DEPARTMENT OF PHYSICS

FINAL YEAR PROJECT, DISSERTATION ORPHYSICS EDUCATION REPORT

NAME: Edmund Erskine, Thomas Greening, Alice Rushton,and Adam Taylor

DEGREE COURSE: Level 3 BSc PhysicsPROJECT TITLE: Siemens Magnet Technology - Industrial Group

Project: Cryogenic Fluids ManagementYEAR OF SUBMISSION: 2015SUPERVISOR: Professor David ChernsNUMBER OF WORDS: 7563

DeclarationAll experimental data found in this project was collected by members of the group. All equipment used in the labwas provided by the University of Bristol via Tom Kennedy, with the exception of the 12V Submersible ElectricPump provided by Whale Pumpsr, and the diodes, which were designed by the group and constructed specificallyfor this project by the Physics Mechanical Workshop. The diode designs were drawn using Autodesk’s AutoCAD2013 software, and the fluid simulations and visualisations were completed using Autodesk’s Simulation CFD 2015software package. Setup diagrams included in the report were drawn using Microsoft Paint and Publisher. Dataanalysis was completed using Microsoft Excel 2013 and and OriginLab’s Origin 9, both provided by the Universityof Bristol.

AcknowledgementsFirstly we would like to thank Professor Cherns for his guidance and support throughout the project. Thanks alsoto Bob Wiltshire for delivering safety training in handling cryogens;the Physics Mechanical Workshop, particularlyAdrian Crimp, for their advice and expertise in realising our designs and Dr Germinal Magro for his suggestions inthe safe use of flammable organic compounds. Many thanks to Tom Kennedy for his invaluable assistance in thelab, practical advice, equipment provision, and good humour throughout the practical stage of our project.

Finally we would like to thank Siemens Magnet Technology, specifically Hannah Hale and Adrian Thomas forhosting our visit, and making this project possible overall.

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Contents

Declaration i

Acknowledgments i

Contents ii

1 INTRODUCTION 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Theory 22.1 Diodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Reynolds Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.3 Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.4 Diode Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.4.1 Tesla Diode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.4.2 Vortex Diode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.5 Navier-Stokes Equations and CFD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.6 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 Experimental 43.1 Initial Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.2 Calibration of the Flowmeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.3 Variable Flow Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.4 Viscosity Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.5 CFD Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4 Results and Discussion 74.1 Variable Flow Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4.1.1 Reynolds Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.1.2 Diodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4.2 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.2.1 Comparison with CFD Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.2.2 Comparison with Other Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.3 Dependence of Diodicity on Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.4 Satisfaction of Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.5 Recommendations for Future Investigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5 Conclusions 13

Bibliography 14

Appendices 16

A Flowmeter Calibration 16

B Diode Designs 17B.1 Vortex Diode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17B.2 Tesla Diode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

C Certification of Ownership 18

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Cryogenic Fluids Management

An investigation into the performance of fluidic diodes with variable viscosity, room-temperature fluids wascarried out in order to predict their success at effectively transporting low-viscosity cryogens. The vortexdiode and Tesla diode were compared with regard to their diodicities at different flow rates of water. Forthe range observed the vortex diode exhibited diodicity up to 7 times greater than the Tesla diode. Therelationship between diodicity and viscosity was investigated using fluid viscosities between 0.24 and 1.19centipoise. CFD simulations were produced to compare to experimental data. The CFD results greatlyimproved understanding despite little correlation between simulation and laboratory data.

1 INTRODUCTION

1.1 Motivation

One of the most powerful tools in modern medicine isthe Magnetic Resonance Imaging (MRI) scanner. Theefficacy of MRI scanners as a non-invasive diagnostictool has meant they are now ubiquitous in modern hos-pitals. MRI scanners use incredibly strong magneticfields, typically around 1.5T1, though studies on ani-mals have involved fields of up to 21.1T2. To gener-ate such strong fields, large superconducting magnetsare employed. Besides greater field strength, supercon-ducting magnets have greater field homogeneity and aremore stable than permanent, resistive alternatives3. Thecoil windings in these magnets are typically constructedof niobium-titanium, which is superconducting belowtemperatures of ≈ 10K4,5. Accordingly, the usual cool-ing agent is liquid helium, which has a boiling point of4.2K6.

Within the MRI scanner, the primary coils of themain magnet are held in a magnet former, then placedin a helium vessel which is sealed within a cryoshieldand vacuum insulated. This is collectively known as thecryostat. Currently, the magnet is cooled by bath cool-ing; half-filling the cryostat with liquid helium. With theaddition of a refrigerator to condense any helium gas thatmay form, the cryogenic temperature (and therefore su-perconductivity of the magnet) can be maintained with-out needing to frequently top-up the liquid helium.

The major drawback to bath cooling is that it uses avery large quantity of liquid helium: a finite and uncom-mon resource. With each scanner using upwards of 2000litres of liquid helium7, increasing scarcity and risingprices provide the motivation to find an alternative, moreconservative, cooling method within MRI scanners. Apromising solution is the implementation of a cryogeniccircuit, to ensure that the cooling agent covers the super-conducting magnet in the most efficient way possible.

A significant hurdle for the development of a circuitsystem is that at cryogenic temperatures, any impurity or

contamination in the fluid will freeze and possibly causemoving components of the system to fail. This problemis exacerbated by the fact that the cryostat is inaccessiblefor maintenance. A solution would require a device thatcan control the flow of a fluid without moving parts orexternal interaction.

Consequently, attention has turned to a family of de-vices known as fluid diodes. These devices are designedsuch that their internal geometry modifies fluid flow: in-hibiting it in one direction whilst permitting it in theother. Fluid diodes are already used widely in industry tosolve a variety of problems such as effluent treatment8,filtration of ballast water9 and controlling the coolingfluid in nuclear reactors10. However, their suitability foruse with cryogens is essentially unknown. Whilst somediode designs have moving parts (for example, Ball-and-Screen Diodes11), there are several which are solid-statedevices, such as the Vortex Diode and the Tesla Valve.A detailed discussion of the design of these two devicescan be found in §2.4.

