Heat Transfer Correlations Between a Heated Surface and Liquid &amp; Superfluid Helium : For Better Understanding of the Thermal Stability of the Superconducting Dipole Magnets in the LHC at CERN

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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

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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

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Avdelning, Institution

Division, Department

Division of Applied Thermodynamics and Fluid Me-chanics

Department of Management and Engineering Linköpings universitet

SE-581 83 Linköping, Sweden

Datum Date 2007-10-19 Språk Language Svenska/Swedish Engelska/English Rapporttyp Report category Licentiatavhandling Examensarbete C-uppsats D-uppsats Övrig rapport

URL för elektronisk version

http://www.ikp.liu.se/mvs/ http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-10124 ISBN — ISRN LITH-IEI-TEK-A--07/00225--SE

Serietitel och serienummer

Title of series, numbering

ISSN

—

Titel

Title

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

Författare

Author

Jonas Lantz

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 supercon-ducting 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 super-conducting, 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 experi-ments 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

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Abstract

This thesis is a study of the heat transfer correlations between a wire and liquid he-lium 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 be-ing built at CERN. The magnets are cooled to 1.9 K and usbe-ing 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 mag-net is very important, since the superconducting wires need to be thermally stable. Thermal stability means that a superconductive magnet can remain superconduct-ing, 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 experi-ments 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 mea-surement 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.

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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

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nor-mal 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

supercon-ducting 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

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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

homoge-neously while being cooled by the nucleate boiling regime. Right: A simulation of the internal temperatures when placing a thermo-couple 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

tem-peratures 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

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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

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Nomenclature

Latin Letters

aF BI Heat transfer coefficient He I film boiling W/m

2_-K _{page 23}

aF BII Heat transfer coefficient He II film boiling W/m

2_-K _{page 31}

aKAP Heat transfer coefficient Kapitza W/m2-KnKAP page 30 aN B Heat transfer coefficient nucleate boiling W/m2_-Kn _{page 23} aN C Heat transfer coefficient natural

convec-tion

W/m2_-K _{page 23}

aT rans Transient heat transfer coefficient W/m2_-KnT rans _{page 22}

A Area m2 _{page 48}

A Gorter-Mellink mutual friction parame-ter

m-s/kg page 28

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

˙

qG Internal heat generation W/m3 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}

qc Critical heat flow in natural convection W/m2 page 23

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6 Nomenclature

qr Recovery heat flow to nucleate boiling W/m2 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 boiling

K page 40

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

He II

K page 18

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

ρn Normal component density kg/m3 page 25

ρs Superfluid component density kg/m3 page 26

Emmisivity - page 73

ηn Normal component viscosity N-s/m2 page 25

ηs Superfluid component viscosity N-s/m2 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

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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

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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 ex-perimental 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, travel-ing 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

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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 supercon-ducting 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.

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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 detec-tors 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.

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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.

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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:

• Current density Jc [A/m2] • 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 superconduc-tive 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.

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14 Introduction

There are several different beam-induced loads possible in the LHC, a few exam-ples 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 heat-ing) 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 supercon-ducting 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 supercon-ducting, 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 liq-uid 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 conduc-tion along the wire.

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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 super-conducting 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

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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 super-conductors 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

mo-tion 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,

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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 su-perconductive materials had to be done by either liquid hydrogen or liquid helium which are more expensive than liquid nitrogen. The warmest superconductive ma-terial 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 superflu-ids; 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 tem-perature 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

4_{helium, 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. The4_{helium atom is a boson and}

it can directly form groups with other4_{helium atoms when the temperature is low}

enough, around 2 K. The 3_{helium 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 3_{helium will form pairs with each other just in the same way as}

electrons form Cooper pairs when a material is becoming superconductive. The

3_{helium pairs make the particles in each nucleus add up to an even number which}

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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 is4_{helium, since it is found in nature whereas}3_{helium is a by-product}

when producing nuclear weapons and is very rare in nature. One other non-trivial aspect is that3helium 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 for4_{helium at low temperatures.}

From now on and throughout the text when talking about helium, it is the isotope

4_{He 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.

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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 cpas a function of temperature.

Figure 2.4. Left: Entropy as a function of temperature. Right: Thermal heat conduc-tivity 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.

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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

impor-tant 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 trans-fer 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.

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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: Heat Generation -Transient Heat Flow Natural Convection Nucleate Boiling Steady-state 7 -S S S S w ? ? 6 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

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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 convec-tion or nucleate boiling can be triggered. An useful expression for calculating the transient heat transfer in He I is:

q = aT rans(TnT rans− TbnT rans) (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.

Values for aT rans are usually set to 180 W/m2-KnT rans and nT ransto 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 convec-tion is forced convecconvec-tion, where the fluid moconvec-tion 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

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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 = aN C(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

convec-tion 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-tateladze [10], to be proportional to the temperature as q(T ) ∝ (T − Tb)n 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:

q = aN B(T − Tb)n (2.3)

where the exponent n normally takes a value around 1.5-2.5. The heat transfer coefficient aN B is in the order of 50 kW/(m2Kn), 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 = aF B_I(T − Tb) (2.4)

The heat transfer coefficient aF BI is in the range of 300 to 1000 W/(m

2_{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

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24 Theory

called qr. An expression for qrcan be set up by using He I film boiling theory and

experimental data as:

qr= q∗

r _ρv

ρv+ ρl

(2.5) 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 qrthere 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.

