Stability of superconductors

Stability of Superconductors

SELECTED TOPICS IN SUPERCONDUCTIVITY Series Editor:

Stuart Wolf Naval Research Laboratory Washington, D.C.

CASE STUDIES IN SUPERCONDUCTING MAGNETS Design and Operational Issues Yukikazu Iwasa INTRODUCTION TO HIGH - TEMPERATURE SUPERCONDUCTIVITY Thomas P. Sheahen STABILITY OF SUPERCONDUCTORS Lawrence Dresner

A Continuation Order Plan is available for this series. A continuation order will bring delivery of each new volume immediately upon publication. Volumes are billed only upon actual shipment. For further information please contact the publisher.

Stability of Superconductors Lawrence Dresner Oak Ridge National Laboratory Oak Ridge, Tennessee

Kluwer Academic Publishers • New York / Boston / Dordrecht / London / Moscow

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To my grandchildren, Matthew, Stephanie, Brian, Jacob, Aaron, Benjamin, Danielle, Chelsea, Haley, Micah, and Max

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Read not to contradict and confute, nor to believe and take for granted, nor to find talk and discourse, but to weigh and consider.

—Francis Bacon

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Preface

The science of superconductivity is split into two parts, each of which is coherent within itself and which are joined by a common interest in superconductors. One part is peopled by solid-state physicists, metallurgists, and materials scientists who try to understand why superconductivity exists and try to use that understanding to make better superconductors. The other part is peopled by physicists, cryogenicists, and engineers who take for granted a superconductor created in the first part and try to answer the question, “How shall I make a magnet from this superconductor?” The workers of the first part, the materials people, deal with the quantum mechanics of the solid state. The workers of the second part, whose field I call applied superconductivity,deal only with classical physics: electricity and magnetism, fluid flow, heat transfer, stress, and strain. It might be thought that classical studies are less adventurous than quantum studies, but applied superconductivity has a novel twist that gives it great intrinsic interest. This twist is the fundamental instability of superconductors: if a superconductor ever loses the property of superconductivity, that is, becomes resistive, this loss tends to become worse, not better; the superconductor departs more and more from the superconducting state. How this comes about, when it happens, and what one can do about it are the subject of this book. I have written this book to appeal to as wide an audience as possible. To do so I have composed two intertwined texts. The simpler text, appearing in unmarked sections, is suitable for a first reading or for use by persons not having extensive training. The more detailed text includes sections marked with a diamond, which contain more advanced material. In addition, notes on special points have been added at the end of each chapter that can be consulted if the reader desires, but ignored without risk of breaking the continuity of explanation. This book should interest senior undergraduates and graduate students in physics and engineering, as well as workers already in or just entering the field of applied superconductivity. ix

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Preface

There is a strong need both for this book and for the series of which it is a part. The burgeoning knowledge of applied superconductivity is now spread through a voluminous literature that includes the international journal Cryogenics, the IEEE Transactions on Magnetics, the recently created IEEE Transactions on Applied Superconductivity, the series Advances in Cryogenic Engineering, and the proceedings of various conferences such as the biennial Applied Superconductivity Conferences, International Cryogenic Engineering Conferences, and Magnet Technology Conferences, to name only a few. These sources and others (e.g., special workshops such as the biennial US/Japan workshops on helium heat transfer and magnet stability) produce many thousands of pages of literature each year. The proceedings of the 1992 Applied Superconductivity Conference, for example, ran to almost 3000 pages, and this year Cryogenics will publish more than a thousand pages. Finding the present state of a specialized topic in this teeming literature is hard for the veteran and positively daunting for the novice. So the time is ripe for creating a smaller, better organized, didactic literature that can guide active researchers, newcomers, and students. These remarks take care of the what, the who, and the why , but there still remains to be discussed the how, or what I call the tone of the book. By tone I mean how much mathematics is used and how rigorous it is. In establishing a comfortable tone, I follow the advice of R. W. Hamming, who chose for the motto of his book on numerical analysis the epigraph, “The purpose of computing is insight, not numbers.” In this book, as in Hamming’s, the purpose of computation (analytical here rather than numerical) is to provide insight. Accordingly, I defer rigor in favor of clarity and keep the mathematics as simple as possible, consistent with satisfactory understanding. We cannot do without mathematics, however, because applied superconductivity is a quantitative science in which we must be able successfully to build magnets. Here I have been guided by the words of Philip Morrison, who said, “The aim of quantity in science is not mere maximum precision but approximations reliable enough to argue from.” (italics mine) Such approximations, based though they are on idealizations and simplifications, often serve us better than inclusive computer programs that solve the same problem. Approximate methods form the backbone of this book. The formulas that result allow relatively easy computation of quantities of interest and so are extremely useful in design. Furthermore, if the reader studies how they have been derived, he will, I hope, learn something of what E. P. Wigner called “the essence of creative . . . thought: methods of work, tools of argument.” I should like to acknowledge here the generosity of Dr. John Sheffield, Director of the Fusion Energy Division of Oak Ridge National Laboratory, who made available to me the time to compose this book and put at my disposal the resources of Oak Ridge National Laboratory. Of those resources, none was more important than the graphics department of the Reports Office, and I would like here to express

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my thanks to artists Margaret Eckerd, Judy Neeley, and Shirley Boatman for the excellence of their drawings. I should like to record here, too, my gratitude to Mr. M. S. Lubell, head of the Magnetics and Superconductivity Section, for his constant support and encouragement, not just in this endeavor, but over two decades of work together. Finally, I should also like to note a more diffuse kind of debt to four of my colleagues, Dr. J. R. Miller, Dr. J. W. Lue, Prof. S. W. van Sciver, and the late Mr. M. O. Hoenig. Over the years, close collaboration with these four colleagues has helped to determine the focus of my work in the field of applied superconductivity, as I think can be seen by the frequency with which their names appear among the references. Lawrence Dresner Oak Ridge, Tennessee