1.2 Objectives

This project is sponsored by Siemens Magnet Technol-ogy. Siemens are the current world-leaders in the MRImarket, with a 40 percent share of total sales in 2014.Their ambition for this project is to look towards the de-velopment of a new cooling system for MRI magnets,with a focus on the conservation of helium and an over-all enhancement of efficiency. The investigation will be-gin with a more open brief: to evaluate the success offluid management devices with low-viscosity cryogenicfluids. After an introductory meeting with the industrialcontacts, Hannah Hale and Adrian Thomas, and comple-tion of a detailed assessment of the available literature,the following objectives were decided upon:

• To design and construct a range of fluid diodes,suitable for testing at both cryogenic and roomtemperatures

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• To design pumping systems suitable for use with arange of fluids

• To characterise and compare the performance ofthese diodes in terms of:

– Pressure versus flow rate

– Fluid viscosity

• To investigate the potential of ComputationalFluid Dynamics (CFD) modelling to simulate theflow in the diodes under various conditions

Initial investigation will be conducted with waterat room temperature, to examine the properties of thediodes. In addition, by investigating low viscosity flu-ids and utilising CFD software, the intention is to makepredictions about the behaviour of liquid helium in fluiddiodes. If successful, and if a suitable circuit/pumpingsystem can be devised, the diodes will then be tested withcryogens (the most probable being liquid nitrogen) in or-der to support or invalidate these predictions.

As the field is relatively under-researched, the com-pletion of the objectives above will satisfy a generalbrief: to lay the foundation for future investigation intothe application of fluid diodes in the management ofcryogens.

2 TheoryThis section outlines the fundamental physical principlesrelevant to the investigation, as well as useful formulaefor describing the fluid mechanics at work in the diodes.

2.1 DiodicityThe efficacy of a fluid diode is typically described by aparamater called the diodicity, D. This is defined as theratio of “the pressure drop in the high-resistance flow di-rection to the pressure drop in the low-resistance flow di-rection for the same mass flow rate”11, expressed math-ematically as:

D =

(∆Pr

∆Pf

)F

(1)

where ∆Pr is the pressure difference across the diode inthe reverse direction (high resistance), ∆Pf is the pres-sure difference in the forward direction (low resistance),and the subscript F signifies a constant flow rate in bothdirections.

2.2 Reynolds NumberAnother parameter to describe the flow conditions in adiode is the Reynolds number: the ratio of inertial forces

to viscous forces within the fluid12. The general expres-sion for the Reynolds number of flow through a pipe is:

Re =ρuDH

µ=

QDH

νA(2)

where ρ is the density of the fluid, u is the meanvelocity perpendicular to the cross-section, µ is the dy-namic viscosity, ν is the kinematic viscosity (ν = µ/ρ),Q is the volumetric flow rate, A is the cross-sectionalarea and DH is the hydraulic diameter (in the case of acircular pipe this is simply the conventional diameter).

A low Reynolds number corresponds to the domi-nance of viscous forces within the fluid, and hence in-dicates laminar flow. Conversely, a high Reynolds num-ber means that inertial forces are dominating, and thatthe flow is turbulent13. Ideally, the flow within diodesis predominantly laminar in the forward direction, witha large amount of turbulence in reverse. This turbulentflow increases the energy lost through frictional forcesincreasing resistance. Therefore, to maintain the flowrate an increase in pressure is required.

2.3 Viscosity

Viscosity is a defining property of fluid flow and is es-pecially influential within diodes, as the formula for theReynolds number, Eq. (2), clearly shows. As mentionedin §1.1, previous investigations into fluidic diodes havenot involved cryogens. This is important as cryogenicfluids typically have very low viscosities; just below theirboiling points liquid nitrogen is roughly 6 times less vis-cous than water14, and liquid helium is around 300 timesless viscous15.

At extremely low temperatures (< 2K), helium en-ters superfluidity, and therefore has no measurable vis-cosity16. However, as previously stated (§1.1), MRIscanners operate at 4.2K7 and so superfluid behaviouris beyond the scope of this investigation. Even withoutconsidering superfluidity, the use (either practically or insimulation) of cryogens within the diodes will involve afar greater Reynolds number than in previous studies.

2.4 Diode Design

2.4.1 Tesla Diode

It is worth noting that fluidic diodes are not new inven-tions. Perhaps the most famous example is the Tesladiode (named after the inventor, Nikola Tesla), whichwas patented in 1920. Fig. 1 shows the internal geome-try:

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Figure 1 Cross-section showing the looped internal structureof the Tesla diode17.

It is clear from the diagram how the bulk of a fluidtraveling from right to left is able to follow an unim-peded, mostly laminar flow, where any energy losses aredue to surface friction. However, if the same fluid ismade to enter from the left, it is forced back against theoriginal flow by the counter-flow loops. In this way, thefluid itself provides the inhibiting force against the flow.

Because the geometry of the Tesla diode is so in-volved, many parameters can be adjusted in order to alterthe diodicity. Work with CFD has found that diodicityincreases in accordance with two variables in particu-lar: increasing the number of loops (as one would ex-pect), but also with decreasing the distance between theloops18.

2.4.2 Vortex Diode

Figure 2 Fluid motion within a vortex diode in the reversedirection (left) and forward direction (right)11.

In comparison to the Tesla diode, the geometry of thevortex diode is less detailed. It consists of a large circu-lar cavity with an axial port and a tangential port. In theforward direction, flow enters the axial port and passesout through the tangential port, largely unimpeded. Inthe reverse direction, however, flow enters the tangentialport and adopts a swirling motion, known as a vortex.As opposed to a free vortex (those found naturally in theform of hurricanes and whirlpools, etc.), a forced vortexis observed within the diode, where the tangential veloc-ity is proportional to the radius of the chamber. Theseprocesses are displayed in Fig. 2.

A parameter conventionally used for describing thegeometry of a vortex diode is the aspect ratio, α . This isgiven by:

α =dh

(3)

where d and h are the diameter and height of the cham-ber, respectively.

A previous experiment by Kulkarni has indicated thatthe diodicity of a vortex diode increases with α , untilα ≈ 6, beyond which there is no further increase. Thereis a similar critical point for Reynolds number that alsomarks an optimal diodicity19,20. These design featureswill be important to consider in the practical element ofthis investigation.

