Heat Transfer Correlations Between a Heated Surface and Liquid & Superﬂuid Helium

For Better Understanding of the Thermal Stability of the Superconducting Dipole Magnets in the LHC at CERN

Jonas Lantz

LITH-IEI-TEK-A--07/00225--SE

Examensarbete Institutionen för ekonomisk och industriell utveckling

Examensarbete LITH-IEI-TEK-A--07/00225--SE

Heat Transfer Correlations Between a Heated Surface and Liquid & Superﬂuid Helium

For Better Understanding of the Thermal Stability of the Superconducting Dipole Magnets in the LHC at CERN

Jonas Lantz

Handledare: Arjan Verweĳ CERN, Accelerator Technology Department Gerard Willering CERN, Accelerator Technology Department Examinator: Dan Loyd IEI, Linköping University

Linköping, 19 October, 2007

Avdelning, Institution Datum Division, Department Date Division of Applied Thermodynamics and Fluid Me- chanics Department of Management and Engineering 2007-10-19 Linköpings universitet SE-581 83 Linköping, Sweden

Språk Rapporttyp ISBN Language Report category —

Svenska/Swedish Licentiatavhandling ISRN Engelska/English Examensarbete LITH-IEI-TEK-A--07/00225--SE C-uppsats Serietitel och serienummer ISSN D-uppsats Title of series, numbering — Övrig rapport URL för elektronisk version http://www.ikp.liu.se/mvs/ http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-10124

Titel Heat Transfer Correlations Between a Heated Surface and Liquid & Superﬂuid Title Helium

For Better Understanding of the Thermal Stability of the Superconducting Dipole Magnets in the LHC at CERN

Författare Jonas Lantz Author

Sammanfattning Abstract

This thesis is a study of the heat transfer correlations between a wire and liquid helium cooled to either 1.9 or 4.3 K. The wire resembles a part of a superconducting magnet used in the Large Hadron Collider (LHC) particle accelerator currently being built at CERN. The magnets are cooled to 1.9 K and using helium as a coolant is very efficient, especially at extremely low temperatures since it then becomes a superfluid with an apparent infinite thermal conductivity. The cooling of the magnet is very important, since the superconducting wires need to be thermally stable.

Thermal stability means that a superconductive magnet can remain superconducting, even if a part of the magnet becomes normal conductive due to a temperature increase. This means that if heat is generated in a wire, it must be transferred to the helium by some sort of heat transfer mechanism, or along the wire or to the neighbouring wires by conduction. Since the magnets need to be superconductive for the operation of the particle accelerator, it is crucial to keep the wires cold. Therefore, it is necessary to understand the heat transfer mechanisms from the wires to the liquid helium.

The scope of this thesis was to describe the heat transfer mechanisms from a heater immersed in liquid and superfluid helium. By performing both experiments and simulations, it was possible to determine properties like heat transfer correlations, critical heat flux limits, and the differences between transient and steady-state heat flow. The measured values were in good agreement with values found in literature with a few exceptions. These differences could be due to measurement errors. A numerical program was written in Matlab and it was able to simulate the experimental temperature and heat flux response with good accuracy for a given heat generation.

Nyckelord Keywords superﬂuid, helium, heat transfer correlation, cooling

Abstract

This thesis is a study of the heat transfer correlations between a wire and liquid helium cooled to either 1.9 or 4.3 K. The wire resembles a part of a superconducting magnet used in the Large Hadron Collider (LHC) particle accelerator currently being built at CERN. The magnets are cooled to 1.9 K and using helium as a coolant is very efficient, especially at extremely low temperatures since it then becomes a superfluid with an apparent infinite thermal conductivity. The cooling of the magnet is very important, since the superconducting wires need to be thermally stable.

Thermal stability means that a superconductive magnet can remain superconducting, even if a part of the magnet becomes normal conductive due to a temperature increase. This means that if heat is generated in a wire, it must be transferred to the helium by some sort of heat transfer mechanism, or along the wire or to the neighbouring wires by conduction. Since the magnets need to be superconductive for the operation of the particle accelerator, it is crucial to keep the wires cold. Therefore, it is necessary to understand the heat transfer mechanisms from the wires to the liquid helium.

The scope of this thesis was to describe the heat transfer mechanisms from a heater immersed in liquid and superfluid helium. By performing both experiments and simulations, it was possible to determine properties like heat transfer correlations, critical heat flux limits, and the differences between transient and steady-state heat flow. The measured values were in good agreement with values found in literature with a few exceptions. These differences could be due to measurement errors. A numerical program was written in Matlab and it was able to simulate the experimental temperature and heat flux response with good accuracy for a given heat generation.

Acknowledgments

This thesis have been conducted at the AT-MCS-SC group at CERN, and I would like to thank my supervisor Arjan Verweĳ for his invaluable support, thoughts and guidance during the year I spent at CERN. I would also like to give my warm thanks to my ”unoﬃcial” supervisor and collegue Gerard Willering who supported me throughout my project, took the time to answer my stupid questions and helped me with a lot of other things, no matter if it was work-related or just ﬁnding the fastest way up a mountain. Dank u wel!

For support in the laboratory and help with the cryogenic systems I would like to give my sincere thanks to Stefano Geminian, Pierre-François Jacquot, Alejandro Bastos Marzal, Jean Louis Servais, and David Richter. Without your help my measurements would not have been done. Un grand merci à tous!

A big thanks also goes to my examiner Dan Loyd, who gave valuable feedback and ideas to my project. Tack!

Finally, I would like to thank all my friends in Geneva for making my time here unforgettable. I am going to miss the snowboarding, hiking, partying and all the other good times we shared during this year. Thank you, Merci, Tack, Takk, Danke, Grazie, Gracias...!

Jonas Lantz Geneva, September 2007

vii

Contents

1 Introduction 9 1.1 CERN, a Short Introduction ...... 9 1.2 The Large Hadron Collider - LHC ...... 9 1.3 The Proton Beam ...... 11 1.4 Detectors ...... 11 1.5 The LHC Dipole Magnets ...... 11 1.6 Superconducting Cables ...... 13 1.7 Problem Formulation ...... 14

2 Theory 15 2.1 Superconductivity ...... 15 2.2 Superfluidity ...... 17 2.3 Liquid Helium ...... 18 2.3.1 Introduction ...... 18 2.3.2 Thermal Properties of Liquid and Superfluid Helium . . . . 19 2.4 Helium as a Classical Fluid, He I ...... 21 2.4.1 Transient Heat Flow ...... 21 2.4.2 Natural Convection ...... 22 2.4.3 Nucleate Boiling ...... 23 2.4.4 Film Boiling ...... 23 2.4.5 Summary of He I Heat Flow ...... 24 2.5 Helium as a Quantum Fluid, He II ...... 25 2.5.1 The Two-fluid Model ...... 25 2.5.2 He II Dissipation Mechanisms ...... 26 2.5.3 He II Heat Transport ...... 27 2.5.4 Kapitza Conductance ...... 30 2.5.5 Transient Heat Flow Mechanisms ...... 30 2.5.6 Film Boiling ...... 31 2.5.7 Summary of He II Heat Flow ...... 32

3 Experimental Setup 33 3.1 Preparation ...... 34 3.1.1 Constantan Wire ...... 34 3.1.2 Thermocouples ...... 34

ix x Contents

3.1.3 Wiring ...... 37 3.1.4 Heat Generation and Heat Flux ...... 37 3.1.5 Signal Ampliﬁcation ...... 37 3.1.6 Data Acquisition System, DAQ ...... 38 3.1.7 Cryostat ...... 38 3.1.8 Cernox Temperature Probes ...... 38 3.1.9 Assembly ...... 39 3.2 Measurements ...... 39 3.2.1 Measured Parameters at 4.3 K, He I ...... 40 3.2.2 Measured Parameters at 1.9 K, He II ...... 43

4 Numerical Model 45 4.1 Derivation of Governing Equations ...... 46 4.1.1 Finite Diﬀerences ...... 46 4.1.2 Boundary Conditions ...... 47 4.1.3 Material Parameters ...... 48 4.1.4 Heating ...... 49 4.1.5 Helium Heat Flow ...... 49 4.2 Matlab Implementation ...... 50

5 Results & Discussion 53 5.1 Results for He I ...... 53 5.1.1 Experimental Results ...... 53 5.1.2 Numerical Results ...... 57 5.1.3 Comparison ...... 59 5.2 Results for He II ...... 60 5.2.1 Experimental Results ...... 60 5.2.2 Numerical Results ...... 64 5.2.3 Comparison ...... 65

6 Conclusions & Future Work 67

Bibliography 69

A Calculations 71 A.1 Estimation of Heating Power to Heat up the Wire ...... 71 A.2 Scaling of Power and Current for Numerical Program ...... 72 A.3 Thermal Radiation Estimation ...... 73

B Graphs 75 B.1 Scale Factors ...... 75 B.2 Experimental Results at 4.3 K ...... 76 B.3 Experimental Results at 1.9 K ...... 79

C Source Code for Matlab Programs 82 C.1 Batch ﬁle ...... 82 C.2 Numerical program ...... 83 List of Figures

1.1 Layout of the CERN particle accelerators, not in scale...... 10 1.2 The cross-section of the twin-aperture LHC dipole magnet. . . . . 12 1.3 The magnetic field produced in the dipole magnet...... 12 1.4 Left: Photo of a Rutherford cable. Center: Photo of the cross- section of one wire, showing the copper matrix and bundles containing the Nb-Ti filaments. Right: Photo of the filaments in each bundle...... 13 1.5 Cross-section of a Rutherford cable...... 13 1.6 Schematic figure of the heat flow from a wire in a cross-section of a Rutherford cable, compare with figure 1.5. Dashed lines corresponds to conduction between wires and solid lines heat flow to liquid helium. Missing in the figure is the heat conduction along the wire...... 14

2.1 A schematic figure of the critical surface for a superconducting Nb- Ti cable...... 15 2.2 Phase diagram for 4helium at low temperatures...... 18 2.3 Left: Density ρ as a function of temperature. Right: Specific heat capacity cp as a function of temperature...... 19 2.4 Left: Entropy as a function of temperature. Right: Thermal heat conductivity as a function of temperature...... 19 2.5 A flow chart describing the different heat transfer regimes in He I. 21 2.6 Surface temperature difference versus time, with varying step heat flux. Valid only for He I...... 22 2.7 A schematic figure that shows at which temperatures and heat fluxes the different steady-state heat flow regimes are active. Note the logaritmic scales...... 24 2.8 The temperature dependency for the density of superfluid and normal components in He II...... 27 2.9 Heat conductivity function f −1 for turbulent He II...... 29 2.10 A flow chart describing the different heat transfer regimes in He II. 29 2.11 Transient and steady state heat transfer in He II...... 32 2.12 Heat flux versus temperature difference in steady-state He II. . . . 32

3.1 Principal sketch of a thermocouple with a metal block used for reference temperature...... 35 3.2 The frame used in the experiments. Black lines represent superconducting current leads and grey lines represent the contantan wires used as heaters...... 39 3.3 Left: A typical temperature versus heat flux graph, with two heat transfer regimes visible: nucleate boiling (NB) and film boiling (FB). Right: A zoom of the lower left corner of the left figure, revealing the natural convection regime (NC)...... 40 2 Contents

3.4 Left: A typical temperature response for He I when ramping the heat generation. Right: A zoom on the lower left corner, showing the natural convection regime and a typical curve fit (notice the different scales)...... 41 3.5 Left: Nucleate boiling with a non-linear curve fit. Right: A curve fit for the film boiling...... 41 3.6 Left: A typical temperature response for He II with the resolution set to ”high” on the amplifier. Right: A measurement with a smaller resolution allowing for a higher temperature signal. Now the entire temperature range is shown (note the different scales)...... 44 3.7 Left: A curve fit on the Kapitza heat transfer regime. Right: A curve fit on the film boiling regime when ramping down the heat generation...... 44

4.1 Principal look of the geometry used in the numerical model and the associated heat ﬂuxes...... 46 4.2 Cross-section of the numerical model, with 4 layers and 8 sections in each layer...... 48

5.1 Left: The temperature when the heat regime changes from nucleate boiling to film boiling, measured by seven thermocouples. Right: The same measurement, after being scaled. Note the different y-scales. 54 5.2 Left: An example showing the nucleate boiling and film boiling regimes for a horizontal and vertical wire. Right: A zoom which shows the natural convection regime...... 56 5.3 Left: The heat flux as a function of temperature. Right: Figure 2.7 from the theory chapter, for comparison...... 57 5.4 Left: The temperature and generated power as a function of time. Right: The temperature as a function of heat flux...... 57 5.5 Left: The internal heat profile of a wire that is heated homogeneously while being cooled by the nucleate boiling regime. Right: A simulation of the internal temperatures when placing a thermocouple on the surface...... 58 5.6 Left: Simulation and experimental measurements plotted in the same graph. Right: A zoom on the natural convection regime. . . . 59 5.7 Left: The measured temperatures at T∗. Right: The scaled temperatures of the same measurement...... 60 5.8 Left: A typical graph of the heat flux as a function of temperature for horizontal and vertical wires. Right: The Kapitza heat regime. 61 5.9 Left: A plot with logaritmic scales, showing both the Kapitza and the film boiling regime. Right: Figure 2.12 from the theory chapter for comparison...... 63 5.10 A comparison of the heat transfer at different temperatures in He II. 64 5.11 Left: The internal heat profile when heating in He II. Right: A comparison between simulation and experimental values...... 65 Contents 3

A.1 Speciﬁc Heat Capacity cp for Constantan at low temperatures. . . 71 A.2 Temperature response in K (left) to the power generation in W (right). 72

B.1 The heat transfer coefficient for natural convection...... 76 B.2 The heat transfer coefficient for nucleate boiling...... 76 B.3 The exponent used in in the temperature difference in nucleate boiling. 77 B.4 The heat transfer coefficient for film boiling...... 77 B.5 Critical heat flux versus critical temperature for natural convection. 78 B.6 Critical heat flux versus critical temperature for nucleate boiling. . 78 B.7 Recovery heat flux versus recovery temperature from film boiling. . 79 B.8 Heat transfer coefficient for Kapitza conductance...... 79 B.9 Exponent in Kapitza conductance...... 80 B.10 Critical heat flux versus critcal temperature for Kapitza conductance. 80 B.11 Critical temperature for Kapitza conductance...... 81 B.12 Scaled critical temperature for Kapitza conductance...... 81

List of Tables

1.1 The main parameters for one of the LHC dipole magnets...... 11

3.1 Response data for AuFe-Chromel thermocouples...... 36 3.2 A summary of the measured parameters in liquid He I...... 42 3.3 The measured parameters in superﬂuid He II...... 43

5.1 The measured steady-state parameters in liquid He I...... 55 5.2 General values for the heat transfer correlations in He I...... 55 5.3 The measured parameters in liquid He II...... 62 5.4 General values for the heat transfer parameters in He II...... 62

B.1 The temperature scale factors in He I...... 75 B.2 The temperature scale factors in He II...... 75

Nomenclature

Latin Letters

2 aFBI Heat transfer coefficient He I film boiling W/m -K page 23 2 aFBII Heat transfer coefficient He II film boiling W/m -K page 31 2 nKAP aKAP Heat transfer coefficient Kapitza W/m -K page 30 2 n aNB Heat transfer coefficient nucleate boiling W/m -K page 23 2 aNC Heat transfer coefficient natural convec- W/m -K page 23 tion 2 nT rans aT rans Transient heat transfer coefficient W/m -K page 22 A Area m2 page 48 A Gorter-Mellink mutual friction parame- m-s/kg page 28 ter B Magnetic field T page 13 c Speed of light in vacuum m/s page 11 cp Isobaric specific heat capacity J/kg-K page 19 d Diameter m page 26 f −1 Heat conductivity function for He II W3/m5-K page 29 h Planck’s constant J-s page 27 He I Liquid phase of helium - page 18 He II Superfluid phase of helium - page 18 I Current A page 37 J Current density A/m2 page 13 k Thermal conductivity W/m-K page 19 L Characteristic length m page 27 l Length m page 49 m Mass kg page 71 m4 Mass of helium atom kg page 27 P Power W page 37 P Pressure Pa page 28 3 q˙G Internal heat generation W/m page 46 Q Heat energy J page 71 q Rate of heat flow W/m2 page 22 q00 Heat flow W/m2 page 48 q∗ Critical heat flow nucleate boiling W/m2 page 23 2 qc Critical heat flow in natural convection W/m page 23

5 6 Nomenclature

2 qr Recovery heat flow to nucleate boiling W/m page 24 R2 Coefficient of determination - page 41 R Resistance Ω page 49 Re Reynolds number - page 27 S Entropy J/K page 26 S Seebeck coefficient V/K page 34 s Specific entropy J/kg-K page 19 sn Normal component specific entropy J/kg-K page 25 ss Superfluid component specific entropy J/kg-K page 26 T ∗ Temperature limit for nucleate boiling K page 40 Tc Temperature limit for natural convection K page 40 Tr Temperature limit for recovery from film K page 40 boiling Tmax Maximum temperature K page 40 T Temperature K page 13 T Thomson coefficient V/K page 34 t Time s page 46 Tb Temperature of helium bath K page 22 Tλ Transition temperature between He I and K page 18 He II U Internal energy J page 25 U Voltage V page 37 v Velocity m/s page 26 Vol Volume m3 page 48

Greek Letters

α Thermal diffusivity m2/s page 46 β Geometrical constant - page 28 ρ Density kg/m3 page 19 3 ρn Normal component density kg/m page 25 3 ρs Superfluid component density kg/m page 26 Emmisivity - page 73 2 ηn Normal component viscosity N-s/m page 25 2 ηs Superfluid component viscosity N-s/m page 26 ∇2 Laplacian Operator - page 46 Π Peltier coefficient - page 34 ρ Resistivity Ω-m page 49 σ Stefan-Boltzmann constant W/m2-K4 page 73

Superscripts

+ Value after next time step - page 47 n Exponent for nucleate boiling - page 23 nKAP Exponent in Kapitza heat transfer - page 30 nT rans Exponent in transient heat transfer - page 22 Nomenclature 7

Subscripts

b Helium bath - page 22 c Critical value - page 13 i,j,k Spatial indices - page 47 l Liquid - page 24 n Normal component - page 25 r Radial direction - page 46 φ Azimuthal direction - page 46 cs Cross-section - page 48 s Superﬂuid component - page 26 v Vapour - page 24 z Axial direction - page 46

Chapter 1

Introduction

1.1 CERN, a Short Introduction

CERN, or European Organization for Nuclear Research, was founded in 1954 and is the world’s largest particle physics centre. The name originally comes from the French ”Conseil Européen pour la Recherche Nucléaire”. It is situated on the Franco-Swiss border outside Geneva and currently includes 20 member states and several observer states and organizations. As stated in the funding convention, CERN does only pure scientiﬁc research:

The Organization shall provide for collaboration among European States in nuclear research of a pure scientiﬁc and fundamental character, and in research essentially related thereto. The Organization shall have no concern with work for military requirements and the results of its experimental and theoretical work shall be published or otherwise made generally available.