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Units

The units used in this book to specify physical quantities are those of the International System (meter, second, kilogram, ampere). The electrical units are rationalized. This means that factors of 4π do not appear in Maxwell’s equations. The unit of magnetic field (H) in the International System is amperes per meter, but it has become customary in applied superconductivity to express fields in terms of the magnetic induction (B) they produce in free space, the unit of which is the tesla (abbr.: T). I will adhere to this custom. In carrying out mathematical derivations, special units are often defined and used. A special system of units is one in which various quantities have been set equal to 1 for convenience. Only a limited number of quantities can be set equal to 1. The criterion for this is that no dimensionless combination of the quantities should exist because dimensionless quantities, which are not necessarily equal to 1, have the same value in all systems of units. The introduction of special units simplifies the appearance of the equations and reduces the amount of writing in a mathematical derivation. However, when a derivation is complete, there will be missing from the final formula in various places powers of the quantities that have been set equal to 1. These must be inserted in the final formula to make it dimensionally homogeneous. There is only one way to do this, so nothing is ever lost by the introduction of special units. I call this reconstituting the formula to regular units. Introducing special units is an alternative to creating appropriate dimensionless variables at the outset of the problem. It allows us to jump into the real work with a minimum of formal preliminaries. But when we are done, we must perform the work of reconstituting our results. How one proceeds is largely a matter of choice, and I prefer special units to dimensionless variables, having grown habituated to them by long practice. To be sure that the matter is clear, I give here a short derivation of the formula connecting the length and the period of a pendulum. Let us introduce special units xiii

xiv

Units

in which the mass m of the pendulum bob, the length h of the pendulum, and the acceleration of gravity g all have the value 1. Then when the pendulum is displaced slightly from the vertical by an angle θ, the tangential restoring force is -sin θ ~ -θ and the tangential acceleration is θ″ ≡ d 2θ/dt 2. The differential equation of the pendulum in special units is then θ″ + θ = 0, the solution of which is θ = A sin t + B cos t. In special units, then, the period ∆t = 2π. To reconstitute this formula, we must add powers of m, h, and g to this last equation so that it is dimensionally homogeneous. Now the left-hand side has the dimensions of time T. The dimensions of m are mass M, those of h are length L, and those of g are LT-2. Then we must add a factor (h/g)1/2 to the right-hand side so that both sides have the dimensions T. Our result then becomes in regular units ∆t = 2π(h/g)1 / 2 .

Numbering of Sections, Equations, etc.

The numbering of text sections is consecutive within each chapter. Accordingly, Section 3 of Chapter 4 is referred to as Section 4.3 in other parts of the text. Equations are numbered consecutively in each section, but the numbering begins anew in the next section. Thus if Section 4.3 contained five equations, the last would be Eq. (4.3.5). The next equation (if it occurred in the next section) would be Eq. (4.4.1). References are noted in the text using authors’ names and year. Otherwise identical references have the short title appended after the year. The references are listed at the end of the book in alphabetic order of authors, then by year, then by alphabetic order of title.

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Contents

Chapter 1. Introduction and Overview 1.1. 1.2. 1.3. 1.4. 1.5. 1.6. 1.7. 1.8.

Early Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Critical Temperature and Critical Field . . . . . . . . . . . . . . . . . . . . . 3 Type-II Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Pinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Composite Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Quenching and Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 The Phase Diagram of Helium . . . . . . . . . . . . . . . . . . . . . . . . . 13 High-Temperature Superconductors. . . . . . . . . . . . . . . . . . . . . . 15

Chapter 2. Material Properties 2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7. 2.8. 2.9.

Specific Heat: The Debye Formula . . . . . . . . . . . . . . . . . . . . . . 19 Specific Heats of Type-II Superconductors . . . . . . . . . . . . . . . . 22 First Law of Thermodynamics for a MagnetizableBody . . . . . . . . . 25 Gibbs Free Energy of a Magnetizable Body . . . . . . . . . . . . . . 27 Specific Heat at Zero Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Contribution of the Magnetic Field to the Specific Heat . . . . . . . 29 Matrix Resistivity; Bloch-Grüneisen Formula . . . . . . . . . . . . . . 30 Magnetoresistivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 The Wiedemann-Franz Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Chapter 3 . Flux Jumping 3.1. 3.2. 3.3. 3.4.

TheCritical-State Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Three-Part Curve of Joule Power . . . . . . . . . . . . . . . . . . . Charging of a Superconductor: Critical-State Model . . . . . . . . . . Charging of a Superconductor: Power-Law Resistivity . . . . . . . . .

35 36 38 39 xvii

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3.5. 3.6. 3.7. 3.8. 3.9. 3.10.

Flux Jumping Defined . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Validity of the Adiabatic Assumption . . . . . . . . . . . . . . . . . . . . . . Stability against an External Magnetic Field . . . . . . . . . . . . . . . . Twisted Filaments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Self-Field Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Final Word on Flux Jumping . . . . . . . . . . . . . . . . . . . . . . . . . . .

42 45 46 48 51 52

Chapter 4. Boiling Heat Transfer and Cryostability 4.1. Fundamentals of Boiling Heat Transfer . . . . . . . . . . . . . . . . . . . 53 4.2. Additional Factors Affecting Boiling Heat Transfer . . . . . . . . . . . . 55 4.3. Cryostability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.4. Cold-EndRecovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.5. Improving Boiling Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.6. Minimum Propagating Zones (I) . . . . . . . . . . . . . . . . . . . . . . . . 64 4.7. Minimum Propagating Zones (II) . . . . . . . . . . . . . . . . . . . . . . . . 68 4.8. The Formation Energy of the Minimum Propagating Zone . . . . . . . 71 4.9. The Maximum Allowable Resistive Fault . . . . . . . . . . . . . . . . . . 72 4.10. Stability of Partly Covered Conductors . . . . . . . . . . . . . . . . . . . . 75 4.11. Transient Heat Transfer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Chapter 5. Normal Zone Propagation 5.1. 5.2. 5.3. 5.4. 5.5. 5.6. 5.7. 5.8. 5.9. 5.10.

Exact Calculation of the Propagation Velocity . . . . . . . . . . . . . . Approximate Calculation of the Propagation Velocity . . . . . . . Comparison with Experiments of Iwasa and Apgar . . . . . . . . . Effect of Transient Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . Traveling Normal Zones (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Traveling Normal Zones (II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Traveling Normal Zones (III) . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Excess Joule Heat Due to Current Redistribution . . . . . . . . . . The Special Case of a Cylindrical Conductor . . . . . . . . . . . . . . . Comparison with Experiment of Pfotenhauer et al . . . . . . . . . .

83 85 87 88 90 92 93 95 97 98

Chapter 6. Uncooled Conductors 6.1. 6.2. 6.3. 6.4. 6.5. 6.6. 6.7.

The Bifurcation Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Group Analysis of the Bifurcation Energy . . . . . . . . . . . . . . . . . . 102 Estimation of the Undetermined Constant . . . . . . . . . . . . . . . . . . 104 Size of the Bifurcation Energy . . . . . . . . . . . . . . . . . . . . . . . . . . 105 The Effect of Transverse Heat Conduction on the Bifurcation Energy 106 Bifurcation Energies of High-Temperature Superconductors . . . . 107 PropagationVelocities of UncooledSuperconductors . . . . . . . . . . . 110

Contents

6.8. 6.9. 6.10. 6.11. 6.12. 6.13. 6.14.