2.5 Navier-Stokes Equations and CFD

The Navier-Stokes equations are a fundamental aspect offluid dynamics; they quantify the motion of viscous flu-ids by applying Newtons second law to fluid motion12.They are incredibly powerful because they relate the ve-locity, temperature and density of a moving fluid. Theyare a set of time-dependent coupled differential equa-tions that are solved approximately in the CFD simula-tion software for a given system. The software makesuse of finite element analysis (FEA)21 which separates adomain into smaller parts called finite elements. In thecontext of the software this is the meshing stage (dis-cussed later in §3.5) that separates the solid and fluidvolumes into a mesh of elements. The Navier-Stokesequations are then solved for this simplified system toproduce an approximate solution which allows the anal-ysis of fluid flows in terms of calculated values such asvelocity, temperature, density and pressure. The solutionis time-dependent so transient analysis over a time frameis possible; the equations are solved repeatedly at differ-ent time steps where the initial conditions are taken fromthe previous time step and the new variables are calcu-lated. These transient analyses allow the investigation ofhow fluid flows evolve over time and in some situationsconverge. Convergence of the Navier-Stokes equationsover iterations is key in the calculation of variables in or-der to ensure a stable solution. Any divergent behaviourresults in significant deviations from the expected resultsand therefore an invalid simulation.

2.6 Assumptions

Throughout the practical element of this investigation,it will be crucial to minimise the number of air bubblesthat could potentially enter the apparatus. This is to en-sure that all pumping force is used to push the fluid, andnot to compress the gas in the bubbles. Eliminating airbubbles will also ensure a constant fluid density. Hence,the investigation may be carried out in accordance withBernoulli’s principle, where each fluid to be tested shallbe regarded as incompressible, with constant density andviscosity.

Since viscosity is temperature dependent, it will beimportant to ensure that isothermal conditions be heldthroughout. This will be relatively simple to achieve, by

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carrying out all measurements at room temperature, butit will be very important to monitor.

3 Experimental

3.1 Initial Development

Figure 3 Prototype glass vortex diode used in initial tests.

A prototype vortex diode was constructed from glasswith an attached U-tube manometer, pictured in Fig. 3.If the U-tube is partially filled with an indicating fluid,an applied pressure difference will drive the levels offluid apart. The magnitude of this pressure differencemay then be directly inferred from the height differ-ence between the two columns of indicating fluid in themanometer. Silicone oil was chosen as a suitable indi-cating fluid since its density (of around 970kg m−3)22 isless than that of water, which allows for greater sensitiv-ity.

Though initial tests of the prototype exhibited suc-cessful diodic behaviour, it was quickly apparent thatvarious modifications needed to be made. For exam-ple, it was difficult to eliminate air bubbles from thepermanently-attached manometer tubing. This was a se-rious issue since the contamination of air changes theoverall density of the indicating fluid, which in turn willgive rise to inaccurate pressure measurements. For thisreason it was decided that further use of a glass manome-ter would be impractical, and was replaced by a digitalmanometer.

The glass vortex diode was replaced with two largervortex diodes. These were built in accordance with the

optimal geometrical guidelines suggested by Kulkarni20,which are discussed in §2.4. Two Tesla diodes were alsodesigned closely in style with the geometry specified inTesla’s 1920 patent17.

Figure 4 Large and small vortex diodes (for detaileddimensions see Appendix B).

Figure 5 Large and small Tesla diodes (for detaileddimensions see Appendix B).

The four new diodes (Figs. 4 and 5) were con-structed from polymethyl methacrylate (PMMA), a ro-bust, readily-available plastic with high chemical stabil-ity, which is commonly used in cryogenic apparatus23.Transparent upper sections were used to both monitorthe flow and allow easy identification of excessive airbubbles or leaks.

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3.2 Calibration of the Flowmeter

Calibration of the flowmeter required the mass of the wa-ter pumped through in 5 seconds to be recorded againstthe rotor frequency of the flowmeter.

With each rotation, a magnet inside one of the ro-tor paddles induced a current in a stationary wire withinthe meter as it moved past, and the frequencies of thesefluctuations in current were recorded by an oscilloscope.A Tf380 universal frequency counter recorded the av-erage rotor frequency over 10 second intervals. Fromthese measurements, a relationship between frequencyand mass flow rate was obtained. The flowmeter becamean essential component of the apparatus because it al-lowed a consistent flow rate to be maintained by moni-toring frequency readings. The ability to maintain a con-stant flow rate is vital in investigations of diodicity, sincethe calculation requires measurements of ∆P that corre-spond to the same flow rate in both directions.

The flow rates were recorded over the full range ofthe pump’s power. The frequency of rotation was in-creased by increments of 5 Hz, with 10 volume mea-surements taken at each frequency.

The optimum position of the flowmeter was foundto be between the pump and the first manometer tube.Placing the flowmeter within the manometer tubes neg-atively affected the pressure readings. The flowmeterwas not placed after the diodes as the vortex createdby the vortex diodes disturbed the flowmeter, producingwildly varying results. Once the relationship betweenflow and frequency had been established, it was testedwith each diode in the system to ensure the calibrationof the flowmeter was accurate for all four diodes, in bothdirections.

3.3 Variable Flow Experiment

Figure 6 Experimental set-up for the variable flowexperiment. M, F and P indicate the digital manometer,flowmeter and pump respectively.

After the flowmeter was calibrated, the four diodes wereplaced in turn into the experimental set-up shown in Fig.6 in order to investigate the diodicity of each. The systemwas filled with room temperature (20◦C) water. Increas-ing the power of the water pump steadily increased theflow rate, and measurements of pressure difference weretaken at ten evenly-spaced intervals. The ∆P values wereobtained as averages over ten readings at each flow rate.This method was repeated for each diode, in the forwardand reverse directions, to collect values of ∆Pf and ∆Prover a range of flow rates.

Precaution had to be taken to ensure the apparatuswas clear of all air bubbles before each test. This wasachieved by flushing the system at a high flow rate be-fore beginning to collect data at lower flow rates.