The research is done in several diﬀerent experiment facilities at CERN, most well known is surely the particle accelerators. Currently, a new particle accelerator is being built, named LHC.

1.2 The Large Hadron Collider - LHC

The Large Hadron Collider, or LHC, is a particle accelerator currently being built at CERN. It is due to begin operating in May 2008, after numerous delays. The accelerator has a circular shape, is about 27 km long and is situated 50-175 meters underground on the border between Switzerland and France, just outside Geneva. When fully operational, it will collide protons with an energy of 7 TeV, traveling very close to the speed of light. Beams of lead nuclei will be also accelerated, smashing together with a collision energy of 1150 TeV. The name LHC comes from the size of the accelerator (Large) and the fact that hadrons (protons or ions) are collided with each other. Some of the goals with the LHC are trying to answer

9 10 Introduction questions like why particles have mass and also try to bring more clarity into the mystery of antimatter. Scientists still do not know what particles have mass, but the answer might be the so-called Higgs mechanism. The Higgs field has at least one new particle associated with it, the Higgs boson, and that particle should be able to give an answer to why particles have mass. Antimatter was once thought to be the perfect reflection of matter, but current research shows that there is a difference between matter and antimatter and the LHC might bring an answer to why.

The LHC consists of more than 9000 magnets whereof the 1232 main superconducting dipole magnets of 15 m length each are installed to guide the beams. The dipole magnets are one of the most crucial parts of the accelerator because the maximum energy of the proton beam is directly proportional to the strength of the dipole magnet field. The LHC dipoles will have a magnetic field of about 8.4 tesla and in order to achieve this, the magnets need to be cooled with superfluid helium to 1.9 Kelvin. About 36 800 tonnes of mass need to be kept at 1.9 K, requiring very large cryogenic systems. If normal magnets were used instead of superconducting magnets, the accelerator ring would have to be 120 km long and use 40 times more electricty [1]. The LHC is not the only particle accelerator at CERN, but by far the biggest in terms of size and energy. Figure 1.1 shows the entire accelerator complex.

Figure 1.1. Layout of the CERN particle accelerators, not in scale. 1.3 The Proton Beam 11

1.3 The Proton Beam

The proton beam being accelerated inside the LHC is not a continuous line of protons, but instead the protons are grouped into almost 3000 bunches with about 7 meters of space between each bunch. There are two proton beams inside the LHC, one beam going clockwise and the other counterclockwise and the beams are crossing each other at certain interaction points, where the collisions occur. In the interaction points, each bunch is squeezed to allow for a greater chance of collision. Each bunch consists of almost 100 billion particles, so in each interaction point there are 200 billon particles at the same time. Since the particles are so small, only about 20 of these particles are expected to collide in each bunch crossing. But, since the beam is traveling with 0.999999991 · c, where c is the speed of light in vacuum, almost 30 million bunches will cross every second, creating 600 millions collisions per second.

1.4 Detectors

In each interaction point a detector is installed. The reasons to collide particles are to ﬁnd out what is inside them and to use the energy available in every collision to create new particles. The detectors measure particle properties such as momen- tum, charge and energy. There are four main detectors, named ATLAS, CMS, ALICE and LHC-b, and two smaller detectors named TOTEM and LHC-f. The amount of data measured at the detectors will be enormous, in total the detectors will handle as much information as the entire European telecommunications network does today!

1.5 The LHC Dipole Magnets

The main magnets in the LHC are dipoles, used to deﬂect the proton beam around the accelerator. The most interesting parameters are summarized in table 1.1.

Nominal current 11796 A Peak ﬁeld in coil 8.76 T Operating temperature 1.9 K Length 15.2 m Mass of cold mass 23.8 tonnes

Table 1.1. The main parameters for one of the LHC dipole magnets.

The LHC consists of 1232 dipole magnets, divided into eight octants with 154 magnets each. By dividing the magnets into eight sections with separate powering and cryogenic systems, the stored magnetic power per dipole circuit is reduced. The magnets need to be cooled to 1.9 K to become superconducting, otherwise it would be diﬃcult to carry the high current which produces the magnetic ﬁeld. 12 Introduction

Figure 1.2. The cross-section of the twin-aperture LHC dipole magnet.

Figure 1.2 shows the cross-section of a LHC dipole magnet, and for this thesis the most interesting thing is the superconducting coils that surrounds the beam screen. The coils are made up out of stacked superconducting cables that together with the iron yoke produces the magnetic ﬁeld, ﬁgure 1.3, needed to bend the particle beam inside the accelerator.

Figure 1.3. The magnetic ﬁeld produced in the dipole magnet. 1.6 Superconducting Cables 13

1.6 Superconducting Cables

The coil in the dipole magents are made of Niobium-Titanium cables, Nb-Ti. These cables are superconducting Rutherford type cables, and consists of 28 or 36 smaller wires which are pressed to a trapezoidal shape, see figures 1.4 and 1.5. The number of wires depends on where in the magnet the cable is placed. Each wireis about 1 mm in diameter and can alone carry about 600 A when superconducting. The wires consists of a multifilamentary Nb-Ti superconductor with a copper matrix. The Nb-Ti filaments have a diameter of about 5 µm and are shown in detail in figure 1.4. About 7600 km of supercondicting cable is used in the LHC, weighing over 1200 tonnes [2].

Figure 1.4. Left: Photo of a Rutherford cable. Center: Photo of the cross-section of one wire, showing the copper matrix and bundles containing the Nb-Ti ﬁlaments. Right: Photo of the ﬁlaments in each bundle.

Figure 1.5. Cross-section of a Rutherford cable.

The cable is only superconducting if three parameters are below their critical values. These parameters are: 2 • Current density Jc [A/m ]

• Magnetic ﬁeld Bc [T]

• Temperature Tc [K] where the subscript c denotes critical value. Order of magnitude of each each parameter is typically 10 kA/mm2, 10 T and 10 K, but the actual value depends on other parameters, which is discussed in section 2.1. Keeping these parameters below their critical values is a central part of the studies of stability of superconductive materials. The inside of a cable is composed of a network of inter-connected cavities ﬁlled with superﬂuid helium at 1.9 K for cooling [3]. Generally, two types of heating the superconducting magnet are possible: beam-induced loads and fric- tional heating due to movement of wires and cables. 14 Introduction

There are several different beam-induced loads possible in the LHC, a few examples are synchroton radiation, proton diffusion, losses of secondary particles and nuclear inelastic beam-gas scattering. For a complete list, see [2]. Secondly, if the magnetic field is too big, the Lorentz force can move the wires or cables, generating fricton. If enough heat is generated in the cable and the cooling is insufficient, the temperature of a part of the superconducting cable can rise above the critical temperature Tc, making it normal conductive. If current is passed through the normal part, joule heating will occur. Joule heating (also known as resistive heating) is caused when electrons flowing in a conductor collide with the atomic ions. Each collision increases the kinetic energy of the ions, resulting in a temperature increase of the conductor. If a small part of the cable turns normal conductive, joule heating can push other parts of the cable to become normal conductive as well, which leads to more and more heating. An entire LHC dipole magnet can become normal conductive in less than one second, meaning that the superconducting abilitity is lost and the operation of the entire accelerator is interrupted.

1.7 Problem Formulation

Thermal stability means that a superconducting magnet still can be superconducting, even though a part of the magnet has become normal conductive due to a temperature increase. This means that the generated heat in a wire in the cable must be transferred to the helium by some sort of heat transfer mechanism, or along the wire or to the neighbouring wires by conduction. Since it is crucial to keep the cables in the magnets superconducting, it is necessary to understand the heat transfer mechanisms from the wires to the liquid helium. The scope of this thesis is to describe the heat transfer mechanisms from a heater immersed in liquid helium and by doing both experiments and simulations, determine properties like heat transfer correlations, critical heat flux limits, and the differences between transient and steady-state heat flow. The simulations should be able to reproduce the results from the experiments.

Figure 1.6. Schematic figure of the heat flow from a wire in a cross-section of a Ruther- ford cable, compare with figure 1.5. Dashed lines corresponds to conduction between wires and solid lines heat flow to liquid helium. Missing in the figure is the heat conduction along the wire. Chapter 2

Theory

This theory chapter focuses on the properties of liquid and superﬂuid helium, but a short introduction to superconductivity and superﬂuidity is given.

2.1 Superconductivity

Superconductivity means that a material has exactly zero electrical resistance. It was first discovered by a Dutch physicist named Heike Kamerlingh Onnes in 1911. A material is only superconductive if the temperature, current density and applied magnetic field are below critical values. This can be characterized by the critical surface, see figure 2.1.

Figure 2.1. A schematic ﬁgure of the critical surface for a superconducting Nb-Ti cable.

When all of these three parameters are below the surface, the material is superconducting and if one or more are above the surface the material will return to the normal conducting state. If a part of the superconductive material gets warmer

15 16 Theory than the rest of the material, that part can become resistive which, if current is passed through, will generate heat. That heat can spread troughout the material, leading to a total loss of superconductivity. This is of course something that is undesirable for magnets since it can cause severe damage if the stored magnetic energy suddenly is released. There are two types of superconductors, type I and II. The difference between the two is that type I expel external magnetic field from its interior, while type II can let the magnetic field into its interior. Type I superconductors can actually be levitated due to this phenomenon, which is called the Meissner effect. For applications such as magnets, type II is used.

It was not until 1957 that a complete theory of how superconductivity works on a microscopic scale was proposed. The BCS theory, by Bardeen, Cooper, and Schrieffer, states that the current going through a superconducting material can be explained as pairs of electrons, called Cooper pairs, interacting by exchang- ing phonons. Phonons are quantized lattice vibrations, much in the same way as photons are quantized electromagnetic radiation. To put it more simply: • Electrical resistance in a material exists because the electrons travelling in the material are scattering due thermal motion of ions and lattice vibrations. • Normally, an electron would not form a pair with another electron since they have the same charge. But if an electron attracts a positively charged ion in the lattice, the attraction can displace the ion. This causes phonons to be emitted, which forms a trough of positive charges around the electron. • A second electron is then drawn to the trough and even though the electrons should repel each other, the force exerted by the phonons is greater making the electrons to form a pair. This is called electron-phonon interaction and the two electrons are referred to as a Cooper pair. • If one of the electrons in the Cooper pair passes an ion in the lattice, the difference in potential between the electron and the ion causes a vibration, which propagates from ion to ion, until the other electron in the Cooper pair absorbs the vibration. The total effect is that one electron emitts a phonon and the other electron absorbs the phonon, and it is this that makes the electrons keep together. • As the first electron goes through a positively charged lattice, the ions in the lattice will be drawn towards it. When the first electron has passed, the lattice returns to its original state. The second electron will then be attracted by the lattice, making the second electron follow the first one.

• If the temperature is higher than the critical temperature Tc, thermal motion of the lattice will break the Cooper pairs and the material will not be superconductive. Superconductivity only occurs at very low temperatures and in special materials, and until the middle of the 1980s it was belived that superconductivity only was possible below 30 K. But with the discovery of Yttrium Barium Copper Oxide, 2.2 Superﬂuidity 17

YBCO, by Maw-Kuen Wu and Paul Chu, the critical temperature changed to a temperature above 77 K, the boiling point of nitrogen. Before that, cooling of superconductive materials had to be done by either liquid hydrogen or liquid helium which are more expensive than liquid nitrogen. The warmest superconductive material today has a critical temperature at 138 K and consists of a thallium-doped, mercuric-cuprate comprised of Mercury, Thallium, Barium, Calcium, Copper and Oxygen.

2.2 Superﬂuidity

Superfluidity is a special phase of matter, and can be described by Bose statistics or BCS theory, depending on the fluid. There are two different types of superfluids; pure and impure. An impure superfluid behaves as if it would be made up by a combination of a component of a fluid with normal properties and a component with superfluid properties. The characteristics of the superfluid component are somewhat unusual: zero viscosity, zero entropy and an apparent infinite thermal conductivity, whereas in the normal fluid there is viscosity, entropy and a finite thermal conductivity. Pure superfluids only consists of the superfluid component. Since superfluids have infinite thermal conductivity, it is impossible to get a temperature gradient in the fluid, much in the same way as it is impossible to get a voltage difference in a superconductor.

Two examples of superﬂuids are the two stable isotopes of helium: 3helium and 4helium, where the former is a fermion and can therefore be explained by BCS theory, and the latter a boson which can be explained by Bose statistics. To be able to become a superﬂuid, the atoms or molecules must be condensed to a point where they all occupy the same quantum state. The 4helium atom is a boson and it can directly form groups with other 4helium atoms when the temperature is low enough, around 2 K. The 3helium atoms on the other hand, can not condensate alone so that each atom occupies the same quantum state since they are fermions, obeying the Pauli exclusion principle. But cooled to a low enough temperature, about 2 mK, the 3helium will form pairs with each other just in the same way as electrons form Cooper pairs when a material is becoming superconductive. The 3helium pairs make the particles in each nucleus add up to an even number which makes it a boson and the atoms can then also occupy the same quantum state. 18 Theory

2.3 Liquid Helium 2.3.1 Introduction The reason to use liquid helium as a coolant is that it can easily transfer huge amounts of heat at temperatures below 4 K. The isotope normally used for cooling applications is 4helium, since it is found in nature whereas 3helium is a by-product when producing nuclear weapons and is very rare in nature. One other non- trivial aspect is that 3helium becomes superfluid first at 2 mK compared to 2.17 K for 4helium. In figure 2.2, a phase diagram for 4helium at low temperatures is presented [4]:

Figure 2.2. Phase diagram for 4helium at low temperatures.

From now on and throughout the text when talking about helium, it is the isotope 4He that is being referred to. It is seen in the phase diagram that helium does not solidify even at 0 K at normal pressure. Unlike any other material, helium needs an external pressure at absolute zero to form a solid state. This is due to the helium molecules large zero point energy, and the lowest energy state is the liquid state. It is hard to distinguish solid helium from liquid helium because the refractive indices of the two phases are almost the same. There is no triple point in helium, meaning that there is no coexistence between solid, liquid and gaseous helium. As seen in the phase diagram, liquid helium can be in two phases; He I or He II. These two liquids are extremely different, the He I phase behaves just like any normal fluid while the He II phase behaves like a superfluid. The two phases will be discussed in sections 2.4 and 2.5, respectively. The line in the phase diagram that separates the He I and He II phases is called the lambda-line, because the shape of the specific heat capacity close to the transition has the shape of the Greek letter λ, as seen in figure 2.3. Associated with this line is the lambda-temperature, Tλ, which is 2.17 K at normal pressure. 2.3 Liquid Helium 19

2.3.2 Thermal Properties of Liquid and Superﬂuid Helium Data for all four ﬁgures are taken from [5, 6, 7], and at 1 bar. The data were compared with each other to make sure that they were correct.

Figure 2.3. Left: Density ρ as a function of temperature. Right: Speciﬁc heat capacity cp as a function of temperature.

Figure 2.4. Left: Entropy as a function of temperature. Right: Thermal heat conductivity as a function of temperature.

As seen in figures 2.3 and 2.4, it is obvious that something happens at around 2.2 K and 4.4 K. It is found that these changes in thermal properties are explained by phase transitions between He I, He II and vapour helium. The transition between He I and He II occurs at 2.17 K and between He I and vapour helium at 4.4 K. One reason for the good cooling capacities of liquid helium is the combination between high specific heat capacity and almost infinite thermal conductivity compared to the Nb-Ti and copper in the cable. 20 Theory

There are lots of factors that can affect the heat transfer from a heated surface to the helium: • The surface conditions play a big role in how much heat can be removed. A well polished surface may not transfer as much heat as a rough surface. • The surface orientation can change the characteristics of the heat transfer because of the gravitational force on the fluid motion. If bubbles of gaseous helium forms on the surface due to high temperature, the buoyancy force helps the bubbles detach. A heated surface facing upwards gives the highest heat flux. • The geometry of the application surrounding the surface may play a role. A a big bath of helium without interfering boundaries will not be a problem, but narrow channels can cause vapor lock or other phenomenons affecting the heat transfer. • The type of helium that surrounds the surface is maybe the most important factor. Since the heat conductivity, specific heat capacity, density and entropy are very different for the two phases, the heat transfer correlations changes dramatically. A typical measurement consists of finding the temperature difference ∆T as a function of the heat flux q. But in engineering applications the situation is the opposite: the heat flux can be calculated from a measured temperature difference. The relation between heat flux and temperature difference is called a heat transfer correlation and is of high importance in superconducting applications, since they play a big role when dealing with stability. A heat transfer correlation that is active on small time scales (in the order of µs to ms) is considered transient, while heat transfer correlations active on larger time scales are considered to be steady-state. Usually, the steady-state heat transfer mechanisms are considered to be less effective than the transient, making short time cooling a powerful tool for cooling.

The two diﬀerent helium phases and the associated heat transfer correlations are explained in detail in the following sections. 2.4 Helium as a Classical Fluid, He I 21

2.4 Helium as a Classical Fluid, He I

In the temperature range of 2.17 K to 4.40 K at normal pressure, liquid helium is in a phase called He I. The upper temperature limit is the boiling point and the lower limit is called the lambda point, where the transition to He II occurs. Liquid He I has a relatively small thermal conductivity compared to He II, but the specific heat capacity is large which means that the heat transfer is mainly dominated by convection. The different types of heat transfer regimes that appear in He I are dependent on the temperature difference between the cooled surface and the helium. A transient heat flow regime appears before the onset of any of the three different steady-state regimes. The different regimes can schematically be seen as: Steady-state

Natural Convection 7 ? Heat - Transient - Nucleate Generation Heat Flow S Boiling S 6 S ? Sw Film Boiling

Figure 2.5. A ﬂow chart describing the diﬀerent heat transfer regimes in He I.