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Propagation with Temperature-Dependent Material Properties . . . 113 The Effect of Current Sharing on the Propagation Velocity . . . . . 115 An Interesting Counterexample . . . . . . . . . . . . . . . . . . . . . . . . 115 The Approach to a Traveling Wave . . . . . . . . . . . . . . . . . . . . 119 The Effect of Heat Transfer to the Potting on the Propagation Velocity 121 The Adiabatic Hot-Spot Formula . . . . . . . . . . . . . . . . . . . . . 122 Thermal Stresses during a Quench . . . . . . . . . . . . . . . . . . . . . . . 124

Chapter 7. Internally Cooled Superconductors 7.1. 7.2. 7.3. 7.4. 7.5. 7.6. 7.7. 7.8.

Stability Margins and Induced Flow . . . . . . . . . . . . . . . . . . . . The One-Dimensional Equations of Compressible Flow . . . . . . Induced Flow in a Long Hydraulic Path . . . . . . . . . . . . . . . . . . . Induced Flow in the Experiments of Lue et al . . . . . . . . . . . . . . Multiple Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Limiting Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discussion of the Isobaric Assumption . . . . . . . . . . . . . . . . . The Lower Stability Margin . . . . . . . . . . . . . . . . . . . . . . . . . . . .

129 132 134 136 141 146 148 151

Chapter 8. Hydrodynamic Phenomena 8.1. 8.2. 8.3. 8.4. 8.5. 8.6. 8.7. 8.8. 8.9. 8.10. 8.11. 8.12. 8.13. 8.14. 8.15.

Neglect of Fluid Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Maximum Quench Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 Thermal Expulsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 Expulsion into an Unheated Part of the Conductor . . . . . . . . . 158 Short Initial Normal Zones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 The Piston Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 The Piston Formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 Slug Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 The Propagation Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 Thermal Hydraulic Quenchback . . . . . . . . . . . . . . . . . . . . . . . . 164 Hydrodynamic Quench Detection . . . . . . . . . . . . . . . . . . . . . . 167 Rational Design of cable-in-Conduit Conductors . . . . . . . . . . . 168 Perforated Jackets: Modified Hydrodynamic Equations . . . . . . . 169 Perforated Jackets: Reduction of the Quench Pressure . . . . . . 172 Perforated Jackets: Effect on the Stability Margin . . . . . . . . . . . 173

Chapter 9. Cooling with Superfluid Helium 9.1. 9.2. 9.3. 9.4.

The Superfluid Diffusion Equation . . . . . . . . . . . . . . . . . . Superconductor Stability: The Method of Seyfert et al . . . . . . . Similarity Solution in a Long Channel . . . . . . . . . . . . . . . . The Temperature Dependence of the Properties of He-II . . . . . .

175 176 178 180

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9.5. The Kapitza Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 9.6. The Two-Dimensional Channel . . . . . . . . . . . . . . . . . . . . . . . . . . 182

Chapter 10. Miscellaneous Problems 10.1. 10.2. 10.3. 10.4. 10.5. 10.6. 10.7. 10.8.

An Uncooled Segment of a High-Temperature Superconductor . . 185 The Critical Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 Vapor-Cooled Leads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .188 The Heat BalanceEquation forVapor-Cooled Leads . . . . . . . . . . . 190 Copper Leads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 Superconducting Leads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 Partly Normal States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 Partly Normal States: Results and Discussion . . . . . . . . . . . . . 195

Appendix A. The Method of Similarity Solutions A.1. Partial Differential Equations Invariant to One-Parameter Families of One-Parameter Stretching Groups . . . . . . . . . . . . . . A.2. Similarity Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3. The Associated Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4. Asymptotic Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.5. Example: The Superfluid Diffusion Equation . . . . . . . . . . . . . A.6. Information Obtainable by group Analysis Alone . . . . . . . . . . .

Appendix B. B.1. B.2. B.3. B.4. B.5. B.6.

199 200 201 201 202 205

Stability of the MPZ

The Ordering Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proof of the Ordering Theorem . . . . . . . . . . . . . . . . . . . . . . . . . Application of the Ordering Theorem to the MPZ . . . . . . . . . . . Lagrangian Formulation and the Stability of Steady States . . . . . . Action Integral of the MPZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stability of the Steady States of an Uncooled Segment of a Superconductor . . . . . . . . . . . . . . . . . . . . . . . . . . . .

207 207 209 209 211 212

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

1 Introduction and Overview

1.1. EARLY RESEARCH The story of superconductivity begins in 1911 with the accidental discovery of superconductivity in mercury by the Dutch physicist Heike Kammerlingh-Onnes. Although the nature of his discovery was not expected, it could have been expected that someone would be doing what he was doing at the time, and it is fair to say that if Kammerlingh-Onnes had not discovered superconductivity when he did, someone else probably would have discovered it during the same epoch of physics. In 1911, Kammerlingh-Onnes stood at the confluence of several main streams of research in nineteenth-century physics. On the theoretical side, the kinetic theory of gases, Maxwell’s equations of electromagnetism, and Thomson’s discovery of the electron, three of the grand achievements of the century, had been joined by H. Lorentz in his electron theory of metals. On the experimental side, the early attempts of Davy and Faraday to liquefy gases had culminated in Cailletet’s, Pictet’s, and Olszewski-Wroblewski’s liquefaction of oxygen, Dewar’s liquefaction of hydrogen, and Dewar’s invention of the indispensable thermos flask known in scientific circles by his name. So among the attractive possibilities open to physicists at Kammerlingh-Onnes’s time was the study of the properties of metals at low temperatures using liquid cryogens in the hope of better understanding the behavior of the Lorentz electron gas. Three years earlier, Kammerlingh-Onnes had succeeded in liquefying helium, the last of the so-called permanent gases to be liquefied. With liquid helium in a dewar flask, Kammerlingh-Onnes had a peerless tool for low-temperature research. He picked as his first target the electrical resistance of metals. He had two reasons for his choice (Mendelssohn, 1968). First, Mendelssohn says, Kammerlingh-Onnes felt that the measurement of electrical resistance would be relatively simple and thus a good place to begin. Second, such measurements might help to decide between rival theories then extant of how the electrical resistance of metals behaves 1