3.4 Viscosity Experiment

Figure 7 Experimental set-up for the viscosity experiment.

Readily available organic fluids, pentane and heptane, to-gether with a 10% solution of acetic acid were chosen tobe tested in the second part of the investigation. Thesefluids in particular were chosen because their viscositiesof 0.24 cP24, 0.39 cP24 and 1.19 cP25 respectively com-pare well to that of distilled water (1.00cP26).

More viscous liquids were trialled but the pumpcould not provide sufficient power to eliminate air bub-

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bles from within the apparatus. Furthermore, it wasdecided that the pumping system used for water wouldbe inappropriate in the viscosity experiment, due to thehighly volatile nature of pentane and heptane, togetherwith the electrical wires in the pump and the flowme-ter, potentially creating a fire-risk. Therefore, an alter-native, safer system was assembled, which simply har-nessed gravity as a pumping force (Fig. 7).

The new system consisted of a 200ml vessel, con-nected above the small Tesla diode via silicone tubing.The digital manometer was connected, as before, imme-diately to either end of the diode. The flow of liquidthrough the system was controlled using a screw-clampabove the diode, which was adjusted for each liquid toensure a consistent flow rate of 25± 1g s−1, despite thefact that the various viscosities and densities would influ-ence the movement of each fluid differently under grav-ity.

The smaller Tesla diode in particular was selected tobe used in the viscosity experiment for a number of rea-sons: the small internal volume meant that the amountof fluid used in each run could be reduced and hencethe fire and fumes risk of pentane and heptane was keptto a minimum. Also, the two-dimensional nature of thediode eliminated gravitational bias (the fluid experiencedthe same downward component of gravitational force foreach orientation). This would not have been the case fordifferent orientations of the vortex diode.

3.5 CFD Simulations

In order to simulate flow through a diode, one must firstcreate a geometry file which can be opened in the chosensimulation software. For each diode, a three-dimensionalscale drawing was created using AutoCAD software byAutodesk, so that the solid elements could be importedinto the simulation software. Autodesk Simulation CFD2015 was chosen for its special capabilities in simulat-ing fluid and thermal flows and its wide use in testingproduct behavior.

Once the solid elements have been imported fromthe geometry file, the fluid elements can be created byselecting the volume to be filled. In order to simulatethe practical elements of this investigation the same vari-ables must be applied to the simulated system as the ex-perimental system. First the materials are defined. Thesolid element was set as PVC, which is not the materialused in the experiment but the simulation only requiresthat the wall roughness values (coefficients of friction)are the same. Next, the fluid element was set to be oneof the experimental fluids: water, pentane, heptane andacetic acid. For the variable flow experiment , the massflow rate boundary conditions were set at the inlet andoutlet ports to cover the range of mass flow rates prac-tically tested through the diodes: between 20g s−1 and140g s−1. For the viscosity experiment two flow rates

were used, 25g s−1 for comparison with the experimen-tal results and 100g s−1 in order to better represent thefull range of flow rates used in the variable flow exper-iment. Simulations of forward and reverse flows mustbe carried out separately, because the re-direction of themass flow rate causes the boundary conditions to change.A mesh of the system was then generated, using the stan-dard auto-meshing tool built into the software. A meshsplits the system into smaller domains which allows theflow to be analysed, and it was found that each mesh wassufficiently refined to produce a convergent solution. Asimulation of fluid flow in the system was then createdby transient analysis with the variables assigned.

Transient analysis works to simulate the system atindividual time-steps with a set number of iterations foreach. For the solution to converge, a small enough time-step is required to capture the flow detail. A time-stepsize of 0.01 seconds was found to converge for all simu-lations with sufficient flow detail and was the most timeefficient. The number of inner iterations of the transientflow equations at each time-step was determined auto-matically by selecting the AutomaticIteration configura-tion option in the flag manager; this allowed for the mosttime efficient solutions by minimizing the total numberof iterations. A stop time and data save intervals wereassigned: each simulation had a minimum stop time of4 seconds to allow the flows to stabilise and accuratelysimulate diodicity. Advection schemes control the nu-merical mechanism of transporting quantities such as ve-locity and pressure through the solution domain, eachof the schemes were tested and advection scheme 5 waschosen for the final set of results for various reasons: itwas found to consistently converge, it produced the ex-pected flow patterns and it is widely recommended forrecirculating flows and pressure drop predictions. Thestandard K-epsilon turbulence model was also used inthe final set of simulations because it was found to bethe most convergent and stable over time in terms of thecalculated variables and flow patterns.

The quality of the simulation was determined bycomparing the detail of the flow pattern with the ex-pected flow pattern and the convergence of globally cal-culated values such as flow velocity, turbulence and pres-sure. Once the quality of a simulation was found to beacceptable, a summary file was saved which contains allthe calculated information at each time-step (specificallythe inlet and outlet bulk pressures and the Reynolds num-ber). From the bulk pressures, a value of diodicity maybe calculated using the average pressure drop from thefinal five time-steps of the simulation.

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4 Results and Discussion

4.1 Variable Flow Experiment

Fig. 8 shows the experimentally-observed dependence of∆Pf on mass flow rate for each of the diodes in the for-ward direction. There is a positive correlation betweenpressure difference and flow rate. However, the trend foreach data-set was found to be non-linear.

A notable feature of the graph is that for each diode,∆Pf converges to zero as F becomes small, as one mightexpect. It is also worth noting that, of the four diodes,the small vortex diode consistently exhibited the greatestpressure difference, over the whole range of flow rates.The small Tesla diode displayed the next highest ∆Pfabove a flow rate of 40g s−1. At 40g s−1, the valuesof ∆Pf for the small and large Tesla diodes were verysimilar: 5.09mbar and 5.13mbar respectively. Below40g s−1 the large Tesla diode exhibited a greater pres-sure difference than the smaller Tesla diode. The largevortex diode consistently produced the smallest ∆Pf overall flow rates.

Figure 8 Experimental relationship between flow rate and ∆Pffor all diodes in the forward direction.

