮𝐮 0𝐳𝐳� This assumption is used in the simplified EMIP geometry shown in Figure 3. Here, the pump is assumed to be infinitely long in the z-direction and infinitely wide in the x-direction with a finite fluid thickness of in the y-direction. Note that the geometry in Figure 3 is cartesian while the geometry in an ALIP is cylindrical. While an analytical solution exists in cylindrical coordinates [26], the added complexity of the axisymmetric solution does not provide additional insight into the physical phenomenon. This approximation is justified by assuming the ratio of channel width a to mean channel radius Ravg is much smaller than unity [7] [9].

Figure 3: Simplified geometry for an infinite length and width EMIP.

The external magnetic field Be is produced by an infinitely thin current sheet with the form in Equation 36. Note that the pre-exponential factor J0 is the magnitude of the effective current sheet, ωB is the angular frequency of the magnetic field, and k0 is the fundamental wave number.

a , z, t = J e ( ) 2 i ωBt−k0z e Equation0 36 𝐣𝐣 � � A consequence of assuming a constant velocity is that the Induction Equation can be solved separately from the Fluid Equations. Furthermore, this implies that the vector potential A must have the same time dependence as the applied currents in Equation 36 [9]. Thus, the form for a solution to A is given in Equation 37 where A the complex amplitude is. For simplicity, A is assumed to have a single component in the x-direction. ̇

(y, z, t) = A(y)e ( ) Equation 37i ω Bt−k0z 𝐀𝐀 ̇ 𝐱𝐱� The simplified differential equation for the vector potential is shown in Equation 38 where λ2 is given by Equation 39 and sm is given by Equation 40.

A A(y) = 0 2y ∂ ̇ 2 2 Equation− λ ̇ 38 ∂ = k (1 + is Rm ) 2 Equation2 39 λ 0 m f Note that Equation 40 is defined as the normalized difference between the synchronous velocity UB and the mean fluid velocity U0. This value is called the mean slip. Observe that as the slip approaches zero, the fluid is moving with the same velocity as the magnetic field and there is no relative motion. This implies that no EMF will be induced and likewise no currents will be induced.

U s = 1 U0 mEquation− 40 B Boundary conditions for Equation 38 are provided in Equation 41 and Equation 42.

A = 0 y ∂ ̇ Equation� 41 ∂ y=0 A = J y ̇ ∂ f 0 �Equation 42μ ∂ y=a� 2 12

Equation 38 can be solved for the complex vector potential A. Then, A can be converted to the magnetic field B which is averaged over the channel width assumed the channel width a is much smaller than the magnetic skin thickness δs [25]. The height averaged complex amplitude of the magnetic field is given in Equation 41.

J 1 B | = (Rm s + i) ak 1 + (Rm s ) μf 0 〈 ẏ 〉 y Equation 43 2 f ∙ m 0 f ∙ m The height averaged complex amplitude of the current density in the fluid is given by Equation 44.

J Rm s | = (Rm s + i) a 1 + (Rm s ) 0 f ∙ m 〈ȷẋ 〉 y Equation 44 2 f ∙ m f ∙ m Thus, the force volume density, also known as the pressure gradient, can be calculated using Equation 45 where * is the complex conjugate operator.

p = { × } Equation 45 ∗ ∇ EM ℜ 𝐣𝐣f 𝐁𝐁 Often it is desired to know the time averaged electromagnetic force. This can be calculated by taking the root-mean-square (RMS) value of jf and B. Thus, the RMS value of the height averaged EM pressure gradient can be calculated using Equation 46.

1 p = | B | 2 ∗ ��EM�� Equationx 46y ẏ y ∇ ℜ�〈ȷ̇ 〉 〈 〉 � The RMS value of the average EM pressure gradient is given in Equation 47 where B , is given in B Equation 48. , is defined as the height averaged amplitude of the total magnetic field.y 0 y 0 1 p = B (U U ) 2 , 2 ��EM�� Equationf y 0 47 B 0 ∇ σ − J B , = k a 1 + (fR0m s ) y 0 μ Equation 48 2 0 � f ∙ m The amplitude of the external field is found by setting Rm s equal to zero. Note that this is the case where there is no relative motion and therefore no induced currents. Thus, the form of B , is given in f ∙ m Equation 49. y e

J B , = kf 0 Equationy e μ 49 0 13

The calculated pressure gradient can then be substituted into the Navier-Stokes Equations in Equation 21 to find the total pressure gradient in the pump [7] [9] [25]. As discussed earlier, in the case of the constant velocity assumption the pressure gradient can be simply split into two components shown in Equation 50. The quantity pEM represents the pressure developed by the EM body force while the quantity ploss represents the frictional losses inside the pump conduit.

p = p p Equation 50 ∆ EM EM − loss Then, the RMS value of the height averaged EM developed pressure can be calculated using Equation 51.

p = p dz L EMEquation� �∇� �51�EM� �� 0 In this case, the time averaged electromagnetic body force is constant everywhere. Thus, Equation 51 becomes trivial and the electromagnetic induced pressure is given by Equation 52. This is the maximum possible pressure that can be developed by an EMIP. This form shows that the pressure is linearly proportional to the length of the pump, relative velocity, and fluid conductivity. However, the pressure is proportional to the square of the height averaged magnetic field.

1 p = B (U U )L 2 , 2 EM Equationf y 0 52B 0 σ − The average EM pressure gradient in Equation 52 can now be written as a function of Rm s as in Equation 53. f m ∙ Rm s p = p , 1 + (Rm s ) f ∙ m EM EM Equationmax ∙ 53 2 f ∙ m 1 k p = B L , 2 , 0 2 EM maxEquation 54 y e μf

A normalized plot of Equation 53 is given in Figure 3 as a function of Rm s . As discussed earlier, when Rmf is small, diffusion dominates advection. In this regime of Figure 4, the pressure output for an f ∙ m ideal EMIP increases linearly. However, as Rmf increases, advection dominates diffusion and the pressure output for an ideal EMIP deceases. This is a common observation in EMIPs [25] [27] [28].

Figure 4: Normalized pressure output of an ideal EMIP as a function of . Note that as passes unity, the developed pressure decreases which is a common observation in EMIPs [25] [27] [28]. 𝑓𝑓 𝑚𝑚 𝑓𝑓 𝑚𝑚 𝑅𝑅𝑚𝑚 ∙ 𝑠𝑠 𝑅𝑅𝑚𝑚 ∙ 𝑠𝑠 The efficiency of an induction pump can be calculated using Equation 55 which is the ratio of electromagnetic power to electrical power.

W = WEM Equationη 55 E The electromagnetic power delivered to the fluid is simply the volume integral of the product between the developed EM pressure gradient and the fluid velocity shown in Equation 56.

W = p dV

EM Equation�〈∇ 56EM 〉 ∙ 𝐮𝐮

The total electrical power can be found by an energy balance on the pump system. As shown in Equation 57, the energy delivered to the pump goes to the total developed EM pressure gradient, resistive heating in the fluid, or other losses. Note that the W , represents losses in generating the magnetic field as well as other losses. R e

W = W + W , + W , + W , Equation 57 E EM R f R c R e Resistive losses in the fluid can be calculated using Equation 58.

( , ) ( , ) W = dV ,( , ) 2 ∗ 𝐉𝐉 f c ∙ 𝐉𝐉 f c R f c Equation� 58 σf 15

For an ideal EMIP, where W , is neglected, the efficiency is purely a function of mean slip as shown in Equation 59 [7] [9] [29]. R e

= 1 s Equation 59 ηEMIP − m 2.3 Annular Linear Induction Pumps 2.3.1 Fundamental Components In principle, an Annular Linear Induction Pump (ALIP) can be derived from an Asynchronous Induction Motor (AIM) [30]. Like an AIM, an ALIP uses polyphase electric coils which establish a magnetic wave that travels down the length of the pump. Figure 5 shows a cross-section of an ALIP channel [6]. Here, the magnetic field is oriented radially across the channel width. Induced currents flow azimuthally in the channel and react with the total magnetic field to produce a Lorentz Force directed in the axial direction. In general, the coils are pancake shaped, slipped over the outer conduit, and set in stacks of comb-shaped puchings called the outer stator cores [6]. The outer and inner cores provide a similar function as the stator in an AIM by providing a low magnetic resistance, or reluctance, path for the magnetic flux lines.

Figure 5: On the left is a cross-section of the pump. A cut-away of a practical ALIP [6].

2.3.2 Analytical Solution of a Finite Length Pump Analysis of the non-dimensionalized Induction Equation in Section 2.2 showed that the time rate of

change of was due to diffusion and advection. For small Rmf, diffusion dominates and the solution will be like that of a stationary conductor [9]. However, for large Rmf, advection dominates causing a � considerable𝐀𝐀 distance for to fully diffuse into the fluid [9]. This suggests that important physical behavior may be missed by neglecting the finite length of the pump. The finite length was considered in � the analytic solution developed𝐀𝐀 by Valdmanis [31]. Figure 6 shows the new geometry and material assumptions. As before, the pump is assumed to be infinitely wide. Note that the effective current sheet is now a finite length L and is constrained to L = 2 p where p is the number of pole pairs.

τ 16

Figure 6: Simplified geometry for a finite length, infinite width, EMIP [25].

A full solution to the problem will not be presented. Rather, the results will be summarized and compared to the ideal solution in Section 2.2. Full details of the solution can be found in Valdmanis's work [31]. Zone One and Zone Three are defined as the entrance and exit of the pump respectively, while Zone Two is defined as the inductor zone. The external magnetic field amplitude solutions for Zone One through Zone Three are given in Equation 60 and Equation 61.

( , ) J B = sin( p) cos ( t) , k 1 3 μf 0 ̇ y e Equation 60 B π ω ( , ) B , = (sin ( ) cos ( ) sin( )) 1 3 𝑓𝑓 0 𝜇𝜇 𝐽𝐽 Equation 61 ̇ y e 𝜔𝜔𝐵𝐵𝑡𝑡 − 𝑘𝑘𝑘𝑘 − 𝜋𝜋𝜋𝜋 𝜔𝜔𝐵𝐵𝑡𝑡 𝑘𝑘 Note that Equation 60 and Equation 61 are in the same form as the external magnetic field in Equation 62 that was derived in Section 2.2. However, now the external field pulsates. Note that if the number of pole pairs is odd, which corresponds to p = m/2 where m is an odd integer, then the pulsation disappears from Zone Two and appears in Zone One and Zone Three. If the number of pole pairs is even, which corresponds to p = n where n is any integer, then the pulsating component appears in the Zone Two and disappears in the Zone One and Zone Three.

( , ) J B = , k 1 3 μf 0 ̇ Equationy e 62

While this model is simplistic, it shows that the finite length effect is purely electromagnetic. Note that this problem was solved under the constant velocity assumption which decoupled the Induction Equation from the Navier-Stokes Equations. Therefore, no fluid effects were considered in this solution. However, a finite length region was considered which produced a pulsating magnetic field that was independent of the flow. This phenomena did not appear in the infinite length solution. Thus, the finite length effect appears to purely arise from electromagnetic phenomenon.

2.3.3 Effects of the Finite Length While the analytical solution in the previous section provided useful insight into the finite length effect, the solution lacked details of a real ALIP. For example, the previous analysis neglected the finite width of the coils and the stator cores. Also neglected in the previous analysis were fluid dynamic effects such as a velocity profile, acceleration, and turbulence which couple the Induction Equation with the Navier-Stokes 17

Equations. Thus, this section will review analytical, numerical, and experimental work on the finite length effect that encompass more detail than the simple analytical solution.

Numerical models have the advantage of considering more detail than the analytical solutions reviewed earlier. In addition to capturing finite length effects, these models can also couple the Induction Equation with the Navier-Stokes Equations [28] [32] [33] [34]. Furthermore, some models have also tried to capture the finite width polyphase coils [33]. Thus, numerical models may provide more insights into the behavior of finite length effects than analytical solutions.

Like the analytical solutions, numerical models of finite length ALIPs also predict a severe braking force [28] [32] [34]. Figure 7 shows a numerical solution of the EM force density as a function of the length of the pump [32]. At the entrance and exit of the pump, large, negative, and oscillating braking forces exist which are qualitatively consistent with the analytical work discussed earlier. Additionally, Figure 7 shows that the braking force at the exit is larger in magnitude than at the entrance.

Roman reported similar numerical results [28]. Figure 8 shows the time averaged force density as a function of pump length. As observed in Araseki's work, braking forces exist at the entrance and exit of the pump. Furthermore, the exit braking force is larger in magnitude than the entrance braking force. Additionally, note that Figure 8 also shows a suppression of the EM force density near the entrance of the pump which increases with the fluid velocity. This is also consistent with the Induction Equation scaling arguments and the analytical finite length effect theory.

Figure 7: Numerical model of the force density along the length of an ALIP. In this model, large negative force pulsations developed at the entrance and exit of the pump, with the exit braking force have a larger magnitude than the entrance braking force [32].

Figure 8: Numerical model of the time averaged force density along the length of an ALIP. This model also predicts large negative braking forces at the inlet and exit of the pump [28].

In addition to predicting large braking forces and magnetic field suppression, numerical models have also predicted the presence of pressure pulsations [32] [33]. Figure 9 plots the deviation of pressure from the mean output as a function of time for a pumping operating at 50 Hz [33]. Also plotted is the pressure frequency spectrum. A distinct peak is observed at 100 Hz. It was found that as the supply frequency changed, the peak in pressure pulsation was always twice that of supply frequency. Thus, these pulsations were called Double Supply Frequency (DSF) pressure pulsations.

Figure 9: Numerical model of pressure pulsations in a finite length ALIP. These pressure pulsations were found to occur at twice the supply frequency of the pump [33].

Numerical models have predicted that DSF pressure pulsation magnitude is a function of slip [32]. Figure 10 shows the pulsation magnitude versus slip for a pump operating at 20 Hz [32]. At low values of slip, the magnitude of the DSF pulsations is observed to reach near 40% of the mean pressure output.

Figure 10: Numerical and experimental comparison of DSF pulsation magnitude as a function of slip [32].

Experimental measurements have confirmed the existence of DSF pulsations predicted by the numerical models [35]. Figure 11 shows a graph of the experimentally measured pressure pulsation and its frequency spectrum. For a pump operating at 20 Hz and the conditions specified in Figure 8, a pressure deviation of 4% was observed with a frequency of 40 Hz.

Figure 11: Experimental measurements of DSF pulsations in a finite length ALIP [35].

Araseki then investigated the magnitude of the DSF pulsation as a function of slip and supply frequency. The experimental results in Figure 12 confirm the dependence of pulsation magnitude on slip [35]. Additionally, for a fixed value of slip, the magnitude of the DSF pulsation increases as the supply frequency decreases. In some cases, the magnitude of the pulsation can reach up to 50% of the mean developed pressure.

Figure 12: Experimental measurement of DSF pulsation magnitude as a function of slip and supply frequency [35]. 20

This experimental study would seem to suggest that reducing the DSF pulsation magnitude could be simply achieved by running the pump at high supply frequencies. However, two criterions may be violated with this method. Firstly, stable operation is ensured by Equation 63 which requires Rm s be less than unity. However, note that increasing the supply frequency will directly increase Rm s . f m ∙ f ∙ m Rm s = s 1 k μfσfωB f ∙ mEquation2 63 m ≤ 0 Secondly, full magnetic diffusion is ensured by Equation 64 which requires the channel width to magnetic skin thickness ratio be many orders less than unity. However, increasing the supply frequency will push this ratio towards unity. Thus, two fundamental criterions may be violated by increasing the supply frequency and the pump may be pushed out of its optimal operating regime [35].

a = a s 1 2 μfσfωB �Equation 64 m ≪ δs However, DSF pulsations can be avoided by tapering the magnetic field at the ends of the pump [7] [29] [35]. This tapering, called coil grading, is accomplished by decreasing the number of turns in the polyphase coils at the entrance and exit of the pump. Three coil grading configurations experimentally studied by Araseki. In this particular ALIP, the pole pitch corresponded to 6 coils and the pump had a total of 36 coils. Therefore, 1/3 to 2/3 of the total pump length could be graded.

Experimental results show that tapering the magnetic field reduces the DSF pulsation magnitude. Figure 13 shows that in the low slip regime of 0.1, linear grading over one-pole length at both stator ends reduced the amplitude of the pressure pulsation to about 1/2 of the non-grading configuration [35]. This was found to have a positive impact on the efficiency of the pump. Figure 14 shows that while coil grading did not impact the pressure-flowrate performance, it did lower the electrical input power [35]). This observation is consistent with the perturbation wave theory discussed earlier. In this configuration, the efficiency of the pump was increased by as much as 10%. Compared to normal efficiencies of ALIPs which can range from 5% to 45%, a 10% increase in efficiency is significant.

Figure 13: The reduction in DSF pulsation magnitude as a function of coil grading at three pump frequencies [35].

Figure 14: The reduction of electrical input power and resulting increase of efficiency as a function of coil grading [35].

2.3.4 Conclusions In total, this literature review identified that the main Edge Effect in an ALIP is the Finite Length Effect. This effect was found to have two factors. The first factor is an Rmf effect, which reduces the total inlet and outlet magnetic fields through advection by the fluid [9] [31] [36] [37]. The second factor is a finite length effect in the magnetic core materials of the ALIP, which causes large, negative, oscillating braking forces to appear at the inlet and outlet of the pump [36]. These effects were found to decrease the η-sm performance of the ALIP.

These oscillating braking forces were found to manifest as pressure pulsations called Double Supply Frequency (DSF) pulsations [32] [33]. Experimental studies showed that tapering the magnetic field at the entrance and exit of the pump not only reduces the magnitude of the DSF pulsation but also improves the efficiency of the pump [35]. This suggests that DSF pulsations may be an additional measurement to quantify the impact of the Finite Length Effect. Additionally, tapering of the inlet and outlet magnetic field was identified as a successful method of increasing pump efficiency.

2.4 Permanent Magnet Induction Pumps 2.4.1 Fundamental Components In principle, a Permanent Magnet Induction Pump (PMIP) can be derived from an AIM [30]. However, a PMIP setups a moving magnetic field in a much different way than an AIM. PMIPs use an array of alternating polarity permanent magnets to establish the magnetic wave. Normally, the only practical arrangement is to fix the magnets on a drum or disc and rotate the array around a loop or helix shaped conduit [6]. Two common arrangements are shown in Figure 15 [38]. On the left is the Drum-Type PMIP which orients the magnetic field radially across the channel width. The magnetic field on the outside of a Drum-Type PMIP can be strengthened by arranging the magnets in a Halbach array shown in the center of Figure 15 [39]. On the right is the Disc-Type PMIP which orients the magnetic field axially across the channel width. The magnetic field can be strengthened by using two arrays separated axially with the channel located in the middle.