2

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as the temperature approaches absolute zero. One school of thought held that the electrons would freeze and stop moving, which implied an infinite resistance. Another school, using thermodynamic reasoning based on Nernst’s theorem, argued that the resistance would approach zero smoothly with falling temperature. As Mendelssohn notes, in 1911 Kammerlingh-Onnes already knew that neither of these ideas applied categorically: early experiments on platinum and gold had showed that at low enough temperatures the resistance approached a finite limit and became independent of temperature. This limit is called the residual resistance and plays an important role in practical matters of magnet design. KammerlinghOnnes rightly attributed the residual resistivity to the scattering of conduction electrons by impurity atoms. What then, he asked, would happen in a very pure material? Mercury was the clear choice for a test metal because it could easily be purified by multiple distillation. When Kammerlingh-Onnes measured its resistance as a function of temperature, he found that it dropped suddenly at 4.15 K to unmeasurably low values (< 10¯ 6 of the room temperature value). Such a sudden drop in resistance came as a great surprise, and when it was found to happen also in other metals (e.g., Pb, Sn, In), it inflamed Kammerlingh-Onnes’s mind with visions of resistanceless magnets producing high magnetic fields. Certain difficulties unforeseen (and indeed unforeseeable) dashed these early hopes for high-field magnets, but in the end they were realized, as described below. In the meanwhile, a question of principle remained: How small was the resistance on the low-temperature side of the discontinuity? Now, measurement alone can never show that a physical quantity is zero, because of the finite precision of our apparatus. Today, based on theories created to explain what we know of superconductivity, we believe the resistance in the superconducting state to be exactly zero. But Kammerlingh-Onnes, with only experiment as his guide, could not be sure. Accordingly, he set about improving his value of the upper bound to the resistance. He fashioned a loop of superconductor, charged it with current, and monitored the magnetic field outside the dewar flask for any decay. During several hours, he saw none, which bespoke a resistance < 10¯11 of the resistance at room temperature. Mendelssohn (1968) mentions such a persistent-mode experiment of much later vintage that lasted for two years and came to an end only because a transport strike interrupted the supply of liquid helium. These early researches, which gave birth to the science of superconductivity, were honored in 1913 by the award to Kammerlingh-Onnes of the Nobel Prize in Physics.

Introduction and Overview

3

1.2. CRITICAL TEMPERATURE AND CRITICAL FIELD The obvious application of superconductivity is to magnets, so it did not take Kammerlingh-Onnes long to discover that the application of too strong a magnetic field destroyed the superconducting state and restored resistivity to the conductor. The level of field at which this happened, only a few hundredths of a tesla, were disappointingly low, and the early vision of high-field superconducting magnets fled. (For comparison, note that the six magnets of the IEA’s Large Coil Task, finished in 1987, weighed approximately 45 tonnes apiece and together produced a field of 9 T, Beard et al., 1988.) The substance of Kammerlingh-Onnes’s early experiments is summarized in Fig. 1.1. Here the absolute temperature is the abscissa and the applied magnetic field is the ordinate. At points below and to the left of the curve, the sample is superconducting while at points above and to the right of the curve, the sample is not. In the science of superconductivity, the nonsuperconducting, resistive state is called the normal state, by virtue, I suppose, of its having been the earlier state to be recognized. The curve that separates the normal from the superconducting region is closely approximated by a parabola: B/B c = 1 – (T/T c)2

(1.2.1)

Though accurate enough for the purposes of this monograph, Eq. (1.2.1) is not exact. The intercept on the temperature axis Tc is called the critical temperature;

Figure 1.1. The phase diagram of superconductors as determined in Kammerlingh-Onnes’s early experiments. At points below and to the left of the curve, the sample is superconducting while at points above and to the right of the curve it is not.

CHAPTER 1

Table 1.1. Critical Fields and Temperatures of Some Superconductors Superconductor In Sn Hg Ta Pb

Critical temperature (K)

Critical field (mT)

3.41 3.72 4.15 4.48 7.18

28.7 30.9 41.2 82.9 80.4

the intercept on the field axis Bc is called the critical field. Table 1.1 gives the critical fields and temperatures of several of the superconductors discovered in the early days of superconductor research. When a field B p is applied to the superconductor, it remains superconducting up to some temperature T p < T c. The pair of values (T p, B p) are the coordinates of some point on the curve separating the superconducting from the normal state. A temperature like Tp lying at an interior point of the curve is often loosely referred to as the critical temperature; a clearer but clumsier nomenclature would be the critical temperature at a field B p. But in most of the literature, the reader is left to decide from the context what temperature is meant by the designation critical. After Kammerlingh-Onnes’s initial discoveries, superconductivity research lay fallow for more than two decades owing to the low values of the critical fields noted in Table 1.1. Then Meissner and Ochsenfeld (1933) discovered that superconductors expelled the applied magnetic field from their interiors. They found that if a material was cooled into the superconducting state and then exposed to an external magnetic field, the field would not penetrate the interior of the material until the field reached the critical value (at the particular temperature). Furthermore, they found that if the external field was first applied and then the temperature was decreased, when the temperature dropped below the critical value (at that field), the sample, as it became superconducting, expelled the field from its interior. In short, superconductors are not just perfect conductors, they are also perfect diamagnets. This phenomenon is called the Meissner effect.

1.3 TYPE-II SUPERCONDUCTORS At first, no one knew that there were two kinds of superconductors. The superconductors discovered by Kammerlingh-Onnes and described above are now known as type-I superconductors or soft superconductors. Today, we know that there is another class of superconductors, known as type-II superconductors or hard superconductors that differ from the type-I superconductors in that they admit the magnetic field into their interiors while still remaining superconducting. It is from

Introduction and Overview

5

Figure 1.2. The phase diagram of type-II superconductors.T he lower left-hand region represents a superconducting state exactly like that of type-I superconductors: zero resistance and perfect diamagnetism. The upper right-hand region represents the normal state. The middle region represents the mixed state. The fields B c1 and B c2 are called the lower and upper critical fields, respectively.

these type-II superconductors that contemporary scientific and commercial superconducting magnets are wound. Shown in Fig. 1.2 is a phase diagram for the type-II superconductors similar to the phase diagram in Fig. 1.1. But now instead of two regions in the field-temperature plane, there are three. The lower left-hand region represents a superconducting state exactly like that of type-I superconductors: zero resistance and perfect diamagnetism. The upper right-hand region represents the normal state. The middle region represents a state, called the mixed state, that requires some detailed description. The magnetic field that a type-II superconductor admits in the mixed state is not uniform, as classical physics requires, but is confined to discrete bundles called fluxoids (see Fig. 1.3). Each bundle has a normal (i.e., nonsuperconducting) core that is threaded by magnetic field. Outside the core, where the material is superconducting, the magnetic field drops off exponentially. The total magnetic flux associated with one core is f _= h/2e = 2.068 x 10¯ 15 Wb, where h is Planck’s constant (6.625 x 10¯ 34 J s) and e is the charge on the electron (1.602 x 10¯ 19 C). Circling each core in the superconducting region, where the field is falling off, is an electric current, whose strength also falls off exponentially with distance from the core. The existence of the triangular flux lattice shown in Fig. 1.3 was first proposed on theoretical grounds by Abrikosov (1957). Then Essmann and Träuble (1967) visualized Abrikosov’s flux lattice directly. They exposed the surface of a type-II