Figure 9 Experimental relationship between flow rate and ∆Prfor all diodes in the reverse direction.

For the reverse direction, shown in Fig. 9, there isa similar positive correlation between ∆Pr and flow rate.The small vortex diode clearly remains the strongest overthe whole range, with the small Tesla diode again pro-ducing the second-highest pressure differences. How-ever, the large vortex diode now exhibits a greater pres-sure difference than the large Tesla diode.

Once again, each data-set converges to zero for smallF . However, the pressure gradient for the small vortexdiode is by far the steepest at this point. The resistanceof this diode in the reverse direction was observed to beso large that the maximum power setting of the pumpwas unable to produce a flow rate of more than 55g s−1.

4.1.1 Reynolds Number

Figure 10 Graphs comparing the relation between Reynoldsnumber to pressure for the Tesla diodes in both directions.

Figs. 8 and 9 (displaying the relationship between flowrate and ∆Pf or ∆Pr for the four diodes) are useful to in-terpret how each diode would perform in a given system.However, the effectiveness of the diodes in comparisonto one another is best represented, not by a comparison offlow rate to ∆P, but by the comparison of Reynolds num-ber to ∆P. This is because the Reynolds number takesinto account the relative hydraulic radii of the diodes,such that the volumetric flow rate is scaled to the volumeof the port through which it enters the system. Two flowswith the same Reynolds number will have a similar flowpattern, as discussed in §2.2.

Fig. 10 shows the clear difference between the effec-tiveness of the small and large Tesla diodes. Both sizesdisplay similar pressure differences for similar flow pat-terns (described by the Reynolds number) in the forwarddirection. However when the flow is reversed the pres-sure drop in the small diode is far greater than the largediode over the same range of Reynolds numbers.

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Figure 11 Comparison of vortex diodes.

Transforming flow rate to Reynolds number for thevortex diodes (Fig. 11) shows even more clearly theeffect of scale - the limited range of flow rates in thesmall vortex diode is no longer a problem as the rangeof Reynolds numbers for the diodes is more compara-ble. In Fig. 11, we can see that the relationship betweenReynolds number and ∆Pf is similar for both diodes,whilst, as with the Tesla diodes, the smaller of the twoachieves much greater success in reverse.

4.1.2 Diodicity

Diodicity can be also calculated from the Reynolds num-ber because Reynolds number is directly proportional toflow rate for both directions, as seen in Eq. (2). Valuesof diodicity could not be calculated directly from the ex-perimental results as matching the flow rate to the cor-responding reading in the opposite direction was nearimpossible. In light of this problem, it was decidedthat a polynomial fit would be plotted to the graphs of∆P against Re for each diode, in the forward direction.The forward direction was chosen for the fit because theReynolds numbers calculated had a greater range. Thebenefit of using a fit calculated over the greatest rangemeans that the fit would not need to be extrapolated tocompare of Reynolds numbers in the reverse direction.The coefficient of determination (denoted R2) was cal-culated for each data-set to evaluate the accuracy of eachfit. Since an R2 value of 1 corresponds to a perfect fit,it was decided that any fit in this experiment would bedeemed acceptable if R2 > 0.995.

In this way, each fit could be used to predict, witha good degree of confidence, ∆P for the same Reynoldsnumber in the forward direction. Taking the recorded Refor the reverse direction, then using the trend-line to cal-culate the corresponding ∆ Pf , an estimate for diodicitycould be calculated at this Reynolds number using Eq.(1). For consistency, the same type of polynomial fit wasapplied to each forward flow.

Figure 12 Diodicity (calculated from polynomial fits toexperimentally-obtained ∆Pf and ∆Pr values) againstReynolds number for all diodes.

From Fig. 12 it can be seen that the vortex diodesdisplay a very slight increase in diodicity with Reynoldsnumber, which is not true of the Tesla diodes. From thisit can be surmised that for Tesla diodes, increased turbu-lence affects flow in both directions proportionally to oneanother. Accordingly, the diodicity values for the Tesladiodes level off. Within the vortex diodes, increased tur-bulence affects the reverse direction more than the for-ward, increasing the diodicity.

The large Tesla diode has the lowest diodicity of thefour diodes, which seems to keep to a maximum value ofD ≈ 2.75 for Re > 5000. Unfortunately, the pump usedin the system was not powerful enough for data collec-tion at Re > 10000 due to the large internal volume.

Neither of the vortex diodes exhibited the steady in-crease in diodicity displayed by the large Tesla diode atlower Reynolds numbers. The small and large vortexdiodes do both appear to have slight increases at higherReynolds number but this trend is far from steady andconsistent. The steady increase shown for the large Tesladiode may have already occurred at lower Reynoldsnumbers for the vortex diodes. However, this could notbe ascertained as the flow would have been too slow toregister on the flowmeter.

We can see that the performance of both Tesla diodesis notably below that of the vortex diodes. The diodicityvalues for the large Tesla diode appear to reach a steadyvalue of about 5.9 above the Reynolds number of 5000.This is notable as 5000 is the value at which the flow go-ing into the diodes is likely to be already turbulent. Thispossibly suggests that once the flow entering the Tesladiodes reaches a certain turbulence the diodicity does notchange.

The vortex diode results are not as consistent, butthey do appear to level off, or have a very small increasein diodicty, as the turbulence of the flow entering in-creases with Reynolds number. This suggested that the

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vortex diodes outperform the Tesla didoes for more tur-bulent flows. R. Narasimha and Sreenivasan suggestedthat this may be due to an effect called ”relaminarisa-tion” when the flow enters the chamber.27

Using the definition of Reynolds number 2, we cansee that if the viscosity is reduced below 1, say to thesame as a cryogenic fluid such as liquid nitrogen, theReynolds number would increase dramatically. With theperformance of the vortex diode being better than theTesla diode at high Reynolds numbers, it implies thatthe vortex diode would also outperform the Tesla diodeswhen using liquid helium, the very low viscosity coolingagent within an MRI machine.