The flow-chart in figure 2.5 shows that when the heat generation has started, the heat flow always goes through a transient heat flow regime. Depending on different factors, a transition to either natural convection, nucleate boiling, or film boiling occurs. Assuming that the natural convection regime follows after the transient regime, depending on the heat flow, a transition to nucleate boiling can occur and from there a transition to film boiling is possible. A reduction of heat generation can trigger a transition back to nucleate boiling from film boiling, but a transition to natural convection never happens.

2.4.1 Transient Heat Flow Transient heating means heating during times up to 1 ms. It is limited by the time it takes until natural convection can be fully established, which means that the transient heat transfer is active under a very short time. Nonetheless, this heat transfer regime is considered to be of high importance when dealing with the stability of superconducting magnets. The heat transfer coefficient is much higher than any of the steady-state heat regimes, meaning that more heat can be removed. This is because the heat transfer is dominated by the specific heat of helium and interfacial conductance (also called Kapitza conductance, which will be discussed in section 2.5.4). If enough energy is dissipated to the helium under 22 Theory a certain time, some helium will vaporize and form either bubbles or even a thin film of gaseous helium. This will make the heat transfer less efficient and force a transition to the next regime. The transient regime is generally considered to be controlled by a time or an energy limit which both are a function of helium properties. Steward [8] made a surface temperature difference versus time plot, where the different heat regimes are marked:

Figure 2.6. Surface temperature diﬀerence versus time, with varying step heat ﬂux. Valid only for He I.

Figure 2.6 shows the different heat transfer regimes and the time and heat flux associated with it. A small heat pulse takes longer time to go from the transient heat regime to steady-state, compared to a large heat pulse. For high heat pulses (and thus high surface temperatures), the active heat transfer regimes quickly changes from transient to film boiling, but for smaller heat pulses natural convection or nucleate boiling can be triggered. An useful expression for calculating the transient heat transfer in He I is:

nT rans nT rans q = aT rans(T − Tb ) (2.1)

In equation 2.1, q is the heat flux per unit area, aT ransthe heat transfer coefficient and T − Tb the temperature difference between the helium and the cooled surface. 2 nT rans T rans Values for aT rans are usually set to 180 W/m -K and n to 4 [4].

2.4.2 Natural Convection The natural convection heat transfer regime starts a fluid motion due to density differences caused by temperature gradients. It is therefore essential to have a gravitational force making the different densities flow. One other type of convection is forced convection, where the fluid motion is generated by an external source like a pump or a fan. There are some cryogenic applications where forced helium flow is present, making it possible to control the mass flow and thus optimizing 2.4 Helium as a Classical Fluid, He I 23 the heat transfer. But the cryogenic system for the LHC is constructed to use stationary helium inside the magnets and therefore the forced helium flow will not be explained here. Further reading about forced helium flow can be found in [9]. In natural convection, the heat transfer correlation can be modeled as a linear function:

q = aNC (T − Tb) (2.2) Normal values for the heat transfer coeﬃcient in natural convection in liquid He I are in the orders of 250-500 W/m2K [4].

2.4.3 Nucleate Boiling

Above a critical heat flux called qc, the heat transfer changes from natural convection to nucleate boiling and the amount of heat that is transferred to the helium dramatically increases. This is because bubbles start to form on surface imperfec- tions and as the heat flux is increased these bubbles start to detach, transferring heat away from the surface in the form of gaseous helium. This heat transfer regime is much more efficient than natural convection because of the latent heat of the gaseous helium inside the bubbles: the more bubbles going away from the surface, the more heat is removed. And as each bubble leaves the surface, liquid helium moves down to the surface for cooling. The heat flux was derived by Ku- n tateladze [10], to be proportional to the temperature as q(T ) ∝ (T − Tb) when the heat flux is above the critical heat flux qc but below the film boiling limit q∗. The maximum amount of heat transferred depend on surface treatment and orientation and must be determined experimentally, but an useful heat transfer correlation is: n q = aNB(T − Tb) (2.3) where the exponent n normally takes a value around 1.5-2.5. The heat transfer 2 n coefficient aNB is in the order of 50 kW/(m K ), and the amount of heat that is possible to remove is about 1000 times more than natural convection [4].

2.4.4 Film Boiling Increasing the heating even more will change the heat transfer regime. Above the heat flux limit q∗, the heat transfer goes into a regime called film boiling, meaning that the bubbles that appeared in the nucleate boiling regime have grown and formed a film of gas on the surface. The amount of heat transferred to the helium is much less than in nucleate boiling, since the liquid helium is not in contact with the surface. Breen and Westwater [11] found that the heat transfer could again be modeled as a linear function:

q = aFBI (T − Tb) (2.4)

2 The heat transfer coeﬃcient aFBI is in the range of 300 to 1000 W/(m K) [4], and is then more or less comparable to natural convection. If the heat ﬂux to the helium is reduced, the nucleate boiling regime will appear agagin at a limit 24 Theory

called qr. An expression for qr can be set up by using He I film boiling theory and experimental data as: r ∗ ρv qr = q (2.5) ρv + ρl where subscript l means liquid and v vapour. Calculating the value of the density ∗ terms at 4.3 K gives the expression qr = 0.35 · q which gives results in good agreement with experimental data [9]. Equation 2.5 is only valid for He I and flat planes, but can be used in other geometries as well. Increasing the heating even more in film boiling will not change the heat transfer regime.

2.4.5 Summary of He I Heat Flow Figure 2.7 shows the three different steady-state regimes in a heat flux versus temperature difference plot. NC stands for the natural convection regime, which changes into nucleate boiling when the heat flux is above a heat flux limit qc. NB corresponds to nucleate boiling and above a heat flux q∗ there is a transition to FB which is the film boiling regime. The film boiling regime is active until the heat generation is lowered. At a heat flux qr there is a transition back to nucleate boiling. This phenomena is called hysteresis, meaning that the result is depentent of the history of the system. When turing of the heat generation completely, the active heat transfer regime stays in nucleate boiling until the temperature difference is zero, i.e. natural convection only occurs when heating up. That is why there is not an arrow going from nucleate boiling to natural convection in the flow chart in figure 2.5.

Figure 2.7. A schematic figure that shows at which temperatures and heat fluxes the different steady-state heat flow regimes are active. Note the logaritmic scales. 2.5 Helium as a Quantum Fluid, He II 25

2.5 Helium as a Quantum Fluid, He II

Below the lambda point, the characteristics of helium are very unusual compared to normal fluids. The He II phase is a superfluid, meaning that it is at a quantum- state of matter. The fact is that its characteristics can only be understood and modeled by using quantum mechanics. He II appears to have an apparent vanishing viscosity and an effective thermal conductivity several orders of magnitude higher than high conducting metals. In 1938, Allen and Misener measured the viscosity of He II to be in the orders of 10−12 Pa-s, when it was flowing through a capillary tube [12]. But when using a rotating cylinder viscometer the viscosity was found to be in the order of 10−5 Pa-s. Both measurements were done correctly and the difference in the values were referred to as the He II viscosity paradox. In the same, year Tisza [13] formulated a model suggesting that He II could be seen as two fluids interpenetrating each other. This became known as the two-fluid model, and it gained a lot of credibility since it could explain many experimental results.

2.5.1 The Two-fluid Model Basically, the two-fluid model describes the helium as two fluids interpenetrating each other, one superfluid and one normal fluid. With that in mind, calling He II a superfluid is not entirely true since it is only one part that is superfluid. Therefore, from now on and troughout the text, helium below the lamda point will be called He II and one of its components will be called superfluid.

The superﬂuid component corresponds to the part of the helium which occupies the ground-state energy level, while the normal component consists of a spectrum of excitations above the ground state. The excitations of the normal component are called phonons and rotons. Phonons are quantized vibrations and are related to the temperature of the system. The higher the temperature, the more phonons are in the system, and since every phonon carries a quantum of vibrational energy, higher temperature also means higher internal energy. A roton is a quasiparticle which is a higher-order excitation than a phonon.

Since the superfluid component is at the ground energy state, its internal energy U is zero even at temperatures above 0 K. Because it does not have any internal energy, it does not contribute to the specific heat or entropy since Cv = (∂U/∂T )v. The two-fluid model could therefore provide an explaination for the He II viscosity paradox: in capillary tubes the normal component is stopped by the viscous interaction with the walls but the superfluid can flow through without losses, which results in a vanishing measured viscosity. On the other hand, when using a cylindrical viscometer the normal component is draged by the viscous surface interaction, which results in a higher measured value.

It is assumed that the normal component behaves as a normal liquid, with density ρn, specific entropy sn and viscosity ηn, where the subscript n stands for normal 26 Theory component. On the other hand, the superfluid component with subscript s has a density ρs, but no specific entropy ss or viscosity ηs as explained earlier. The He II fluid properties are a linear combination of the two components, making the total density of the He II as the sum:

ρ = ρn + ρs (2.6)

For the entropy S of He II, the relation looks like:

ρs = ρnsn (2.7) since the superfluid component does not carry any entropy. It is seen in figure 2.4 that the entropy is very temperature dependent, decreasing with decreasing temperature. The specific entropy s is assumed to be constant in He II, taking the value sλ which is the specific entropy at the lambda temperature. This makes the the entropy (the product ρs) temperature dependent by changes in the normal 5.6 components density. The entropy goes approximately as T between 1 and Tλ and therefore the following expression for the ratio between the normal componets density and the density for He II can be written:

ρ T 5.6 n = (2.8) ρ Tλ This implies that the amount of superfluid increases as the temperature decreases and at 1 K, about 99% of the He II is made up of the superfluid component. It also implies that the He II density is made out of only the normal components density at Tλ, which feels intuitive, since the transition to normal liquid helium occurs there. As seen in figure 2.8, the density of the superfluid helium ρs goes to zero when the temperature approaches Tλ. Figure 2.8 is a direct consequence of equation 2.8.

2.5.2 He II Dissipation Mechanisms The two fluid model assumes that the superfluid component has no viscosity, but the He II does not stay dissipationless. Three different dissipation mechanisms can be identified: vortex tangle, normal component turbulence and mutual friction.

When the superﬂuid has reached a critical velocity vsc, quantized vortex lines forms. Experiments have found that the critical velocity is only dependent on the diameter of the vessel containing the liquid helium. As an approximation the velocity can be calculated as

1/4 vsc ≈ 0.003d (2.9) where d is the diameter of the helium vessel. As the diameter dependence is going as the power of 1/4, the critical velocity will increase with increasing diameter, but for practical applications there is an upper limit of about 3 mm/s. The vortices of 2.5 Helium as a Quantum Fluid, He II 27

Figure 2.8. The temperature dependency for the density of superfluid and normal components in He II. the superfluid appears to be created at the boundaries and move as a tangle with the superfluid flow. The dynamics of these vorticies can be explained by classical hydrodynamics with one exception: its circulation. The circulation can only take multiple values of h/m4, where h is Planck’s constant and m4 the mass of the helium atom [14].

In the same way as for the superfluid component, the normal component is also associated with a critical velocity. Since it is a classical fluid it has a transition from laminar to turbulent flow when exceeding a certain Reynolds number. The Reynolds number is defined as Re = ρvnL/ηn, where ρ is the density of He II, vn the normal components velocity, L a characteristic length and ηn the viscosity of the normal component. A Reynolds number above 2000 is usually considered turbulent flow, but it may depend on experimental conditions.

Finally, the third dissipation mechanism is called mutual dissipation and is re- leated to the velocity difference between the two fluid components. When the difference have reached a critical velocity, the two components starts to interact with each other. The interaction is caused by the superfluid vortex tangles who scatter the normal components thermal exitations [15].

2.5.3 He II Heat Transport

For the cooling of superconducting magnets the most important material parameter in superfluid helium is the thermal conductivity. The effective thermal conductivity is very high in superfluid helium, even higher than high-conducting metals. Heat dissipated by a surface can excite some part of the superfluid, taking it out of its ground-energy state and transform it to a normal fluid component. The 28 Theory heat transport is described by the two-fluid model as a thermal counter-flow process where the normal component carry the entropy and temperature to a cold sink, and the superfluid component flows back so that the net mass transport is zero. The amount of heat transported away from the surface with the normal fluid component can be written as: q = ρsT vn (2.10)

Below a critical heat flux qc, the heat flux is proportional to the temperature difference between the surface and the helium. At the critical heat flux limit the velocity difference between the superfluid and the normal component is large enough for mutual friction to start. Chase [16] found that for channels of a helium vessel with a diameter larger than 1 mm, the mutual friction starts for heat fluxes at about 1 mW/cm2. Below the critical heat flux there is no mutual friction and the thermal counterflow is laminar. Assuming steady-state conditions and constant cross-section of the vessel containing the helium, a relation between the pressure gradient and normal fluid velocity can be written as:

βη v ∇P = − n n (2.11) d2 where β is a numerical constant taking the value β = 12 for parallel plates and β = 32 for circular tubes. A relation between the temperature and the pressure gradient can be written as: ∇P ∇T = (2.12) ρs Combining equations 2.10, 2.11 and 2.12 gives the relation between the heat ﬂux and the temperature gradient:

d2ρ2s2T q = − ∇T (2.13) βηn

Note that equation 2.13 is only valid for q < qc, otherwise the heat flux could be increased indefinitely by increasing the diameter (q ∝ d2). For higher heat fluxes an extra term is added to account for the mutual friction. If the normal components flow still is laminar, equation 2.13 can be rewritten to express the temperature gradient as a function of the heat flow:

βηn Aρn 3 ∇T = − 2 2 2 q − 3 4 3 q (2.14) d ρ s T ρss T The second term on the right hand side in equation 2.14 takes into accout the mutual friction interaction, and A is the Gorter-Mellink mutual friction parameter which can be experimentally determined [9]. The term dominates the temperature gradients at medium and high heat fluxes, since it does not have a diameter dependence and have heat flux with cubic power. Therefore, the first term is often neglected and the expression can be simplified to:

∇T = −f(T )q3 (2.15) 2.5 Helium as a Quantum Fluid, He II 29 with Aρn −f(T ) = 3 4 3 (2.16) ρss T Several experimental measurements of heat ﬂux versus temperature gradients have allowed a determination of the function f(T ) and the Gorter-Mellink mutual friction parameter A. The function f −1(T ) is called the heat conductivity function for He II and is shown in ﬁgure 2.9. The function reaches a maximum at 1.9 K which means that the most heat can be transported in the helium at that temperature. This is why cryogenic systems like the LHC operate at 1.9 K.

Figure 2.9. Heat conductivity function f −1 for turbulent He II.

But the heat needs to be transferred from the surface to the helium, and the amount of heat that is possible to transfer is determined by which type of heat transfer regime that is active. Figure 2.10 describes the different types available in He II. After a heat generation a transient heat flow regime starts, followed by either Kapitza conductance or film boiling which are the steady-state regimes in He II.

Transient Heat Flow Steady-state

Second- Kapitza Sound Conductance @ Heat - Kapitza @R - 6 Generation Conductance ? @@R @@R Gorter- Film Mellink Boiling

Figure 2.10. A ﬂow chart describing the diﬀerent heat transfer regimes in He II. 30 Theory

2.5.4 Kapitza Conductance Kapitza conductance, or Kapitza heat transfer, occurs at the interface between a solid and the helium. It was first discovered by Kapitza in 1941, when trying to study the heat flow from copper in He II. He found that the liquid helium did not have any measurable temperature gradients while the temperature of the copper increased. This discontinuity was defined as:

”The interfacial thermal boundary conductance which occurs between any two dissimilar materials where electronic transport does not contribute.” [7]

The effect is only visible at cryogenic temperatures, and it strongly decreases with increased temperature (∝ 1/T 3). Kapitza conductance can be measured in He I, but its contribution to the heat flow is too small to be significant and is therefore often neglected. Mathematically, the Kapitza conductance is defined as: q aKAP0 = lim (2.17) ∆Ts→0 ∆Ts

In equation 2.17, ∆Ts is the temperature difference between the surface and the helium, and aKAP0 the heat transfer coefficient where the subscript 0 refers to the limit ∆Ts → 0 [7]. When the limit is small ”enough”, aKAP0 can be described n with the simple relationship aKAP0 = αT , where values of α and n are found experimentally. However, several experiments have found that aKAP0 can vary as much as 2-3 orders of magnitude between samples [9]. Surface orientation and conditions are believed to play a big role and great care must be taken when designing applications. For practical use, the following heat transfer correlation can be used when calculating the Kapitza heat flux:

nKAP nKAP qKAP = aKAP (T − Tb ) (2.18)

Although the Kapitza conductance is deﬁned by experiments, considerable work has been done to try to explain the physics behind it. Acoustic Mismatch The- ory predicts the lower limit for the Kapitza conductance coeﬃcient aKAP , and Phonon Radiation Limit the upper limit, but agreement with experimental values are poor [9]. These theories fall outside the scope of this thesis, but are explained in detail in [9] and [7].

2.5.5 Transient Heat Flow Mechanisms There are mainly two transient heat transfer mechanisms in He II, one is referred to as ideal, non-turbulent, second sound or Landau regime, and the other is called turbulent or Gorter-Mellink regime. The first regime, second sound, comes from the fact that He II is able to transfer more than one type of sound. Normal sound propagation - or first sound - in a fluid is density variation caused by local pressure gradients. Second sound on the other hand is a propagation of thermal waves as a result of fluctations in local entropy. In non-turbulent He II a wave equation for 2.5 Helium as a Quantum Fluid, He II 31 second sound can be written as:

2 2 ∂ s s ρs 2 2 = ∇ T (2.19) ∂t ρn For small temperature perturbations the second sound velocity, and thus the temperature wave, is about 20 m/s between 1 and 2 K [9]. Second sound waves can only carry a limited amount of energy [17], so for most engineering applications where transient heat flow is used, the transient flow is dominated by internal convection. In this case the heat transfer can be described by the Gorter-Mellink regime, looking like: ∂T 1 1/3 ρC = ∇ · ∇T (2.20) p ∂t f(T ) where f(T ) is the heat conductivity function for He II defined earlier. Because of the huge thermal conductivity of He II in the second sound and Gorter-Mellink regime, Kapitza conductance is belived to be responsible for the temperature difference before the onset of steady-state heat transfer. In fact, the transient heat regime in He II can be seen as the same as the following Kapitza steady-state regime, with one important difference: the maximum heat transfer coefficient is much higher in transient heat transfer than in the steady-state case. An arbitrary limit for the transient heat flux is around 100 kW/m2 while for the steady state case it is around 35-50 kW/m2, although the transient limit varies with the time and energy input [4]. An example is given in figure 2.11.