6

CHAPTER 1

Figure 1.3. The triangular fluxoid lattice of a type-II superconductor in the mixed state. Each fluxoid has a normal core that is threaded by a quantum of flux. An electric current circles each core in the superconducting region.

superconductor to a puff of cobalt vapor produced by exploding a cobalt wire in the same (cold) chamber as the sample. The cobalt atoms, being ferromagnetic, followed the field lines in their outward flight from the wire and converged on the normal cores where the cores ended on the sample surface, revealing with great clarity the triangular lattice predicted by Abrikosov. If a current is established in the sample perpendicular to the applied magnetic field (Fig. 1.4), it flows in the superconducting region between the normal cores. This current, called the transport current, by flowing through a region in which there is a magnetic field, feels a sidewise volume force J x B, where J is the transport current density and B is the local field. By the equality of action and reaction, the flux lattice feels a volume force -Jx B. (At this point, it is worth stating explicitly that the flux lattice is not just the triangular array of normal cores, but the entire continuous array of normal cores, circulating currents, and inhomogeneous magnetic field that fills the whole superconductor.) This volume force causes the flux lattice to move as indicated in Fig. 1.4. As fluxoids reach one surface of the sample, they disappear, while new fluxoids appear at the opposite surface, so that a steady flow through the sample is maintained. If one moves with a velocity v through a region of magnetic field B, one feels an electric field E = v x B. Now the velocity of the transport current relative to the flux lattice, is parallel to J x B, so that in the rest frame of the transport current there is an electric field E that has the direction (J x B) x B. The electric field E thus always has a component in the direction opposite to that of J (see Fig. 1.5). So the

Introduction and Overview

7

Figure 1.4. A sketch showing the volume forces that a transport current and the fluxoid lattice exert on each other.

motion of the flux lattice induces an electric field that opposes the flow of transport current, in short, a back emf. When an external voltage source, e.g., a battery, is connected across a type-II superconductor, the transport current increases until the back emf just balances the applied voltage. At that point, further increase in the transport current ceases and a steady state prevails. But the battery must be left connected to maintain the transport current and it must supply a steady power J·E d(vol) to overcome the back emf. To the battery, then, the sample shows resistance (called the flux-flow resistance). Fortunately for the practical applications of type-II superconductors, the flow of the flux lattice is impeded by solid-state defects such as impurities, vacancies, interstitial atoms, dislocations, grain boundaries, and precipitates. If in their motion across the sample, the fluxoids encounter a defect, they are attracted to it.1 If enough such defects are present the entire flux lattice may get snagged on them and be

Figure 1.5. Vector relations among the magnetic field B, the current density J, the velocity v ~ J x B of the transport current relative to the fluxoid lattice , and the electric field E present in the rest frame of the transport current.

8

CHAPTER 1

unable to move. In such a case, the flux-flow resistance vanishes, and the transport current, once established, persists just as in a type-I superconductor.

1.4. PINNING The immobilizing of the flux lattice by defects and impurities is called pinning. If a transport current (current density J) is flowing in the superconductor and if it is exposed to a magnetic field B transverse to the current (as is typically the case in a conductor in a solenoid), the volumetric Lorentz force JB acting between the current and the fluxoids is restrained by the pinning. But there is a limit to how great a force the pinning centers can sustain, and if the product JB becomes too large, the flux lattice is torn loose and begins to move. Flux-flow resistance appears, and for practical purposes the superconducting ideal of lossless current flow is vitiated. In a fixed background field B, the largest current density that a type-II superconductor can sustain without the appearance of flux-flow resistance is called the critical current density Jc. The rise in resistivity in a superconductor when the critical current density is surpassed is very rapid, and almost immediately beyond the critical current density, the flux-flow resistivity exceeds by a large margin the residual resistivity of ordinary conductors like copper. So the critical current density can be looked upon as a kind of boundary between two states, one superconducting and the other resistive. In this respect there is a loose analogy between the type-I and type-II superconductors. But it should be emphasized that there is in fact an important difference. When type-I superconductors first become resistive, the superconducting state has been destroyed, but when type-II superconductors first become resistive the superconducting state of quantized fluxoids still persists . Experiments, among which the earliest were those of Kim, Hempstead, and Strnad (1963), show that the pinning force JcB can be considered independent of B to a rather good approximation. These authors have proposed on the basis of their experiments the empirical formula Jc = const/(B + Bo)

(1.4.1)

where Bo is a constant of the order of a few hundredths of a tesla. At high fields it can be neglected and the critical current density taken inversely proportional to the field. Not only does the critical current density Jc decrease with increasing field, but it also decreases with increasing temperature. This is because the pinning energy decreases with increasing temperature (cf. Eq. (2.4.8)). The higher the temperature, therefore, the smaller the Lorentz force Jc B required to tear the entire lattice loose and initiate steady flux flow.

Introduction and Overview

9

Even when the pinning force is strong, the flux lattice can creep due to thermal fluctuations. Since the pinning energy is comparable with the thermal energy kT, where k is Boltzmann’s constant and T is the temperature, thermal fluctuations occasionally lift a pinned fluxoid from its pinning site, after which it drifts under the action of the Lorentz force until it encounters another site. Such flux creep is accompanied by the appearance of a small flux-flow resistivity. Figure 1.6 shows a surface in a three-dimensional space whose axes are temperature T, external magnetic field B, and current density J. Points below the surface correspond to a firmly pinned fluxoid lattice and the absence of resistance to current flow. Points above the surface correspond to flux flow and the presence of resistance. Inside the larger surface, near the origin, is a second much smaller surface that marks the boundary of type-I behavior. At points below that surface, the superconductor exhibits a Meissner effect and is a type-I superconductor. The bounding curve of the smaller surface in the B-T plane is the same as the phase boundary in Fig. 1.2 that separates the Meissner from the mixed state. The current on this lower surface is determined by the Silsbee condition that the sum of the self-field and the background field always be less than the critical field that separates the Meissner from the mixed state (Silsbee, 1916). On the other hand, the current on the upper surface is determined by the condition that the flux lattice remain pinned, i.e., that there be no flux-flow resistance. The bounding curve of this surface in the B-T plane need not necessarily be the phase boundary in Fig. 1.2 that separates the mixed state from the normal state, and in fact for some of the new high-temperature ceramic superconductors (Tc ~ 100 K ) it is not. For some of those superconductors, there is yet another curve, called the irreversibility curve, that traverses the region of the B-T plane corresponding to the mixed state: on one side of it Jc is finite, on the other Jc = 0. It is the irreversibility curve that determines the practical performance of the high-temperature superconductors rather than the phase boundary between the mixed and normal states. On the other hand, for the older, low-temperature superconducting metallic alloys and compounds (e.g., NbTi and Nb3Sn; Tc = 9.6 K and 18 K, respectively), the irreversibility curve and the phase boundary are practically indistinguishable and usually no distinction is made between them. The difference in behavior between the two classes of superconductors is caused, of course, by the quite large difference in the amplitude of the thermal agitation of their fluxoid lattices. The surface in Fig. 1.6, the so-called critical surface, is the basic datum on which the considerations of this monograph are based. With it in hand we can turn to the stability problem; but first we must begin with a short description of the structure of modern superconducting wires.