Figure 15: A two-dimensional depiction of two types of PMIPs. On the left is a Drum-Type PMIP. On the right is a Disc-Type PMIP. In the middle is a Halbach array arrangement which can be used in the Drum-Type PMIP configuration [38] [39].

In contrast to an ALIP, the channel of a PMIP is square. A square channel maximizes the flow area while satisfying the magnetic skin thickness constraint. This constraint requires the ratio of channel thickness to magnetic skin thickness a/ to be much smaller than unity. This ensures that the magnetic field is roughly uniform across the channel thickness. Thus, the channel thickness is minimized in the radial and s axial directions of the Drumδ-Type and Disc-Type PMIPs respectively.

PMIPs have several advantages over ALIPs. Firstly, PMIPs can achieve larger values of peak flux density than ALIPs [6] [40] [41]. Thus, the pressure output of a PMIP is theoretically larger for a similar sized ALIP. Secondly, PMIPs have a nearly unity power factor due to the drive being an induction motor [6] [41]. Therefore, most of the power is converted to useful work. Lastly, PMIPs are simple to construct and have relatively low capital costs [38]. Thus, PMIPs can be built with easily and quickly.

However, PMIPs do have limitations when compared to ALIPs. Firstly, rectangular channels have pressure limitations relative to circular conduits [6] [40]. Secondly, PMIPs are not axisymmetric like ALIPs due to the square conduit. Therefore, PMIPs suffer the finite width effect [41] [42]. Lastly, PMIPs having heavy rotating discs that require balancing to limit vibrations and bearing wear.

2.4.2 Analytical Solution of a Finite Width Pump An analytical solution to a finite width PMIP can be derived using the Induction Equation with similar assumptions in Section 2.2 [17] [42] [43] [44]. Firstly, the geometry is reduced to Figure 16. Note that full penetration of the magnetic field into the fluid is ensured by requiring the channel width to magnetic skin thickness ratio a/ be much less than unity. Therefore, the effects of the magnetic field over the height of the duct can be averaged [25]. Additionally, the applied magnetic field in a PMIP is fixed during s operation and can be measuredδ a priori. Thus, no external current sheets are needed to produce the magnetic wave.

Figure 16: Simplified geometry of a Permanent Magnet Induction Pump. 23

In this analysis, the finite length effect and the finite dimensions of the magnets are neglected. Here, the moving magnetic field is assumed to travel as a pure sinusoidal wave along the z-direction. It was found that for a 12-magnet Disc-Type double-array, with a pole pitch of 6.25 cm, and a mean channel radius to magnet width ratio of 2.375, the applied magnetic field can be approximated as a sinusoidal wave as shown in Figure 17 [38].

Figure 17: Normalized radial magnetic field distribution of a double-array Disc-Type PMIP. Here the mean channel radius to magnet width ration is 2.375. Γ values of 3, 7, 31, and 125 correspond to 12, 6, 4, and 2 magnets respectively [38].

Radial effects are also neglected in this analysis. This is justified by requiring that the mean channel radius to be larger than the magnet width. Experimental measurements in Figure 18 have shown that this is an appropriate assumption for magnet width to mean radius ratios of around two [42] [45]. Therefore, the radial effects of magnet speed and fluid velocity are ignored and the effective length of the pump is approximated by Equation 65.

Figure 18: Radial magnetic field distribution of a double-array Disc-Type PMIP with a mean channel radius to magnet width ratio of 2 [42].

L = R Equation 65 e 2π avg 24

Lastly, the magnetic field is assumed to have a single component in the y-direction, but is allowed to vary in the x-direction. Thus, with these assumptions, the applied magnetic field can be written in the form of Equation 66.

( ) = B , (x)e Equation i66ω t−kz 𝐁𝐁𝐞𝐞 e y 𝐲𝐲� Details of the solution to the problem in Figure 15 can be found in [17] [42] [43] [44]. The results from two cases of the derivation are summarized. The first case considers a magnetic field that is uniform across the width of the duct. Thus, Equation 66 is now written as Equation 67 where B0 is the effective, height averaged magnetic field, and is a constant.

= B e ( ) Equationi ω 67t− kz 𝐁𝐁e 0 𝐲𝐲� Using this form results in a developed EM pressure in Equation 61. Note that this equation is exactly the same as Equation 52 from Section 2.2. However, a new factor Kat,1 appears which accounts for the finite width of the pump. The new correction factor is defined in Equation 69.

1 p = B (U U )L K 2 , 2 EM fEquation0 B 68 0 e at 1 σ − a k tanh 2 K = 1 , 2 a 0 �2λ � at 1 𝔑𝔑 � 2 � − �� λEquation 69 λ The second case considers a magnetic field that follows the shape of a cosine function. Note that a cosine function was found to approximate the flux distribution within the width of a rectangular magnet. Thus, Equation 66 is now written as Equation 70 where B0 is the effective, height averaged magnetic field amplitude and Γ is the shape factor which describes the uniformity of the magnetic field in the x- direction.

= B cos( x)e ( ) Equation 70i ωt−kz 𝐁𝐁e 0 Γ 𝐲𝐲� Using this form results in the same developed pressure as in Equation 68, but with a new attenuation factor in Equation 71. Note that by taking the limit as Γ approaches zero, Kat,2 collapses to Kat,1.

a a a a a tanh cos + sin k 1 sin 2 cos 2 2 2 2 K = + , +2 2 a a 0 4� 2Γ � �Γ � �λ �Γ( �+ �Γ) � Γ �Γ �� at 2 2 2 𝔑𝔑 � 2 2 � − 2 2 �� λ Γ Equation 71 Γ λ Γ Note that Equation 68 is identical to the EM pressure output from an ideal EMIP. Thus, Kat,1 and Kat,2 in Equation 69 and Equation 70 respectively are what determine the performance of a PMIP. Figure 19 plots a normalized Rm s = 0 over a range of normalized duct widths. One can see that as the duct becomes

f ∙ m 25

wider, Equation 69 asymptotically approaches unity and the pump performance approaches its theoretical maximum.

Figure 19: The normalized attenuation coefficient Kat,1 as a function of normalized duct width = 0.

𝑅𝑅𝑚𝑚𝑓𝑓 ∙ 𝑠𝑠𝑚𝑚 Practically speaking, the pump duct width must become very wide to approach unity. Table 1 presents the calculated duct widths in centimeters to achieve Kat,1 values of 0.9 and 0.99 at an Rm s of 0.1. Note that as Rm s increases towards unity, the required duct width decreases. However, in practice Rm f m s is kept as small as possible. ∙ f m f ∙ ∙ m Table 1: Duct width in centimeters required to achieve the specified normalized attenuation coefficient for two different values. 𝑅𝑅𝑚𝑚𝑓𝑓 ∙ 𝑠𝑠𝑚𝑚 Rm s K , = 0.90 K , = 0.99

0.1f m at 1 20 at 1198 1∙ 9.62 90.6

As noted in Section 2.2, the fluid magnetic Reynolds number Rmf must be kept small in order for diffusion to dominate advection. This can be demonstrated by sweeping Equation 69 over a range of Rm s numbers. Note that the fluid magnetic Reynolds number is multiplied by the mean slip sm since the fluid sees a reduced magnetic velocity due to the relative motion. Figure 20 shows that as Rm s f ∙ m approaches a value of 1.7, Kat,1 approaches unity rapidly. Beyond, Rm s of 1.7 the solution breaks f m down which suggests Equation 69 is only valid up to this limit. This limit of Rm s = 1.7 is ∙ f ∙ m independent of both the fluid conductivity σf and wave number k0, suggesting that it could be a f m fundamental criterion specific to this solution. ∙

Next, the behavior of Kat,2 with Γ is shown in Figure 21. As noted earlier, Kat,2 approaches its maximum value at = 0 and collapses to Kat,1. Additionally, it should be noted that it does not make physical sense for the magnetic field to go negative anywhere. Thus, the smallest value Equation 70 can have is zero on the edges.Γ This means that the product b can only be as large as π. One can see from Figure 18 that the performance of a PMIP can be reduced by half when the field falls to near zero at the edges. Γ ∙ This can also be shown in Figure 22. Here, Γ is set by a given duct width b such that the field drops to zero on the edges. The duct width b is then reduced while holding Γ constant. One can see that as the width b becomes smaller, the magnetic field variation across b decreases, and the solution approaches the 26

uniform solution where Γ is zero. Note that the effective height averaged magnetic field was not altered in this analysis, only the uniformity of the magnetic field itself. Thus, this suggests that PMIP performance may be improved by selecting magnets which are much wider than the duct itself.

Figure 20: The normalized attenuation coefficient Kat,1 as a function of normalized duct width plotted at various numbers. As approaches 1.7, Kat,1 rapidly approaches units. 𝑅𝑅𝑚𝑚𝑓𝑓 ∙ 𝑠𝑠𝑚𝑚 𝑅𝑅𝑚𝑚𝑓𝑓 ∙ 𝑠𝑠𝑚𝑚

Figure 21: The normalized attenuation coefficient Kat,2 as a function of shape factor Γ.

𝑏𝑏 27

2.4.3 Effects of the Finite Width The finite width effect is a result of current continuity [46] [47]. Qualitatively, this is shown in Figure 23. Since current, or charge, must be conserved, the induced currents form continuous loops in the fluid. However, since the channel is not axisymmetric, the current loops form in the x-z plane. Note that for a pumping direction in the positive z-direction, only the x-component of the induced current loop produces a useful force. Conversely, the z-component produces a force which is directed inwards and is of no use for pumping. In the ideal case where the duct is infinitely wide, the current loops effectively only have an x-component since they reconnect at infinity.

Figure 23: Qualitative distribution of currents in a finite width duct with no electrically conducting walls.

In the previous section, the analytical solution neglected the conduit walls. Therefore, the induced current was solely contained in the fluid as in Figure 23. However, this is not generally true. Some induced current may flow into the conduit walls as shown in Figure 24. Thus, the conduit walls may be useful in maximizing the induced current's x-component and satisfying current continuity.

Figure 24: Qualitative distribution of currents in a finite width duct with electrically conducting walls.

In literature, this effect has been studied for MHD flows confined to a square channel under the influence of a uniform and stationary magnetic field. The qualitative results of the analytical solution are given in Figure 25 [48]. In the case where the A-A walls have infinite conductivity, the currents tend to pass into the walls. In the other extreme, where the A-A walls have zero conductivity, the currents will bypass the walls and connect in the fluid. Therefore, the wall conductivity has an important role in determining the current distribution in a square channel pump.

Figure 25: Qualitative current distribution in a rectangular under the influence of a stationary and uniform magnetic field [48]. On the left is the case where the side walls have a finite conductivity. On the right is the case where the side walls have zero conductivity.

However, as noted in the energy balance from Section 2.2, the conduit currents will cause resistive heating as shown in Equation 72. Thus, the increase in performance due to more uniform currents may be negated due to the increase resistive losses, decreasing the pump efficiency. No analytic work could be found which investigates this optimization.

W = dV , 2 ∗ 𝐉𝐉c ∙ 𝐉𝐉c R c Equation� � � 72 σc Wall effects have been modeled in FLIPs numerically as shown in Figure 26 [49]. Note that while FLIPs use polyphase coils instead of permanent magnets, the pump conductivity is like a PMIP. Therefore, the results will provide some insight into the finite width effects of a PMIP. In this study, a parameter called ε, defined as the ratio of side wall to fluid conductivity was swept over a range of values assuming uniform and non-uniform velocity profiles. σc⁄σf

Figure 26: Geometry used in a FLIP numerical model. While the results do not directly represent a PMIP, the channel geometry is similar and the results do provide insight into the finite width effects of a PMIP [49]. 29

Figure 27 shows the effect of the conductivity ratio ε on pump performance [49]. For increasing ε, the performance of the pump is predicted to increase. Furthermore, one can see that the efficiency of the pump is predicted to increase with increasing wall conductivity ratio. This suggests that the side walls may be promoting the currents to reconnect in the walls as was shown in the analytical investigations. However, a subtly should be noted. Observe that the magnetic flux in the sidewalls is small in magnitude. Thus, the induced currents in the walls and their associated resistive heating will also be small.

Figure 27: Geometry used in a FLIP numerical model. While the results do not directly represent a PMIP, the channel geometry is similar and the results do provide insight into the finite width effects of a PMIP [49].

Very little experimental literature could be found investigating finite width effects in PMIPs. A single study exists which investigates the impact of copper side bars on a drum-type PMIP using GaInSn [50]. Figure 28 shows a sketch of the experimental setup. Four configurations were studied to investigate the impact of a ferromagnetic yoke and copper side bar combinations.

Figure 28: Experimental setup of a Drum-Type PMIP used to pump GaInSn. Four configurations were used to study the impact of outer ferrous yokes and copper side bars [50].

Figure 29 shows that the presence of side copper bars increased the maximum developed pressure regardless of the ferromagnetic yoke [50]. Note that the electrical conductivity ratio of copper to GaInSn at 20 oC is approximately 0.17. Assuming the original walls were made of a material with zero conductivity, this work seems to experimentally confirm that higher conductivity side walls improve 30

PMIP performance. However, it should be noted that the author was not clear on the materials of the walls or whether the bars were simply affixed to an existing wall of finite conductivity.

Figure 29: Experimental pressure output of the Drum-Type PMIP with, 1 no ferromagnetic yoke or copper side bars, 2 with a ferromagnetic yoke and no copper side bars, 3 no ferromagnetic yoke and with copper side bars, and 4 with a ferromagnetic yoke and copper side bars [50].

2.4.4 Conclusions In summary, this literature review identified that the main Edge Effect in a PMIP is the Finite Width Effect. This effect is a consequence of current continuity which requires that current, or charge, must be conserved [46] [47]. Since the pump conduit is not axisymmetric, the induced current loops have a component which provides no useful pumping force. The current loop effect is exacerbated as the width of the duct becomes small [42] [50] relative to the loops. Literature also identified that the sidewall conductivity may be important in determining the current distribution in the pump channel and therefore the p-Q performance of the pump [48] [49]. It has been shown that the efficiency of a Flat Linear Induction Pump (FLIP), a close cousin to a PMIP, can be greatly increased with higher conductivity walls [49].

2.5 Literature Review Summary This literature review has identified the primary Edge Effects and key measurements to quantify these effects in ALIPs and PMIPs. Firstly, the literature review identified that the primary Edge Effect in the ALIP is the Finite Length Effect. This effect was found to be caused by two factors. The first factor is an Rmf effect, which carries the field down the length of the pump due to fluid advection [9] [31] [36] [37]. The second factor is a finite length core effect, which causes large, negative, oscillating braking forces to occur at both ends of the pump [36]. These effects were found to decrease the η-sm performance of the ALIP. Additionally, it was shown that the braking forces manifested themselves as Double-Supply- Frequency (DSF) pressure pulsations [32] [33]. These pulsations were reduced by tapering the magnetic field at the inlet and outlet of the pump which increased the η-sm performance of the ALIP.

Lastly, the literature review identified that the primary Edge Effect in the PMIP is the Finite Width Effect. This effect was found to be caused by current continuity, which requires that current must be conserved. The non-axisymmetric geometry of the pump means that some component of the induced current loop will contribute no useful pumping force. This effect increases as the width of the pump becomes small relative to the length [42] [50]. Literature also identified that the sidewall conductivity is important in determining the current distribution in MHD channel flows [48]. This sidewall conductivity effect was shown to greatly impact the p-Q and η-sm performance of a FLIP [49]. However, little experimental studies and no numerical studies have been completed studying this effect for a specific PMIP. 31

In total, this literature review identified the main Edge Effects impacting the performance of ALIPs and PMIPs. It was found that these effects all reduce the η-sm performance of the pumps and may also reduce the p-Q of the pump. Key experimental measurements were identified to quantify these Edge Effects and several methods of reducing their impact were also identified.

3 Numerical Calibration of a PMEMFM 3.1 Model Development To verify the accuracy of FLUENT in solving MHD flows, a simple model was investigated first. As previously discussed in Section 2.1, PMEMFMs are instrument used to measure liquid-metal flowrates. PMEMFMs using small diameter flow conduits and comparatively large magnets can approach the ideal limits of the analytical theory described in Section 2.1.

This work modeled a PMEMFM using the FLUENT/ANSYS Electric Potential MHD solver. The basic geometry of the model is shown in Figure 30. The flow conduit was modeled as a 316-stainless steel tube with a 12.7 mm outer diameter and a 9.4 mm inner diameter. This conduit is positioned between two grade N42 NdFeB magnet cubes. These magnets are held in place using non-magnetic materials.

Figure 30: Three-dimensional depiction of the PMEMFM geometry. The flowmeter conduit was modeled as 12.7 mm outer diameter and 9.4 mm inner diameter 316-stainless steel tube centered between a pair of cube magnets. The magnets used are a pair of grade N42 NdFeB cubes. Magnet sizes of 2.5 cm and 5 cm were investigated in this work.

The magnetic field produced by the pair of NdFeB magnet cubes was used as an input for the FLUENT simulation. These data were calculated using an analytical solution of the magnetic field around an unyoked magnet [51]. A custom Matlab script then produced the required MAG-DATA input files needed for the FLUENT simulation. MAG-DATA files contain uniformly spaced magnetic vector information within a Cartesian coordinate system. FLUENT linearly interpolates magnetic field information from the MAG-DATA file and projects it onto a 3D mesh prior to performing a simulation.

An important aspect to the Electric Potential method is that it assumes that the applied external magnetic field is not affected by the presence of induced currents or secondary magnetic fields [52]. This assumption was verified by calculating the magnetic Reynolds number Rmf given in Equation 73 where μf is the magnetic permeability of the fluid, σf is the electrical conductivity of the fluid, di is the hydraulic diameter of the conduit, and U is the mass averaged sodium velocity.

� Rm = d U Equation 73 f μfσf i� Values for Rmf at the velocities relevant to this study are shown in Figure 31. For the investigated flowrates, Rmf < 1 for all conditions. Therefore, it can be assumed that the induced magnetic field had a small effect on the flowing sodium compared to the externally applied magnetic field. 33

Figure 31: Calculated magnetic Reynolds number in the experimental conditions of the PMEMFM. Note that for the measured conditions, Rmf is always less than unity.