2.5.6 Film Boiling According to [9], boiling heat transfer in He II is the least understood heat transfer process in He II, but maybe the most important since its properties can lead to catastrophic events in cryogenic systems. It is believed that the film can be made up out of He I, a vapour or both. This triple phenomenon brings all types of helium in close contact with the surface, and because the He I film is very unstable this process can rapidly change to a gaseous film layer. The low thermal conductivity of the insulating film makes the heat transfer much less effective, in the order of 100 times smaller than Kapitza conductance. The heat flux limit back to Kapitza conductance from film boiling qr has been proven difficult to calculate, and the predictions from current models are poor compared with experimental data. This is believed to be because of the complex mechanisms active in the He II film boiling regime. Measurements have found that the ratio qr/aKAP takes a somewhat constant value of 23 K, suggesting the excistence of a recovery temperature Tr. Unfortunately, no further understanding about the effect is available. For engineering applications the heat transfer correlation can be simplified with a linear function looking like:

q = aFBII (T − Tb) (2.21) 2 The heat transfer coeﬃcient aFBII is in the order of 250 to 1000 W/m K [4]. 32 Theory

2.5.7 Summary of He II Heat Flow As seen in ﬁgure 2.11, both the transient and the steady-state Kapitza heat transfer are much better than the ﬁlm boiling in terms of removing heat.

Figure 2.11. Transient and steady state heat transfer in He II.

Figure 2.12. Heat ﬂux versus temperature diﬀerence in steady-state He II.

Figure 2.12 shows the two steady-state heat transfer regimes, where KAP is the Kapitza conductance and FB the ﬁlm boiling regime. Recovery to Kapitza conductance occurs at a lower heating power. This is due to hysteresis. Chapter 3

Experimental Setup

The scope of the experimental part was to measure both transient and steady-state heat transfer parameters. Despite numerous attempts to measure the transient heat flow regimes in both He I and He II, a signal could not be found. This is believed to be because of one or several reasons: • The generated heat inside the wire was not enough to produce a measurable temperature difference. • The amplifier was too slow, meaning that the signal could not be amplified because it was too fast. • The thermocouples were too slow. • The thermocouples were cooled by the liquid helium and could not measure a voltage difference. • The generated heat was transferred to the helium too fast during transient heat transfer. • The noise level was too high, making the signal disappear in the noise. Therefore, only steady-state heat flow could be measured. To be able to measure and understand the different heat flow regimes, an experimental setup was designed in laboratory 163 at CERN. The whole purpose of the experiment was to measure the amount of heat flowing from a wire to liquid helium. A normal conductive wire was used, the reason not to use a superconducting wire was that if a normal resistive wire was used instead, it could be used as a heater at the same time. The material that was chosen as a heater was Constantan, because it has both a high and fairly temperature independent resistivity. It is an alloy with 55% Copper and 45% Nickel and is very often used as a resistance wire. A 2 µm tin-silver coating was applied on the Constantan wire to match the surface conditions on a superconducting wire which is used in the LHC dipole magnets. To be sure that the measured parameters were correct and reproduciable, 4 different Constantan wires were used, with new thermocouples for each wire. The tests were performed

33 34 Experimental Setup with the same settings in temperature and heat generation. The wire was ﬁxated in a frame and several thermocouples were attached to it for measuring the surface temperature. By measuring the surface temperature and knowing the heating power, the heat transfer to the helium could be calculated.

3.1 Preparation 3.1.1 Constantan Wire A Constantan wire with a diameter of 1.016 mm was used, having a resistance of 0.6062 Ω/m. To match the surface conditions of a superconducting LHC dipole wire, a silver-tin coating was applied using a electrolyte bath. The thickness was carefully measured to be 2 µm which is in the same order of magnitude as the coating on the superconducting wire.

3.1.2 Thermocouples In 1821 the German-Estonian physicist Thomas Seebeck discovered that a voltage is produced when a junction of two dissimilar metals are exposed to a temperature gradient. This became known as the Thermoelectric effect. The measured voltage is relative to the temperature gradient compared to a reference temperature, therefore thermocouples can only measure temperature differences, not absolute temperature - unless the reference temperature is at exactly 0 K. Generally there are three different effects to take into account when dealing with thermocouples: Seebeck effect, Peltier effect, and Thomson effect.

• The Seebeck effect is the voltage generated by the temperature difference along the thermocouple wire. The difference in voltage with respect to the temperature is called the Seebeck coefficient S, or thermoelectrical sensitivity.

• The Peltier effect is the opposite of Seebeck effect, it describes how the temperature changes due to variation of voltage. The amount of heat current carried per each unit charge is described by the The Peltier coefficient Π, which is different for different types of material.

• The Thomson eﬀect describes if there is an absorption or evolution of energy when current moves from one end of a wire to the other. The amount of heat is proportional to both current and temperature, and the proportionality constant is known as the Thomson coeﬃcient T. One example of evolution of energy is copper: if the low potential side is at a cold sink and the high potential side is at a heat sink, the current will move from the heat to the cold sink, thus going from a high to a low potential, and there is an evolution of energy. On example of the opposite is Nickel, which will absorb energy.

Since the voltage needs to be measured, the connection between a voltmeter and the thermocouples will also create a junction exposed to a temperature gradient and thus produce another voltage. Fortunately, if these connections are made 3.1 Preparation 35 at a known reference temperature, the voltages for each connection will cancel each other. In ﬁgure 3.1 there is a voltage produced at V1 due to the junction of two dissimilar metals, but there is also a voltage produced at V2, equal in magnitude but opposite in polarity, witch results in that the only voltage measured by Vmeasured is the voltage generated at Vout. Setting up an expression for the measured voltage yields:

TRef TT ip TRef TGauge Z dT Z dT Z dT Z dT V = S (T ) dx+ S (T ) dx+ S (T ) dx+ S (T ) dx out Cu dx Chromel dx AuF e dx Cu dx TGauge TRef TT ip TRef (3.1)

Figure 3.1. Principal sketch of a thermocouple with a metal block used for reference temperature.

Equation 3.1 can be simpliﬁed to:

TT ip TRef TT ip Z Z Z Vout = SChromel(T )dT + SAuF e(T )dT = SChromel(T ) − SAuF e(T ) dT

TRef TT ip TRef (3.2) Note that the voltage produced by the copper wire cancels in a mathematical sense, but in reality noise will be added. Magnetic noise is produced when current is flowing through a conductor, but twisting a cable-pair can cancel most of the effect. If the Seebeck coefficients for both materials and the reference temperature are known, the the only unknown is the measured temperature, which then can be calculated from voltmeter. If the Seebeck coefficients are fairly constant on the expected temperature range, then equation 3.2 can be simplified to:

Vout TT ip = TRef + (3.3) SA − SB 36 Experimental Setup

The diﬀerence SA −SB is often given by the supplier of the thermocouple in charts for dV/dT and EMF at diﬀerent temperatures.

The design critera for the thermocouples was that they would have a high sensitivity at very low temperatures and that the thermal conductivity in the wires would be low, so that they would not interfere with the heat flow from the Constantan wire. Thermocouple wires made out of Chromel and Gold-Iron (0.07% Iron) with a diameter of 0.127 mm, were found to be the best choise. According to tables from the manufacturer, the sensitivity was 9-16 µV/ K in the temperature range 1.2-10 K, as shown in table 3.1. Each wire had an insulation which was carefully removed on both ends. Then one end of the Chromel wire was electrically welded to one end of the Au-Fe wire in Argon gas to make sure that the junction is free from air contamination. After that, copper wires with a diameter of 0.14 mm were welded to each end of the thermocouple. These copper wires were then connected to the wiring going out to the data acquisition system. The welds between the copper wires and the thermocouple were placed on a copper heat sink, to make sure that all welds had the same temperature. The copper heat sink was insulated with Kapton tape to make sure that the thermocouple welds were electrially shielded to prevent current from going through the copper heat sink instead of the thermocouple. The heat sink is assumed to have the same temperature as the helium that surrounds it, which is reasonable since it was large both in size and mass and copper has a very good thermal conductivity. Since the temperature of the helium was known, and thus the temperature of the copper block, the reference temperature in equation 3.3 was known and the measured temperature could easily be calculated. As seen in table 3.1, the sensitivity of the thermocouple at liquid helium temperatures is around 10 µV/K. Since the temperature increase will be in the order magnitude of a few degrees, the signal coming from the thermocouples will be very small. Therefore, the signal had to be amplified before being stored in the data acquisition system. This was done with a Keithley amplifier, described in section 3.1.5. Temperature [K] Emf [µV] dV/dT [µV/K] 1.2 -5299.6 8.98 2 -5292.0 10.1 3.2 -5278.9 11.6 4.2 -5266.8 12.6 10 -5181.8 16.0 20 -5014.0 17.0 30 -4846.4 16.6 40 -4681.5 16.5

Table 3.1. Response data for AuFe-Chromel thermocouples.

Attaching the thermocouples to the Constantan wire was everything but easy, because they were very thin and brittle. The thermocouples were ﬁxated on the Constantan wire using a silver epoxy composition called E-Solder 3025 (Von Roll 3.1 Preparation 37

Inc., USA). The thermal conductivity of the silver epoxy was very high, which made it ideal for connecting the thermocouple to the wire. Very small amounts were used to make sure that as little heat as possible was transferred through the silver epoxy to the helium instead of to the thermocouple. The wire and thermocouples were then heat treated in an oven in 100◦C for 30 minutes to let the silver epoxy harden. Finally, a heat insulating epoxy was carefully put on the thermocouple to prevent them from getting cooled by the helium. The epoxy used was Stycast FT-2850, which had a low thermal conductivity compared to the silver epoxy and was considered to protect the thermocouples from direct cooling from the helium.

3.1.3 Wiring Every thermocouple needed two wires each, and another two wires were used to measure the voltage across the wire. Each wire-pair was also twisted to reduce electro-magnetic interference. Magnetic noise is always produced when current is ﬂowing through a conductor, and twisting two cables can cancel large amounts of the eﬀect.

3.1.4 Heat Generation and Heat Flux By measuring the voltage over the Constantan wire, the generated heating power could be deduced by P = UI, where P is the power, U is the voltage and I the current going through the wire. The amount of current passing through the wire was controlled by an Oxford current source. It was possible to ramp the current with a rate as high as 1200 A/min, but a ramp rate of 18 A/min was found to be suitable. One important assumption was that the heat generated inside the wire was completely lost to the helium, i.e. the amount of heat needed to heat up the wire was very small compared to the heat ﬂux to helium. A calculation in section A.1 gives that the energy needed to heat the wire 1 K at 4.3 K is about 0.3 mJ, while in the same time about 10 J is generated inside the wire and lost to the helium.

3.1.5 Signal Amplification Since the signal coming from the thermocouples was very small, in the order of µV , the signal needed to be amplified. This was done with a Keithley 182 Sensitive Digital Voltmeter (Keithley Instruments, Inc. Cleveland, Ohio USA). The voltmeter had an analog input/output and amplification could be done up to 1000 000 times depending on resolution and input signal. For the most times, 100- 1000 times were enough. One parameter that had to be set was the intergration time. Integration time affects the usable resolution, the amount of reading noise, as well as the ultimate reading rate of the instrument. Three different times were available: 3, 20 and 100 ms. If speed was most important in a measurement, then 3 ms would be a good choise. On the other hand, if the signal was noisy and the signal does not really depend on time, then 100 ms was suitable. A compromise 38 Experimental Setup between noise performance and speed was 20 ms, which was also chosen. It was also possible to set both digital and analog filtering, but these were turned off since additional filtering could be done later in Matlab if necessary. The usable resolution was 6.5 digits and the accuracy of the output ± 0.15% of the output + 1mV at room temperature [18].

3.1.6 Data Acquisition System, DAQ For the data acquisition, a Nicolet Vision (Nicolet Instrument Technologies Inc, Madison Wisconsin USA), was used. For steady state measurements a sample rate of 1000 samples/s was considered to be enough, even though the Nicolet Vision could handle up to 100 000 samples/s. The reason not to use a higher sample rate was that the ﬁles would be really big in size and a resolution of 0.1 ms was considered enough for steady-state measurements. It had 16 analog channels and a 16 bit 100 kS/s digitizer with a 20 kHz analog bandwidth. According to the manufacturer, the analog output error was less than 0.05% [19].

3.1.7 Cryostat The cryostat used in the experiments could automatically go to either 1.9 or 4.3 K in subcooled helium at 1 bar, meaning that both liquid He I and superﬂuid He II could be used in the tests. It was also possible to manually regulate the pressure in the heat exchanger so that temperatures above 1.9 K but below 2.17 K could be used in the experiments. Three diﬀerent temperature probes were used to measure the temperature inside the cryostat. These were used to verify the initial temperature measurements of the thermocouples.

3.1.8 Cernox Temperature Probes For real-time measurement of the helium temperature, two Cernox temperature- sensing elements were used. Cernox, short for Ceramic Nitride-Oxide, is a thin film resistance temperature sensor commercialized by Lakeshore Cryotronics, Inc. The sensor is fabricated from zirconium reactively sputtered in a nitrogen-oxygen atmosphere. The resulting thin film is comprised of conducting zirconium nitride embedded within a zirconium oxide non conducting matrix. Cernox temperature sensors have many specifications desirable in a temperature sensor including high sensitivity, excellent short-term and long-term stability, small physical size, fast thermal response and small calibration shifts when exposed to magnetic fields or ionizing radiation [20].

One of the two Cernoxes was attached to the copper heat sink, while the other measured the liquid helium temperature. In this way it was possible to see if there was any diﬀerence in temperature between the copper heat sink and the liquid helium. A 1 µA current source was connected in series with the two Cernoxes and the resulting voltage was stored in the DAQ. 3.2 Measurements 39

3.1.9 Assembly A frame was produced to keep the Constantan wire in place inside the cryostat, as shown in ﬁgure 3.2. The material used was G10 which is an all-around laminate made out of a continuous glass woven fabric base impregnated with an epoxy resin binder. Superconducting wires were connected to the constantan wire and used as current leads. The frame could be rotated 90◦ inside the cryostat so that the wire could be in either a horizontal or vertical position. Shown in ﬁgure 3.2 is the G10 frame, the copper heat sink, cernox temperature sensors, one thermocouple, two resistive Constantan wires, and two superconducting current leads.

Figure 3.2. The frame used in the experiments. Black lines represent superconducting current leads and grey lines represent the contantan wires used as heaters.

3.2 Measurements

Diﬀerent measurements were performed in the experimental setup. The possible variables to change in the measurements were:

• Helium temperature, He I or He II.

• Heating power generated in the sample, controlled by the current source.

• Heating power generation rate, controlled by the current ramp rate.

• Wire position, alignement in either horizontal or vertical position. 40 Experimental Setup

How the heat transfer parameters were calculated are described in sections 3.2.1 and 3.2.2. The data stored in the DAQ was the temperature voltage from the thermocouples, the voltage over the wire and helium temperature measured from the Cernoxes. The stored data was then imported into a custom-written LabView program for calculation of temperatures and then exported to a ASCII-ﬁle. Finally, Matlab was used to import the ASCII-ﬁles and to analyse the results.

3.2.1 Measured Parameters at 4.3 K, He I A typical measurement was done by chosing a ramp rate, a maximum current and then measuring the resulting temperature as a function of the generated heat. Usually, the current was ramped to the set value and kept there for a few seconds before ramping down again. Plotting the temperature response versus the heat flux gives a graph where it is possible to identify the different heat regimes. From ∗ ∗ such a figure the critical heat fluxes qc, q , qr and temperatures Tc,T ,Tr,Tmax could directly be found. The critical heat fluxes and temperatures are taken when the transition begins from one regime to the other.

Figure 3.3. Left: A typical temperature versus heat flux graph, with two heat transfer regimes visible: nucleate boiling (NB) and film boiling (FB). Right: A zoom of the lower left corner of the left figure, revealing the natural convection regime (NC).

As described earlier in the theory about He I heat transfer in section 2.4, the natural convection and film boiling regimes can be modeled as constant (aNC or aFBI ) times a linear temperature difference, see equations 2.2 and 2.4. The nucleate boiling regime on the other hand is modeled as a constant (aNB) times the temperature difference to the power of n, as described by equation 2.3. So, for natural convection and film boiling a linear curve fit was used, but for nucleate boiling a non-linear equation had to be used instead. Figures 3.4 and 3.5 shows the area where the heat transfer correlations where calculated. A Matlab script was written which used a temperature versus time graph to identify the heat transfer regime. Then the program extracted temperature and heat flux data as an input for the curve fit. The models directly gave the aNC , aNB, n, and aFBI values.

Calculating aFBI for film boiling was done when the temperature was decreasing due to ramping down of power, because the transition between nucleate boiling 3.2 Measurements 41 and film boiling is more or less a step function but the transition back to nucleate boiling goes much slower due to hysteresis. The accuracy of the fitted curves can be measured by the coefficient of determination R2. An R2 value of 1.0 indicates that the line produced by the model perfectly fits the data. Every curve fit had a R2 value of at least 0.98. In figure 3.4 NC stands for natural convection, NB for nucleate boiling and FB for film boiling and the vertical lines between nucleate boiling and film boiling are transitions regimes.

Figure 3.4. Left: A typical temperature response for He I when ramping the heat generation. Right: A zoom on the lower left corner, showing the natural convection regime and a typical curve ﬁt (notice the diﬀerent scales).