10

CHAPTER 1

Figure 1.6. Surfaces in J-B-T space separating different kinds of behavior of a type-II superconductor. Above the larger surface, the superconductorexhibits flux-flow resistivity; below it is superconducting. Above the smaller surface, but below the larger surface, the superconductoris in the mixed state; below the smaller surface it exhibits type-I superconductivity (Meissner effect). Beyond the base curve in the B-T plane, the superconductor is normal.

1.5. COMPOSITE SUPERCONDUCTORS Modern commercial wires made of low-temperature superconductors (almost exclusively NbTi or Nb3Sn) consist of many fine filaments of superconductor buried in a matrix of ordinary metal (almost always copper, but for certain special uses CuNi). Such superconductors are called composite superconductors. The number of filaments varies from a few thousand to as much as a hundred thousand, their diameters varying from a few tens of micrometers to a few micrometers. This construction is convenient for several reasons. First, if the superconducting filaments should be driven out of the superconducting region of Fig. 1.6, the current can safely switch to the copper. Since the normal-state resistivity of superconductors is much higher than the residual resistivity of copper, the Joule heat produced in the copper is much lower than if the current continued to flow in the supercon-

Introduction and Overview

11

ducting filaments. Then there are measures that can prevent the magnet from destroying itself by overheating, which is what would happen if the magnet were wound from the pure superconductors themselves. Second, the disposition of the superconductor in the form of fine filaments avoids a magnetic instability called a flux jump that is at least one of the causes of the superconductor being driven out of the superconducting state into the normal state. We shall discuss flux jumping in detail in Chapter 3. Third, the disposition of the superconductor in the form of fine filaments enables them to bend around rather small radii without having their critical current density degraded by the mechanical strain they suffer. This is a very serious consideration in the case of Nb3Sn composites, where tensile strains of only a few tenths of a percent cause substantial reductions in their current-carrying capacity. This is because Nb3Sn is a rather frangible material in which cracks appear that interrupt the path of current flow in the superconductor. Strain degradation is much less of a problem in NbTi composites because NbTi is ductile; these composites can stand tensile strains of several percent without significant degradation of their current-carrying capacity. This is the reason that most superconducting magnets producing fields less than roughly 8 T are wound with NbTi composite conductors. Typical examples of such magnets are those used for focusing beams of charged particles from accelerators and those used in magnetic resonance imaging (MRI) machines. When higher fields are desired, or when it is desired to operate at temperatures higher than that of liquid helium (4.2 K), the critical current density of NbTi becomes so low that it is necessary to use Nb3Sn. The fourth and last reason for the disposition of the superconductor in the form of fine filaments in a copper matrix is ease of manufacture. For example, NbTi composites are manufactured by stacking NbTi rods in a hexagonal array of holes in a copper billet and then drawing the billet to fine wire. The wires may be restacked and the process repeated several times, the drawings being interspersed with anneals to keep the copper from work-hardening too much. In the case of Nb3Sn composites the most common method is the bronze method, in which the first drawings are of Nb rods in a CuSn (bronze) matrix. Later, the NbCuSn composite wires are surrounded by a Ta diffusion barrier and stacked in a Cu billet. The end result is a composite of Nb wires each encased in an annulus of CuSn that is surrounded by a layer of Ta and all buried in a matrix of copper. The last step is to heat treat the wires: the Sn diffuses into the Nb, reacting with it and forming Nb3Sn; the Ta barrier prevents the mobile Sn from diffusing into the copper and thereby raising its residual resistivity. After the heat treatment such conductors must be handled very carefully because of the great strain sensitivity of the Nb3Sn.

12

CHAPTER 1

1.6. QUENCHING AND STABILITY Now let us perform a thought experiment in which we wind a solenoid out of such a multifilamentary composite conductor, immerse it in a dewar of liquid helium, and charge it with current. As the current rises, we hear the magnet creaking and groaning. Suddenly, with a great whoosh, all the helium is expelled from the dewar forming a cloud of water vapor in the room. Fearful of the results, we immediately reduce the current in the magnet to zero. What happened? The groaning and creaking we heard was caused by the motion of the currentcarrying conductor in response to the Lorentz force J x B exerted on it by the magnet’s own self-field. That is to say, any turn of the magnet carrying a current density J feels a volumetric force J x B caused by the field B created by the current in the other turns. For a solenoid the net result is a tendency to expand in the radial direction and contract in the axial direction. While this expansion is going on, individual strands of wire may slip a little from one position to another. When this happens, the Lorentz force does work, and when the slipping segment of wire comes to rest, either because of friction or impact, this work is converted to heat. Now because the specific heats of materials are very small at low temperatures, this heat of slipping, even though small, is often enough to raise the local temperature above the critical temperature. Now a bit of the wire is normal, i.e., resistive. The current traversing it, though it switches to the copper from the superconductor, generates Joule heat at the normal site, which further heats the locale. More conductor goes normal from this heating, and a process of thermal runaway, called a quench, begins. Now that we have figured out what happened, we ask ourselves what we can do to avoid such quenches. Four things come to mind at once: hold the conductor more tightly to prevent sudden slipping, add more copper to the conductor to reduce the Joule power in the normal state, cool the conductor better, or reduce the operating current in the magnet. Sometimes these solutions conflict with one another. For example, one way of holding the conductor more tightly is to pot it in epoxy, but the epoxy then retards the transfer of heat from the conductor to the helium. Nevertheless, this method is used quite frequently to stabilize small magnets (stored energy < 100 kJ, roughly). Even so, quenches occur with potted magnets, for occasionally the epoxy cracks at points of high stress, and the mechanical energy released is enough to quench the magnet. Typically, if the magnet is recharged after such a quench, it will not quench again or it will quench because of a crack at another point. This is because once a crack develops at the first point, no sudden energy release occurs at that point. Instead, the edges of the crack move gradually under the action of the Lorentz force as the magnet is charged, and the heat generated has time to be conducted away before it can raise the local temperature beyond the critical temperature. With such magnets, after a few