4.2 Simulations

The flow patterns inside the diodes were investigated us-ing a range of tools in the simulation software. Fig. 13shows a top-down contour plot of velocity magnitudewithin the large Tesla diode. The forward and reversedirections are shown below and above, respectively. Itis clear from the diagram that the fluid reaches muchhigher velocities in the forward direction compared tothe reverse direction, as expected.

A set of traces was used to investigate the flow pat-tern inside the vortex diode (see Fig. 14) Again, the dif-ference in flow pattern for each direction is clear. In theforward direction the flow enters through the axial portinto the chamber and flows out radially, in the reversedirection the flow enters through the tangential port intothe chamber and creates a forced vortex.

Figure 13 Tesla diode contour plot of velocity magnitudes.

Figure 14 Simulated trace particles shown travelling throughthe vortex diode in both directions, velocity magnitude shownby the colour.

4.2.1 Comparison with CFD Model

Figure 15 Simulation versus experimental data in the forwarddirection.

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Figure 16 Simulation versus experimental data in the reversedirection.

Figure 15 shows how the simulations of the diodes (datapoints with connecting lines) compare to the experimen-tal data collected (data points with associated error barscollected in the laboratory). The relationship betweenReynolds number and pressure is similar for both thesimulation and experimental result: a steadily increas-ing pressure possibly with a polynomial or exponentialtrend. The simulation’s most noticeable drawback ap-pears to be the reduction of the size of the diodes giv-ing an exaggerated increase in ∆Pf . The experimentaldata for the large Tesla diode appears to have a good cor-relation with the CFD data until the Reynolds numberreaches 8000. From here the simulation suggests thatthe pressure remains the same for a period and then in-creases again. This constant pressure drop being main-tained over such an increase in the Reynolds numberis not exhibited in any other experimental result, or in-deed any other simulation result for any of the diodes.This suggests that this data point is a bad result, pos-sibly due to decreased stability in the solution with ahigher Reynolds number. The following data point af-ter the ”bad result” is also lower than the experimentalresult, which is expected if the solution loses its validityat higher Reynolds numbers.

In the reverse direction there is a clear difference inthe respective correlations between the experimental andCFD data for the Tesla diode and the vortex diode, shownin Fig. 16. The simulated pressure drops for the smallTesla diode show the most agreement with the experi-mental data. The results for the larger Tesla diode belowa Reynolds number of 5000 also show good correlationwith the experimental results. The divergence from theexperimental results above this value could be explainedby the turbulent nature of the flow at these points.

When Re < 4000, the simulation for both Tesladiodes are in agreement with the experimental results,but above this point the simulations display pressure thatis lower than the experimentally-determined values. The

critical Reynolds number for a flow in a pipe to be tur-bulent is 400028; for any Reynolds number below theflow is likely to be exhibiting laminar characteristics.When the flow enters the chamber the hydraulic radiusincreases, which reduces the Reynolds number and insome cases will produce a laminar flow. This is ”relami-narisation”. This drop in turbulence and Re is underesti-mated in the CFD model. To correct this the plots wouldneed to be shifted to the left. However, for Re < 4000the flow in the port is already laminar and would not un-dergo relaminarisation. The difference between the CFDmodel and the experimental data is that the relaminarisa-tion is measured experimentally, but is underestimated inthe CFD model. The chaotic nature of turbulence meansthat any deviation from the correct model, in this caselaminar flow, will result in drastic changes in the flowpattern and therefore any calculated ∆P.

Another explanation for the differences suggested bythe data is the uniform nature of the flow at the inlet.In the laboratory set-up, the fluid has already travelledthrough pipes, the flowmeter and the manometer junc-tion. This could result in a non-uniform flow enter-ing the diode which would in turn result in more tur-bulence within the system. This would suggest thatthe Reynolds numbers calculated for the practically-obtained data could be too low. Correcting this calcu-lation would shift the experimental data to the right inFig. 9, moving them closer to the CFD data. The onlyway to account for this in the simulation would be tosimulate the entire system, however this increased com-plexity would likely result in a less stable solution and adeviation from the measurement of the true effectivenessof the isolated diode.

In Fig. 17, an area where there is a rapid changein colour gradient or the vector direction indicates areaswhere turbulent flow is likely to arise. These turbulentpockets, called eddies, are not present in the CFD simu-lation for the bulk of the vortex diode chamber, indicat-ing some degree of relaminarisation. As the flow patternsuggests, the greatest tangential velocities are found onthe outside of the chamber. This implies that the entirevolume of water in the chamber is behaving as a forcedvortex.

Another factor in the differences between the CFDand experimental results for the vortex diode is the lackof gravity in the simulated system. In the laboratory, alldata was collected with the axial port vertical and thetangential port horizontal; this results in an increased ∆Prbecause more energy is required for the fluid to flow outof the axial port. In the forward direction, the opposite istrue: the flow into the axial port accelerates as it travelsinto the chamber, requiring less energy and resulting ina smaller pressure drop. Any attempt to simulate gravityin the vortex diode system resulted in an unstable, diver-gent solution so the quantitative effect of gravity on thepressure drop is not known.

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Figure 17 A snapshot of the different vector components inthe vortex diode, a red or purple colour signifies a fast flowrate and a blue colour signifies a slower flow rate.

A comparison of the CFD and experimental diodicityvalues showed that there is significant deviation betweenthe results. The simulated diodicity values are lower thanthe experimental values, this is because the simulatedforward pressure drops are in general larger and the sim-ulated reverse pressure drops are in general smaller thanthe experimental values. The simulated results did agreewith the experimental and literature results in that all thediodes diodicity values increase with increased Reynoldsnumber.

4.2.2 Comparison with Other Work

Kulkarni19 also experimented with different sized vor-tex diodes. However, the experimental data is not di-rectly comparable as the vortex diodes that were usedwere of slightly different chamber diameters. Nonethe-less, plots of the relationship between Reynolds numberand pressure drop were made for a vortex diode of di-ameter of 90mm. The large vortex diode in this experi-ment had a chamber diameter of 86.9mm,and an aspectratio of α = 6 (as suggested by Kulkarni in his designpaper20), a comparison of the pressure drops for differ-ent flows can be made. Kulkarni’s larger diode exhibiteda very similar pattern of increasing pressure drop withReynolds number for both directions. In accordance withour findings in Fig. 11, the larger diameter chamber sizeproduced a lower pressure drop for the same Reynoldsnumber again in both flow directions. This similarity inthe relationships is encouraging as it confirms the vortexdiodes built for this experiment behaved in a consistentmanner to the diode built in Kulkarni’s experiment.