Figure 3.5. Left: Nucleate boiling with a non-linear curve fit. Right: A curve fit for the film boiling. 42 Experimental Setup

In the experiments, ramp rates of 1, 6, 12 and 18 A/min were ﬁrst tested, to see if there was a ramp rate dependence on the temperature response. It was found that there was no measurable diﬀerence and therefore 18 A/min were used to save time. Summing up, the following parameters could be found or calculated from measurement data:

Parameter Description qc,Tc Heat flow and temperature limit for natural convection q*, T* Heat flow and temperature limit for nucleate boiling qr,Tr Heat flow and temperature limit for recovery from film boiling aNC Heat transfer coefficient for natural convection aNB, n Heat transfer coefficient for nucleate boiling and exponent

aFBI Heat transfer coeﬃcient for ﬁlm boiling Tmax Maximum temperature

Table 3.2. A summary of the measured parameters in liquid He I. 3.2 Measurements 43

3.2.2 Measured Parameters at 1.9 K, He II In the same way as for He I, in He II the current was ramped to a specified value and kept there for a few seconds before ramping down again. Initially a temperature could not be measured, but it was found that the thin tin-silver coating had become superconducting, which made the current go trough the coating instead of the Constantan. This was solved by applying a magnetic field of 0.2 T, which were found to be the critical magnetic field for the superconductive surface coating. Fortunately, the magnetic field-dependent temperature error in the thermocouples is very low, in the order of a few percent at a magnet field of 5 T and even lower at 0.2 T [21]. For He II more heating power was required to get a change in heat transfer regime, due to the good cooling properties of the Kapitza heat transfer. Therefore, a ramp rate of 240 A/min was used instead, since the current needed to go to film boiling was much higher than in He I. In figure 3.6 the Kapitza conductance and film boiling regimes are visible, and the transition regimes in between. Note that the temperature reaches a maximum of about 29 K in the left graph, this is not the actual temperature but instead the maximum limit of the current settings on the amplifier. Increasing the range on the amplifier allows for higher temperature readings but lower resolution and it was decided that resolution was more interesting than temperatures above 30 K. In the same way as for He I, val- ∗ ∗ ues for q , qr, T , Tr and were read directly out of a temperature versus heating power graph, and curve fitting was used to calculate aKAP , nKAP , and aFBII . In figure 3.6, KAP means Kapitza conductance and FB film boiling.

The measured parameters in He II were:

Parameter Description q*, T* Heat flow and temperature limit for Kapitza heat flow qr,Tr Heat flow and temperature limit for recovery from film boiling aKAP , nKAP Heat transfer coefficient for nucleate boiling and exponent

aFBII Heat transfer coeﬃcient for ﬁlm boiling Tmax Maximum temperature

Table 3.3. The measured parameters in superﬂuid He II. 44 Experimental Setup

Figure 3.6. Left: A typical temperature response for He II with the resolution set to ”high” on the ampliﬁer. Right: A measurement with a smaller resolution allowing for a higher temperature signal. Now the entire temperature range is shown (note the diﬀerent scales).

Figure 3.7. Left: A curve fit on the Kapitza heat transfer regime. Right: A curve fit on the film boiling regime when ramping down the heat generation. Chapter 4

Numerical Model

To validate the experimental data with theory, a numerical model of the experiment was written. The wire can be seen as a solid cylinder in a helium bath with heat generation in the wire due to current ﬂowing through it. The model is designed to simulate the following cases:

• The heat ﬂow to helium, due to a higher surface temperature on the wire.

• Internal temperature gradient in the wire.

• Heat conduction in the axial direction of the wire, to see if there are any end eﬀects.

• Variable power generation, both in time and current.

• The interference between thermocouples and the surface when measuring the temperature.

• A semi-adiabatic case, where the complete surface of the wire was insulated with Stycast.

A three dimensional time dependent model was chosen. Since the geometrical shape of the wire is a cylinder, the governing heat equations were expressed in a cylindrical coordinate system. The numerical scheme used was the explicit Eu- ler ﬁnite diﬀerence method. There were three reasons for chosing the explicit method. First, transient models are not as accurate in the implict method since the truncation error can be larger due to large time steps. There is no constraint on the time step in the implicit method. Secondly, lots of matrix manipulations are needed in every time step, and since it was the transient solution that was computed, each time step will take longer time to compute compared to the explicit method. Thirdly, the explicit method is much easier to program, and since the model already was in three dimensions and time, it was an easy choise.

45 46 Numerical Model

4.1 Derivation of Governing Equations

The model consists of heat conduction inside the wire and heat transfer in various ways from the surface to the liquid helium. The derivation starts with the normal heat conduction equation expressed with a Laplacian operator, ∇2. q˙ 1 ∂T ∇2T + G = (4.1) k α ∂t

In equation 4.1 is T temperature, q˙G internal heat generation, k thermal conductivity, α thermal diﬀusivity and t time. Since it is formulated with a Laplacian operator, it is independent of coordinate system and can easily be transformed into a suitable coordinate system. For a general three-dimensional time-dependent problem in cylindrical coordinates, equation 4.1 becomes:

1 ∂ ∂T 1 ∂2T ∂2T q˙ 1 ∂T r + + + G = (4.2) r ∂r ∂r r2 ∂φ2 ∂z2 k α ∂t where r, φ and z are cylindrical coordinates and it is assumed that the thermal conductivity k is constant over each computational node.

Figure 4.1. Principal look of the geometry used in the numerical model and the associated heat ﬂuxes.

In figure 4.1 the geometry of the model is shown, with arrows indicating the different heat flow directions. The subscripts r, φ and z indicate radial, azimuthal and axial direction, respectively.

4.1.1 Finite Differences To replace the partial derivatives with an algebraic expression, finite differences based on Taylor’s series expansions were used. Since equation 4.2 is a parabolic partial differential equation it lends itself to a marching solution [22], and in this 4.1 Derivation of Governing Equations 47 case the marching variable was the time t. The discretization scheme used was Forward Time Central Space (FTCS), which means that the expression for time will be discretized with a forward difference while the spatial expressions will be central differences. This makes the time difference first order accurate and the spatial differences second order accurate. In the following equations the indices i, j, k representes spatial indices in r, φ, z directions, and T + represents temperature at the next time step. The first term on the left hand side of equation 4.2 can be written as: 1∂r ∂T ∂2T + r = r ∂r ∂r ∂r2 1T − T T − 2T + T i+1,j,k i−1,j,k + r i+1,j,k i,j,k i−1,j,k + O((∆r)2) (4.3) r 2∆r (∆r)2

Second term: 1 ∂2T 1 T − 2T + T = i,j+1,k i,j,k i,j−1,k + O((∆φ)2) (4.4) r2 ∂φ2 r2 (∆φ)2

Third term: ∂2T T − 2T + T = i,j,k+1 i,j,k i,j,k−1 + O((∆z)2) (4.5) ∂z2 (∆z)2 And ﬁnally the right hand side:

+ 1 ∂T 1 T − Ti,j,k = i,j,k + O(dt) (4.6) α ∂t α dt

Rearranging and using that the thermal diffusivity can be expressed as α = k/ρcp, the final expression for three-dimensional transient heat flow in cylindrical coordinates becomes:

+ dt h nTi+1,j,k − Ti−1,j,k Ti,j,k = Ti,j,k + k + ρcpr 2∆r T − 2T + T 1 T − 2T + T + r i+1,j,k i,j,k i−1,j,k + i,j+1,k i,j,k i,j−1,k + (∆r)2 r (∆φ)2 T − 2T + T o i + r i,j,k+1 i,j,k i,j,k−1 +q ˙ r (4.7) (∆z)2 G

Equation 4.7 is the fully discretized equation of equation 4.1, and was used in the numerical model.

4.1.2 Boundary Conditions Equation 4.7 needed to be modified to suit the boundaries in the radial direction. When i = 1 there was no heat conduction in azimuthal direction, since there was only one volume in that layer. The radial heat conduction is connected with all sections in the neighbouring layer, but in axial direction the heat flow worked in the same way as for the interior sections. Since the radial coordinate r = 0 when 48 Numerical Model i = 1, equation 4.7 was modified with an expression for the volume instead of the coordinate r. The heat equation looks like:

n dt h n X T2,j,k − T1,1,k T + = T + k ∆φ∆z + 1,1,k 1,1,k ρc Vol 2 p 1,1,k j=1 T − 2T + T o i + Vol 1,1,k+1 1,1,k 1,1,k−1 +q ˙ Vol (4.8) 1,1,k (∆z)2 G 1,1,k

The n-value in the sum is the number of computational nodes for each layer in radial direction. The expression for the heat equation on the outer layer is similar to the one for the interior layer, but diﬀer in one way: a term containing the heat ﬂow to helium. The heat equation for the surface boundary looks like:

+ dt h nTi+1,j,k − Ti−1,j,k Ti,j,k = Ti,j,k + k + ρcpr 2∆r T − 2T + T 1 T − 2T + T + r i+1,j,k i,j,k i−1,j,k + i,j+1,k i,j,k i,j−1,k + (∆r)2 r (∆φ)2 T − 2T + T o i + r i,j,k+1 i,j,k i,j,k−1 +q ˙ r − q00A (4.9) (∆z)2 G surf

00 The only difference between equations 4.9 and 4.7 is the last term q Asurf which contains the heat flow to helium. The heat flow is counted as negative even though it flows in positive r-direction. See section 4.1.5 on how to model the heat flow. In figure 4.2 a cross-section of the numerical model is shown with the computational nodes.

Figure 4.2. Cross-section of the numerical model, with 4 layers and 8 sections in each layer.

4.1.3 Material Parameters

Material parameters like density and thermal heat conductivity are very temperature dependent and could vary troughout the model. The material parameters were calculated at every computational node from fourth or ﬁfth order polynomials with data taken from [6] and [9]. 4.1 Derivation of Governing Equations 49

4.1.4 Heating

The heating was calculated by taking the resistivity for Constantan ρcons times the length of the cable l and divided by the cross sectional area Acs to get the resistance Rs: ρconsl Rs = (4.10) Acs

Since the current I going in to the cable is known, the heat Ps can be calculated by be the well known formula: 2 Ps = I Rs (4.11)

Ps is the total heat going into the wire, and must therefore be scaled to match each volume in the model. This is done by taking the total heat times the volume of each computational node and dividing it with the total volume:

Volsec q˙G = Ps (4.12) Voltot The heat generated in the surface coating is neglected since its volume is much smaller than the volume of Constantan and much less resistive.

4.1.5 Helium Heat Flow As described earlier in section 2.3.1, the heat flow to helium can occur in a number of ways. It depends on both temperature and time, which makes it a bit difficult to model. In He I, the heat flow is believed to first start in the transient regime, followed by natural convection, nucleate boiling and ending with film boiling. In He II the heat flow regimes are a transient regime, Kapitza conductance and then film boiling. The transition between the different regimes occur when either a temperature, energy or time limit have been reached. In literature the values for the limits can be found to differ by a factor of ten, which adds an uncertainty to the model. But if these values are seen as variables that can be used to fit experimental data, the model is very useful.

He I In He I the heat flow starts with the transient regime which has the highest heat transfer coefficient. The transition to natural convection is reached after a time limit in the order of microseconds or if a specified energy have been transferred from the cable during that time. The natural convection regime transfers heat poorly and the limit to nucleate boiling is reached at around 10 W/m2-K, according to [4]. Natural convection is started because of density differences in the fluid due to temperature gradients, which makes the fluid flow and transport heat. In the nucleate boiling regime, the surface is so hot that bubbles of gaseous helium forms and collapses into the liquid helium. For steady state conditions, nucleate boiling is the most efficient heat flow. The limit for the transition to film boiling is about 5-15 kW/m2-K i.e. about 1000 times more than natural convection. Film boiling occurs when the bubbles from the nucleate boiling grows bigger and form 50 Numerical Model a film that covers the surface. The heat transfer is poor since the heat has to be transferred from the surface to the gaseous helium and then to the liquid helium. Turning of the heat generation leads to a transition back to nucleate boiling at about a third of the limit between nucleate boiling and film boiling [9].

According to numerous references [4, 9, 23], the diﬀerent heat ﬂow regimes can be modeled using the following expressions:

Transient regime: 00 nT rans nT rans qT rans = aT rans(Ts − Tb ) (4.13)

Natural convection: 00 qNC = aNC (Ts − Tb) (4.14)

Nucleate boiling: 00 n qNB = aNB(Ts − Tb) (4.15)

Film boiling: q00 = a (T − T ) (4.16) FBI FBI s b

He II In superfluid He II there is a transient regime, followed by Kapitza conductance or film boiling. The transient regime and the Kapitza conductance physically works in the same way, but the transient heat flow is much more efficient and limited by a energy or time limit, while the Kapitza conductance is limited by a maximum heat flow. Film boiling works in the same way as in He I, a layer or gaseous helium or liquid He I forms over the surface and decreases the heat flow to helium. The heat flow limit between Kapitza conductance and film boiling is in the order of 35000 W/m2K. The heat flow to He II have been discussed in detail in section 2.5. The different regimes can be modeled in the following ways:

Kapitza conductance:

00 nKAP nKAP qKAP = aKAP (Ts − Tb ) (4.17) Film boiling: q00 = a (T − T ) (4.18) FBII FBII s b

4.2 Matlab Implementation

The model was implemented in a Matlab script and linked to a batch file, so that several different setups could be calculated after each other. The different parameters that could be changed were helium temperature, heat flow coefficients like aKAP or aNB, length and width of the wire, number of computational nodes 4.2 Matlab Implementation 51 in the model, heat generation and different surface conditions. The source code is found in appendix C.

Chapter 5

Results & Discussion

This chapter is divided into two sections, one for He I and one for He II. Each section is then divided into one experimental and one numerical part, followed by a comparison between the results. The measured heat transfer parameters are presented in tables, but some are also shown in graphs. The complete set of measurement data is found in appendix B.

5.1 Results for He I

5.1.1 Experimental Results Since several experiments have been done, a large number of thermocouples have been used. From the beginning eight thermocouples were used, numbered from 1 to 8. Unfortunately, thermocouple number 3 died during cooldown which ex- plains the empty spot in some of the graphs. It was interesting to know how accurate they were, and therefore a plot comparing the measured temperature at a given heat flux was produced. Figure 5.1 shows the temperature T ∗ when the heat regime switches from nucleate boiling to film boiling and as seen, the measured temperatures vary with almost 1 K between the thermocouples. This could not be because of any end-effects causing the ends of the wire to be warmer, because there are two wires which are being tested at the same time, where thermocouples 1 to 4 are attached to the first wire and thermocouples 5 to 8 on the second wire. If there were end-effects that would mean that thermocouples 1 and 4 and 5 and 8 would be equally warm, but now they represent each extreme instead. This means that either the thermocouples measure different temperatures due to some measurement error, or that the surface temperature on the wire varies a lot. One experiment where the surface conditions were significantly changed but the thermocouples where left untouched, showed no difference in measured temperatures. Thus, the reasonable explanation was that the thermocouples measure different temperatures due to some measurement error. This error could because a varying contact with the Constantan surface (likely), that some thermocouples where cooled more than the others by the helium (likely), a bad weld between the

53 54 Results & Discussion thermocouple wires (unlikely), or diﬀerent temperatures for the thermocouple- copper wire weld on the copper heat sink (unlikely).

However, the measurements are not useless. Assuming that the heat flux is the same for all thermocouples when going to film boiling, taking an average of each thermocouple measurement and then scaling each thermocouple with the total mean temperature, gives scale factors for every thermocouple. Scaling the measured temperatures gives a more homogenous temperature distrubution. The scale factors are found in appendix B.1. In figure 5.1 the temperature differecence before and after scaling is shown. Before scaling the difference is 0.9 K between the highest and lowest measurement, but scaling decreases the difference to 0.1 K which was an acctepted difference.

Figure 5.1. Left: The temperature when the heat regime changes from nucleate boiling to ﬁlm boiling, measured by seven thermocouples. Right: The same measurement, after being scaled. Note the diﬀerent y-scales.

The position of the wire, i.e. if it was in a horizontal or vertical position was expected to have an influence on some of the heat transfer correlations. This was because the three steady-state heat transfer regimes all have some sort of gravity dependency: natural convection needs gravity to make the changes in density of the fluid to flow, nucleate boiling and film boiling produces gaseous helium which is much lighter than liquid helium and will then flow uppwards. How big the difference between horizontal and vertical position would be was difficult to estimate before the measurements were done.

In table 5.1 is the complete set of measured parameters in He I, with values for median, mean, standard deviation, maximum and minimum values and the number of samples. By just looking at the values it is clear that there was a diﬀerence between the horizontal and vertical postion, just as expected. General values for the heat transfer coeﬃcients are found in literature [9, 24] and are presented in table 5.2. 5.1 Results for He I 55

Parameter Position Median Mean Std Max Min Samples aNC horizontal 550 590 215 1735 318 107 vertical 351 376 112 585 207 51 aNB horizontal 9120 9750 4798 18180 1750 65 vertical 9980 11100 5826 23760 3770 41 n horizontal 1.51 1.52 0.15 1.80 1.26 65 vertical 1.41 1.38 0.077 1.57 1.26 41

aFBI horizontal 433 516 235 915 226 35 vertical 510 506 128 775 264 33 qc horizontal 50 58 31 154 10 107 vertical 28 30 10 55 11 51 q∗ horizontal 4411 4454 252 4870 3911 65 vertical 4834 4805 156 5002 4502 41 qr horizontal 2619 2557 155 2774 2274 35 vertical 2364 2394 107 2637 2228 33 Tc horizontal 4.35 4.35 0.031 4.42 4.29 107 vertical 4.33 4.33 0.016 4.37 4.3 51 T ∗ horizontal 4.90 4.92 0.23 5.34 4.58 65 vertical 4.88 4.93 0.26 5.49 4.6 41 Tr horizontal 5.33 5.64 0.66 7.03 4.85 35 vertical 5.72 5.92 0.64 7.47 5.07 33 Tc scaled horizontal 4.35 4.36 0.030 4.43 4.29 107 vertical 4.40 4.32 0.24 4.63 3.89 51 T ∗ scaled horizontal 4.92 4.92 0.035 4.98 4.83 65 vertical 4.94 4.94 0.023 4.97 4.87 41 Tr scaled horizontal 5.75 5.75 0.42 6.59 5.14 35 vertical 5.80 5.91 0.37 6.75 5.37 33

Table 5.1. The measured steady-state parameters in liquid He I.