Introduction and Overview

13

quenches, the magnet can be charged to its full operating current without any further quenches. This process is called training and is typical of potted magnets. Another strategy to stabilize magnets is to add more copper to the conductor, reserve some void space in the winding for the infiltration of liquid helium, and keep the current density down. This strategy keeps the Joule power in the normal state low and allows good cooling of the conductor. It is possible to provide enough cooling so that after a normalizing event (e.g., conductor motion) the heat produced by the normalizing event and the Joule power can be transferred away from the superconductor and the superconductor can be cooled back down to the superconducting state (called recovery ). Magnets stabilized in this way are called cryostable. They have the enormous advantage of being unconditionally stable, that is, they always recover the superconducting state after any perturbation and so can be operated without interruption. Coupled ineluctably to this advantage is the disadvantage of large size and consequently high cost because of the low current density in the winding. This cost disadvantage may be multiplied when the large size of the magnet forces the apparatus of which it is a part also to be large in scale, as may be the case with fusion machines. To offset this disadvantage, we can build magnets with less copper in the conductor that operate with higher current densities than cryostable magnets. These magnets are metastable : they recover from small thermal perturbations but quench if exposed to large thermal perturbations. The idea is to stabilize them just against the perturbations that actually occur in the operation of the magnet. Since the spectrum of thermal perturbations is hardly known at all for most magnets, there is some guesswork in the design of metastable magnets, and failure of prototypes to meet specifications is not uncommon.

1.7. THE PHASE DIAGRAM OF HELIUM So far we have mentioned only the liquid phase of helium as a coolant for the low-temperature superconductors. Historically, it was the first to be used, of course, but it is not the only possibility. Shown in Fig. 1.7 is the phase diagram of helium in the low temperature region. The point C lying roughly at 5.1 K and 0.22 MPa is the critical point and the curve OC is the saturation line separating the liquid phase from the vapor phase. The point P1 at 4.2 K and 0.1 MPa represents helium boiling at atmospheric pressure and is the point that corresponds to liquid helium in an open dewar. Most materials have a triple point at which solid, liquid, and gaseous phases coexist. Helium, however, does not solidify at pressures less that 4 MPa even at zero temperature. Instead, below about 2.2 K, it enters another liquid phase, called superfluid helium or He-II. The line LL´ , called the lambda line, is the phase boundary between ordinary liquid helium (He-I) and the superfluid phase. The point L, lying at about 2.2 K and 5 kPa, is called the lambda point. Superfluid helium has

14

CHAPTER 1

Figure 1.7. A sketch of the phase diagram of helium at low temperatures. Point C is the critical point. Instead of a triple point, as is usual in other materials, helium has two liquid phases with vastly different properties. (Redrawn from an original appearing in Dresner (1984, “Superconductor”) with permission of Butterworth-Heinemann, Oxford, England.)

rather different properties from ordinary liquid helium, and indeed from most other materials. For the purposes of this book it suffices to point out that superfluid helium does not obey Fourier’s law of heat conduction in which the heat flux is proportional to the temperature gradient. Instead, in superfluid helium, the heat flux is proportional to the cube root of the temperature gradient (when the heat flux is high enough, as it almost always is in magnet applications). This means that superfluid helium can support a very high heat flux for very small temperature gradients, and this is one of the reasons that it is considered desirable as a coolant for superconducting magnets. A second reason is that at the low temperatures at which it exists, NbTi composite conductors are capable of satisfactory operation up to fields of 12 T, rather than the 8 T typical of NbTi magnets cooled with boiling helium. The French tokamak (a kind of fusion machine) Tore-II Supra is cooled with superfluid helium having a temperature of 1.8 K and a pressure of 0.1 MPa (point P2 in Fig. 1.7). Magnets have been built that are cooled with supercritical helium, typically helium at a temperature of 4 K and pressures around 1 MPa (point P3). The conductors of such magnets consist of superconducting composite wires braided into cables and placed in a strong jacket, which contains the helium. In some of these conductors, the cable is held tightly by the jacket, which is usually swaged

introduction and Overview

15

around it, and the supercritical helium circulates through the interstices of the cable. This kind of conductor, to which a considerable fraction of this book is devoted, is called a cable-in-conduit conductor. Because the strands of the cable are unrestrained except where they cross each other, wire motion is possible in such conductors, though it appears not to cause quenches. In other kinds of internally cooled conductors, the cable is soldered together, and the helium circulates through a tube down its center. This brief overview of the stability problem shows that it has many aspects: we can have potted, externally cooled, and internally cooled conductors, cooled with either boiling, superfluid, or supercritical helium.

7.8. HIGH-TEMPERATURE SUPERCONDUCTORS So far we have spoken only of low-temperature superconductors, for which helium is the only coolant. (Actually, this is not perfectly accurate because recently Nb3Sn magnets that were conductively cooled by cryocoolers have been operated at temperatures around 11 K.) The newly discovered high-temperature superconductors, whose critical temperatures are ~ 100 K hold out the possibility of operation in liquid nitrogen (atmospheric boiling temperature, 77 K). Because their behavior is somewhat different from that of the low-temperature superconductors, some description of the differences is now in order. In the first place, the high-temperature, copper oxide superconductors are granular, in contrast to the low-temperature superconductors NbTi and Nb3Sn. Grain boundaries at which adjacent grains are misaligned with respect to their internal structure are regions of weak superconductivity (so-called “weak links”). These weak links act as bottlenecks to limit the current that can be transported over macroscopic lengths of conductor. In fact, the intragranular critical current density can be three to five orders of magnitude as great as the intergranular critical current density (Larbalestier, 1991). In the design of superconducting magnets, which is a principal interest of this book, the much lower intergrain critical current density is what concerns us. The higher intragrain critical current density is of interest in superconducting electronics, which makes use of thin films of superconductor grown on substrates that orient the grains with their internal structures parallel to one another. But even in the design of magnets, there are cases where the existence of a high intragrain critical current produces noticeable effects (Kwasnitza and St. Clerc, 1993). In the second place, the copper oxide superconductors are anisotropic with respect to the direction of the magnetic field—that is to say, the critical current density depends on the direction of the magnetic field with respect to the crystal axes. The copper oxide superconductors consist, roughly speaking, of conducting planes of copper and oxygen atoms separated by intercalated, nonconducting planes of atoms. In the conventional crystallographic terminology, the a- and b-axes of the

16

CHAPTER 1

Figure 1.8. The critical surface of NbTi (based on data of Wilson, 1983; redrawn from an original provided courtesy of Clarendon Press, Oxford, England.)