Several mesh types and inflations layers were investigated in this work. Figure 32 shows that four inflation layer thicknesses were investigated for the Cut-Cell (CC) mesh assembly and two inflation layer thicknesses were investigated for the Tetrahedral (TET) mesh assembly.

Figure 32: Two different mesh types and several inflation layer thicknesses were investigated. Note that CC is a Cut-Cell mesh type and TET is a Tetrahedral mesh type.

The behavior of the sodium in the flow conduit was calculated in FLUENT using the realizable k turbulence model with enhanced wall functions. At the solid-liquid wall interface, a coupled boundary condition was implemented. The Electric Potential MHD solver was used to study the impact of the− ϵ external magnetic field given in the MAG-DATA files. Figure 33 shows the results of the steady-state induced voltage distribution on the outer surface of the flow conduit.

Figure 33: Example of the steady-state voltage distribution on the outer surface of the PMEMFM flow conduit. These results were obtained at a sodium temperature of 250 oC, mass averaged velocity of 1.5 m/s, cube magnets of 5 cm in length, whose faces were separated by 10 cm.

A cross-section of the flow conduit at the maximum axial induced voltage distribution is down in Figure 34. Note that the distribution does not vary greatly near the top and bottom of the conduit. Figure 35 shows the bottom-to-top voltage profile from Figure 34. Note that the shaded region indicates the wall of the tube. Shunting effects described in Section 2.1 will decrease the voltage within the tube wall. With measurement leads affixed to the outer wall of the conduit, this effect will reduce the overall output of the PMEMFM.

Figure 34: Cross-section of the PMEMFM flow conduit at maximum induced axial voltage. Again, these results were obtained at a sodium temperature of 250 oC, mass averaged velocity of 1.5 m/s, cube magnets of 5 cm in length, whose faces were separated by 10 cm.

1.5

0.5 ] - [

OD 0 V/V -0.5

-1

-1.5 -1.5 -1 -0.5 0 0.5 1 1.5

r/rOD [-]

Figure 35: Bottom-to-tope voltage profile in the PMEFM conduit from Figure 33. The shaded region above indicates the wall of the conduit.

The mesh sensitivity study shown in Figure 36 show how the calculated voltage output of a PMEMFM is affected by the size of the mesh used in a numerical study. The resolution of the mesh was enhanced by reducing the size of the nodes and elements in the mesh. A size was deemed to be appropriate when FLUENT simulations yielded consistent, accurate, and computationally efficient results.

0.0012

0.0010

0.0008

0.0006 Output [V] 0.0004

0.0002

0.0000 0 2 4 6 8 10 12 14 16 18 Mesh Nodes [#] x106

Figure 36: The results of the mesh sensitivity study using a TET mesh and assuming 250 oC sodium at a velocity of 1 m/s.

Figure 37 shows the impact the mesh types have on the modeled output of the PMEMFM. It can be seen that both mesh types yield comparable results over a range of sodium velocities. Note that these results were obtained with a sodium temperature of 250 oC, 5 cm cube magnets, and a magnet spacing of 10 cm. The tetrahedral mesh shown below was much more computationally expensive then the cut-cell mesh.

1400 Tetrahedral Mesh 1200 Cut Cell Mesh 1000

V] 800 μ 600

Output [ 400

200

0 0 0.25 0.5 0.75 1 1.25 1.5 U [m/s]

As noted earlier, the Electric Potential method is only valid for Rmf < 1. This is shown in Figure 38 which demonstrates the impact of sodium velocity on the modeled output of the PMEMFM. It was observed that at high flow rates, when the magnetic Reynolds number Rmf was larger than 0.2 and the Hartmann number was approximately 150, the output of the PMEMFM model began to deviate from linear theory. Note that these results were obtained with a sodium temperature of 250 oC, 5 cm cubed magnets, and a magnet spacing of 10 cm. Therefore, it was determined that the Electric Potential MHD module was not valid for solving the output of a PMEMFM beyond those conditions. Nevertheless, low flow rate calibrations can still be useful for PMEMFM’s used in future systems.

0.006 All Flowrates 0.005 U < 1 m/s

0.004

0.003

Output [V] 0.002

0.001

0.000 0 1 2 3 4 5 6 7 8 U [m/s]

Figure 38: The PMEMFM output solved by FLUENT’s Electrical Potential MHD solver deviates from linear theory at high flowrates. This deviation begins when the magnetic Reynolds number Rmf is approximately 0.2.

3.2 Experimental Setup An experimental flowmeter was installed in one of the University of Wisconsin – Madison’s three sodium technology test facilities. Figure 39 shows a two-dimensional schematic of the test facility and associated diagnostic flow loop. Flow rates in the main flow loop are measured using an experimentally calibrated PMEMFM. A branch off the main flow loop directs sodium to a diagnostic flow loop which is then split into Test Section and Bypass streams. Flow rates in both the Bypass stream and Test Section stream are measured using experimentally calibrated PMEMFMs.

Figure 39: Two-Dimensional schematic of the University of Wisconsin – Madison’s sodium test facility and test section used for PMEMFM testing. 38

An initial calibration constant Ktheory for the Bypass and Test Section reference flowmeters was calculated using the theory described in the previous section. To account for K1’s temperature dependence, the inlet temperature to each flowmeter was continuously monitored and updated in real time. K3’s temperature dependence was neglected after experiments revealed that the magnets stayed at ambient temperature during operation. The calibration constant for each PMEMFM at three different temperatures is given in Table 2.

Table 2: Theoretical calibration constants at three temperatures. o T in C Ktheory in μ-V/m/s 200 846.11 260 841.69 300 838.63

Prior to testing, these reference PMEMFM's were experimentally calibrated. A Micromotion F025A Coriolis flowmeter was installed on the outlet of the test section and the flowrates were varied from 0.1 GPM to 0.4 GPM at 200 oC, 260 oC, and 300 oC [53]. Compared to the Coriolis flowmeter readings, the reference PMEMFM's generally over predicted. This error could be caused by inaccurate magnet spacing, inaccurate magnetic field measurements, or any of the other effects not accounted for in ideal flowmeter theory. To correct for these unaccounted effects, an empirical correction factor, Kcorrected, for each flowmeter was calculated from Figure 40 using a linear model with a least squares regression method [53].

Figure 40: Calibration curves of the theoretical reference PMEMFM output versus the measured Micromotion F025A Coriolis Flowmeter flowrate [53].

These two calibration constants are given in Table 3. With the theoretical correction factor Ktheory and the empirical correction factor Kcorrected a final simplified form to convert the measured potential to volumetric 39

flowrate is given in Equation 74. With these correction constants, it was found that over the range of calibrated flowrates, the PMEMFM relative error never exceeded 3.5% [53].

Table 3: Correction constants for each

Flowmeter Kcorrected Bypass 1.449 Test Section 1.480

V u = K K ∆ M ��x� Equation 74 theory corrected A PMEMFM was constructed to collect experimental data. The two major components of the flowmeter in Figure 41 are the conduit and the magnet assembly. The conduit was constructed from 316L stainless steel. Any components welded to the conduit, such as thermal stand-offs and measurement leads, were also constructed of 316L stainless steel. The thermal stands-offs fix the conduit to a rigid frame. Also fixed to this frame is the magnet assembly. The entire magnet assembly was constructed from non- ferromagnetic materials such as 6061 aluminum and 316L stainless steel. Furthermore, the magnet assembly can test several magnet sizes and magnet face spacing configurations. For example, a maximum magnet cube size of 5 cm can be installed into the magnet holders. With this configuration the magnet face-to-face distance can set to a minimum and maximum of 7.6 cm and 12.7 cm respectively.

Figure 41: Experimental flowmeter setup. The configuration pictured has a pair of 2.5 cm magnet cubes with their faces separated by 10 cm.

Two 316L stainless steel 1/32 inch diameter measurement leads were used to measure the induced voltage. These leads were hand welded to the conduit near the magnet centerlines. The welds were then machined to the same diameter as the measurement leads. The position of the leads relative to the conduit and the magnet centerlines was referenced post-fabrication. Positioning of the leads relative to the conduit centerline is shown in Figure 42 while positioning of the leads relative to the magnet centerline is shown in Figure 43. A sectional view guide is given in Figure 44 for reference.

Figure 42: Measurement lead positioning reference to the centerline of the conduit.

Figure 43: Measurement lead positioning reference to the centerline of the magnets.

Figure 44: Distances to flowmeter grounding points and a sectional diagram. 41

Rare-earth magnets (N42-NdFeB) were used in the experimental work. According to the magnet manufacturer [54], these magnets have a residual magnetism 1.32 T, a max temperature of 80 oC, and a curie temperature of 310 oC. The experimental work tested two cube magnets sizes of 2.5 cm and 5 cm. The centerline strength of each magnet was experimentally measured and checked against theory to verify the accuracy of the manufacturer's specifications. Experimental measurements were made using an F.W. Bell Model 4048 Gauss Meter and theoretical centerline predictions were computed with Equation 75 which is valid for an "M" by "N" cube magnetized through thickness "T" with a residual magnetism of Br [55] [56]. From Figure 45 it was found that the centerline magnetic field strength for each magnet was as expected.

( ) = arctan arctan 𝑟𝑟 2 4 + + 2( + ) 4( + ) + + 𝐶𝐶𝐶𝐶 𝐵𝐵 𝑀𝑀 𝑁𝑁 𝑀𝑀 𝑁𝑁 𝐵𝐵 𝑧𝑧 � 2 2 Equati2 −on 75 2 2 2� 𝜋𝜋 𝑧𝑧√ 𝑧𝑧 𝑀𝑀 𝑁𝑁 𝑇𝑇 𝑧𝑧 � 𝑇𝑇 𝑧𝑧 𝑀𝑀 𝑁𝑁

Figure 45: Experimentally measured and theoretically calculated centerline magnetic field measurements as a function of distance from the face of the magnet.

3.3 Results Four configurations were experimentally studied. In each configuration, fluid velocities were adjusted by the pump and varied in equal increments from 0.1 m/s to 1.7 m/s. Fluid temperatures in the flowmeter were kept with 10 oC of 250 oC and the magnet temperature was kept within 1 oC of ambient temperature.

The four configurations tested used a pair of 2.5 cm cube magnets with a face-to-face spacing of 7.6 cm to 10 cm and a pair of 5 cm cube magnets with a face-to-face spacing of 10 cm and 12.7 cm. Experimental and numerical results for each configuration are plotted in Figure 46.

Figure 46: Experimental and Numerical results of PMEMFM calibration.

In an ideal flowmeter, there will be no offset voltages ΔV0. However, a real flowmeter signal may be corrupted by effects such as electric line noise and the thermoelectric effect. Care was taken to reduce or eliminate signal corruption. Firstly, signals were transmitted in a twisted pair of wires with a grounded sheath. Secondly, a zero-flowrate signal was measured at 250 oC to account for thermoelectric effect in the flowmeter and measurement leads. Lastly, the flowmeter signals were measured in two polarity configurations to eliminate constant offset voltages in the measurement system. With the implementation of these procedures, the data was assumed to follow a linear model y = A x which was fitted using a Linear Regression analysis. Coefficients for each experimental dataset, their associated 95% confidence intervals, and the reduced chi-squared value are shown in Table 4. ∙

Table 4: Experimentally obtained slopes using a = model and Linear Regression analysis. Configuration Slope A 𝑦𝑦 𝐴𝐴 ∙ 𝑥𝑥 reduced-χ2 Magnet Size, Spacing μ-V/m/s 2.5 cm, 7.6 cm 437.19 ± 0.82 1.27 2.5 cm, 10 cm 228.13 ± 0.57 0.84 5 cm, 10 cm 1038.4 ± 1.2 1.53 5 cm, 12.7 cm 675.04 ± 0.98 1.50

Similarly, the numerical data was also assumed to follow a linear model y = A x which was fitted using a Linear Regression analysis. These coefficients, along with theoretically calculated coefficients are given in Table 5 below. ∙

Table 5: Comparison of experimental and numerical slopes to their theoretical values. Configuration Experimental Numerical Theoretical Magnet Size, Slope A Slope A Error Slope A Error Spacing μ-V/m/s μ-V/m/s % μ-V/m/s % 2.5 cm, 7.6 cm 437.19 436.01 0.27 442.24 1.15 2.5 cm, 10 cm 228.12 226.36 0.77 228.14 0.01 5 cm, 10 cm 1038.49 1002.35 3.48 964.65 7.11 5 cm, 12.7 cm 675.04 651.77 3.45 618.02 8.45

3.4 Conclusions During the reporting period, this work developed a numerical method for calibrating a PMEMFM using FLUENT. This method showed that the magnetic field of a pair of unyoked magnets can be calculated analytically and imported into FLUENT. This file can then be used with to predict the output voltage of a PMEMFM using the EM-potential method in FLUENT’s MHD solver. This output voltage was compared to experimental measurements and shown to be accurate up to sodium flowrates of 2 m/s.

This result is significant for two reasons. Firstly, this work shows that PMEMFMs can be numerically calibrated. Therefore, considerable expense can be avoided by calibrating a PMEMFM using a reference flowmeter. Moreover, this model was developed in the ANSYS/FLUNET suit which is used widely across engineering disciplines. Secondly, this work shows that the FLUENT EM-potential method is a valid approach to solving some MHD problems. This suggested that it may also be useful for calculating the p-Q performance of EMIPs.

4 Experimental Measurements of ALIP Performance and the Finite Length Effect 4.1 Technical Approach Literature showed that the Finite Length Effect is caused by two factors. The first factor is a suppression of the externally applied magnetic field caused by an Rmf effect [9] [36] [37]. The second factor is caused by the finite length of the inner and outer cores which cause large inlet and outlet braking forces [28] [33] [36]. These effects will be quantified through pressure-flowrate, efficiency-slip, and pressure pulsation magnitude. Literature has shown that the Finite Length Effect does not greatly impact the pressure- flowrate performance, but does have a negative impact on the efficiency of the pump [36]. Therefore, the effectiveness of a pump entrance or exit configuration will be evaluated on the change in measured efficiency. Literature has also showed that the Finite Length Effect is manifested as a pressure pulsation [36]. Therefore, the effectiveness of a pump entrance or exit configuration will be evaluated on the relative magnitude of this pulsation.

The first metric to quantify ALIP performance is pressure-flowrate (p-Q) performance. Pressure is defined as the measured pressure difference between two points across a pump. Note that this definition also includes frictional losses between those two points. Therefore, the developed EM pressure will be reduced by frictional pressure losses as shown in Equation 76.

p = p p Equation 76 Δ Δ EM − Δ loss Flowrate is defined as the mass-averaged flowrate through the ALIP. Note that this definition of flowrate does not account for fluid density changes across the pump. However, the density may change considerably from the pump inlet to the pump outlet due to resistive heating by the induced currents. Since the inlet temperature will be held at a known constant value, the flowrate will be referenced here. But, since the main flowmeter is located downstream from the pump, the flowrates will be to be corrected for any density changes using Equation 77.

(T ) Q = Q ( ) PMEMFMT inlet ρ Equation 77 PMEMFM ρ inlet While p-Q performance is useful, it does not fully characterize the pump. The second metric is efficiency- slip (η-sm) performance. Quantitatively, η is defined by Equation 78 which is the ratio of mechanical power, WM, to electrical power, WE.

W = WM Equationη 78 E Mechanical power is defined in Equation 79 and quantifies the energy converted to useful pumping. Finding the mechanical power is straightforward since it uses the data collected during p-Q measurements. WE is defined as the total electrical power drawn by the pump. This quantity can be directly measured for the ALIP and can be calculated by speed and torque measurements in the PMIP.

W = p Q Equation 79 M ∆ EM Mean slip is defined in Equation 80 where U0 is the mass averaged velocity and UB is the synchronous velocity defined as U = 2 f where τ is the pole-pitch and f is the frequency. Calculation of slip is relatively straightforward since the mass averaged velocity U0 can be calculated from flowrate B measurements and the synchronousτ velocity UB can be calculated by knowing the frequency and pole- pitch of the pump.

U s = 1 U0 mEquation− 80 B Another metric used to quantify ALIP performance is pressure pulsation magnitude. As discussed in the literature review, pressure pulsations were found to be a manifestation of the pulsating braking force caused by the finite length of the stator. Thus, a reduction of pressure pulsation magnitude is taken to be a reduction in the impact of the finite length effect. In this work, the total pressure measured at a point is assumed to take the form of Equation 81 where pm is the mean absolute pressure and δp is the deviation from the absolute. The magnitude of the pressure pulsations δp are then normalized by the mean pressure output of the pump p shown in Equation 82.

Δ p = p + p Equation 81 m δ p p = p δ Equation�δ�� 82 ∆

Performance testing will be split into ten campaigns. Campaign 1 and Campaign 2 are shown in Table 6. These campaigns will form the baseline p-Q and η-sm data that will be used to evaluate the effectiveness of the pump configurations in Campaign 3 through Campaign 10. Note that Campaign 1 will make baseline pressure pulsation measurements while Campaign 2 will not. This is due to the temperature limitations of the high frequency pressure transducers. Campaign 1 will also verify the dependence of pressure pulsation frequency on pump supply frequency. This aspect will not be revisited in the following test campaigns.

Table 6: ALIP Testing Campaign 1 and 2. These campaigns will form the baseline p-Q and η-sm data that will be used to evaluate the effectiveness of the modified pump configurations. Test Measurement Temperature Supply Frequency Campaign (Y/N) [oC] Voltage [V] [Hz] �𝛅𝛅𝛅𝛅�� 100 140 120 140 1 Y 200 100 210 120 140 100 140 120 140 2 N 400 100 210 120 140

Table 7 summarizes Campaign 3, 4, and 5 which will test coil grading Configuration 1, 1-1, and 1-2 respectively. Similarly, Table 8 summarizes Campaign 6, 7, and 8 which will test coil grading Configuration 2, 2-1, and 2-2 respectively. Note that the coil configurations are described in Section 4.5.2. All measurements will be performed at 200 oC so that pressure pulsations can be measured by the high frequency pressure instrumentation. Only a single frequency is tested since the pressure pulsation frequency dependence will be deduced from Campaign 1.