Parameter Value Unit 2 aNC 500 W/m -K 2 n aNB 50000 W/m -K n 1.5 - 2 aFBI 250 W/m -K 2 qc 10 W/m q∗ 5000-15000 W/m2 ∗ 2 qr ≈ 0.35 · q W/m

Table 5.2. General values for the heat transfer correlations in He I. 56 Results & Discussion

A comparison between the tables shows that the measured vales are in very good agreement with the general values. There is only one exception, the aNB coefficient which is about five times lower. The measured values scatter from 1750 to 23760 W/m2-Kn, with a mean value around 10000 W/m2-Kn. This big difference could be be because the temperature difference is to the power of n, and that the thermocouples which measure different temperatures for the same power generation. The natural convection heat flux limit qc is a bit high and the limit between nucleate boiling and film boiling q∗ was in the lower range but they were still ∗ resonable. The recovery heat flux qr was about 0.5 · q compared to the general ∗ ∗ ∗ value of 0.3 · q , but that could be because q was so low. If q was higher then qr might take place at 0.3 · q∗.

It might be difficult to see the most striking differences between the horizontal and vertical wire in a table with lots of numbers, so therefore a figure with both postitions are plotted:

Figure 5.2. Left: An example showing the nucleate boiling and ﬁlm boiling regimes for a horizontal and vertical wire. Right: A zoom which shows the natural convection regime.

For a reminder of where to find the different heat transfer regimes, review figure 3.3. In figure 5.2 it is seen that the horizontal wire goes to film boiling at about 4400 W/m2 compared to 4800 W/m2 for the vertical wire. The recovery back to nucleate boiling occurs as 2800 W/m2 for the horizontal wire and 2400 W/m2 for the vertical wire. The natural convection heat flux limit appears to be higher for the horizontal wire compared to the vertical: 100 W/m2 versus 40 W/m2. These values are just examples, but shows the difference between the two positions. To be able to see all heat transfer regimes at the same time, a logaritmic plot was produced, figure 5.3. At low temperatures the noise was big compared to the signal, but the natural convection regime was still visible. The shape was in very good agreement with the example given in figure 2.7, but the transitions between different heat transfer regimes were not as exact. 5.1 Results for He I 57

Figure 5.3. Left: The heat ﬂux as a function of temperature. Right: Figure 2.7 from the theory chapter, for comparison.

5.1.2 Numerical Results The calculated heat transfer correlations from the experimental values were taken as inputs for the numerical program. The results from the numerical program were both the internal and surface temperatures, the amount of heat flow from the wire to the helium, and which heat regimes which were active during the simulation. As an example, figure 5.4 shows one simulation where the input current is 10 A and ramped with 18 A/min. Note that in the left figure the power generation is ramped upp (dashed lines) and then turned off, whereas in real measurements the power generation is also ramped down. This flaw in the simulation program gives an erroronous shape for the curve in the time axis, but does not matter for the plot to the right because it is time independent.

Figure 5.4. Left: The temperature and generated power as a function of time. Right: The temperature as a function of heat ﬂux.

Figure 5.5 shows the internal heat profile in the wire, and it is seen that the temperature varies from a bit over 4.5 K in the center of the wire to almost 4.3 K at the surface, which is the helium temperature. This simulation is only valid for 58 Results & Discussion a homogeneous material such as Constantan, whereas for a superconducting wire with a copper matrix and bundles of Nb-Ti filaments, a different geometry must be used for the internal heat profile. This is because the difference in thermal properties for the different materials.

One other interesting thing to know is wether the thermocouple changes the temperature inside the wire. The thermocouple could act as an insulation since it covers a part of the Constantan surface from the helium. If there was a large variation in temperature, then the thermocouple might measured a temperature that was too high. Therefore, a simulation of a thermocouple-Constantan wire geometry was simulated. A small temperature difference compared to the rest of the wire was found, in the orders of 0.05-0.1 K directly under the thermocouple. Assuming that all thermocouples look the same in terms of insulating properties, this temperature difference could still not explain the variation in measurement data. In the right plot in figure 5.5 a thermocouple has been simulated on the middle-top side of the plot, and the temperature variation is really visible. Ap- parently the heat conduction in radial direction is much higher than the axial conduction, which most certainly depends on the heat flow from the surface to the helium.

Figure 5.5. Left: The internal heat proﬁle of a wire that is heated homogeneously while being cooled by the nucleate boiling regime. Right: A simulation of the internal temperatures when placing a thermocouple on the surface. 5.1 Results for He I 59

5.1.3 Comparison A figure with both a measurement from an experiment and a simulation using the values calculated from the measurement are presented in figure 5.6. The agreement was very good, the things that the numerical model can not simulate was the instable film boiling regime and the transitions between the different regimes. And, since measurement data was taken as input the limits for the critical heat fluxes can differ a bit. This was especially shown in the natural convection regime, 2 where the qc value has been determined to be 40 W/m , but it could as well have been in the range of 20-30 W/m2, which would change the shape of the simulation curve. In this example the maximum temperature was simulated to be 11.20 K and the measured maximum temperature was found to be 11.15 which is a very good agreement. On the other hand, the temperatures in the film boiling regime differs with 0.5-1 K between simulation and experiment. This is explained by the instability of the film boiling regime, because the temperature fluctuations gives a non-linear heat flow whereas it was modeled as a linear function. The results from the simulation of the nucleate boiling regime was a very good match of the experiment, without the noise.

Figure 5.6. Left: Simulation and experimental measurements plotted in the same graph. Right: A zoom on the natural convection regime. 60 Results & Discussion

5.2 Results for He II

5.2.1 Experimental Results The same method for the experiments in He I was used when doing experimetes in He II: a current was chosen and then ramped up and down. Because of the great cooling capacity of He II, more power was needed to get a transition between the two steady-state heat regimes. For He I a heating power of 1 W was neeeded to get to film boiling, but in He II more than 30 W was needed. The equaivalent current in both cases were 7 and 35 A, respectively. Ramping to 35 A took a long time with the same ramp rates as used in He I, which meant that the whole He II bath was heated up during the experiment. Therefore, a ramp rate of 240 A/min was used instead, meaning that ramping up could be done in about 10 seconds compared to 130 seconds if 18 A/min would have be used. Even though the ramp rate was high, a difference in helium temperature of about 0.05 K was measured. Since this affects the reference temperature on the copper heat sink, this was compensated for when calculating the measured surface temperature. The temperatures measured by the thermocouples varied even more for He II, with a temperature difference of almost 10 K for T∗ which was assumed to occur at the same heat flow. This was most probably due to the superfluid helium component that can cool certain parts of the thermocouple that was not covered by Stycast. The large variation in measured temperature makes it necessary to scale the temperatures just as with the temperatures in He I, but this time the result was not as good. The scaled temperatures scatter between 9.5 and 12.5 K and this makes the accuracy of the calculated heat transfer coefficients poor. In the Kapitza heat transfer regime the

Figure 5.7. Left: The measured temperatures at T∗. Right: The scaled temperatures of the same measurement. difference between horizontal and vertical position of the wire was small, but the differences became noticeable in film boiling. It was expected that the vertical wire would have a higher value for q∗ than the horizontal, but the measurements showed the opposite. This was somewhat strange, since it would be expected that the mechanism that controlls the transition to film boiling would work in the same way for both He I and He II. Looking on the maximum and minimum values for 5.2 Results for He II 61 q∗ for both the horizontal and vertical wire in table 5.3, it was seen that the maximum value for the vertical wire was higher than the lowest value for the horizontal wire. Even though the median and mean values says different, this could mean that the heat flux limit was the same for both positions.

The right plot in figure 5.8 shows the temperature response when ramping in the Kapitza regime. The difference between ramping up and down the heat generation is really visible and reasons for this was most probably hysteresis, but could also be because a temperature increase of the helium which makes the heat transfer worse when ramping down. On the whole, heat transfer measurements in He II were difficult to perform, especially film boiling.

Figure 5.8. Left: A typical graph of the heat ﬂux as a function of temperature for horizontal and vertical wires. Right: The Kapitza heat regime.

Bauer [4] made a literature survey for the He II heat transfer coefficients for copper. The values were scattered, with low values of aKAP corresponding to high values of nKAP , and high values of aKAP corresponding to low values of nKAP . The difference in measurement results depends on different surface treatments. For an order of magnitude of normal values, see table 5.4. Data is also taken from [9]. 62 Results & Discussion

Parameter Position Median Mean Std Max Min Samples aKAP horizontal 1445 1703 903 3763 722 41 vertical 1089 1388 750 3183 419 42 nKAP horizontal 1.9 1.93 0.113 2.23 1.8 41 vertical 1.97 2.03 0.109 2.3 1.79 42

aFBII horizontal 750 801 222 953 621 13 vertical 882 807 297 1119 320 13 q∗ horizontal 118050 120180 6332 130510 111270 30 vertical 109040 111410 4820 124780 107860 19 qr horizontal 48732 50987 16201 78550 30109 25 vertical 46560 46210 19260 94584 23555 17 T ∗ horizontal 11.09 11.44 2.79 16.94 7.85 30 vertical 11.00 10.70 2.84 16.00 6.83 19 Tr horizontal 30.11 31.51 3.51 33.75 26.12 25 vertical 27.55 28.12 2.09 26.91 20.6 13 T ∗ scaled horizontal 11.44 11.44 0.56 12.31 10.33 30 vertical 10.72 10.70 0.63 11.99 9.4 19

Table 5.3. The measured parameters in liquid He II.

Parameter Value Unit 2 n aKAP 130-840 W/m -K nKAP 4-1 - 2 aFBII 250 W/m -K q∗ 35000-50000 W/m2 Tr 23 K

Table 5.4. General values for the heat transfer parameters in He II. 5.2 Results for He II 63

Even higher values for Kapitza heat transfer were found for other materials: 2 n 2 n aKAP = 2000 W/m -K and n = 4 for Pt-Co, and aKAP = 2240 W/m -K and n = 2.72 for stainless steel. The measured values for Kapitza heat transfer are in that range, with high values of aKAP and relatively low values for nKAP . There is also a small difference between horizontal and vertical position of the wire. The values for the heat transfer coefficient in film boiling are also a bit high compared to those found in literature, but the biggest difference is the heat flux limit to film boiling: 100000 W/m2 compared to normal values of 35000-50000 W/m2. This difference could be because of the tin-silver coating on the heater, which makes the Kapitza heat transfer regime active for higher heating power. This is of course a good thing, the film boiling regime drastically reduces the heat transfer which makes the temperature rise and for a superconducting magnet that could be dis- ∗ astrous. The recovery heat flux qr occurs at about 40% of q which is almost the same as for He I, and the recovery temperature Tr is about 27 K compared to a suggested value of 23 K. A good comparison with literature values is not possible since the film boiling regime in He II is still not well understood, as discussed in chapter 2.5.6.

Using logaritmic scales and plotting the heat flux as a function of the temperature difference in figure 5.9, the experimental results can be compared with the theoretical curve in figure 2.12 from the theory chapter. The noise was high when the temperature difference was low, because the range of the amplifier was set to high to be able to measure the complete temperature spectrum. The agreement with the theoretical curve was good, but the transition regime between Kapitza and film boiling does not follow a step function but rather a gradual increase in heat flux as the temperature increases.

Figure 5.9. Left: A plot with logaritmic scales, showing both the Kapitza and the ﬁlm boiling regime. Right: Figure 2.12 from the theory chapter for comparison. 64 Results & Discussion

One other interesting thing was to know how much the He II temperature effect the heat transfer. The temperature in the cryostat was regulated by changing the pressure in the heat exchanger and tests at 1.9, 2.0 and 2.1 K were done. It was found that as the temperature increases less heat could be removed, which was expected since the effective themeral conductivity of superfluid helium has a peak at 1.9 K and then decreases. In figure 5.10 below, it was seen that the transition to film boiling happens at around 115 kW/m2 for the 1.9 K-curve, but at 100 kW/m2 for the 2.1 K-curve. The instability that occurs just before the transition to film boiling could be because a layer of He I film has formed, making the heat transfer fluctuate. When enough heat has been transferred from the surface, gaseous helium forms in the layer and the transition to film boiling occur.

Figure 5.10. A comparison of the heat transfer at diﬀerent temperatures in He II.

5.2.2 Numerical Results

The same plots which were found in He I were also produced for He II, but only the internal heat profile is shown here. The left plot in figure 5.11 shows the internal heat profile in the wire, and it was seen that the temperature varies from more than 10 K in the center of the wire to about 3.5 K at the surface. This could be because the Kapitza conductance was able to transfer large amounts of heat, while there was a high heat generation inside the wire. The figure was taken when steady-state was reached. The difference in internal heat profile between He I and He II was big, but this was because more heat was generated in the He II simulation. If the same amount of heat was generated in both models, the He II model would have a much smaller temperature gradient than the He I model due to the good cooling of Kaptiza conductance. 5.2 Results for He II 65

5.2.3 Comparison Film boiling in He II is difficult to model, especially when both He I and vapour helium forms the film. The large temperature gradients in the film layer also adds difficulty when trying to construct a model. In fact, theoretical models have only met with a resonable degree of success [9]. As stated in the theory chapter, an engineering correlation can look like the linear equation 2.21, but it does not by any means explain the complex mechanisms in the film layer. With that in mind, when looking at the right plot in figure 5.11 the agreement was still quite good, at least at low temperatures. The transition regimes between Kapitza and film boiling were difficult to model and the film boiling curves differ at temperatures higher than 25 K, but the Kapitza conductance was simulated with good agreement. In the figure the experimental values were taken as input to the simulation.

Figure 5.11. Left: The internal heat proﬁle when heating in He II. Right: A comparison between simulation and experimental values.

Chapter 6

Conclusions & Future Work

The measured parameters in He I were in good agreement with literature values, some were a bit low but the order of magnitude were nonetheless the same. Possi- ble reasons for this could be measurement errors in the thermocouples or that the surface coating in some way changed the heat transfer characteristics - making the values diﬀerent than other general values. The simulated temperature and heat ﬂow response curves were in very good agreement witch the experimental values, witch gives more credibility to the measured values and the methods used in the simulation.

The results from the He II measurements were not as good as the He I results, with scattering values and a really high critical heat flow limit between nucleate boiling and film boiling once again. The surface coating could have made the critical heat flow limit higher than normal materials. This is of course a good thing, because as long as the Kapitza conductance heat transfer regime stays active, more heat can be transported to the helium and the temperature of the heater stays relatively low. When the film boiling starts there was a temperature increase due to the bad heat transfer through the film layer. The numerical program could simulate the Kapitza conductance in very good agreement with experimental values, but when the temperature increased above 25 K the agreement gets poor. This was most probably because of the simplified expression used for the film boiling regime.

In general, the simulation program works very good, it gave results that were very much alike the measured values. One thing with the program that could be modified is to make it two-dimensional, it is not worth the extra computational time for the heat conduction in the axial direction. It could be shown that heat flow in the axial direction is very low compared to the heat flow in the radial axis. And if there is no need to simulate the thermocouple on the surface, the heat equation could be reduced to a one-dimensional equation, reducing calculation time even more.

To be able to measure the temperature more accurately and maybe most im-

67 68 Conclusions & Future Work portant during short time scales, thermocouples should not be used. Although a good thermal contact exists between the thermocouple and the Constantan wire, the thermal signal was too small to measure during transient time scales. In ad- dition, the thermocouples need to be insulated very well from the helium bath to reduce the measurement errors that comes from direct cooling of the helium. This is especially true for He II, where the superﬂuid component can penetrate even the smallest voids. One alternative solution could be to have a wire which have a very sensitive temperature dependent resistance. Knowing the current going through the wire, it is possible to measure the voltage over the wire and calculate the resistance as a function of power input. Schmidt [25] used a single Nb-Ti wire with a diameter of 36 µm, and was able to measure the temperature in the wire by applying a current density J > Jc. Since Jc is a function of T, the temperature could be determined from I and the voltage U measured over the wire. This solution was not chosen in this project since it was believed that the thermocouples would work for short heat pulses. For steady-state measurements thermocouples can be used, if the insulation problem can be solved. As long as one know that there will always be a measurement error Bibliography

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[5] C. F. Barenghi R. J. Donnelly. The observed properties of liquid helium at the saturated vapor pressure. Journal of Physical and Chemical Reference Data, Volume 27, Issue 6, 1998.

[6] R.D. Mccarty V. Arp and B.A. Hands. Cryopak Software. Cryopak.

[7] S. W. v. Sciver. Forced ﬂow He II cooling for superconducting magnets - design considerations. Cryogenics 38, 1998.

[8] W.G. Steward. Transient helium heat transfer / phase 1 - static coolant. International Journal of Heat and Mass Transfer, Vol.21, p.863, 1978.

[9] S. W. v. Sciver. Helium Cryogenics. Plenum Press, 1986.

[10] S. S. Kutateladze. Statistical science and technical publications, 1952.

[11] B.P. Breen and J.W. Westwater. Eﬀect of diameter of horizontal tubes on ﬁlm boiling heat transfer. Chem. Eng. Prog., vol.58 (7), 1962.

[12] J.F. Allen and A.D Misener. The fountain eﬀect. Nature, Vol. 141, p243, 1938.

[13] L. Tisza. The λ-transition explained. Nature, Vol. 141, p913, 1938.

[14] R. J. Donnelly. Superﬂuid turbulence. Scientiﬁc American, 1988.

[15] Vinen W.F. Theory of the mutual friction. Proc. R. Soc A242, p 493, 1957.

[16] Chase C.E. Thermal conduction in liquid helium 2. Phys. Rev., Vol 31 No. 5, p 1898, 1963.

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[17] Shimazaki T. Murakami M. and Iida T. Second sound wave heat transfer, thermal boundary layer formation and boiling: highly transient heat transport phenomena in He II. Cryogenics, Vol. 35 No. 10, p 645-651, 1995. [18] Keithley Instruments Inc., Cleveland, Ohio USA. Model 182 Sensitive Digital Voltmeter Speciﬁcations Rev. B.

[19] Nicolet Instrument Technologies Inc., Madison Wisconsin USA. Visions User’s Guide version 3.6, volume 2. [20] K. A. Prashant and B. Arunkumar. Introduction to cryogenic instrumentation in SM18. CERN internal note, 2004.