A plot of flow rate against pressure drop for a vortexdiode with an axial port diameter of 10mm, was madeby Priestman in 198729. The plot displays a very simi-lar polynomial-type relation to that observed in this ex-periment displayed in Fig. 9. The port diameter of thelarge vortex diode used in this experiment was 12.4mm(shown in Appendix B), similar to the size of the diodeused in Preistman’s experiment. As stated earlier oursimulation results suggested that the entire volume of thevortex diode was behaving as a forced vortex diode, in-dicated by the decreasing tangential velocity towards the

centre of the chamber. Kulkarni’s plot of tangential ve-locity against the normalised radius used similar sizeddiodes within the range of velocity flow that was inves-tigated in this experiment. However, in contrast to oursimulation of what appears to be a forced vortex through-out the chamber, Kulkarni’s plot suggests that the vor-tex behaves as a free vortex (with the tangential velocityincreasing towards the centre) before transitioning to aforced vortex. This clear discrepancy in the simulationscasts further doubt over the accuracy of the simulationscomputed in this experiment.

4.3 Dependence of Diodicity on ViscosityThe second half of the investigation attempted to evalu-ate the success of the small Tesla diode at low viscosi-ties. This diode was chosen for reasons highlighted in§3.4, but the selection was further supported by the re-sults of the variable flow investigation; in terms of in-hibiting reverse flow, the small Tesla diode was the mostsuccessful, apart from the small vortex diode. How-ever, the small vortex diode was too strong for the lower,gravity-induced flow rates used in the viscosity experi-ment. Therefore the small Tesla diode was selected asthe best alternative.

Due to a number of practical challenges, by way ofprocuring suitable equipment and trialling various meth-ods of data-collection, only four substances were usedto practically observe the effect of viscosity on diodicity.Fig. 18 displays the measured diodicities together witha CFD simulated relationship that was obtained for thesame range.

Figure 18 Relationship between diodicity and viscosity forexperimentally-obtained data and CFD simulations.

A feature of the graph that is immediately obvious isthe lack of correlation between the practically-observeddata points. It is likely that this may be attributed to in-consistencies in the method. For example, the gravity-induced flow rate was too low to totally eliminate air

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bubble contamination within the diodes. As previouslydiscussed, this type of contamination causes any pres-sure reading to be an inaccurate measure of diode be-havior. The strong temperature-dependence of viscositycould also be a source of error, if the temperature of thesystem was not monitored to an acceptable degree.

It is hence difficult to draw a conclusion about thedirect effect of viscosity on diodicity using practical re-sults alone. For this reason, the system was simulatedusing CFD and these results are plotted on the graph forcomparison. The simulation predicts a gradual increasein diodicity as viscosity decreases.

For lower viscosities, the resistance due to surfacefriction between the fluid and the diode walls is reduced.Therefore, an overall decrease in ∆P is expected to oc-cur in both directions, but to different degrees; resistancedue to surface friction is larger in the reverse directionbecause the counter-flow loops cause the molecules tospend a longer time in contact with the diode walls. Sofor a smaller ∆Pf in comparison to ∆Pr, the overall diod-icity will increase.

Figure 19 Relationship between Reynolds number andviscosity for the experimental fluids.

Fig. 19 displays calculated values of Reynolds num-ber for each of the four experimental fluids. These valuesconfirm the inverse proportionality between Re and vis-cosity, given by Eq. (2). Since Re indicates the propor-tion of inertial forces to viscous forces within the fluid(§2.2) it can be physically understood how Re (and henceturbulent, inertial forces) would increase as the contribu-tion of viscous forces tends to zero.

Data collected previously in the experiment (Fig. 16)indicates that the CFD software has underestimated theamount of turbulence in the Tesla diode; a smaller ∆Pris predicted than was observed experimentally. Hence,the CFD simulated trend in Fig. 18 may contain thesame discrepancy. If this is the case, then the lower vis-cosities (corresponding to higher Reynolds numbers andthe dominance of turbulent flow) may in fact give rise togreater diodicities.

We may assess the accuracy of the CFD predicted

trend by considering the diodicity of the small Tesladiode with water. Water has a viscosity of 1.00cP andhence a Reynolds number of 3067.4 in this system.Fig. 12 indicates that for this Reynolds number, the diod-icity of the small Tesla diode should lie between 8 and10. Since the CFD simulated trend predicts that D ≈ 2for this Re = 3067.4, the simulation may be underesti-mating the diodicity by up to a factor of 5.

Despite this limitation of the CFD software, a fur-ther simulation was run at the visocisty of liquid helium(0.0033 cP15). A flow rate of 100g s−1 was chosen be-cause previous data has shown that, in general, higherflow rates exhibit greater pressure differences and morereliable results. The simulation predicted that if the smallTesla diode was used to manage liquid helium, it wouldexhibit a diodicity of 2.65.

In the context of diodicity, this result is quite small.However, it is important to remember that a correctionmust be made for the under-estimation of turbulence bythe software. This is done, as before, by comparingReynolds number to the diodicities shown in Fig. 12.

It was calculated that the Reynolds number for liq-uid helium in this case is around 465,000. This valueis very high and is unfortunately not included in therange of Reynolds number observed practically. How-ever, Fig. 12 shows a fairly flat relationship betweenReynolds number and diodicity for the small Tesla diode,which seems to remain constant towards higher Re; forRe > 6000, D≈ 6. In this way, it is reasonable to assumethat for low viscosity, high Reynolds number fluids, theeffect of diodicity may still be observed.

At this stage it is useful to consider how a vor-tex diode might behave with the management of low-viscosity fluids. Previous experimentation highlightedthe small vortex diode as the most effective diode of allfour tested. It is therefore sensible to assume that, evenfor lower-viscosity fluids, it would exhibit greater diod-icities than the small Tesla diode.