Table 7: Testing campaigns for coil grading Configurations 1, 1-1, and 1-2. More details on the coil grading can be found in Section 4.5.2. Measurements are performed at 200 oC so that pressure pulsations can be measured by the high frequency pressure instrumentation. Only a single supply frequency is studied since the frequency dependence will be deduced from Campaign 1. Test Coil Temperature Frequency Supply Voltage Campaign Configuration [oC] [Hz] [V] 140 3 1 200 120 210 140 4 1-1 200 120 210 140 5 1-2 200 120 210

Table 8: Testing campaigns for coil grading Configurations 2, 2-1, and 2-2. More details on the coil grading can be found in Section 4.5.2. Measurements are performed at 200 oC so that pressure pulsations can be measured by the high frequency pressure instrumentation. Only a single supply frequency is studied since the frequency dependence will be deduced from Campaign 1. Test Coil Temperature Frequency Supply Voltage Campaign Configuration [oC] [Hz] [V] 140 6 1 200 120 210 140 7 1-1 200 120 210 140 8 1-2 200 120 210

Table 9 summarizes Campaign 8 and Campaign 9 which will test the Entrance Coil Shifting and Exit Coil Shifting. Note that the shifting configurations are described in Section 4.5.2. All measurements will be performed at 200 oC so that pressure pulsations can be measured by the high frequency pressure instrumentation. Only a single frequency is tested since the pressure pulsation frequency dependence will be deduced from Campaign 1.

Table 9: Testing campaigns for Entrance and Exit coil shifting configurations. More details on the coil shifting can be found in Section 4.5.2. Measurements are performed at 200 oC so that pressure pulsations can be measured by the high frequency pressure instrumentation. Test Shifting Temperature Frequency Supply Voltage Campaign Configuration [oC] [Hz] [V] 140 9 Entrance 200 120 210 140 10 Exit 200 120 210

4.2 ALIP Specifications The experimental ALIP in Figure 47 is a custom designed and built by CMI-Novacast. Nominal operating conditions of the pump are shown in Table 10 while some relevant geometric information about the pump is provided in Table 11.

Figure 47: Custom designed and built CMI-Novacast model LA125 ALIP used in experimental testing.

Table 10: Rated conditions for CMI-Novacast’s LA125 ALIP. Property Value Unit Maximum Voltage 480 V Maximum Current 40 A Frequency 120 Hz Maximum Temperature 600 oC Maximum System Pressure 15 bar

Table 11: Some relevant geometric properties for CMI-Novacast’s LA125 ALIP. Property Value Unit Number of Coils 12 - Poles 4 - Stator Length 300 mm Channel Width 3.9 mm Mean Channel Radius 19.4 mm

At the request of UW-Madison, CMI-Novacast provided two modifications to their standard ALIP. Firstly, the pump is set on a pair of wheels and rails to allow the outer-stator and polyphase coils to be repositioned relative to the inner-stator core. With the inner-stator fixed, the pump coils can be positioned 75 mm to the right or left of center. These three positions are shown as a sketch in Figure 48.

Figure 48: Qualitative sketch of coil shifting configurations of the experimental ALIP.

The second modification from the standard LA125 are taps on coils one, two, and three as well as coils ten, eleven, and twelve. Figure 49 shows a two-dimensional sketch of a tapped coils. Taps are located on the 1st, 20th, 31st, or 80th turn to allow for variable coil grading at the pump inlet and outlet. These taps will reduce the number of turns through which the electric current passes, effectively reducing the applied magnetic field near that coil.

Figure 49: Sketch of the tapped coils installed in the first three and last three coil locations. Each of the six coils have taps at the 1st, 20th, 31st, or 80th turn.

A cut-away of the ALIP is shown in Figure 50. This figure shows the position and number of coils in the pump. Note that Coil 1 is the first coil on the left near the entrance and Coil 12 is the last coil on the right near the exit. The first three, Coil 1-3, and last three, Coil 10-12, have the taps shown in Figure 11. Therefore, the pump can be graded over a full pole-pitch.

Figure 50: Cut away of the experimental ALIP showing each of the 12 polyphase coils in the pump. Note that the first three coils and last three coils are tapped.

4.3 Experimental Facility A medium-scale-component sodium test facility was constructed with the purpose of measuring ALIP performance. The partially completed facility in Figure 51 is designed to accommodate heat exchanger, flowmeter, and pump testing at temperatures of 600 oC and flowrates of 8.5 m3/hr. The main components 50

of the facility include a heater bank, large experimental taps, a Venturimeter that doubles as a throttle, a cooler, pressure taps, an EM flowmeter, and the experimental ALIP.

Figure 51: The Medium-Scale-Component Sodium Test Facility constructed to measure the performance of the experimental ALIP. Note that the reservoir in the lower left of the frame will be placed in its own catch-pan.

A two-dimensional sketch of the facility in Figure 52 notes the major components. The heater bank and large experimental taps will not be used directly for pump testing. However, they are included to increase system flexibility for future experiments. In total, the heater bank consist of four radiant-mode cartridge heaters which can provide a total power of 25 kW. This bank will be used to heat the sodium flow for heat exchanger testing which will tapped into the facility through the large experimental taps.

The cooler is a simple counter-flow, tube-and-tube heat exchanger which uses air. The cooler is sized to remove the maximum rated power of the ALIP at 3.4 kW, assuming the sodium flow is at 8.5 m3/hr, 200 oC, and an inlet air temperature of 21 oC. The cooler will be used to keep the pump inlet temperature constant.

Pressure taps are located at the inlet and outlet of the pump to measure the developed pressure difference. These taps can accommodate two forms of pressure measurement instrumentation. The first form is mean pressure and mean pressure difference measurements. The second form is high frequency pressure pulsation measurements. Flowrate measurements will be made with a calibrated Electromagnetic flowmeter located downstream of the pump. This instrument will be the primary device used to measure pump flowrate.

Lastly, the facility has a custom designed and manufactured Venturimeter. This instrument forms a dual purpose. Firstly, the Venturimeter provides secondary flowrate measurements to the main electromagnetic flowmeter. Secondly, the Venturimeter acts as a flow throttle when a special insert is placed in the throat. More details of the Venturimeter and throttle are provided in the following instrumentation section. This pathway was chosen versus a traditional valve primarily because a wide range of flow-coefficients can be achieved with this configuration.

Figure 52: Two-dimensional sketch of the Medium-Scale-Component Sodium Test Facility constructed for ALIP performance testing.

4.4 Instrumentation 4.4.1 Permanent Magnet Electromagnet Flowmeter Primary flowrate measurements will use a calibrated Permanent Magnet Electromagnet Flowmeter (PMEMFM) similar to the experimental one described in Section 3.2. Figure 53 shows a photo of a completed and insulated PMEMFM in the Small-Scale-Component Sodium Test Facility. This flowmeter uses a pair of grade N42 NdFeB cube magnets with a side length of 5 cm. These magnets are mounted in aluminum holders which are attached to a mounting frame by threaded rods. The magnets are then separated by 13 cm about the centerline of the flow conduit. Two 1.6 mm diameter 316L stainless steel measurements leads are welded diametrically to the flow conduit. Note that the measurements leads are orthogonal to the magnets. Both the measurement leads and magnets are mutually orthogonal to the flow conduit.

Figure 53: PMEMFM used to measure sodium flowrates in a 27 mm inner diameter conduit.

Table 12 presents the relevant properties of the PMEMFM that will be used in the experimental ALIP facility. Using these properties and the theory in Section 2.1, a correlation coefficient relating the flowmeter signal to mean velocity can be calculated. This coefficient is also presented in Table 12.

Table 12: Relevant properties of PMEMFM used in experimental ALIP facility. Property Units Value Magnet Type - Grade N42 NdFeB Magnet Length, Width, Height cm 5 Magnet Spacing cm 13 Bmin kG 0.7672 Conduit Material - 316SS Inner Diameter cm 26.6 Outer Diameter cm 33.4 Correlation Coefficient (250 oC) μ-V/m/s 1857

To ensure accuracy of the PMEMFM reading, the flowmeter will be calibrated using a reference flowmeter. In this case, the calibration will use a Foxboro M83F vortex shedding flowmeter. The reported relative error of the measured flowrate is 2%. By the time of this report, calibration curves and error quantification of the PMEMFM were not generated due to construction delays with the ALIP test facility.

4.4.2 Venturimeter Installed in UW’s ALIP test loop is a Venturimeter. The intention of this Venturimeter is to provide secondary flow measurements to the Electromagnetic Flowmeter and to throttle the loop for pump performance curves. Therefore, this section will focus on the methodology behind the Venturimeter calibration, pressure recovery analysis, as well as the instrumentation and associated measurement errors. An additional discussion is devoted to Venturimeter and flow coefficient for the standard and throttled- Venturi.

A Venturimeter correlates mean volumetric flowrate to a pressure drop induced by a reduction of flow area. Ideally, for inviscid, incompressible, and irrotational flow at a constant temperature, this is given by the Bernoulli Equation in Equation 83 where gravitational effects are ignored. Here, ρ is the fluid density, Q is the volumetric flow rate, while At and Ae are the cross-sectional areas of the throat and entrance respectively.

1 1 1 p = Q 2 A A 2 ∆ Equationρ � 832 − 2� t e Calibration curves for a Venturimeter are generated by measuring mass flowrate and a differential pressure between the throat and entrance regions. These measurements are non-dimensionalized in the form of a Discharge Coefficient ‘C’ and a Reynolds Number ‘Re’ evaluated in the entrance region. The Discharge Coefficient is defined in Equation 84 where m is the mass flowrate, β is the throat-to-entrance dimeter ratio, and ΔP is the measured pressure drop between the entrance and throat regions. ‘C’ represents the non-dimensional pressure drop between thė entrance and throat of the Venturimeter. The Reynolds Number evaluated in the entrance region is defined in Equation 85 where De is the entrance diameter, and μ is the fluid viscosity. ‘Re’ defines the ratio of inertial to viscous flow forces. Effectively, the Reynolds Number is a non-dimensional flowrate.

m 1 C = A 2 P 4 ̇ − β Equationt � 84 ∆ ρ D m Re = Ae Equation 85 ̇ eμ In addition to measuring flowrates, the ALIP Loop Venturimeter will also double as a throttling valve. Valves are characterized by a quantity called a Flow Coefficient or ‘Cv’ value shown in Equation 86 where SG is the specific gravity of the fluid and ΔP is the measured pressure drop across the valve. The Flow Coefficient describes the volume of water that will flow through the valve with a pressure drop of 1 psi.

SG Cv = Q P Equation� 86 Δ For a Venturimeter, the throat is fixed in diameter. Therefore, a Flow Coefficient can be calculated from pressure recovery data. What is more interesting in the ALIP Loop Venturimeter is the Flow Coefficient with the throat-insert installed. In this case, the pump speed is fixed while the flowrate and pressure drop are measured as a function of throat-insert retraction. For simplicity, this will be given as the “Number of Turns”.

UW’s ALIP test loop Venutrimeter was constructed following the specifications provided in the ISO 5167-4 2003(3) standard [57]. Final dimensions of the Venturimeter in imperial units are provided in Figure 54.

Figure 54: As-fabricated dimensions of the Sodium Pump Loop (Loop 4) Venturimeter with entrance and exit regions. All dimensions were made in accordance to the ISO 5167-4 2003(3) specifications [57]. 54

The Venturimeter’s internal features were fabricated by Electron Discharge Machining (EDM). This process was chosen for its superior accuracy and surface finish when machining complex internal features such as the internally tapered bores. However, EDM poses geometric constraints. In this case, the internal bore features could not be machined from a single piece since the EDM wire cutting the convergent section would “clip” the divergent section. Thus, the Ventuirmeter needed to be machined in the two pieces shown in Figure 55. Boss and receiver ends were machined into either end of both pieces. These features were indicated off the internal bores which guaranteed that all features would be concentric to a common axis of rotation.

It was hoped that EDM would produce a highly accurate product in a shorter period compared to what could be accomplished in-house. Unfortunately, both the accuracy and time were not met. Specifically, the throat diameter on the divergent section was not machined to the specified tolerance and was found to be non-concentric. Therefore, the throat diameter needed to be post-machined such that no lip existed once the convergent-throat and divergent sections were attached. Originally, the throat diameter was specified as 11.43 +/- 0.0127 mm. However, because of manufacturing errors the new throat diameter was machined to 11.59 +/- 0.0254 mm. Figure 54 reflects the dimensional changes caused by this error in imperial units.

Figure 55: Finished two-piece Venturimeter produced by Electron Discharge Machining (EDM). On the left is the convergent section. Note the pressure tapping at the bottom of the piece. On the right is the divergent section. Both pieces received boss-and- receiver features which were referenced off the internal features. These features guaranteed concentric alignment of both pieces to a common axis of rotation.

In addition to serving as a flow measuring device, the Venutirmeter will also double as a throttle. Throttling will be achieved by restricting the flow area into the throat. This is accomplished by inserting a round, ported piece into the Venturimeter throat. As this piece is retracted, more porting and more flow area will be exposed, increasing the total flowrate. This ported piece is called a ‘Throat-Insert’. A two- dimensional drawing of the throat-insert is shown in Figure 56.

The throat-insert shown in Figure 57 was manufactured to have a sliding fit in the throat. Roughly 0.18 mm of clearance exists between the diameter of the throat-insert and the throat itself. This clearance will prevent the throat-insert from becoming bound and stuck. Furthermore, it will also prevent damage to the throat walls which may impact the Venturimeter calibration. While the throat-insert to throat fit has 55

clearance, the cone is tapered to match the angle of the convergent section. Therefore, when fully inserted, the cone will seat in the convergent section and block all flow.

Figure 56: Two-dimensional drawing of the throat-insert and two positions. On top is the throat-insert itself. In the middle is the 'fully closed' position of the throat-insert while on the bottom is the 'fully open' position.

Figure 57: Finished throat-insert.

The experimental loop and installed Venturimeter are shown in Figure 58. Note that for calibration, the Venturimeter was oriented horizontally despite the experimental orientation being vertical. Water was used as a surrogate fluid to calibrate the Venturimeter. Compared to sodium, water is easier to work with since it does not present a chemical hazard. Therefore, the water loop simplifies testing and troubleshooting of the Venturimeter prior to final installation in the experimental sodium loop. 56

Figure 58: Water loop used for calibration of the Sodium Pump Loop (Loop 4) Venturimeter. This loop can achieve a maximum volumetric flowrate of 5.9 m3/hr at 30 oC.

Pressure differentials across the Venturimeter were measured with a Siemens SitransP pressure transducer. The model used during calibration has a measurement span of 16 mbar to 1600 mbar. Two effects impact the accuracy of pressure measurements. The first is a “Linear Characteristic” which is given in Equation 87. Here, r is the ratio between the maximum span and the set span shown in Equation 88. The total span of the instrument is 1600 mbar and the set span used for the majority of testing was 675 mbar resulting in an r value of 2.37.

, = 0.001 for r 10 Equation 87 σ∆p lin ≤ total span r = set span Equation 88

A temperature effect must also be considered which is given in Equation 89. The total accuracy of the differential pressure measurements can be found using Equation 90. For a total span of 1600 mbar and a set span of 675 mbar, the total accuracy of the pressure measurement was found to be approximately 0.45% of the differential pressure measurement.

, = 0.001 r + 0.002 Equation 89 σ∆p T ∙

= , + , 2 2 σ∆p Equation�σ∆p lin 90 σ∆p T

Mass flowrates were measured with a Foxboro model CFS20 Coriolis Mass Flowmeter. The measurement span of the 25 mm nominal flowtube size instrument ranged from 1.8 kg/min to 180 kg/min. Note that the setup in Figure 58 can reach maximum volumetric flowrates of 5.9 m3/hr at 30 oC. The total accuracy of the measured liquid mass flowrate is given in Equation 91 where Zi is the “zero 57

instability” of the instrument, and m is the mass flowrate. The Zi value for the 25 mm nominal flowtube sized instrument was specified as 0.00907 kg/min by the manufacturer. ̇ Zi[kg/min] = 0.001 + m[kg min] σṁ Equation 91 ̇ ⁄ Bulk temperature was measured by a single K-type thermocouple near the pump exit. The accuracy of this measurement was assumed to be 0.1 oC. Static pressure was also measured near the outlet of the pump by a manual gauge. Pressure measurements ranged from 0.70 bar to 3.80 bar. Fluid properties were evaluated assuming a nominal pressure of 2.25 bar with an uncertainty of 1.55 bar. It will be shown that the magnitude of the static pressure uncertainty will be negligible in the error analysis.

The experimental calibration curves and their associated errors are shown in Figure 59. Note that the measured water Reynolds Numbers are out of the range of the ISO standard. For the ALIP Loop Venturimeter, the applicable Reynolds number range is between 2 x 105 to 1 x 106 [57]. Under these conditions, the discharge coefficient is specified with a constant value of 0.995. However, since the testing conditions are out of range, the measured discharge coefficient does not take on this constant value.

0.99

0.98

0.97

0.96 ] - 0.95 C [ 0.94

0.93 Dataset 2 0.92 Dataset 3 0.91 0 10 20 30 40 50 60 70 80 90 100 110 Re [-] x 103

Figure 59: Discharge Coefficient 'C' versus Reynolds number 'Re'. Note that the Reynolds numbers are lower than the specified values in the ISO standard and therefore 'C' does not take on the predicted constant value.

Uncertainties must be considered in the four principle measurements of temperature, differential pressure, absolute pressure, and mass flowrate. Also considered were uncertainties in geometric quantities such as the inlet diameter De and throat diameter Dt. The absolute error of both geometric quantities was taken to be 0.0254 mm. These uncertainties were propagated through the calculated quantities of volumetric flowrate, Discharge Coefficient, and Reynolds number. The results of the uncertainty analysis are 58

presented in Table 13. Note that the absolute pressure measurement had a negligible impact on the uncertainty of these quantities.

Table 13: Relative contributions of the measured variable uncertainties in the calculated calibration curves. Calculated Quantity Measured Quantity Units Q [m3/s] Re [-] C [-] De [m] 0 0 0 Dt [m] 0 0 0.76 m [kg/s] 0.96 0 0.05 ΔP [Pa] 0 0 0.19 P [Pa] 0 0 0 T [oC] 0.04 1 0

Plotting the Discharge Coefficient as a function of Re-0.4 produced Figure 60. Two distinct linear regions appear near Re = 60,000, which corresponds to a water volumetric flowrate of roughly 8.7 GPM. This change in slope may be explained by the short entrance length of the Venturimeter. At 0.57 m long, fully developed turbulent flow is only ensured up to Reynolds numbers of 10,000. Beyond that, the flow is not guaranteed to be fully developed, which may explain the shift in slopes. Using the two fits shown in Figure 60, a conservative accuracy of 3% was calculated over the full range of tested Reynolds numbers.