[21] Lakeshore. Technical Speciﬁcations Thermocouple Wire. Lake Shore Cry- otronics Inc. [22] J.D Andersson. Computational Fluid Dynamics. McGraw-Hill, 1995. [23] F. Sonnemann. Resistive Transition and Protection of LHC Superconduc- tive Cables and Magnets. PhD thesis, Rheinisch-Westfälischen Technischen Hochschule Aachen, 2001. [24] C Schmidt. Review of steady-state and transient heat transfer in pool boiling helium I. Stability of Superconductors, 1981. [25] C Schmidt. Transient heat transfer to liquid helium and temperature measurement with respone in the microsecond region. Applied Physics Lett., 1978. Appendix A

Calculations

A.1 Estimation of Heating Power to Heat up the Wire

Not all heat generated in the wire is transferred to the helium, some are used to heat up the wire. Therefore, it is necessary to know how much heating power is used inside the wire and how much that is lost to the helium. The amount of heat energy Q needed to increase the temperature with ∆T on a material with mass m and speciﬁc heat capacity cp can be written as:

Q = mcp∆T (A.1) The speciﬁc heat capacity of Constantan at low temperatures is highly temperature dependent, as seen in ﬁgure A.1.

Figure A.1. Speciﬁc Heat Capacity cp for Constantan at low temperatures.

71 72 Calculations

To get an estimation of how much energy is needed to raise the temperature of the wire consider the following example: The temperature is raised from 4.3 K to 5.3 K, an increase with 1 K. The calculation then becomes 2 Q = mcp∆T = ρV cp∆T = 8700 · 0.0005 · 0.07 · π · 0.6 · 1 = 0.000301 J (A.2) where the density is set to 8700 kg/m3, the wire has a diameter of 1 mm, a length of 70 mm, and the speciﬁc heat value for 5 K is used. It is found that approximately 0.3 mJ is needed to heat up the wire 1 K. At even lower temperatures the needed energy is lower since the speciﬁc heat is lower. Now, consider an experiment where the temperature is raised from 4.25 K to 5.25 K with a current ramp rate of 18 A/min.

Figure A.2. Temperature response in K (left) to the power generation in W (right).

The heating power follows a second degree polynomial curve since it is calculated as P = UI2, but as an overestimation it can be seen as a linear slope. Calculating the area under the power curve from 5 to 27 seconds, ﬁgure A.2, gives: (0.9 − 0) · (27 − 5) = 9.9 Ws = 9.9 J (A.3) 2 Even though some crude approximations have been made, comparing the 9.9 J that is put inside the wire by the current with the 0.3 mJ needed to heat up the wire, it is safe to say that almost all of the heat generated inside the wire is lost to the helium.

A.2 Scaling of Power and Current for Numerical Program

Since the geometry of the numerical model is diﬀerent from the experimental model, the power going into the numerical model must be scaled. To do this, it A.3 Thermal Radiation Estimation 73 is convenient to use the power density [W/m3] as it should be the same for both models. The power density can be expressed as: P P exp = num (A.4) Volexp Volnum where subscript exp stands for experimental model and subscript num for numerical model. In both models the diameters of the wire are the same, which makes the only diﬀerence in volumes are the length of the wires. Using that the diameter is the same in both models and solving for the numerical power, equation A.4 becomes: Volnum lnum Pnum = Pexp = Pexp (A.5) Volexp lexp All that is needed now is an expression for the power expressed with current, since the power in the experimental setup is controlled by a current source. Since the resistance R for the wire is known, the well known equation A.6 is a suitable choise:

2 Pnum = I R (A.6) Combining equations A.5 and A.6 and solving for I, gives: s lnum Pexp I = ± lexp (A.7) R Equation A.7 is the expression for the corresponding current in the numerical model for a given power from the experimental model. Note that equation A.5 is enough if only the scaled power is wanted, but since the power is ramped up by the current, an upper limit for the current is needed and that can be calculated by equation A.7.

A.3 Thermal Radiation Estimation

Since all bodies above absolute zero emitts thermal radiation, it is neccesary to know how much heat is lost by radiation. The amount of heat that radiates is governed the Stefan-Boltzmann law: q E (T ) = r = σT 4 (A.8) b A In equation A.8, σ is the Stefan-Boltzmann constant which is approximately 5.67 · 10−8 W/m2K4. The radiant net heat transfer from one body to another is given by the emissivity and the temperature diﬀerence: q r = σ(T 4 − T 4) (A.9) A 1 2 Assuming a perfect blackbody ( = 1), unit area, the helium temperature at 1.9 K and the surface temperature at 10 K, the total heat ﬂux becomes:

−8 4 4 2 Qr = 1 · 5.67 · 10 (10 − 1.9 ) ≈ 0.57 mW/m (A.10) 74 Calculations

Comparing 0.57 mW/m2 with normal heat ﬂux values for Kapitza conductance (or even ﬁlm boling) which are in the range of 0.5-50 kW/m2, makes it obvious that it is possible to neglect the radiation component. Appendix B

Graphs

B.1 Scale Factors

Position 1 2 4 5 6 7 8 Horizontal 1.1068 0.9797 0.9619 0.9375 0.9838 1.0090 1.0208 Vertical 1.5690 0.9285 0.6926 0.5332 0.8162 0.9908 1.2609

Table B.1. The temperature scale factors in He I.

Position 1 2 4 5 6 7 8 Horizontal 1.5761 0.8928 0.8221 0.7059 0.9840 1.0311 1.3832 Vertical 1.5978 1.0098 0.7799 0.5796 1.0162 1.0376 1.3751

Table B.2. The temperature scale factors in He II.

75 76 Graphs

B.2 Experimental Results at 4.3 K

Figure B.1. The heat transfer coeﬃcient for natural convection.

Figure B.2. The heat transfer coeﬃcient for nucleate boiling. B.2 Experimental Results at 4.3 K 77

Figure B.3. The exponent used in in the temperature diﬀerence in nucleate boiling.

Figure B.4. The heat transfer coeﬃcient for ﬁlm boiling. 78 Graphs

Figure B.5. Critical heat ﬂux versus critical temperature for natural convection.

Figure B.6. Critical heat ﬂux versus critical temperature for nucleate boiling. B.3 Experimental Results at 1.9 K 79

Figure B.7. Recovery heat ﬂux versus recovery temperature from ﬁlm boiling.

B.3 Experimental Results at 1.9 K

Figure B.8. Heat transfer coeﬃcient for Kapitza conductance. 80 Graphs

Figure B.9. Exponent in Kapitza conductance.

Figure B.10. Critical heat ﬂux versus critcal temperature for Kapitza conductance. B.3 Experimental Results at 1.9 K 81

Figure B.11. Critical temperature for Kapitza conductance.

Figure B.12. Scaled critical temperature for Kapitza conductance. Appendix C

Source Code for Matlab Programs

C.1 Batch ﬁle This is the batch ﬁle for running the numerical program.

% Batch file for running cylindrical_coordinates.m

% (c) Jonas Lantz, CERN AT-MCS-SC

clear clc close all foldername = char(datestr(now,’yyyy-mm-dd’));

setups = 3; % Number of setups T_setup = [1.9 4.3 4.3]; % Simulation temperature [K] dt = [1e-7 1e-5 1e-5]; % Time step [s] duration = [1 1 1]; % Simulated time [s] t_Joule = [0.9 0.9 0.9]; % Duration of joule heating [s] square_pulse = [0 0 0]; % Boolean value, 1 = square pulse, 0 = ramp NL = [3 3 3]; % Number of nodes in radial direction [-] (min 3) NS = [3 3 3]; % Number of nodes in theta direction [-] (min 3) NZ = [3 3 3]; % Number of nodes in axial direction [-] (min 3)

Pexp = [29.3 3 4]; % Power in experiment [W] (2 W ~ 7 A, 4 W ~ 10A) Ramp_rate_min = [2400 1200 18]; % Ramp rate [A/min]

% Helium I parameters, limits etc aNC = [372 500 500]; % [W/(m^2 K)] aNB = [4510 5e4 5e4]; % [W/(m^2 K^2.5)] nNB = [1.26 2.5]; % [-] aFB_I = [664 500 500]; % [W/(m^2 K)] % Default value is 500 aTrans = [180 180 180]; % [W/(m^2 K)] nTrans = [4 4 4]; % [-] HF_lim_NC = [32.7 10 10]; % [W/m^2] HF_lim_NB = [4592 5e3 5e3]; % [W/m^2] lim_FB_NB = [0.505 0.35 0.35]; % Factor for FB -> NB [-] E_lim_trans = [15 15 15]; % [J/m^2]

%Helium II parameters, limits etc aKap = [2270 200 200]; % [W/(K^nKap m^2)] nKap = [1.89 4 4]; % [-] aFB_II = [798 250 250]; % [W/(K m^2)]

82 C.2 Numerical program 83

HF_lim_Kap = [108816 35000 35000]; % [W/m^2]

for e = 1:setups [t] = cylindrical_coordinates_v2(T_setup(e), dt(e), duration(e), t_Joule(e), ... square_pulse(e), NL(e), NS(e), NZ(e), Pexp(e), Ramp_rate_min(e),... aNC(e), aNB(e), nNB(e), aFB_I(e), aTrans(e), nTrans(e),HF_lim_NC(e),HF_lim_NB(e),... lim_FB_NB(e), E_lim_trans(e), aKap(e), nKap(e), aFB_II(e), HF_lim_Kap(e)); end

%% Sending email when done setpref(’Internet’,’SMTP_Server’,’mail’) setpref(’Internet’,’E_mail’,’[email protected]’); sendmail(’[email protected]’,’The matlab program is finished.’)

C.2 Numerical program The numerical program used in the simulations are presented here. Simulation results are saved in both .txt and .mat ﬁles and can then be read by a suitable program. The plotting routines are omitted.

% cylindrical_coordinates.m % Calculation of 3D heat transfer from a constantan wire with a part of a % section connected to a thermocouple and the rest exposed to liquid helium. % Using cylindrical coordinates and explicit euler finite differences

% (c) Jonas Lantz, CERN AT-MCS-SC

function[t] = cylindrical_coordinates_v2(T_setup, dt, duration, t_Joule, ... square_pulse, NL, NS, NZ, Pexp, Ramp_rate_min,... aNC, aNB, nNB, aFB_I, aTrans, nTrans, HF_lim_NC,HF_lim_NB,... lim_FB_NB, E_lim_trans, aKap, nKap, aFB_II, HF_lim_Kap)

%% Simulation conditions

%Time tic time_iterations = duration/dt; % Number of iterations [-] time_iterations_Joule = t_Joule/dt; % Number of iterations when joule heating is on [-]

%Thermocouple option thermocouple = 0; % Boolean value, 1 = no cooling on one surface section

%Stycast option stycast = 0; % Boolean value, 1 = outer layer is made out of Stycast

%Geometry of the strand length_str = 0.001; % Length of strand [m] dz = length_str/NZ; % Length of each section [m] diameter_str = 0.001; % Diameter of strand [m]

radius = diameter_str/2; % Radius of strand [m] A_str = radius^2*pi; % Area of the whole strand [m^2] dr = radius/(NL-1); % Width of layers in radial direction [m] dth = (2*pi)/NS; % Theta angle between sections [rad]

%Temperature T_int = T_setup; % Initial temperature of constantan [K] T_He = T_setup; % Initial temperature of helium [K]

%Helium parameters Vol_He = 10000.00000001; % Volume of helium per section [m^3]

%Save data display_interval = 1000; % Specifies when to display data. save_interval = 10; % Specifies when to save data to file. interval = 1; % Help variable 84 Source Code for Matlab Programs

empty_m = [ ]; % Help matrix for empty line file_time= char(datestr(now,’yyyymmddTHHMMSS’)); % Specifies the format of the filename file1 = file_time; file_mat = strcat(file1,’.mat’); file2 = strcat(file1,’_T’,’.txt’); % Concatenate filename with file extension

foldername = char(datestr(now,’yyyy-mm-dd’)); % Specifies the format of the folder to write cd data warning off last mkdir(foldername); cd .. filename_T = fullfile(’data’,foldername,file2); % Full filename filename_mat = fullfile(’data’,foldername,file_mat);% Filename for .mat file filename_settings = strcat(’_settings_’,file_time);

cd data cd(foldername)

cell_setting = {’File name: ’, filename_mat; ’Thermocouple: ’,num2str(thermocouple); ’Stycast: ’,num2str(stycast); ’Square Pulse: ’,num2str(square_pulse); ’Power exp: ’,num2str(Pexp); ’Ramp_rate_min :’,num2str(Ramp_rate_min); ’Temperature :’,num2str(T_setup); ’NL: ’,num2str(NL); ’NS: ’,num2str(NS); ’NZ: ’,num2str(NZ); ’Simulation time: ’,num2str(duration); ’N◦ Iterations: ’,num2str(time_iterations); ’Heating time: ’,num2str(t_Joule); ’aNC: ’,num2str(aNC); ’aNB: ’, num2str(aNB); ’nNB: ’,num2str(nNB); ’aFB_I: ’,num2str(aFB_I); ’nTrans: ’,num2str(nTrans); ’HF_lim_NC: ’,num2str(HF_lim_NC); ’HF_lim_NB: ’,num2str(HF_lim_NB); ’lim_FB_NB: ’,num2str(lim_FB_NB); ’aKap: ’, num2str(aKap); ’nKap: ’,num2str(aKap); ’aFB_II: ’,num2str(aFB_II); ’HF_lim_Kap: ’,num2str(HF_lim_Kap)};

xlswrite(filename_settings, cell_setting)

cd .. cd ..

%Constantan density dens_const = 8890; % Density of Constantan [kg/m^3]

%Heating calculations Rho_Cons = 4.36*10^-7; % Resistivity for Constantan [Ohm m] Ramp_rate = Ramp_rate_min/60; % Ramp rate of power supply [A/s] Rs = Rho_Cons*length_str/A_str; % Calculation of resistance [Ohm] Cs = sqrt(Pexp*(length_str/0.07)/Rs); % Experimental heating power [W]

if square_pulse == 0 Cs_t = 0; % Current in strand at time t [A] else Cs_t = Cs; end

%% Spatial calculations C.2 Numerical program 85

% Cross-sectional area [m^2] Az(1) = pi*(dr/2)^2; % Area of "core" section [m^2] Az(2:NL) = dth*dr*dr*((1:NL-1)); % Area of each sections in the middle layers [m^2] Az(NL) = 0.5*dth*dr*dr*(NL-5/4); % Area of the sections in the outer layer [m^2]

% Radial area [m^2] Ar(1:(NL-1)) = ((dr/2)+dr*(0:NL-2))*dth*dz; Ar(NL) = radius*dth*dz;

%Volume of each section, one value per layer [m^3] vol_sec(1:NL) = Az(1:NL)*dz;

%Volume of the entire wire [m^3] vol_wire = A_str*length_str;

%Calculation of distance in radial direction from center to each sections (line) [m] distance_r(1,1:NL) = 0; distance_r(2:NL) = dr*(1:NL-1); distance_r(NL) = dr*(NL-5/4); for j = 1:NS for k = 1:NZ distance_radial(:,j,k) = distance_r; %#ok end end

%% Initial Values and preallocation

T(1,1,1:NZ) = T_int; % Initial temperature of the strand [K] T(2:NL,1:NS,1:NZ) = T_int; % Initial temperature of the strand [K] THe(1,1:NS,1:NZ) = T_He; % Initial temperature of the helium [K] k_cons(1,1,1:NZ) = 0.8; % Initial thermal conductivity [W/mK] from Excel sheet k_cons(2:NL,1:NS,1:NZ) = 0.8; % Initial thermal conductivity [W/mK] from Excel sheet Cp_Cons(1,1,1:NZ) = 0.48; % Initial thermal specific heat [J/Kg*K] from Excel sheet Cp_Cons(2:NL,1:NS,1:NZ) = 0.48; % Initial thermal specific heat [J/Kg*K] from Excel sheet HF_He(1,1:NS,1:NZ) = 0; % Initial heat flow to helium [W/m^2] Cp_He(1,1:NS,1:NZ) = 1; % Initial thermal specific heat [J/Kg*K] from Excel sheet heat(1:time_iterations) = 0; % Initial heat transfer mode %E_limit_count(1,NS,NZ) = 0; % Initial value for transient heat transfer [J/m^2] stay_in_NC = 0; % Help variable for natural convection %% Starting the loop

%Display some information info1 = ([’Doing ’, num2str(time_iterations), ’ time iterations’, ’, ... simulated time is ’ num2str(duration),’ seconds’]); info2 = ([’With ’, num2str(NL), ’ * ’, num2str(NS), ’ * ’, num2str(NZ), ... ’ spatial iterations (r, th, z) for each time iteration’]); disp(info1) disp(info2)

%Write header in file dlmwrite(filename_T , info1, ’delimiter’, ’%s’, ’-append’,’newline’, ’pc’) dlmwrite(filename_T , info2, ’delimiter’, ’%s’, ’-append’,’newline’, ’pc’) dlmwrite(filename_T , empty_m, ’delimiter’, ’%s’, ’-append’, ’newline’, ’pc’, ’roffset’,1)

%Here we go... for t = 1:time_iterations for i = 1:NL for j = 1:NS for k = 1:NZ

%Setting the boundary conditions on the left boundary if k == 1 T(i,j,k) = T(i,j,k+1); end

%------% Material Characteristics %------86 Source Code for Matlab Programs

% Heat capacity constantan [J/(Kg*K)] % y = 0,000032x4 - 0,000472x3 + 0,009965x2 + ... % 0,081451x + 0,023679 Formula from Excel sheet. Cp_Cons(i,j,k) = T(i,j,k).^4.*0.000032-T(i,j,k).^3.*0.000472+... T(i,j,k).^2*0.009965+T(i,j,k).*0.081451+0.023679; Cp_Cons(1,2:NS,k) = NaN;

% Thermal conductivity constantan [W/mK] % Formula from Excel sheet. % y = 0,00214x2 + 0,45712x - 1,15325 for T < 20 % y = 0,128x + 6,225 for T > 20 if T(i,j,k) < 20 k_cons(i,j,k) = 0.00214*T(i,j,k).^2+0.45712*T(i,j,k)-1.15325; else k_cons(i,j,k) = 0.128*T(i,j,k)+6.225; end

k_cons(1,2:NS,k) = NaN;