This hypothesis, however, requires further experi-mentation with a system that allows much greater flowrates to be achieved, both consistently and safely, ifvolatile low-viscosity fluids are tested.

In summation, it has been shown that diodicity isstill an observable effect for low viscosity fluids. This isbecause the contribution of viscous forces becomes lessimportant at high Reynolds numbers. Instead, turbulentforces dominate and diodicity sees a slight increase dueto the decrease in ∆Pf caused by the reduced effect ofsurface friction.

However, this relationship was not observed inpractical experimentation due to inconsistencies in themethod and the possibility that the system was not main-tained at a constant temperature.

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4.4 Satisfaction of ObjectivesAs discussed previously, there were 4 objectives set atthe beginning of the project

• To design and construct a series of fluid diodes ofdifferent models, suitable for testing at both cryo-genic and room temperatures

Testing with cryogenic fluids was not achieved due to thenotable complications involved in producing a pumpingsystem capable of withstanding cryogenic temperatures.PMMA was used to construct the diodes used in roomtemperatures, as it was robust, transparent and readilyavaliable.

• To design pumping systems suitable for use with arange of fluids

A pumping system for water was assembled with the useof an electronic pump, and a gravity feed system wasconstructed to cater for hazardous liquids.

• To characterise and compare the performance ofthese diodes in terms of:

– Pressure versus flow rate– Fluid viscosity

It was found that comparing Reynolds number to pres-sure was more appropriate than flowrate. The compar-ison of the performance of the diodes was completed,determining the small vortex diode the most effective.

• To investigate the potential of ComputationalFluid Dynamics (CFD) modelling to simulate theflow in the diodes under various conditions

The CFD modelling produced some comparable resultsfor the variable flow experiment but other deviationsfrom the model suggested that the model needed signif-icant improvement if any comparison is to be made forcryogens in future systems.

4.5 Recommendations for Future Investi-gation

Despite the achievements detailed in the previous sec-tion, there are still many aspects of this project whichrequire further investigation.

Perhaps most obviously, this investigation failed toundertake any tests with cryogens. Supporting the workalready completed with practical tests at cryogenic tem-peratures should be a priority of any future projects. Ad-ditionally, the results of the viscosity experiment wereinconclusive and so more research is needed in this area.The starting point should be the design of a better pump-ing system than the gravity feed used here, so that vis-cosity tests can be conducted in the vortex diode, whichhas been shown to be by far the more effective of the twomodels used.

Whilst CFD proved useful, several improvementscould be made in the modelling process. The capabil-ity of computer simulations in this project was severelylimited by a lack of computing power, such that gener-ating data took a long time, and was limited to only onecomputer. CFD is a powerful tool and could be incredi-bly useful in this field, but the complexity and breadth ofCFD means it is probably best treated as the focus of anindependent study, rather than as a supporting aspect ofa practical investigation.

5 Conclusions

The vortex diode design was shown to be more effectivethan the Tesla diode in terms of diodicity. This supe-riority was especially clear at high Reynolds numbers.Extrapolating the data towards the even higher Reynoldsnumbers of low-viscosity fluids suggests that the vortexdiode would be the most effective for use with cryogens.

The management of low-viscosity fluids was simu-lated for the small Tesla diode and the CFD results pre-dicted that diodicity sees a slight increase towards lowerviscosities. However, the simulated diodicities were stillquite low; it was predicted that liquid helium would giverise to a diodicity of 2.65.

The CFD modelling was a valuable tool for under-standing the physics at work within the diodes. However,the results failed to correlate with much of the experi-mental data, so improvements to the CFD model mustbe made before predictions can be made for low viscos-ity fluids.

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Bibliography[1] C. Wang, T. Chang, M. Rong, Y. Dai, Z. Ni, L. Li and Q. Wang, Optimal Design for High-Field MRI Super-

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[7] T. Baig, Z. Yao, D. Doll, M. Tomsic and M. Martens, Conduction cooled magnet design for 1.5T, 3.0T and7.0T MRI systems, Superconductor Science & Technology, 2014, 27, 125012.

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AppendicesA Flowmeter Calibration

Figure 20 Linear relationship between mass flow rate and flowmeter rotor frequency. F = 1.812ν−5.269 Where F is the massflow rate in g s−1 and ν is the rotor frequency, in Hertz. The y-intercept shows that a minimum flow rate of 5.3 g s−1 is requiredto turn the paddles of the flowmeter. However, a flow rate of 5g s−1 could not produce a reliable reading on the manometer, asits sensitivity was too low. For this reason a minimum flow rate of 20g s−1 was chosen for data collection.

Each system’s actual flow rate was within 1% of the predicted flow rate when the flow rate was set to 100g s−1

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B Diode Designs

B.1 Vortex Diode

Figure 21 Schematic diagram showing the dimensions (in mm) of the large and small vortex diodes.

B.2 Tesla Diode

Figure 22 Schematic diagram showing the dimensions (in mm) of the large and small Tesla diodes.

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C Certification of OwnershipThis Project Report/Dissertation presented as part of, and in accordance with, the requirements for the Final Degreeof BSc at the University of Bristol, Faculty of Science.

We hereby assert that we own exclusive copyright in the item named below. we give permission to the Universityof Bristol Library to add this item to its stock and to make it available for consultation in the library, and for inter-library lending for use in another library. It may be copied in full or in part for any bona fide library or researchworked, on the understanding that users are made aware of their obligations under copyright legislation, i.e. that noquotation and no information derived from it may be published without the authors’ prior consent.

Authors Edmund Erskine, Thomas Greening, Alice Rushton, and Adam TaylorTitle Siemens Magnet Technology - Industrial Group Project: Low Temperature Fluid

ManagementDate of submission 30/04/2015

Signed:

Full names: Edmund Erskine, Thomas Greening, Alice Rushton, and Adam Taylor

Date: 30/04/2015

This project/dissertation is the property of the University of Bristol Library and may only be used with due regardto the rights of the author. Bibliographical references may be noted, but no part may be copied for use or quotationin any published work without the prior permission of the author. In addition, due acknowledgement for any usemust be made.

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