0.99 y = -7.8846x + 1.0626 0.98 R² = 0.975 y = -1.8873x + 0.9905 0.97 R² = 0.9712 ] - 0.96 C [

0.95

0.94

0.93 0.005 0.01 0.015 0.02 0.025 0.03 Re-0.4 [-]

Figure 60: Discharge Coefficient 'C' versus Re-0.4. The change in behavior as a function of Reynolds number can be distinctly observed. An arbitrary cutoff of Re = 60,000 was chosen. Using the two fits, a conservative flowrate error estimate of 3% was calculated.

Pressure recovery data were also measured during the calibration. Due to the construction of the Venutirmeter, the closest outlet pressure tapping was located 0.92 m from the end of the divergent section. These frictional losses were corrected using the Darcy Friction Factor. A nominal absolute roughness of 0.0075 mm was used in the calculation of the friction factor. Since the actual absolute roughness was unknown, the uncertainty was taken to be +/- 0.0075 mm.

Figure 61 shows that for mass flowrate above approximately 0.5 kg/s, the pressure recovery is roughly a constant value of 87%. At low flowrates, below 0.5 kg/s, or roughly 8 GPM in water, the non-recoverable pressure losses increased markedly. In this range however, the Bernoulli pressure is low in magnitude meaning that the total pressure losses are also low.

0.35

0.3

0.25

0.2 ] - ξ [ 0.15

0.1

0.05 Dataset 1 Dataset 2 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 m [kg/s]

Figure 61: Pressure recovery coefficient ξ as a function of mass flowrate.

Like the calibration data, an uncertainty analysis was carried out on the pressure recovery data. In addition to considering measurement error in temperature, differential pressure, absolute pressure, mass flowrate measurements, the analysis also considered uncertainties in the inlet diameter De and throat diameter Dt. As stated earlier, the uncertainty in the absolute roughness of the pipe was assumed to be 0.0075 mm with a nominal value of 0.0075 mm. Table 14 shows the results of the uncertainty analysis.

Table 14: Relative contributions for the calculated uncertainties in the pressure recovery curves. Calculated Quantity 3 Measured Quantity Units Q [m /s] Re [-] ΔPloss [Pa] ξ [-] De [m] 0 0 0.02 0 Dt [m] 0 0 0 0.47 ε [m] 0 0 0.96 0.31 m [kg/s] 0.94 0 0 0.04 ΔP [Pa] 0 0 0 0.17 P [Pa] 0 0 0 0 T [oC] 0.06 1 0.02 0

Figure 62 plots the Flow Coefficient Cv in imperial units as a function of mass flowrate. Over most of the range of mass flowrates, the Cv take an approximate value of 15.5 GPM/psi0.5. Therefore, the Ventuirmeter will allow roughly 15.5 GPM of flow through it with a 1 psi pressure drop.

Next, Flow Coefficient data were collected with the throat-insert was installed. Flowrate and pressure drop were measured as a function of throat-inset retraction, given in “Number of Turns”, and Cv values were calculated for each measurement. Figure 63 shows that the throat-insert Cv reaches a maximum value of approximately 5 GPM/psi0.5. Note that this is a significant drop compared to the calculated flow coefficient of the Venturimeter. Also note that there is a gap in Flow Coefficients from 5 GPM/psi0.5 to 10 GPM/psi0.5. This suggests that an additional throat-inset with additional porting may need to be constructed to reach the full range of pressure-flowrate output of the experimental pump.

] 14 0.5 13

Cv [GPM/psi 11

10 Dataset 1 9 Dataset 2 8 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 m [kg/s]

Figure 62: Flow Coefficient 'Cv' as a function of mass flowrate. Over most mass flowrates, the flow coefficient takes on a value of roughly 15.5 GPM/psi0.5.

5.5 5 4.5 4 ]

0.5 3.5 3 2.5 2 Cv [GPM/psi 1.5 1 0.5 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Nturn [-]

Figure 63: Flow Coefficient 'Cv' with the throat-insert installed. These data are plotted against the throat-inset retraction which is given in 'Number of Turns'.

4.4.3 Magnetic Field Single components of the ALIP’s magnetic field were measured using air-cored pick-up coils shown in Figure 64. Pick-up coils have several advantages over Hall-Sensors. Firstly, depending on the wire coating, pick-up coils can tolerate higher temperatures than Hall-Sensors. For example, coils constructed using enamel coated wire can be rated to roughly 200 oC and a coils constructed using a special ceramic coated wire can be rated to roughly 535 oC, depending on the temperature rating of the other construction materials. In contrast, most commercial Hall-Sensors have a maximum operating temperature of 150 oC or lower. Secondly, air-cored pick-up coils can measure the large magnitude magnetic fields that were expected in some pump configurations. Commercial Hall-Sensors with the required temperature ratings were found to be rated to 1,000 Gauss or 0.1 T. However, the pump magnetic field magnitudes were expected to exceed this nominal value. Lastly, pick-up coils are cost-effective and simple to construct.

One disadvantage of a flat pick-up coil is its ability to only measure a single component of the magnetic field. However, this was not seen as a major disadvantage as only a single component of the magnetic field was of interest for this work. Another disadvantage of a pick-up coil is that it averages the magnetic field magnitude over the coil area. This effect was addressed by reducing the dimensions of the pick-up coil to be much smaller than the dimensions of the pump.

Figure 64: Air-cored pick-up coil used to measure the radial magnetic field distribution in the ALIP channel.

The pick-up coil shown in Figure 64 was constructed using enamel coated magnet wire following the dimensions in Table 15. The wire was wrapped around a central core 150 times. Then, the outer diameter was recorded. After twisting the leads, the coil was removed from the central core. The core and coil were then filled with an adhesive to increase the coil’s integrity.

Table 15: Relevant dimensions of the air-cored pick-up coil in Figure 52. Property Units Value Inner Diameter mm 1.3 Outer Diameter mm 5 Height mm 0.635 Number of Turns - 150 Wire Gauge (AWG) - 38 Nominal Wire Diameter mm 0.127

Each pick-up coil was calibrated under a known magnetic field magnitude and frequency. This was accomplished by using the Disc-Type PMIP that will be discussed in Section 5. In this procedure, the probe tip of a F.W. Bell Model 4048 Gauss Meter was fixed immediately next to the pick-up coil. Note that the dimensions of the Gauss Meter probe tip are roughly the same as the pick-up coil. These two instruments were then inserted into the PMIP magnetic field by a jig. Using the jig, the magnetic field magnitude was changed by changing the location of the instruments in the pump. Furthermore, the magnetic field frequency was changed by adjusting the pump speed. Four frequencies of 18 Hz, 36 Hz, 54 Hz, and 72 Hz were tested.

Peak values of the magnetic field were measured by the Gauss Meter while the time-varying voltage signals from the pick-up coil were recorded with an oscilloscope. The voltage data was then filtered by an FFT to extract the primary magnetic field frequency and then integrating Equation 92. Equation 92 is Faraday’s Law which relates the induced voltage in a coil of wire to the magnetic field B’s time-rate-of- change.

B V = nA t Equation 92 ∂ − eff ∂ The factor nA in Equation 92 is the product of the number of coil turns n by the effective area Aeff. Here, the effective area is defined in Equation 93 where Do and Di are the outer and inner diameter of the coil 63

respectively. Equation 93 is simply the area of the coil using an average between the outer and inner diameter. Using the properties given in Table 15, a factor nA for the coil in Figure 64 was found to be 13.3 cm2.

A = (D + D ) 16 2 eff Equationπ o93 i

From the inverted pick-up coil data, peaks were extracted and averaged at each frequency. These data were assumed to be normally distributed and the error of these measurements was assumed to be the standard deviation. Then, the peak magnitude Gauss-Meter measurements were plotted against the peak pick-up coil as shown in Figure 65. The relative error of the Gauss-Meter was reported to be 3%. A calibration curve was fitted to these data using a linear regression method assuming a y = A x model. The correction coefficient A was then used to correct the pick-up coil output. ∙ 4.5 B = 13.638 Bcoil 4 3.5 3 2.5 2 1.5 Gauss Meter [kG] 1 0.5 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Pick-up Coil [kG]

Figure 65: Calibration curve for the pick-up coil in Figure 52 used to measure the radial magnetic field component in the ALIP channel.

4.4.4 Differential Pressure and Pressure Pulsations Differential pressure measurements will be made with a set of Rosemount 3051T pressure transducers with oil-filled diaphragms. This device has a performance error which is reported to be 0.0014 the span of the sensor. In this case, the total sensor span is roughly 414,000 Pa. In addition to the performance error, there will be a resolution error caused by the pressure diaphragms in the stand-offs. This error is estimated to be approximately ± inH20 or 500 Pa. Using these values, the total absolute error can be calculated using Equation 94 and is estimated to be roughly 766 Pa.

[Pa] = (0.0014 total span [Pa]) + 500 [Pa] Equation 94 2 2 σ∆p � ∙ 64

High frequency pressure measurements will be made using PCB Piezotronics Model 112A05 piezoeletric pressure transducers shown in Figure 66. Relevant characteristics of this instrument are given in Table 16. The frequency of the pressure pulsations are expected to be on the order of 200 Hz. The rise time in Table 16 is sufficiently short that these pressure pulsations should be resolved.

Figure 66: Photo of PCB Model 112A05 pressure transducer.

Table 16: Relevant properties of PCB Model 112A05 pressure transducer. Property Value Units o Tmax 315 C Resolution 28 Pa Resonant Frequency 200 kHz Rise Time 2 μ-s

The pressure transducer will be recessed in a thermal stand-off. A sketch of the thermal stand-offs and a photo of the machined mount is shown in Figure 67. PCB noted that a channel length may limit the frequency range of the sensor. Equation 95 can be used to calculate the resonant frequency given sound speed c and channel length L. Thus, thus for a 7 cm channel length the resonant frequency 29 kHz which is approximately a 12 μ-s rise time. This rise time is still well below the expected pressure pulsation magnitude.

Figure 67: Sketch of thermal stand-offs for the PCB high frequency pressure sensors. Note that the sensor is recessed from the flow roughly 8.6 cm. Also shown are the manufactured mounts for the PCB sensors. These mounts were machined into a Cone- and-Thread fitting plug. 65

1 c F = 4 L Equationres 95

Two PCB model 112A05 pressure transducers were initially calibrated in water at high pressure to verify that they were functioning properly. This initial calibration is shown in Figure 68. Two fits were performed using a linear regression analysis assuming a y = A x + B model. While this calibration showed the sensor were operating properly, the pressure pulsations in the experimental ALIP are expected to be on the order of 0.15 bar. Therefore, a calibration at lower ∙pressures will need to be performed.

7 Serial 28064 y = 0.0279x + 0.2055 6 Serial 28054 y = 0.0245x - 0.0557

5 ]

DC 4

3 Signal [V Signal 2

0 0 25 50 75 100 125 150 175 200 225 Pressure [bar]

Figure 68: Initial calibration of two PCB model 112A05 pressure transducers in water. Note that a new calibration will need to be performed at lower pressures since the expected ALIP pressure pulsation magnitude is on the order of 0.15 bar.

The PCB pressure sensors use a brass ring to the seal the sensor in the mount. To address concerns about material compatibility of the seal ring, the sensor seals were tested in sodium prior to installment in the experimental facility. Figure 69 shows the vessel fabricated stand and mount. A sensor was installed in the vessel and 150 oC sodium was in contact with the sensor for 25 hours at 1 bar gauge pressure. No visible corrosion was observed on the seal ring. These conditions are not fully representative of the actual testing conditions which are expected to reach 200 oC and 3.5 bar or higher. Therefore, a test at higher temperatures and pressure will need to be conducted.

Figure 69: Test vessel used to evaluate the seal ring compatibility in sodium. On the left is the mount for the pressure sensor while on the right is the stand after heating and insulation.

4.4.5 Frequency, Power, and Efficiency Pump efficiency is defined as a key metric to characterize the performance of a pump and to quantify the impact of Edge Effects. However, literature has shown that the expected changes in pump efficiency may be smaller than 10%. Thus, it is important that the reported error of the calculated efficiency be much less than that for the results to have any practical meaning.

The definition of efficiency used in this work is given in Equation 96 where p is the measured pressure difference between the pump inlet and outlet, Q is the mass averaged flowrate, and WE is the input power. In the case of the ALIP, WE can be measured directly. Each of the measurementsΔ in Equation 96 has an associated error which will need to be propagated through the calculation.

p Q = W ∆ Equationη 96 E At the time of this report, power measurements were not completed in the ALIP. However, prior to testing the measurement errors were estimated in order to verify that the expected changes in efficiency could be measured. The electrical input power will be measured using a Power Analyzer. A power analyzer directly measures the voltage and current waveforms and internally performs the integration to calculate power. Therefore, the uncertainty of the power measurements is quite low. For example, a Yokogawa WT300E Power Analyzer reports an accuracy of 0.1% the reading plus 0.05% of the instrument range.

In practice the full current of the machine is not directly passed through the power analyzer. Rather, current transducers are used to send a smaller current signal to the power analyzer. The current transducers have an uncertainty associated with the output signal sent to the power analyzer. For example, a Flex-Core model ACT current transducers has a reported accuracy of 0.25% of the Full Scale reading. Since actual current and voltage measurements are unknown, the power measurement uncertainty is assumed to have a conservative accuracy of 0.3% of the power reading plus 0.05% of the instrument range.

Propagating these errors through Equation 96 results in a roughly 2% to 3% error in the efficiency calculation. This is in an acceptable range for the expected change in efficiency. Table 17 reports the relative contribution of each of the measurement errors. One can see that the pressure differentials dominate the efficiency uncertainty.

Table 17: Relative contributions of each measured quantities uncertainty towards the calculated efficiency. Measured Calculated η at Units Quantity 140 V, 175 V, 210 V p Pa 0.99 Q m3/s 0.01 Δ WE W 0

4.5 Results 4.5.1 Duct Pressure Losses Pressure differential measurements across the pump were described in the Section 4.1 as a primary metric to quantify Edge Effects in this work. These measurements account for both the developed pressure and frictional losses between those two points. However, the analytical solutions of an ideal EMIP described in the literature review only calculate the developed pressure. Therefore, comparisons of theoretical EMIP calculations and experimental pressure measurements can only be done with knowledge of pressure losses in the pump conduit.

Since the exact geometry of the pump torpedo and conduit were unknown, the frictional pressure losses were characterized experimentally. Like the Venturimeter calibration described in Section 4.4.2, the pressure losses in the ALIP were measured using water. To ensure fully developed turbulent flow over the tested range of flowrates, an entrance length was attached to the inlet of the pump as shown in Figure 70. Also shown in Figure 70 are the entrance length and relative positioning of the pressure taps. Note that these taps are in the same position on the experimental ALIP test facility discussed in Section 4.3.

Figure 70: Entrance length and relative positioning of the pressure taps on the experimental ALIP. Note that the pressure taps are in the same location as the experimental ALIP test facility.

Flowrate and pressure measurements were completed using the same methods described in the Venturimeter calibration in Section 4.4.2. Figure 71 plots the frictional pressure losses as a function of mass flowrate. Then, a curve was fitted to these data using a linear regression analysis and assuming a y = A x + B x + C model. 2 ∙ ∙ 0.16 Δp[bar] = 0.0446 m[kg/s]2 + 0.0131 m[kg/s] - 0.0016 0.14

0.12

0.1

0.08 p [bar] Δ 0.06

0.04

0.02

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 m [kg/s]

Figure 71: Friction pressure losses in ALIP duct as a function mass flowrate.

4.5.2 Magnetic Field Measurements Coil Grading was identified in literature as a potential method to reduce the impact of the Finite Length Effect. Coil Grading is a method which involves reducing the number of turns in the coils located immediately at the entrance and exit of the pump. The result of this is a reduction of the radial magnetic field component in these locations, which has been cited to reduce the impact of the Finite Length Effect [32].

In the experimental ALIP, coil grading was achieved through tapped coils shown in Figure 49. These taps allow the electrical connections to be made at Turn 1, Turn 50, Turn 61, and Turn 80. As stated earlier, reducing the number of turns in a given coil will effectively reduce the magnitude of the radial magnetic field. However, the effective dimensions of the coil will also change depending on which taps are used. For example, a coil tapped at Turn 61 and Turn 80 will have a larger effective inner diameter than a coil tapped at Turn 1 and Turn 80. This will also impact the radial magnetic field magnitude. Relevant dimensions of the coils are provided in Table 13.

Table 18: Relevant dimensions of the tapped coils in the ALIP. Property Units Value Coil Outer Diameter mm 53.7 Inner Coil Diameter mm 133.7 Nominal Wire Thickness mm 2.04 Coil Width mm 13 Mean Radius – Turn 1 mm 27.87 Mean Radius – Turn 50 mm 93.63 Mean Radius – Turn 61 mm 113.67 Mean Radius – Turn 80 mm 131.66

Several coil grading configurations were investigated in this work. These configurations are presented in Table 19 with the numbering scheme and qualitative locations shown in Figure 50. Note that under each coil column lists the taps used for that configuration.

Table 19: Experimentally measured ALIP coil grading configurations. Grading Voltage Current Coil A1 Coil B2 Coil C3 Coil A10 Coil B11 Coil C12 Configuration [V] [A] No Grading 99 20 1 80 1 80 1 80 1 80 1 80 1 80 1 66 20 61 80 50 80 1 50 1 50 50 80 61 80 1-1 83 20 61 80 50 80 1 50 1 80 1 80 1 80 1-2 81 20 1 80 1 80 1 80 1 50 50 80 61 80 2 63 20 50 61 61 80 50 80 50 80 61 80 50 61 2-1 81 20 50 61 61 80 50 80 1 80 1 80 1 80 2-2 78 20 1 80 1 80 1 80 50 80 61 80 50 61

Prior to ALIP installation, the radial magnetic field component was measured as a function of pump length. These measurements were made using a pick-up coil mounted to a long rod. The position of the coil was referenced by the beginning of the pump conduit. These reference positions were then corrected so that zero marked the start of the inner core. This is shown in Figure 72 which plots the radial magnetic field component as a function of pump length with no modification to the coils. A sketch of the pump’s inner core, outer core, and coils is provided for reference.

Figure 72: Radial magnetic field distribution as a function of ALIP length with no coil grading.

Radial magnetic field measurements were also made to an ALIP with two primary coil grading configurations shown in Table 19 as Configuration 1 and Configuration 2. In these configurations, the coil grading was applied to both sides of the pump. In Configuration 1-1 and Configuration 1-2, the coil grading Configuration 1 was only applied to the inlet and outlet of the pump respectively. This is also the same for coil grading Configuration 2. Figure 73 and Figure 74 shows the results of the radial magnetic field of configuration’s one and two compared to the case of no grading. Both configurations result in a reduction of radial magnetic field near the inlet and outlet of the pump. Note that Configuration 1-1 and Configuration 2-1 only apply the grading to the pump inlet while Configuration 1-2 and Configuration 2- 2 only apply the grading to the pump outlet.