% Calculation of Cp for Helium, from the CUDI manual [J/K*g] if THe(1,j,k) < 2.17 % Cp for Helium when T < 2.17 K Cp_He(1,j,k) = 2.12+0.000678.*THe(1,j,k).^12.159; elseif THe(1,j,k) >= 2.17 && THe(1,j,k) < 2.5 % Cp for Helium when 2.17< T < 2.5 K Cp_He(1,j,k) = -274.45.*THe(1,j,k).^3+1961.6.*THe(1,j,k).^2... -4673.2.*THe(1,j,k)+3712.9; elseif THe(1,j,k) >= 2.5 && THe(1,j,k) < 4.3 % Cp for Helium when 2.5 < T < 4.3 K Cp_He(1,j,k) = 0.9163.*THe(1,j,k).^2-4.484.*THe(1,j,k)+7.68; elseif THe(1,j,k) >= 4.3 && THe(1,j,k) < 15 % Cp for Helium when 4.3 < T < 15 K Cp_He(1,j,k) = 5.2+1489.4.*THe(1,j,k).^-4.06; elseif THe(1,j,k) >= 15 % Cp for Helium when T > 15 K Cp_He(1,j,k) = 5.2; else disp(’Error in computing Cp for Helium, T is Inf’) end Cp_He(1,j,k) = Cp_He(1,j,k)/1000; %To get units in [J/K*Kg]

% Density of Helium, from HePac [kg/m^3] % from execel sheet if THe(1,j,k) <= 4.3 rho_He(1,j,k) = 0.196*THe(1,j,k).^3-6.7247*THe(1,j,k).^2+26*THe(1,j,k)+121.2; elseif THe(1,j,k) > 4.3 && THe(1,j,k) < 4.4 rho_He(1,j,k) = -1053*THe(1,j,k)+4651.6; elseif THe(1,j,k) >= 4.4 && THe(1,j,k) < 10 rho_He(1,j,k) = -0.116*THe(1,j,k).^3+2.940*THe(1,j,k).^2-25.57*THe(1,j,k)+... 83.14; % y3 = -0,116x3 + 2,940x2 - 25,57x + 83,14; elseif THe(1,j,k) >= 10 && THe(1,j,k) < 20 rho_He(1,j,k) = 2.44; elseif THe(1,j,k) >= 20 && THe(1,j,k) < 40 rho_He(1,j,k) = 1.26; else rho_He(1,j,k) = 0.5; end

% Check for imaginary material parameters or negative temperatures if isreal(Cp_Cons(i,j,k)) == 0 || isreal(k_cons(i,j,k)) == 0 || ... isreal(Cp_He(1,j,k)) == 0 || isreal(rho_He(1,j,k)) == 0 || ... T(i,j,k) < 0 || THe(1,j,k) < 0% || T(i,j,k) > 200 end_me_now = 1; else end_me_now = 0; end

%------% Stycast option %------if stycast == 1 k_cons(NL,j,k) = 0.064; % [W/mK], source: Lakeshore Cp_Cons(NL,j,k) = 2.25; % [J/KgK] source: cryogenics handbook end C.2 Numerical program 87

%------% Heat calculations %------%Generated heat due to resistance [W] %Ramping up the current, or going directly to a square if chosen if Cs_t < Cs Cs_t = Ramp_rate*dt*t; Ps_unscaled = Cs_t^2*Rs; % Heating due to current [W] else Ps_unscaled = Cs_t^2*Rs; % Heating due to current [W] end

% Scaling the heating to match the volume of each section scaling_factor(i) = (vol_sec(i)/vol_wire); %#ok

if t < time_iterations_Joule Ps(i,j,k) = Ps_unscaled*scaling_factor(i); %#ok Ps(1,2:NS,k) = NaN; else Ps(i,j,k) = 0; end

%The heat is transferred through different regimes depending on the helium. %If the helium is of type II, then the heat is transferred through a %Kapitza regime until a certain heat flux limit and then a film boiling regime %starts.

%If the helium is of type I, then the heat is transferred through a %transient regime, then a natural convection regime, %then a nucleate boiling regime and finally a film boiling regime.

%The lambda point for helium is at 2.17 K if THe(1,j,k) < 2.17 if t < time_iterations_Joule if HF_He(1,j,k) < HF_lim_Kap || heat(t) ~= 2 HF_He(1,j,k) = aKap*(T(NL,j,k).^nKap-THe(1,j,k).^nKap); heat(k,t+1) = 1; %#ok else HF_He(1,j,k) = aFB_II*(T(NL,j,k)-THe(1,j,k)); heat(k,t+1) = 2; end else if HF_He(1,j,k) < HF_lim_Kap HF_He(1,j,k) = aKap*(T(NL,j,k).^nKap-THe(1,j,k).^nKap); heat(k,t+1) = 1; %#ok elseif HF_He(1,j,k) > HF_lim_Kap && heat(t) ~= 1 HF_He(1,j,k) = aFB_II*(T(NL,j,k)-THe(1,j,k)); heat(k,t+1) = 2; end end else if t < time_iterations_Joule if E_limit_count(1,j,k) < E_lim_trans HF_He(1,j,k) = aTrans*(T(NL,j,k)^nTrans-THe(1,j,k)^nTrans); E_limit_count(1,j,k) = E_limit_count(1,j,k) + HF_He(1,j,k)*dt; heat(t+1) = 3; if HF_He(1,j,k) < HF_lim_NC || heat(t) == 3 || stay_in_NC == 1 % NC HF_He(1,j,k) = aNC*(T(NL,j,k)-THe(1,j,k)); heat(t+1) = 4; if T(NS,NL,NZ) < 4.4 stay_in_NC = 1; else stay_in_NC = 0; end elseif HF_He(1,j,k) >= HF_lim_NC && HF_He(1,j,k) < HF_lim_NB && heat(t) ~= 6%NB HF_He(1,j,k) = aNB*(T(NL,j,k)-THe(1,j,k)).^nNB; heat(t+1) = 5; elseif HF_He(1,j,k) >= HF_lim_NB || heat(t) == 6 % FB HF_He(1,j,k) = aFB_I*(T(NL,j,k)-THe(1,j,k)); 88 Source Code for Matlab Programs

heat(t+1) = 6; end else if HF_He(1,j,k) < HF_lim_NC %NC HF_He(1,j,k) = aNC*(T(NL,j,k)-THe(1,j,k)); heat(t+1) = 4; elseif HF_He(1,j,k) >= HF_lim_NC && HF_He(1,j,k) < HF_lim_NB*lim_FB_NB... || heat(t) == 5 % NB HF_He(1,j,k) = aNB*(T(NL,j,k)-THe(1,j,k)).^nNB; heat(t+1) = 5; if HF_He(1,j,k) >= HF_lim_NB*lim_FB_NB % To avoid heat flow "spikes" HF_He(1,j,k) = HF_lim_NB*lim_FB_NB; end elseif HF_He(1,j,k) >= HF_lim_NB*lim_FB_NB && heat(t) ~= 5 % FB HF_He(1,j,k) = aFB_I*(T(NL,j,k)-THe(1,j,k)); heat(t+1) = 6; end end

end

%% Thermocouple option % It is possible to simulate a thermocouple in the He-layer % by removing the cooling in that section. if thermocouple == 1 HF_He(1,1,2) = 0; end

%% Temperature calculations % Quite extensive, but as follows:

%1) Inner boundary, i = 1 %2) 2nd layer, i = 2 (since i-1 also is special) %3) Inner layers, until NL-1 %4) Outer layer with HF to helium, NL %5) Helium layer, 1

%% Inner boundary, i = 1 %Calculating the heat conduction from the center (i =1, j = 1) to the %surrounding sections (i = 2, j = 1:8) if i == 1 if j == 1 if k > 1 && k < NZ for r = 1:j T_cond_2_1(r) = (T(2,r,k)-T(1,1,k)); %#ok end T_cond_2_1 = sum(T_cond_2_1*dth*dz/2); %[Km]

%Temperature on inner boundary T(1,1,k) = T(1,1,k)+(dt/(dens_const*Cp_Cons(1,1,k).*vol_sec(1)))... .*((k_cons(1,1,k).*(T_cond_2_1+vol_sec(1)*(T(1,1,k+1)-... 2*T(1,1,k)+T(1,1,k-1))/(dz^2)))+Ps(1,1,k)); end end end

%% 2nd layer, i = 2 if i == 2 if j == 1 if k > 1 && k < NZ %Special case for i = 2, circular boundary (1 - first section) T(i,j,k) = T(i,j,k) + dt/(dens_const*Cp_Cons(i,j,k).*vol_sec(i))... .*(k_cons(i,j,k).*((((T(i+1,j,k)-T(1,1,k))/2)*dth*dz)... +distance_radial(i,j,k).*(((T(i+1,j,k)-2*T(i,j,k)+T(1,1,k)))/dr)*dth*dz... +(dr*dz/distance_radial(i,j,k)).*((T(i,2,k)-2*T(i,j,k)+T(i,NS,k)))/dth... +distance_radial(i,j,k).*dth*dr*((T(i,j,k+1)-2*T(i,j,k)+T(i,j,k-1)))/dz)... +Ps(i,j,k)); end end C.2 Numerical program 89

end if i == 2 if j > 1 && j < NS if k > 1 && k < NZ %Special case for i = 2, Heat conduction equation - interior nodes T(i,j,k) = T(i,j,k) + dt/(dens_const*Cp_Cons(i,j,k).*vol_sec(i))... .*(k_cons(i,j,k).*((((T(i+1,j,k)-T(1,1,k))/2)*dth*dz)... +distance_radial(i,j,k).*(((T(i+1,j,k)-2*T(i,j,k)+T(1,1,k)))/dr)*dth*dz... +(dr*dz/distance_radial(i,j,k)).*((T(i,j+1,k)-2*T(i,j,k)+T(i,j-1,k)))/dth... +distance_radial(i,j,k).*dth*dr*((T(i,j,k+1)-2*T(i,j,k)+T(i,j,k-1)))/dz)... +Ps(i,j,k)); end end end if i == 2 if j == NS if k > 1 && k < NZ %Special case for i = 2, circular boundary (NS - last section) T(i,j,k) = T(i,j,k) + dt/(dens_const*Cp_Cons(i,j,k).*vol_sec(i))... .*(k_cons(i,j,k).*((((T(i+1,j,k)-T(1,1,k))/2)*dth*dz)... +distance_radial(i,j,k).*(((T(i+1,j,k)-2*T(i,j,k)+T(1,1,k)))/dr)*dth*dz... +(dr*dz/distance_radial(i,j,k)).*((T(i,1,k)-2*T(i,j,k)+T(i,j-1,k)))/dth... +distance_radial(i,j,k).*dth*dr*((T(i,j,k+1)-2*T(i,j,k)+T(i,j,k-1)))/dz)... +Ps(i,j,k)); end end end

%% Interior layers, 3:NL-1 if i < NL && i > 2 if j == 1 if k > 1 && k < NZ %Heat conduction equation - circular boundary (1 - first section) T(i,j,k) = T(i,j,k) + dt/(dens_const*Cp_Cons(i,j,k).*vol_sec(i))... .*(k_cons(i,j,k).*((((T(i+1,j,k)-T(i-1,j,k))/2)*dth*dz)... +distance_radial(i,j,k).*(((T(i+1,j,k)-2*T(i,j,k)+T(i-1,j,k)))/dr)*dth*dz... +(dr*dz/distance_radial(i,j,k)).*((T(i,2,k)-2*T(i,j,k)+T(i,NS,k)))/dth... +distance_radial(i,j,k).*dth*dr*((T(i,j,k+1)-2*T(i,j,k)+T(i,j,k-1)))/dz)... +Ps(i,j,k)); end end end if i < NL && i > 2 if j > 1 && j < NS if k > 1 && k < NZ %Heat conduction equation - interior nodes T(i,j,k) = T(i,j,k) + dt/(dens_const*Cp_Cons(i,j,k).*vol_sec(i))... .*(k_cons(i,j,k).*((((T(i+1,j,k)-T(i-1,j,k))/2)*dth*dz)... +distance_radial(i,j,k).*(((T(i+1,j,k)-2*T(i,j,k)+T(i-1,j,k)))/dr)*dth*dz... +(dr*dz/distance_radial(i,j,k)).*((T(i,j+1,k)-2*T(i,j,k)+T(i,j-1,k)))/dth... +distance_radial(i,j,k).*dth*dr*((T(i,j,k+1)-2*T(i,j,k)+T(i,j,k-1)))/dz)... +Ps(i,j,k)); end end end if i < NL && i > 2 if j == NS if k > 1 && k < NZ %Heat conduction equation - circular boundary (NS - last section) T(i,j,k) = T(i,j,k) + dt/(dens_const*Cp_Cons(i,j,k).*vol_sec(i))... .*(k_cons(i,j,k).*((((T(i+1,j,k)-T(i-1,j,k))/2)*dth*dz)... +distance_radial(i,j,k).*(((T(i+1,j,k)-2*T(i,j,k)+T(i-1,j,k)))/dr)*dth*dz... +(dr*dz/distance_radial(i,j,k)).*((T(i,1,k)-2*T(i,j,k)+T(i,j-1,k)))/dth... +distance_radial(i,j,k).*dth*dr*((T(i,j,k+1)-2*T(i,j,k)+T(i,j,k-1)))/dz)... +Ps(i,j,k)); end end end 90 Source Code for Matlab Programs

%% Outer layer, with heatflow to helium if i == NL if j == 1 if k > 1 && k < NZ %Temperature on outer boundary, with heat flow to Helium (1 - first section) T(NL,j,k) = T(NL,j,k)+(dt/(dens_const*Cp_Cons(NL,j,k).*vol_sec(i)))... .*(k_cons(NL,j,k).*((distance_radial(NL,j,k)-dr/4)*dth*dz.*... ((T(i-1,j,k)-T(i,j,k))/dr+(dr*dz/distance_radial(i,j,k))*... ((T(i,j+1,k)-2*T(i,j,k)+T(i,NS,k))/dth)+distance_radial(i,j,k)*... dth*dr*((T(i,j,k+1)-2*T(i,j,k)+T(i,j,k-1))/dz)))... +Ps(i,j,k)-HF_He(1,j,k)*Ar(NL)); end end end if i == NL if j > 1 && j < NS if k > 1 && k < NZ %Temperature on outer boundary, with heat flow to Helium T(NL,j,k) = T(NL,j,k)+(dt/(dens_const*Cp_Cons(NL,j,k).*vol_sec(i)))... .*(k_cons(NL,j,k).*((distance_radial(NL,j,k)-dr/4)*dth*dz.*... ((T(i-1,j,k)-T(i,j,k))/dr+(dr*dz/distance_radial(i,j,k))*((T(i,j+1,k)... -2*T(i,j,k)+T(i,j-1,k))/dth)+distance_radial(i,j,k)*dth*dr*... ((T(i,j,k+1)-2*T(i,j,k)+T(i,j,k-1))/dz)))+Ps(i,j,k)-HF_He(1,j,k)*Ar(NL)); end end end if i == NL if j == NS if k > 1 && k < NZ %Temperature on outer boundary, with heat flow to Helium (NS - last section) T(NL,j,k) = T(NL,j,k)+(dt/(dens_const*Cp_Cons(NL,j,k).*vol_sec(i)))... .*(k_cons(NL,j,k).*((distance_radial(NL,j,k)-dr/4)*dth*dz.*... ((T(i-1,j,k)-T(i,j,k))/dr+(dr*dz/distance_radial(i,j,k))*((T(i,1,k)-... 2*T(i,j,k)+T(i,j-1,k))/dth)+distance_radial(i,j,k)*dth*dr*... ((T(i,j,k+1)-2*T(i,j,k)+T(i,j,k-1))/dz)))+Ps(i,j,k)-HF_He(1,j,k)*Ar(NL)); end end end if k == NZ T(i,j,k) = T(i,j,k-1); end

%% Helium temperature %Calculating the Helium temperature THe(1,j,k)= THe(1,j,k)+(dt*HF_He(1,j,k)*Ar(NL))./(Vol_He*Cp_He(1,j,k)*rho_He(1,j,k));

%% Ending the spatial loops & checking for obvious errors %Check for imaginary material parameters or negative temperature if end_me_now == 1; break end end %end k loop

if end_me_now == 1; break end end %end j loop

if end_me_now == 1; break end end %end i loop

if end_me_now == 1; disp(’ ’) disp(’WARNING’) disp(’Imaginary material parameter or irrelevant temperature, aborting...’) disp(’ ’) disp(’ ’) break C.2 Numerical program 91

end

%% Saving data and ending time loop if mod(t,save_interval) == 0 display_me = ([’Iteration number: ’,num2str(t),’, Elapsed time: ’,num2str(toc), ’ seconds’,... ’, Simulated time: ’,num2str(dt*t), ’ seconds.’]);

if mod(t,display_interval) == 0 disp(display_me) end

T_save(:,:,:,interval) = T(:,:,:); THe_save(1,:,:,interval) = THe(1,:,:); HF_He_save(1,:,:,interval) = HF_He(1,:,:); heat_save(interval) = heat(t);

%Write to file dlmwrite(filename_T, display_me, ’delimiter’, ’%s’, ’-append’,’newline’, ’pc’) dlmwrite(filename_T, T_save(:,:,:,interval), ’delimiter’, ’\t’, ’-append’,’newline’, ’pc’) dlmwrite(filename_T, empty_m, ’delimiter’, ’%s’, ’-append’, ’newline’, ’pc’, ’roffset’,1)

save(filename_mat,’T_save’,’HF_He_save’,’heat_save’,’dt’,’save_interval’)

interval = interval+1; end end %ending t loop

%% Displaying some data on the screen disp([’Max of T is ’, num2str(max(max(max(max(T_save))))), ’K’]) disp([’Max of THe is ’, num2str(max(max(max(max(THe_save))))), ’K’]) disp([’Final heat density is ’,num2str((Ps_unscaled/vol_wire)*10^-9), ’ W/mm^3’]) toc disp(’’) disp(’’)

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c Jonas Lantz