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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 van-ishing 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 10−5Pa-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 viscos-ity 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

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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 tem-perature. The specific entropy s is assumed to be constant in He II, taking the value sλwhich is the specific entropy at the lambda temperature. This makes the

the entropy (the product ρs) temperature dependent by changes in the normal components density. The entropy goes approximately as T5.6 _{between 1 and T}

λ

and therefore the following expression for the ratio between the normal componets density and the density for He II can be written:

ρn ρ = T Tλ 5.6 (2.8)

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 ρsgoes 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

vsc≈ 0.003d1/4 _(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

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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 parame-ter in superfluid helium is the thermal conductivity. The effective thermal conduc-tivity 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

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28 Theory

heat transport is described by the two-fluid model as a thermal counter-flow pro-cess 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:

∇P = −βηnvn

d2 (2.11)

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:

∇T = ∇P

ρs (2.12)

Combining equations 2.10, 2.11 and 2.12 gives the relation between the heat flux and the temperature gradient:

q = −d

2_ρ2_s2_T

βηn ∇T (2.13)

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:

∇T = − βηn d2_ρ2_s2_Tq − Aρn ρ3 ss4T3 q3 (2.14)

The second term on the right hand side in equation 2.14 takes into accout the mu-tual friction interaction, and A is the Gorter-Mellink mumu-tual friction parameter which can be experimentally determined [9]. The term dominates the tempera-ture 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:

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2.5 Helium as a Quantum Fluid, He II 29 with −f (T ) = Aρn ρ3 ss4T3 (2.16) Several experimental measurements of heat flux versus temperature gradients have allowed a determination of the function f (T ) and the Gorter-Mellink mutual fric-tion parameter A. The funcfric-tion 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−1for 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. Heat Generation -Heat Generation -Kapitza Conductance Second-Sound @ @ R @ @ R - @ @ R Gorter-Mellink Transient Heat Flow

Kapitza Conductance Film Boiling Steady-state ? 6

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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 con-tribute.” [7]

The effect is only visible at cryogenic temperatures, and it strongly decreases with increased temperature (∝ 1/T3_{). 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:

aKAP₀ = lim

∆Ts→0

q

∆Ts

(2.17)

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

with the simple relationship aKAP0 = αT

n_{, 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:

qKAP = aKAP(TnKAP − T_bnKAP) (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

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second sound can be written as:

∂2_s ∂t2 = s2_ρ s ρn ∇ 2_T (2.19)

For small temperature perturbations the second sound velocity, and thus the tem-perature 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 con-vection. In this case the heat transfer can be described by the Gorter-Mellink regime, looking like:

ρCp∂T ∂t = ∇ · 1 f (T )∇T 1/3 (2.20) 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 dif-ference 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 boil-ing 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 engi-neering applications the heat transfer correlation can be simplified with a linear function looking like:

q = aF B_II(T − Tb) (2.21)

The heat transfer coefficient aF BII is in the order of 250 to 1000 W/m

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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 trans-fer 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 con-ductance occurs at a lower heating power. This is due to hysteresis.

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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

Heat Transfer Correlations Between a Heated Surface and Liquid &amp;amp; Superfluid Helium : For Better Understanding of the Thermal Stability of the Superconducting Dipole Magnets in the LHC at CERN

Heat Transfer Correlations Between a Heated

Surface and Liquid & Superfluid Helium

Jonas Lantz

Heat Transfer Correlations Between a Heated

Surface and Liquid & Superfluid Helium

Jonas Lantz

Abstract

Acknowledgments

Contents

List of Figures

List of Tables

Nomenclature

Latin Letters

Greek Letters

Superscripts

Subscripts

Chapter 1

Introduction

1.1

CERN, a Short Introduction

1.2

The Large Hadron Collider - LHC

1.3

The Proton Beam

1.4

Detectors

1.5

The LHC Dipole Magnets

1.6

Superconducting Cables

1.7

Problem Formulation

Chapter 2

Theory

2.1

Superconductivity

2.2

Superfluidity

2.3

Liquid Helium

2.3.1

Introduction

2.3.2

Thermal Properties of Liquid and Superfluid Helium

2.4

Helium as a Classical Fluid, He I

2.4.1

Transient Heat Flow

2.4.2

Natural Convection

2.4.3

Nucleate Boiling

2.4.4

Film Boiling

2.4.5

Summary of He I Heat Flow

2.5

Helium as a Quantum Fluid, He II

2.5.1

The Two-fluid Model

2.5.2

He II Dissipation Mechanisms

2.5.3

He II Heat Transport

2.5.4

Kapitza Conductance

2.5.5

Transient Heat Flow Mechanisms

2.5.6

Film Boiling

2.5.7

Summary of He II Heat Flow

Chapter 3

Experimental Setup

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