Figure 73: Radial magnetic field measurements of Configuration 1. Note that Configuration 1-1 and Configuration 1-2 only apply the coil grading to the pump inlet and outlet respectively.

Figure 74: Radial magnetic field measurements of Configuration 2. Note that Configuration 2-1 and Configuration 2-2 only apply the coil grading to the pump inlet and outlet respectively.

In addition to coil grading, the coils and outer core can be shifted 75 mm left or right of the centered position. These configurations are denoted as Entrance Shifting and Exit Shifting. Figure 75 shows the radial magnetic field measurements of the Entrance Shifted configuration compared to the measurements of the centered configuration. A sketch of the pump’s inner and outer core are shown for reference. Similarly, Figure 76 shows the radial magnetic field measurements of the Exit Shifted configuration.

Figure 75: Radial magnetic field measurements of the Entrance Shifted Configuration compared to measurements in the centered configuration.

Figure 76: Radial magnetic field measurements of the Exit Shift Configuration compared to measurements in the centered configuration.

4.6 Conclusions and Future Work During the reporting period, a modified ALIP was constructed and installed in an experimental facility to investigate the impact of the Finite Length Effect on p-Q and η-sm performance. The modifications included tapped coils at the inlet and outlet of the pump. These coils allow the investigation of magnetic field tapering which was identified as an approach to reduce the impact of the Finite Length effect. Additionally, the pump outer-core and coils can be shifted relative to the inner-core. This modification will also study the impact of the core length on the pump performance.

Several instruments were developed to characterize the ALIP p-Q and η-sm performance as well as the magnetic field distribution and any pressure pulsations. Two instruments were developed for ALIP flow measurements. Firstly, a large PMEMFM was developed to measure the mass averaged flowrate in the flow loop. Secondly, a Venturimeter was developed as a secondary method to the PMEMFM to measure mass averaged flowrates. The Venturimeter also doubled as a throttle with the addition of an insert placed in the throat. As noted, these instruments have the advantage of being relatively simple and can operate in sodium flows at high temperatures.

Two instruments were developed for ALIP pressure measurements. First, oil-filled pressure transducers on thermal stand-offs were developed to measure differential pressures across the pump. Second, piezoelectric pressure transducers were deployed to measure high frequency pressure pulsations. The piezoelectric sensors were calibrated used water to verify their linearity. Additionally, preliminary tests were completed to verify the sensor integrity when exposed to sodium.

ALIP magnetic fields were characterized by using a custom built air-cored pick-up coil. These sensors were found to have superior temperature characteristics compared to Hall Sensors. Furthermore, the pick- up coils could measure the large magnitude magnetic fields found in the ALIP while many commercial Hall Sensors could not. Lastly, air-cored pick-up coils are cost effective and simple to construct. These coils were used to characterize the radial-component of the applied magnetic field as a function of pump axial length. These measurements were completed for coil and coil-shifting configurations.

Frictional pressure losses in the ALIP conduit were measured using water as a surrogate. If was necessary to perform these measurements experimentally due to the exact geometry of the torpedo being unknown. The results show that the ALIP conduit losses are quite small. Moreover, these results are necessary when comparing analytical calculations of the developed pump pressure to experimental measurements of the pump pressure differential.

By the time of this reporting, p-Q and η-sm performance curves for any coil or coil-shifting configuration were not obtained. Similarly, normalized pressure pulsation curves were also not obtained. This objective was not completedΔ in time for the final report due to unanticipated problems in the ALIP control system and delays in the experimental facility construction. That being said, the issues with the ALIP have been resolved and the test facility is nearing completion. Ongoing work includes completion of the experimental facility. Once completed, p-Q and η-sm performance curves will be measured for all configurations outlined in the testing campaigns. Additionally, normalized pressure pulsation curves as a function of mean slip will also be measured. Δ

Future work with the instrumentation includes calibration of the Venturimeter in sodium to verify the initial water calibration. Furthermore, the PMEMFM will also be calibrated in sodium using a reference flowmeter. The first PMEMFM calibration will at low temperatures using a reference flowmeter. A second PMEMFM calibration will occur at high temperatures using the Venturimeter. Additionally, work will continue with generating a calibration curve of the piezoelectric at low pressures. Furthermore, the seals of the sensors will be tested in sodium at more prototypic operating conditions to verify their integrity.

5 Experimental Measurements of PMIP Performance and the Finite Width Effect 5.1 Technical Approach Literature showed a significant lack of knowledge of the Finite Width Effect in PMIPs. Based on literature of MHD flows in square channels and Flat Linear Induction Pump (FLIP) work, it was found that the Finite Width Effect is caused by current continuity, effectively reducing the useful component of current used for pumping. Not only is the width of the pump important, but the material of the walls as well. MHD channel flow indicated that the conductivity of the sidewalls is important in determining the current distribution in the channel which will also likely impact the performance of PMIPs.

Therefore, like the ALIP work, the PMIP work will the same performance metrics. Those being pressure- flowrate and efficiency-slip performance. Since these performance parameters have the same definitions, they will not be repeated in this section. Rather, the reader is referred to Section 4.1 for more details.

This work did not determine a practical method of directly measuring the current distribution in the pump fluid or conduit walls. Therefore, the impact of the Finite Width Effect will be quantified through p-Q and η-sm performance. These data will then be compared to analytic calculations of PMIP pressure output which account for the Finite Width Effect.

Performance testing will be split into three campaigns shown in Table 20. These campaigns will form baseline p-Q and η-sm performance measurement. These measurements will be used to evaluate analytic expressions and numerical models which will account for the Finite Width Effect. These measurements will also address a gap in the knowledge of PMIP efficiency in any configuration.

Table 20: Testing campaigns to evaluate PMIP performance and the impact of the Finite Width Effect. Temperature Magnet Campaign [oC] Frequency [Hz] 60 1 200 90 120 60 2 300 90 120 60 3 400 90 120

5.2 PMIP Specifications The experimental PMIP is shown in Figure 25. This pump was designed and built by M.G. Hvasta [38] [44] and is based on the work of Bucenieks [39] [40] [41] [42] [50]. The extensive work completed by Hvasta provided the motivation behind using the same PMIP configuration. Specifically, this PMIP is a Disc-Type configuration which uses two magnet arrays placed over a horse-shoe shaped flow channel.

Figure 77: Photo of UW Disc-Type PMIP installed in UW-Madison’s Small-Scale Component Sodium Test Facility.

A blow apart of the magnet arrays are shown in Figure 78 [38]. Each array contains a carbon steel backing plate, retainer plate, and face plate. The steel backing holds the magnets to the disc and strengthens the magnetic field. The retainer plate holds the magnets in place. And the face plate protects the magnets from foreign debris.

Figure 78: Blow apart of the magnet array’s used in UW-Madison’s Disc-Type PMIP [38].

Characteristics of the magnet arrays and conduit are given in Table 21. Note that a 12-magnet configuration with a 16 cm pole pitch was found to approximate a sinusoidal wave [38]. Not only does this match a critical assumption in the analytical theory, but it also improves the p-Q performance over configurations with less magnets. Additionally, the SmCo magnets have an operating temperature of 250 oC. Thus, pumps that use these magnets can reach higher temperatures without the worry of demagnetization.

Table 21: Properties of UW-Madison Property Units Value Array Diameter cm 30.5 Array Spacing cm 3.37 Average Channel Radius cm 12 Channel Width cm 5 Channel Height mm 6.35 Disc Spacing mm Magnet Type - SmCo Number of Magnets - 12 Pole Pitch cm 16 Duct Material - 316SS

5.3 Experimental Facility PMIP testing will be conducted in a small-scale-component sodium test facility shown in Figure 79. This facility is designed to accommodate small component testing such as plugging meters, cold traps, and oxygen cells. Forced convection materials corrosion studies can also be accommodated in this facility with two separate experimental legs. Maximum operating temperatures of 650 oC can be achieved with flowrates up to 10 m3/hr. The main facility components relevant to pump testing are a test section with the associated sample holder, pressure taps, an EM flowmeter, and the experimental PMIP.

Figure 79: A two-dimensional sketch of the Small-Scale Component Sodium Test Facility used during PMIP testing.

Power measurements required more extensive modifications to the facility compared to power measurements from the ALIP. This is due to an in-line torque transducer that must be installed in the drive train. Figure 80 shows a two-dimensional sketch of the proposed motor and pump frame modifications while Figure 81 shows a photo of the final mounting frame. Note that shaft couplers are located on their side of the torque transducer to allow for compliance of shaft misalignment. More details on the torque transducer will be discussed in the instrumentation section. 77

Figure 80: Proposed drive-train to accommodate an in-line torque transducer for power measurements.

Figure 81: Photo of PMIP drive-train used in power measurements.

Throttling was achieved by a similar method used in the ALIP test facility. Note that small-scale- component sodium test facility features a test section which can receive a square sample holder. By inserting increasing width samples into the holder, the effective flow area can be reduced. Figure 82 shows the square sample holder with the throttling inserts installed. Note that the insert widths range from 2.5 mm to 6.35 mm. When fully inserted, the 6.35 mm samples take up the full width of the test section.

Figure 82: Picture of the sample holder used to throttle the Small-Scale-Component Sodium Test Facility.

Pressure tap locations in the small-scale-component sodium test facility are not ideally located for pump testing. One of the pressure taps is located 77 cm to the right and 60 cm above the pump outlet. This results in a 119 cm flow distance between the two points. Thus, the frictional losses for this PMIP will be mischaracterized by additional frictional and gravitational losses through the conduit resulting in a decreased Δp. To account for the vertical displacement, a zero reading at no flow and constant temperature will be recorded. To account for the additional flow distance, an equivalent lengths method will be used to estimate the frictional losses.

5.4 Instrumentation Much of the instrumentation used in experimental measurements of the PMIP was adapted from the instrumentation developed for experimental measurements of the ALIP. For example, the PMIP’s magnetic field distribution was measured using the same pick-up coils. PMIP flowrates and pressure differentials were measured using the same PMEMFMs and pressure transducers respectively. For brevity, these descriptions will not be repeated in this section. Rather, the reader is referred to Section 4.4 for more details on specific instrumentation. Thus, this section will focus on new instrumentation not previously discussed Section 4.4.

5.4.1 Frequency, Power, and Efficiency Pump efficiency is defined as a key metric to characterize the performance of a pump and to quantify the impact of Edge Effects. The definition of efficiency used in the PMIP work is given in Equation 97 where Δp is the measured pressure difference between the pump inlet and outlet, Q is the mass averaged flowrate, and WE is the input power. In the PMIP, WE is defined as Equation 98 were ω is the angular velocity and τ is the applied torque.

p Q = W ∆ Equationη 97 E W = Equation 98 E ω ∙ τ At the time of this report, power measurements were not completed in the PMIP. However, prior to testing the measurement errors were estimated in order to verify that the expected changes in efficiency could be measured. Several vendors produce highly accurate torque transducers which can also measure angular velocity. Interface force measurement systems produce an appropriately sized torque transducer with a reported absolute measurement error of 0.2% the rated sensor torque. Angular velocity measurements have a reported error of 0.1% the rated speed. With a maximum locked-rotor torque of 30 N-m, the sensor was sized to 50 N-m which is 1.5x the maximum motor torque. This was done to avoid damage to the sensor while providing the high accuracy torque measurements. With this instrument, the absolute torque measurement error is 0.1 N-m and the absolute angular speed measurement error is 0.1571 rev/min.

Propagating these errors through Equation 97 and Equation 98 results in a roughly 3.5% error in the efficiency calculation. The analysis showed that the errors were larger for low torque and differential pressure measurements. This is shown in Table 22 which shows the relative contribution of measurement errors towards the calculation. Note that the differential pressure measurement and torque measurement contribute roughly equally to the measurement error. 79

Table 22: Relative contributions of each measured quantities uncertainty towards the calculated efficiency. Measured Relative Contribution Units Quantity to Total Error p Pa 0.54 Q m3/s 0 Δ τ N-m 0.46 ω rad/s 0

This torque sensor will be installed in-line with the PMIP drive train. Figure 83 shows a two-dimensional sketch of the PMIP drive-train. To account for shaft misalignments, flexible shaft couplings will be used on either end of the torque sensor to attach it to the PMIP motor and belt drive.

Figure 83: Two-dimensional schematic of PMIP drive-train that will be used to measure pump power.

5.5 Results 5.5.1 Duct Pressure Losses Pressure differential measurements across the pump were described in the Section 4.1 as a primary metric to quantify Edge Effects in this work. These measurements account for both the developed pressure and frictional losses between those two points. However, the analytical solutions of an ideal EMIP described in the literature review only calculate the developed pressure. Therefore, comparisons of theoretical EMIP calculations and experimental pressure measurements can only be done with knowledge of pressure losses in the pump conduit.

Frictional pressure losses in the PMIP duct in Figure 84 were characterized experimentally. Like the Venturimeter calibration described in Section 4.4.2, the pressure losses in the ALIP were measured using water. Water was used as a surrogate fluid since it does not pose a chemical hazard and is there easier to work with. Like the procedure discussed in Section 4.4.2, the loop was filled by gravity and trapped air evacuated by bleeding the loop with the pump on. Air voids were determined to be evacuated by visual inspection.

Figure 84: PMIP duct configuration used during frictional pressure loss measurements.

To ensure fully developed turbulent flow over the tested range of flowrates, an entrance length was attached to the inlet of the pump conduit shown in Figure 85. This section was sized to ensure fully developed flow. However, the flow may be disturbed near the inlet due to a treaded connection between the entrance length and pipe nipple. Relevant dimensions of the PMIP duct are provided in Figure 85.

Figure 85: Two-dimensional sketch of PMIP with some relevant dimensions.

Flowrate and pressure measurements were completed using the same methods described in the Venturimeter calibration in Section 4.4.2. Figure 86 plots the pressure losses in the PMIP conduit as a function of mass flowrate. A y = A x + B x + C model was fitted to these data using a linear regression analysis. 2 ∙ ∙ 81

0.6 Δp[bar] = 0.1528 m[kg/s]2 + 0.0673 m[kg/s] - 0.0077 0.5

0.4

0.3 p [bar] Δ 0.2

0.1

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 m [kg/s]

Figure 86: Experimental frictional pressure loss measurements in water.

5.5.2 Magnetic Field Measurements During the literature survey, the axial magnetic field’s radial and axial distributions were found to be important in characterizing the performance and impact of the Finite Width Effect in a PMIP. These distributions were characterized experimentally using the same pick-up coils described in Section 4.4.3. Figure 87 shows a sketch of the locations these measurements were made. Note that the radial distribution measurements were made 11 mm above the duct centerline. This was done because the duct walls obstructed the centerline. Measurements were not made elsewhere where the centerline could be accessed because it was desired to include the impact of the walls on the measurements.

Figure 87: Sketch of magnetic field measurement locations in the experimental PMIP.

Figure 88 presents the axial magnetic fields distribution as a function of axial height. Note that the measurements are centered on the between the two magnets. Observe that over the channel height of ±3.2 mm, the axial magnetic field is roughly uniform. This provides justification towards averaging the axial magnetic field over the channel height.

0 z [mm] -5

-10

-15

-20 2.5 3 3.5 4 4.5 5 5.5 B [T]

Figure 88: Axial magnetic field magnitude as a function of axial distance between the PMIP discs. Note that in the conduit region of ±3.2 mm the distribution is roughly uniform.

Figure 89 presents the axial magnetic field distribution as a function of pump radius. Note that these measurements are centered on the magnet centerline. Observe that over the channel width of ±25 mm, the axial magnetic field is roughly symmetric to the magnets centerline. This provides justification towards ignoring radial effects in this particular PMIP configuration. Note that since these measurements were no completed in the pump centerline, a fitted cosine function will not produce an accurate shape factor Γ discussed in Section 2.4.2.

Figure 89: Axial magnetic field magnitude as a function of radial distance between the PMIP discs. Note that in the region of the conduit region of ±25 mm the distribution is roughly symmetric. 83

5.5.3 Performance Curves Initial PMIP p-Q curves were taken at 300 oC and 600 oC at pump speeds of 600 rev/min, 1200 rev/min, and 1800 rev/min. With these pump speeds, the magnet frequency was calculated to be 60 Hz, 120 Hz, and 180 Hz respectively. Figure 90 presents the collected p-Q performance data. Note that the total pressure output p includes the frictional losses in the PMIP duct.

Figure 90: Initial p-Q performance curves of the experimental

To compare these data to the theoretical calculations from Section 2.4.2, the experimental data be corrected to account for the frictional losses in the PMIP duct as shown in Equation 99. These losses were modeled using FLUENT/ANSYS. The frictional pressure loss correlation is given in Equation 100. More details on this model will be discussed in Section 6.

p = p + p Equation 99 max ∆ ∆ loss p [bar] = 0.1108 m[kg s] + 0.0716 m[kg s] Equation 1002 ∆ loss ∙ ̇ ⁄ ∙ ̇ ⁄ Recall from Section 2.4.2 that two finite width correction coefficients were found in literature. These so called attenuation coefficients reduce the maximum output pressure as shown in Equation 101. The first coefficient K , in Equation 102 accounts for the finite width of the duct under a uniform magnetic field. K The second coefficientat 1 , in Equation 103 accounts for the finite width of the duct under a cosine shaped magnetic field. at 2

1 p = B (U U )L K 2 ,{ , } 2 EM f Equation0 B 1010 e at 1 2 σ − a k tanh 2 K = 1 , 2 a 0 �2λ � at 1 𝔑𝔑 � 2 � − �� λEquation 102 λ 84

a a a a a sin cos tanh cos + sin k 1 2 2 2 2 2 K = + , +2 2 a a 0 4� 2Γ � �Γ � �λ �Γ( �+ �Γ) � Γ �Γ �� at 2 2 2 𝔑𝔑 � 2 2 � − 2 2 �� λ Γ Equation 103 Γ λ Γ The initial experimental data and their associated theoretical calculations are plotted in Figure 91 and Figure 92. Note that the flowrate has been non-dimensionalized as Rm s where Rmf is the magnetic Reynolds number evaluated in the fluid and sm is the mean slip. The definition of Rm s is given in f ∙ m Equation 104 where dch is the channel height and dgap is the non-magnetic gap height. f m ∙

Figure 91: Corrected experimental PMIP pressure output at 300 oC compared to theoretical calculations.

Figure 92: Corrected experimental PMIP pressure output at 600 oC compared to theoretical calculations.

U d Rm s = s k d μfσf B ch f ∙ m Equation 104 ∙ m 0 gap Note in Figure 91 that the experimental PMIP output pressure is linear with Rm s as predicted by the theoretical calculations. Observe that of the two theoretical calculations, K , over predicts the pressure f m K K ∙ output more than , does. This is expected, as the derivation of , assumesat 1 a uniform magnetic field at 2 at 1 85

while the derivation of K , assumes a realistic cosine shape. However, both theoretical calculations fail to accurate predict the output pressure of the experimental PMIP even though they consider finite width at 2 effects. One reason for this could be attributed to ignoring the finite conductivity of the sidewall in the theoretical calculations. Recall from the literature review in Section 2.4.3 that the sidewalls were fundamental in determining the current distribution in square channel MHD flows. This could likely explain why even the K , coefficient is over estimating the PMIP output pressure. Unfortunately, no literature exists which analyticallyat 2 calculation the pressure output of a PMIP with finite conductivity sidewalls.

Note in Figure 92 that the experimental PMIP output pressure is non-linear with Rm s . This is counter to what is expected in the theoretical calculations. This effect may be caused by an inaccurate pressure f m loss curve. Perhaps the frictional pressure losses have a functional dependence on temperature∙ that was not observed in the FLUENT/ANSYS analysis that will be discussed in Section 6. A more likely explanation may be an experimental measurement error of pressure or flowrate. Therefore, these conditions will be revisited with a second dataset to rule out this possibility.

5.6 Conclusions and Future Work During the reporting period, several instruments were developed to characterize the performance of a PMIP. Much of this instrumentation was adapted from the instruments developed for ALIP testing. However, the power measurement instrumentation did differ significantly. For PMIP power measures, a new pump frame was constructed to accommodate the addition of a rotary torque sensor. This instrument allowed the pump power measurements to be decoupled from the electric motor inefficiencies.

Several key experimental measurements of the PMIP were completed. Firstly, the frictional pressure losses in the PMIP duct were experimentally characterized using water as a surrogate fluid. These measurements were necessary to allow direct comparison of experimentally measured and analytically calculation pump output.

Secondly, the magnetic field in the PMIP was measured using air-cored pick-up coils. These measurements showed that the axial distribution of the magnetic field’s z-component was roughly uniform over the channel thickness. This provides justification in the analytical calculations of averaging quantifies over the channel thickness. Additionally, measurements of radial distribution of the magnetic field’s z-component showed that the field is roughly symmetric. This provides justification in the analytical calculations of ignoring radial effects of the magnetic field. Moreover, it was found that the z- component followed an approximate cosine shape in the region of the duct.

Lastly, p-Q performance measurements of the PMIP were completed at 300 oC and 600 oC at 60 Hz, 120 Hz, and 180 Hz. These measurements were then compared to analytical calculations of pump performance which accounted for the Finite Width Effect. These comparisons showed that the attenuation coefficient which accounted for the finite width and non-uniformity of the magnetic field yielded the best results. However, both analytic calculations were not successfully in accurately predicting the pump output. Therefore, this suggests other Finite Width Effects, such as the finite conductivity of the sidewalls, may play a role in accurately determining the pump output.

By the time of this reporting, η-sm performance curves were not obtained. This objective was not completed in time for the final report due to delays in identifying an appropriate power measurement method for the PMIP. That being said, proper instrumentation has been secured and the new PMIP frame 86

is completed. Ongoing work includes installing the instrumentation into the new PMIP frame and performing new performance measurements under the conditions described in the test campaigns.

Future work includes updated magnetic field measurements as a function of radial position. These measurements will be performed on the duct centerline to obtain a shape-factor. These measurements will also be performed as a functional of axial distance. Lastly, work will continue on developing an analytic attenuation coefficient which accounts for the finite conductivity of the sidewalls in a PMIP. This coefficient will be verified against the newly obtained experimental p-Q measurements.

6 Numerical Modeling of PMIP Performance 6.1 Model Development As described in Section 2.4, PMIPs operate using the same principles as other EMIPs like ALIPs and FLIPs [38] [50] [58]. Literature has shown that the pressure output of an MMP can be accurately predicted using analytical techniques [38] [45] [50]. However, the analytical models reviewed in Section 2.4.2 rely upon many assumptions making it difficult to study transient or spatially dependent properties, such as the fluid velocity or Lorentz-force distribution. Moreover, the spatially and temporally varying velocity profiles and magnetohydrodynamic (MHD) effects can be difficult, if not impossible, to experimentally measure using current technology. Therefore, development of computational techniques to quantify these parameters is required to more fully understand Edge Effect phenomena that could affect pump performance.

Previous work by Koroteeva et al [59] showed that numerical analysis could accurately model PMIP performance and capture a range of magnetohydrodynamic effects within an PMIP. However, while Koroteeva used a combination of CFX and MAXWELL numerical solvers, this work will outline an alternative method of numerically modeling the performance of a Disc-Type PMIP using only the MHD- module within FLUENT.

The output of the PMIP was modeled using V19 of the ANSYS Workbench platform. Two PMIP models were constructed to investigate the impact of different meshing techniques, boundary conditions, and FLUENT settings. These details can be found in Table 23. Both models used ‘double precision’ accuracy and the default ‘absolute convergence criteria’ settings. Additionally, both models used a ‘no-slip’ fluid boundary condition and a ‘coupled’ electric boundary condition. Lastly, inlet, outlet, and external wall electric boundary conditions were set to ‘electrically insulating’. All subsequent MHD calculations were performed using the ‘Electric Potential’ method within the FLUENT MHD-module.

Table 23: An overview of the meshing and modeling methods used in this work. Parameter Model 1 Model 2 Size Function Proximity and Curvature Proximity Size Function Faces and Edges Sources Mesh Assembly Meshing Cut Cell Tetrahedrons Settings # Inflation Layers 5 Inflation Growth Rate 1.2 # Nodes 3158154 1225942 # Elements 2861437 4379904 Mid-Plane Symmetry BC No Yes Viscous Model k-ω SST k-ε Realizable MHD Method Electrical Potential FLUENT Inlet BC Mass-flow-inlet Velocity-inlet Settings Outlet BC Outflow Pressure-Velocity Coupling SIMPLE Coupled Solution Methods All 2nd-Order Upwind

Figure 93 show the sodium flow path between the two experimental pressure transducers. Note that the PMIP duct has an inner cross-section measuring 2 inches by 0.25 inches. The walls were made of 316- stainless steel that were 0.1 inches thick. To reduce computational expense, MHD calculations were performed within the ‘active channel’ shown in Figure 94. The active channel captures the duct within the circular region of the PMIP magnet discs.

Figure 93: A schematic showing the sodium flow-path between the experimental pressure transducers. These transducers are located at the inlet and outlet of this geometry. Note that all units are presented in inches.

Figure 94: The mid-plane of the ‘active channel’ modeled using the FLUENT MHD module.

Figure 95 shows the full 3D ‘cut cell’ mesh used in Model 1. Figure 96 shows the symmetric ‘tetrahedral’ mesh used in Model 2. A symmetry boundary condition was used to further reduce the computational expense.

Figure 95: Detailed look at the cut cell used in Model 1.

Figure 96: Detailed look at the symmetric tetrahedral mesh used in Model 2.

As noted in Section 2.4.2, the magnetic field produced by a Disc-type PMIP can be approximated using the following Equation 105. Note that Γ is the ‘cosine factor’ that gives the correct magnitude of the magnetic field in the radial direction, r is radial position within the duct, ri and ro are the inner and outer radii of the duct respectively, ω is the angular velocity of the rotating magnetic field, t is time, nmag is the number of magnets on an MMP disc, ϕ is the angular position with respect to the duct, and γ is an odd integer greater than unity that accounts for the non-sinusoidal nature of the magnetic field within an MMP.

r + r n (r, , t) = B cos r sin t 2 2 γ o i mag ϕ 𝐵𝐵 ϕ 0 �Γ � Equation− � 105� �� ω − �

Figure 97 shows the external magnetic field on the mid-plane of the model assuming γ=3 [-]. This field was input to FLUENT with a MAG-DATA file that was generated using a MATLAB script. The script used Equation 105 to calculated the magnetic field when the radial position was less than or equal to the outer radius of the duct and set B = 0 T for all points where r ≥5.90 inches. Additionally, the magnetic 90

field was assumed to be constant along the Z-axis. The MAG-DATA file was then imported into FLUENT. This file filled a rectangular region of space corresponding to X = ±18 inch, Y = ±18 inch, and Z = ±1 inch. The MAG-DATA file written with 2501 x 2501 x 2 points along the X, Y, and Z-axes respectively, therefore rectilinear spacing between points was 0.0144 inches by 0.0144 inches by 1 inch. At locations between the uniformly spaced MAG-DATA values, FLUENT automatically calculated the magnetic field using linear interpolation.

Figure 97: Contours of the Bz magnetic field in Tesla. Note that the z-axis is directed out of the page. The magnets on the PMIPs were arranged to produce an alternating polarity magnetic field.

The Electric Potential method and Magnetic Induction method are the two ways to model MHD effects using the FLUENT MHD-module [60]. This study used the electric potential method which assumes that the externally applied magnetic field was not impacted the induced currents or secondary magnetic fields [52]. This assumption was verified by calculating the magnetic Reynolds number Rmf over the flowrates of interest like in Section 3. The definition of the magnetic Reynolds number is given in Equation 106 where DH is the hydraulic number defined in Equation 107. Here, the quantity a is the width of the PMIP duct perpendicular to the magnetic field and the quantity b is the height of the PMIP duct parallel to the magnetic field. Rm = D U Equation 106 f μfσf H� 2 = + Equation 𝑎𝑎107𝑎𝑎 𝐷𝐷𝐻𝐻 𝑎𝑎 𝑏𝑏

Values for Rmf at the velocities relevant to this study are shown in Figure 98. For this experiment Rm < 1 for all experimental conditions. Therefore, it can be assumed that the induced magnetic field had a f small effect on the flowing sodium compared to the externally applied magnetic field.

Figure 98: The magnetic Reynolds number of sodium traveling through the PMIP duct as a function of velocity.

Solving the transient formulation of this problem in FLUENT is computationally expensive. Moreover, it can be challenging to create a MAG-DATA file for a travelling magnetic field using Equation 105 when γ>1. Furthermore, transient analysis add complexity in selecting appropriate time steps for a given problem. However, these problems can be avoided if one recognizes that the theoretical Lorentz-force only depends on the relative velocity between the fluid and the magnets.

Two calculations were performed for every flow rate to obtain the maximum pump output pressure as shown in Equation 108. The first calculation involved modeling the frictional pressures losses across the active channel without an applied magnetic field. The second calculation involved modeling the total pressure across the active channel with an applied magnetic field. There are denoted as and p respectively. The second, larger pressure differential p resulted from the combined 𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿−𝐴𝐴𝐴𝐴 hydrodynamic and MHD forces. Δ𝑝𝑝 Combined Combined Δ Δ p = p p Equation 108 ∆ max ∆ combined − ∆ loss−AC Once simulations for the active channel were complete, the hydrodynamic pressure losses across the total system between the pressure transducers in Figure 93 were calculated. This calculation was compared to similar results in previous work [45]. Representative pressure losses across the active channel are compared to pressure losses for the total system in Figure 99. After p was calculated, the output of the MMP could be calculated using Equation 109. loss ∆ p = p p Equation 109 ∆ ∆ max − ∆ loss 92

Figure 99: The calculated hydraulic losses across the total system in Figure 92 and the active channel in Figure 93.

6.2 Results The performance of the PMIP was modeled for 60 Hz, 120 Hz, and 180 Hz operation at 600 °C. Sample data for 180 Hz PMIP operation can be found in Table 24. A comparison of the FLUENT modeling results and the experimental data is provided in Figure 100. It was found that the maximum absolute difference between the simulations and experiments was about 6.7 psi, which corresponded to a maximum error of roughly 24%. Additionally, it was found that FLUENT models produced similar results, despite different meshes, boundary conditions, and program settings being used.

Table 24: Sample FLUENT data for 180 Hz.

Model [m/s] [m/s] [kg/s] [psi] [psi] [psi-s/m] [psi] [m/s] [psi] 𝐯𝐯𝐦𝐦𝐦𝐦𝐦𝐦 𝐋𝐋𝐋𝐋 𝐋𝐋𝐋𝐋𝐋𝐋𝐋𝐋−𝐀𝐀𝐀𝐀 𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂 𝒕𝒕𝒕𝒕𝒕𝒕 𝐋𝐋𝐋𝐋𝐋𝐋𝐋𝐋 𝐫𝐫𝐫𝐫𝐫𝐫 22.74 𝐯𝐯1 0.262𝐦𝐦̇ 𝚫𝚫𝐩𝐩0.105 𝚫𝚫𝐩𝐩2.551 2.446𝑲𝑲 𝚫𝚫0.382𝐩𝐩 21.74𝐯𝐯 52.793𝚫𝚫𝚫𝚫 22.74 2 0.523 0.331 5.413 2.541 0.984 20.74 51.715 22.74 4 1.047 1.169 11.463 2.574 2.848 18.74 45.388 22.74 6 1.570 2.453 17.917 2.577 5.592 16.74 37.559 1 22.74 8 2.093 4.126 24.749 2.578 9.216 14.74 28.787 22.74 10 2.617 6.182 31.937 2.575 13.720 12.74 19.096 22.74 12.5 3.271 9.327 41.426 2.568 20.589 10.24 5.712 22.74 15 3.925 12.973 51.453 2.565 28.833 7.74 -8.972 22.74 1 0.262 0.094 2.606 2.511 0.382 21.74 54.22 22.74 5 1.308 1.684 14.820 2.627 4.110 17.74 42.50 2 22.74 10 2.617 6.052 31.975 2.592 13.720 12.74 19.31 22.74 15 3.925 12.687 51.257 2.571 28.833 7.74 -8.93 Note: Calculations assume magnetic field is static. K = ( P P ) v (see Section7.1).

tot Δ Combined − Δ Loss−AC ⁄ LM 93

Figure 100: A comparison of the calculated differential pressures and the experimental measured values. The x-error bars corresponds to ±3.6% while the y-error bars correspond to ±0.1254 psid.

In addition to predicting PMIP output, the numerical modeling technique can be used to study other physics as shown in Figure 101 through Figure 104. For example, during testing it would be challenging or impossible to measure the electrical current distribution or Lorentz-force gradients. By benchmarking the FLUENT simulation against experimental data, PMIP designers can be confident that the calculated magnitudes shown in the figures below are accurate.

Figure 101: A contour plot showing the magnitude of the induced electrical current density on the mid-plane in A/m2.

Figure 102: A vector plot showing the induced electrical currents on the mid-plane in A/m2.

Figure 103: A contour plot showing the magnitude of the Lorentz-force on the mid-plane in N/m3.

Figure 104: A vector plot showing the Lorentz-force on the mid-plane in N/m3. Note that the vectors are directed against the flow since the magnetic field was assumed to be stationary.

6.3 Conclusions and Future Work This work has shown that the steady-state numerical modeling techniques used in the PMEMFM modeling can also be used to measure the output of a PMIP. Specifically, it was shown that steady-state EM Potential solver within FLUENT’s MHD-module can produce reasonable accurate results. It is hoped that these simple techniques can enable other researchers to rapidly and accurately verify the future PMIP designs.

However, this steady-state approximation may not be applicable to PMIPs with a small number of magnets. For a large number of magnets, the active area of then channel covered by them is roughly constant. For a small number of magnets, this could fluctuate by a large amount. For example, if the PMIP in this study had four magnets, the number of magnets contributing to the pump output would vary from 2-3 or a difference in output of 50%. Therefore, it is advisable to repeat FLUENT calculations for several magnet locations that are incrementally sweep over one full rotation. Then, an accurate result may be produced by averaging each of the models. This technique was used successfully by Koroteeva et al [59].

Lastly, ongoing experimental work is expected to produce additional PMIP performance data. New numerical models will be completed at the same conditions of the experimental data. These data will be used as a further comparison to verify the accuracy of this technique.

7 Numerical Modeling of ALIP Performance 7.1 Model Development As demonstrated Section 3 and Section 6, the Electric Potential solver within FLUENT’s MHD-module can be used to accurately model the performance of PMEMFMs and PMIPs. Like PMIPs, the analytical models reviewed in Section 2.3.2 rely upon many assumptions making it difficult to study transient or spatially dependent properties, such as the fluid velocity or Lorentz-force distribution. As noted earlier, spatial and temporal properties such as velocity profile and MHD effects may be difficult, or even impossible, to experimentally measure with current technologies. Therefore, development of computational techniques to quantify these parameters has value in aiding the understanding of Edge Effect phenomena that could affect pump performance.

The output of the ALIP was modeled using V19 of the ANSYS Workbench platform. A simplified pump geometry is shown in Figure 105. The overall length of the pump was modeled as 90 cm with an active region length of 50 cm. The annular gap was modeled as 1 cm while the inlet and outlet diameters were modeled as 10 cm. This geometry was imported into FLUENT to study the impacts of arbitrary magnetic field profiles on pump performance.

Figure 105: Model of the simple ALIP geometry.

Several magnetic field configurations in the active region can be applied to the geometry in Figure 105. These include a constant baseline field, a saw-tooth profile, and sinusoidal profile. Table 25 provides a sample input to model the magnitude of the field in these configurations. These codes can be modified to produce an arbitrary profile and investigate a wide range of coil gradings. These profiles are presented in Figure 106 through Figure 108.

Table 25: Sample input to model the magnetic field in the active region of the ALIP. Constant Saw Tooth Sinusoidal if abs(z) < 0.25 if abs(z) > 0.25 if abs(z) > 0.25 B_0 = 0.25; B_0 = 0; B_0 = 0; else else else B_0 = 0.25 B_0 = 0.25-abs(z); B_0 = 0.25*cos(z*pi/(2*0.25)); end end end

Figure 106: ALIP model with a constant 0.25 T magnetic field within the active region.

Figure 107: ALIP model with a sawtooth grading profile. Here the peak field is 0.25 T.

Figure 108: ALIP model with a sinusoidal grading profile. Here the peak field is 0.25 T.

7.2 Results Some initial results of the current distributions inside the model ALIP with a constant, uniform, 0.25 T magnetic field are shown in Figure 109 and Figure 110. Unfortunately, by the time of this report, the pressure output results have not converged, likely due to meshing. That being said, these initial results show the capabilities of the model at providing details into the physics that may be otherwise extremely challenging to measure.

Figure 109: Contour plot of the initial current distribution results in a general ALIP with a constant, uniform, 0.25 T field.

Figure 110: Logarithmic contour plot of the initial current distribution results in a general ALIP with a constant, uniform, 0.25 T field.

Figure 111 and Figure 112 show the initial results of the current distribution inside the general ALIP with a sinusoidal magnetic field with a magnitude of 0.25 T. The contour plots of the current distribution is similar to Figure 110. Ongoing work continues in achieving convergence of the p-Q performance results.

Figure 111: Contour plot of the initial current distribution results in a general ALIP with a sinusoidal magnetic field with 0.25 T magnitude.

Figure 112: Logarithmic contour plot of the initial current distribution results in a general ALIP with a sinusoidal magnetic field with 0.25 T magnitude.

7.3 Conclusions and Future Work During the reporting period, the steady state numerical modeling techniques successfully applied to the PMEMFM and PMIP work were also adapted to the ALIP. Flow geometries were constructed and several magnetic field shapes were applied. While the method looked promising for the ALIP work, the numerical models have not yet converged at the time of this report. Work will continue after the reporting period to achieve model convergence and accurate model results. Additionally, once model convergence is achieved, the coil grading configurations described in Section 7.1 will be modeled. The relative change in performance will then be compared to the relative change in performance data collected experimentally. While the results did not converge, the methods described in Section 6 are expected to produce accurate results that can enable other researchers to rapidly and accurately verify the future ALIP designs.

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8 Summary of Conclusions During the course of this work, progress was made in understanding the impact of Edge Effects on EMIP performance. A comprehensive literature review and theoretical analysis of Edge Effects in ALIPs and PMIPs produced several conclusions. Under ALIPs, the literature review identified the Finite Length Effect as the primary Edge Effect. This effect was found to be caused by fluid advection [9][31][36][37], or Rmf effects, and finite length core effects [36]. Overall, the Finite Length Effect was found to decrease the η-sm performance and cause pressure pulsations called DSF pulsations [32] [33]. These pulsations were reduced by tapering the magnetic field at the inlet and outlet of the pump which also increased the η- sm performance [32].

Under PMIPs, the literature review identified the Finite Width Effect as the primary Edge Effect. This effect was found to be caused by current continuity. Overall the Finite Length Effect was found to decrease p-Q performance [42] [50]. The literature review also identified that the sidewall conductivity may be important in determining the current distribution in the pump [48]. In FLIPs, a close cousin to PMIPs, the sidewall conductivity effect was shown to greatly impact the p-Q and η-sm performance of a FLIP [49]. However, little experimental studies and no numerical studies have been completed studying this effect for a specific PMIP.

To investigate these effects, experimental data was collected and numerical models were constructed of prototypic pumps. Under the numerical models, the accuracy of FLUENT’s MHD solvers were checked with simple MHD flows. A numerical method for calibrating a PMEMFM was developed using FLUENT. This method showed that a PMEMFM using a pair of unyoked magnets could be accurately modeled using the EM-potential method in FLUENT’s MHD solver. This output was checked against experimental measurements at sodium flowrates up to 2 m/s. This result is significant as it shows that PMEMFMs can be calibrated numerically and that the EM-potential method in FLUENT could be useful for EMIP modeling.

Under the experimental investigations, a modified ALIP was constructed and installed in an experimental facility to investigate the impact of the Finite Length Effect on p-Q and η-sm performance. These modifications allowed the inlet and outlet magnetic field to be modified by coil grading. Additionally, the modifications allowed the outer-core and coils to be repositioned relative to the inner-core and coils.

Several instruments were developed to characterize the ALIP and PMIP p-Q performance, η-sm performance, magnetic field distributions, and pressure pulsations. Two instruments were developed for flow measurements. These include a large PMEMFM and a Venutirmeter. Both instruments are used to measure mass averaged velocities. An added feature to the Venturimeter is that it also doubled as throttling valve with the help of an additional throat-insert piece.

Two instruments were developed for pressure measurements. These include oil-filled pressure transducers on thermal stand-offs and piezoelectric pressure transducers for high frequency pressure pulsations. The piezoelectric sensors were calibrated used water to verify their linearity and preliminary tests were completed to verify the sensor integrity when exposed to sodium.

Magnetic fields were characterized by using a custom built air-cored pick-up coil. These sensors were found to have superior temperature characteristics compared to Hall Sensors and could measure the large magnitude magnetic fields found in the pumps. Additionally, these sensors were cost effective and simple to construct. 101

Several key experimental measurements were completed in the ALIP. First, frictional pressure losses were measured in the conduit using water as a surrogate. The results show that the ALIP conduit losses are quite small. Additionally, the results allow comparisons between analytical pump output calculations and experimental pump pressure differentials. Additionally, the radial component of the magnetic field was measured as a function of axial position. These measurements were completed for all pump configurations. By the time of the report, p-Q curves and -sm curves were not obtained for any pump configuration. This was due to construction delays with the ALIP test facility and unexpected issues with the pump control system. With that being said, the ALIP test facility is nearing completion and the ALIP control system problems have been resolved. Ongoing work will continue to measure the performance characteristics of the ALIP at all magnetic field configures described earlier.

Key experimental measurements were also completed in the PMIP. First, frictional pressure losses were also measured in the conduit using water as a surrogate. These measurements allow comparisons between the analytical pump output calculations and experimental pump pressure differentials. Additionally, the axial component of the magnetic field was measured as a function of radial position and axial position. These measurements showed that over the height of the conduit the fields are approximately uniform. Additionally, these measurements show that the fields are roughly symmetric along the radial position.

PMIP p-Q curves were obtained at 300 oC and 600 oC at 60 Hz, 120 Hz, and 180 Hz. These measurements were compared to analytical calculations of pump performance. While the analytic calculations that accounted for finite width effects came close to predicting the actual pump output, they still failed at producing accuracy results. This suggests that other Finite Width Effects, such as the finite conductivity of the sidewalls, may play a role in accurately determining the pump output. By the time of the report, η- sm curves were not obtained due to problems with identifying the proper instrumentation to measure PMIP power input. Initial methods that were explored included using a power analyzer to measure the input power to the motor leaders. However, it was recognized that this would also include motor inefficiencies, which can be significant at low speeds and torques. These measurements would therefore not accurately characterize the pump power input. With that being said, proper instrumentation has been identified and ongoing work continues at installing the new setup.

A numerical model of the PMIP was constructed using the same steady-state numerical modeling techniques used in the PMEMFM modeling. The pressure output of a PMIP was modeled using the steady-state EM Potential solver within FLUENT’s MHD-module and compared to experimental data. These results show that the developed modeling method can produce reasonably accurate results for a PMIP.

The same numerical modeling techniques used for the PMEMFM and PMIP were adapted to a general ALIP. In this model, two magnetic field shapes were investigated including a uniform magnetic field and a sinusoidal magnetic field. Initial results show the detail current distributions inside the pump. However, p-Q performance results could not be obtained by the time of this report due to convergence issues of the model. This is likely caused by meshing problems. Ongoing work continues at achieving convergence of the models.

In total, this work has identified the key Edge Effects impacting the performance of ALIPs and PMIPs. This work has developed significant instrumentation towards measuring pressure, flowrates, and power in these EMIPs. Additionally, this work has developed a numerical method to quickly and accurately model 102

liquid metal flowmeters voltage output and EMIP pressure output. In conclusion, it is hoped that the completed experimental and numerical results provide inspiration for future work with Edge Effects ALIPs and PMIPs.

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9 References

[1] T. Cochran, H. Feiveson and F. Von Hippel, "Fast Reactor Development in the United States," Science and Global Security, Vols. 17:2-3, pp. 109-131, 2009. [2] R. Ashley, R. Beeley, F. Fillmore, W. Hallet, B. Hayward, T. Gershun and J. Lundholm, "SRE Fuel Element Damage," 1959. [3] R. Atz and R. Baker, "Pumps," in Sodium-NaK Engineering Handbook, vol. 4, O. Foust, Ed., New York, Gordon and Breach Science Publishers Inc., 1972. [4] D. Nixon, "Survey of Sodium Pump Technology," 1963. [5] J. Balkwill and J. Brindley, "Bearings and Seals," in Sodium-NaK Engineering Handbook, vol. 4, O. Foust, Ed., New York, Gordon and Breach Science Publishers Inc., 1972. [6] L. Blake, "Electromagnetic Pumps for Liquid Metals," Journal of Nuclear Energy, Part B: Reactor Technology, vol. 1, pp. 65-76, 1959. [7] R. E. Schwirian, "Analysis of Linear Induction or Traveling Wave Electromagnetic Pump of Annular Design," 1965. [8] J. Verkamp and R. Rhudy, "Electromagnetic Alkali Metal Pump Research Program," 1966. [9] E. S. Pierson and W. D. Jackson, "The MHD Induction Machine," 1966. [10] R. Campana, E. Diehl and R. Faught, "Components," in Liquid Metal Handbook, H. Grantz, Ed., The Atomic Energy Commission, 1955. [11] H. Elrod and R. Fouse, "An investigation of electromagnet flowmeters," Transactions of the American Society of Mechanical Engineers, vol. 74, 1952. [12] W. Gray, "Magnetic Flowmeter Calibration Results (KAPL-613)," 1951. [13] J. Shercliff, The Theory of Electromagnetic Flow-Measurement, Cambridge University Press, 1962. [14] W. C. Gray and E. Astley, "Liquid metal magnetic flowmeters," Journal of the Instrument Society of America, vol. 1, pp. 15-23, 1954. [15] J. Mausteller, F. Tepper and S. Rodgers, Alkali Metal Handling and Systems Operating Techniques, Gordon and Breach, 1968. [16] U. A. E. Commission, "Permanent Magnet Flowmeter for Liquid Metal Piping Systems (Technical Report RDT C 4-5T)," 1971. [17] L. G. Vasil'yev and A. I. Kozhainov, "Magnetohydrodynamics in Marine Engineering," 1967. [18] J. Chavanne, O. Chubar, P. Elleaume and P. Van Vaerenbergh, "Nonlinear numerical simulations of permanent magnets," Proceedings of EPAC, pp. 2316-2318, 2000. [19] S. Kim and C. Doose, "Temperature compensation of NdFeB permanent magnets," 1998. [20] U. Jeong, Y. Kim, J.-M. Kim, T.-J. Kim and S. Kim, "Experimental evaluation of permanent magnet probe flowmeter measuring high temperature liquid sodium flow in the ITSL," Nuclear Engineering and Design, vol. 265, pp. 566-575, 2013. [21] M. A. Ziniuk, C. C. Scott and T. Meshii, "Permanent Magnet-Type Flowmeters and Their Application in the Enrico Fermi Atomic Power Plant," 1968. [22] B. Axcell and W. Walton, "Thermoelectric effects in miniature permanent magnet probes used for velocity measurement in flowing sodium," Experimental Thermal and Fluid Science, vol. 6, pp. 309-323, 1993. 104

[23] B. R. Munson, T. H. Okiishi, W. W. Huebsch and A. P. Rothmayer, Fundamentals of Fluid Mechanics, Seventh ed., Wiley, 2013. [24] C. Borghi, A. Cristofolini and W. Fabbri, "Study of the Design Model of a Liquid Metal Induction Pump," IEEE Transactions on Magnetics, vol. 34, no. 5, pp. 2956-2959, 1998. [25] L. Goldsteins, "Experimental and numerical analysis of behavior of electromagnetic annular linear induction pump," 2015. [26] H. R. Kim, "The theoretical approach for viscous and end effects on an MHD pump for sodium circulation," Annals of Nuclear Energy, vol. 62, pp. 103-108, 2013. [27] H. Ota, K. Katsuki, M. Funato, J. Taguchi, A. W. Fanning, Y. Doi, N. Nibe, M. Ueta and T. Inagaki, "Development of 160 m3/min Large Capacity Sodium-Immersed Self-Cooled Electromagnetic Pump," Journal of Nuclear Science and Technology, vol. 41, no. 4, pp. 511-523, 2004. [28] C. Roman, "Studies of the Annular Linear Induction Pumps for Sodium Circuits Use in Nuclear Plants," IEEE, pp. 1-10, 2013. [29] L. R. Blake, "Conduction and Induction Pumps for Liquid Metals," in British Nuclear Energy Conference, 1957. [30] S. Nasar, Handbook of Electric Machines, McGraw-Hill, 1987. [31] J. Valdmanis, "On theory of longitudinal finite length effect in linear induction magnetohydrodynamic machine," Magnetohydrodynamics, pp. 101-106, 1966. [32] H. Araseki, I. R. Kirillov, G. V. Preslitsky and A. P. Ogorodnikov, "Double-supply-frequency pressure pulsation in annular linear induction pump Part I: Measurement and numerical analysis," Nuclear Engineering and Design, vol. 195, pp. 85-100, 2000. [33] I. Kirillov and D. Obukhov, "Completely Two-Dimensional Model for Examining the Characteristics of a Linear Cylindrical Induction Pum," Technical Physics, vol. 5, pp. 999-1005, 2005. [34] L. Lazarenko, "Analysis of Longitudinal Fringe Effects in Linear Induction MHD Machines," Magnetohydrodynamics, no. 1, pp. 91-98, 1971. [35] H. Araseki, I. R. Kirillov, G. V. Preslitsky and A. P. Ogorodnikov, "Double-supply-frequency pressure pulsation in annular linear induction pump Part II: Reduction of pulsation by linear winding grading at both stator ends," Nuclear Engineering and Design, vol. 200, pp. 397-406, 2000. [36] I. Kirillov, "Longitudinal End Effect in Cylindrical Induction MHD Machines," Magnetohydrodynamics, no. 2, pp. 109-116, 1972. [37] J. Gieras, G. Dawson and A. Eastham, "A New Longitudinal End Effect Factor For Linear Induction Motors," IEEE Transactions on Energy Conversion, Vols. EC-2, no. 1, pp. 152-169, 1987. [38] M. G. Hvasta, W. K. Nollet and M. H. Anderson, "Designing moving magnet pumps for high- temperature, liquid-metal systems," Nuclear Engineering and Design, vol. 327, pp. 228-237, 2018. [39] I. Bucenieks, "Perspectives of Increasing Efficiency and Productivity of Electromagnetic Induction Pumps for Mercury Basing on Permanent Magnets," in International Conference on Nuclear Engineering, 2006. [40] I. Bucenieks, "High Pressure and High Flowrate Induction Pumps with Permanent Magnets," Magnetohydrodynamics, vol. 39, no. 4, pp. 411-417, 2003. 105

[41] I. Bucenieks and K. Kravalis, "Efficiency of EM Induction Pumps with Permanent Magnets," Magnetohydrodynamics, vol. 47, no. 1, pp. 89-96, 2011. [42] I. Bucenieks, K. Kravalis and R. Krishbergs, "Pressure-Flow Rate Characteristics of the Pumps with Permanent Magnets," Magnetohydrodynamics, vol. 47, no. 1, pp. 97-104, 2011. [43] Y. Y. Liyelpeter, "Motion of conducting bodies in a magnetic field," 1968. [44] M. Hvasta, "Designing and Optimizing a Moving Magnet Pump for Liquid Sodium Systems," 2013. [45] M. G. Hvasta, Designing & Optimizing a Moving Magnet Pump for Liquid Sodium Systems (PhD Thesis), 2013. [46] E. Pierson, R. Hanitsch, T. Huehns and H. Mosebach, "Predicted and Measured Finite-Width Effects in Linear Induction Machines," IEE Transactions on Power Apparatus and Systems, Vols. PAS-96, no. 4, pp. 1081-1086, 1977. [47] H. Bolton, "Transverse edge effect in sheet-rotor induction motors," Proceedings of IEEE, vol. 116, no. 5, pp. 725-731, 1969. [48] J. Hunt, "Magnetohydrodynamic flow in rectangular ducts," Journal of Fluid Mechanics, vol. 4, no. 21, pp. 577-590, 1965. [49] S. Tado and S.-i. Kamiyama, "Analysis of Transverse Edge Effect of Linear Induction Pump with Electrically Conducting Side Walls," Nuclear Science and Technology, vol. 20, no. 5, pp. 379-388, 1983. [50] I. E. Bucenieks, "Perspectives of Using Rotating Permanent Magnets in the Design of Electromagnetic Induction Pumps," Magnetohydrodynamics, vol. 36, no. 2, pp. 151-156, 2000. [51] X.-f. Gou, Y. Yang and X.-j. Zheng, "Analytic Expression of Magnetic Field Distribution of Rectangular Permanent Magnets," Applied Mathematics and Mechanics, vol. 25, no. 3, p. 297, 2004. [52] U. Jeong, T.-J. Kim, Y. H. Kim, H. H. Son, G. Hyeok and S. J. Kim, "Numerical evaluation of SmCo permanent magnet flowmeter measuring sodium flow in a low flow rate regime using FLUENT/MHD module," Journal of Nuclear Science and Technology, vol. 53, no. 2, pp. 173-183, 2016. [53] M. Weathered, J. Rein, M. Anderson, P. Brooks and B. Coddington, "Characterization of Thermal Striping in Liquid Sodium With Optical Fiber Sensors," Journal of Nuclear Engineering and Radiation Science, vol. 3, no. 4, 2017. [54] K. Magnetics, Neodymium Magnet Physical Properties. [55] I. Magnetics, Magnetics 101 - Design Guide. [56] MagnetShop, Magnet Design Guide. [57] ISO, "Part 4: Venturi tubes," in Measurement og fluid flow be means of pressure differential devices inserted in circular cross-section conduits running full, Vols. ISO 5167-4:2003(E), International Standard Organization, 2003, pp. 1-24. [58] M. Butzek and I. E. Bucenieks, "Proposed Mercury Pump for ESS," in ICANS-XVI 16th Meeting of the International Collaboration on Advanced Neutron Sources, Dusseldorf-Neuss, Germany, 2003. [59] E. Koroteeva, M. Scepanskis, I. Bucenieks and E. Platacis, "Numerical modeling and design of a disk-type rotating permanent magnet induction pump," Fusion Engineering and Design, vol. 106, pp. 85-92, 2016. [60] A. Inc., ANSYS Fluent Advanced Add-On Modules (Release 18.2), 2017. 106

[61] I. S. Organization, "Part 4: Venturi tubes," in Measurement of fluid flow by means of pressure differential devices inserted in circul cross-section conduits running full, International Standards Organization, 2003, pp. 1-24.