crystal lie in the conducting planes and the c-axis is perpendicular to them. When the magnetic field is parallel to the a- b plane, the critical current density is several times as large as when the magnetic field is perpendicular to the a-b plane (i.e., parallel to the c-axis). In bulk material, in which the grains are randomly oriented, this anisotropy is not evident. But randomly oriented grains accentuate the weaklink behavior. To suppress the weak-link behavior when Ag/BSCCO conductor is manufactured, the grains are partially aligned by rolling or pressing the conductor into a tape, thereby increasing the critical current density, but at the same time reintroducing anisotropy. The critical current density is then larger when the magnetic field is parallel to the tape surface than when it is perpendicular to it. In the third place, the dependence of the critical current on magnetic field is somewhat different for the copper oxide superconductors than it is for the low-temperature superconductors. A glance at Fig. 1.8, the critical surface for NbTi, shows the nearly hyperbolic behavior of the Jc -B curve required by Eq. (1.4.1). The situation depicted by Fig. 1.9, the critical surface of Ag/BPSCCO (Sato et al., 1993), is quite different. At low temperatures (4.2–20 K, for example), the critical current

Introduction and Overview

17

Figure 1.9. The critical surface of a Ag/BPSCCO high-temperature superconductor(Sato et al., 1993). (Redrawn from an original appearing in Sato (1993) with permission of Butterworth-Heineman, Oxford, England.)

decreases only gradually with increasing magnetic field up to quite large fields (~ 20–30 T). This means that the constant Bo in Eq. (1.4.1), though only of the order of a few hundredths of a tesla for the low-temperature superconductors, may be tens of teslas at low temperatures for the copper oxide superconductors. (Note that both Figs. 1.8 and 1.9 imply that Bo decreases with increasing temperature!) The insensitivity of Jc of the copper oxide superconductors to B at low temperatures holds out the promise of using them at low temperatures to make high-field magnets. On the other hand, the sharp drop of Jc of the copper oxide conductors with B at high temperatures is at present a stumbling block to the construction of magnets cooled with liquid nitrogen (77 K). Finally, the temperature ranges over which the high-temperature superconductors are used are both higher and more extensive than those over which the low-temperature superconductors are used. For example, a NbTi conductor cooled

18

CHAPTER 1

by liquid helium (4.2 K) in a field of 8 T has a critical temperature of 5.6 K. Over the restricted temperature range 4.2–5.6 K, which is the range of importance in determining superconductor stability, it is quite acceptable to assume that the specific heats of the matrix and the superconductor, the matrix resistivity, and the matrix thermal conductivity are all constants independent of temperature. On the other hand, a Ag/BPSSCO conductor cooled to 20 K by a cryocooler in the same 8 T field has a critical temperature of ~60 K, judging from Fig. 1.9. Over the range 20–60 K, the specific heats, the matrix resistivity, and the matrix thermal conductivity change by large amounts, making the assumption of constant properties no longer tenable. Notes to Chapter 1 1

Materials become superconducting because their free electrons interact by locally deforming the atomic lattice that makes up the solid. This interaction is attractive and causes the formation of bound pairs of electrons called Cooper pairs. The Cooper pairs are the current carriers in the superconducting state. In the neighborhood of a defect or impurity, the lattice is changed, the electron–electron interaction is altered, and the Cooper pairs are destroyed.At low temperatures, the superconducting state has a lower free energy than the normal state (that is why it forms). Therefore, the flux lattice can achieve the lowest free energy by locating the normal cores at the defects rather than by destroying the superconducting state at a point where it otherwise could exist.

2 Material Properties

2.1. SPECIFIC HEAT: THE DEBYE FORMULA From the overview of the stability problem presented in Sections 1.6 and 1.7, we can see that the issue of quench or recovery depends on the matrix resistivity, the specific heats of the matrix and the superconductor, the rate of heat transfer from the normalized superconductor to the coolant, and the shape of the critical surface. To understand the resolution of the stability issue, we must know the detailed behavior of all these quantities. We begin with the specific heat of the matrix. In 1819, Dulong and Petit, at the end of a series of measurements, concluded that all elements had the same heat capacity. An explanation for this behavior was found later in Maxwell’s 1860 discovery of the principle of equipartition of energy. But the satisfaction this explanation brought was short-lived. In 1872, H. F. Weber measured the specific heat of diamond between 0°C and 200ºC and found, contrary to expectations based on equipartition, that the specific heat increased by a factor of 3 with increasing temperature. I read somewhere (though I cannot recall the reference) that this discrepancy was known to Maxwell (who died in 1879) and that he, quite correctly, felt it represented a crisis in classical physics, which could offer no explanation. Similar exceptional behavior was observed in silicon and boron. When the liquid cryogens became available, studies of the specific heats of solids showed that the specific heats of diamond, silicon, and boron, heretofore considered exceptional, were not so and formed a part of a universal scheme in which the specific heats of solids fall from the Dulong–Petit limit at high temperatures to quite low values at low temperatures. As just mentioned, classical physics offers no explanation for this decrease, but quantum mechanics does, as was shown by Einstein in 1907. Einstein’s theory was quite successful in accounting for the low-temperature behavior of solids, but a small residual discrepancy remained: the Einstein theory predicted a far more rapid decrease of the specific heat at extremely 19

20

CHAPTER 2

low temperatures than was observed. This defect was remedied by improvements to Einstein’s theory made in 1912 by Debye. Einstein’s and Debye’s theories are a well-known part of modern physics and will not be described in detail; only Debye’s result will be quoted here. The interested reader can find the details of the Einstein and Debye theories in many places; two good references are Born (1946) and Richtmyer (1947). According to Debye, the molar specific heat Cv of solids is given by Cv /rR = 12[(3/ z ³ )

{t ³ /(et – 1)} dt] – 9z /(e z – 1); z = θ/T

(2.1.1)

where r is the number of atoms per molecule, R is the universal gas constant (8.317 J mol-1 K-1), T is the absolute temperature, and q is the so-called Debye temperature, an empirical parameter related to the phonon spectrum of the solid. (Phonons are the quantized vibrations of the solid lattice.) The integral in brackets has been tabulated by Abramowitz and Stegun (1965), among others.

Figure 2.1. A log-log plot of C v / r versus thedimensionless temperature T/θ according to Debye’s theory. See Eq. (2.1.2) and the text after Eq. (2.1.4) for the definition of b. (Redrawn from an original appearing in Dresner (1993, “Stability”) with permission of Butterworth-Heineman, Oxford, England.)

Material Properties

21

Figure 2.1 shows a log-log plot of Cv /r calculated using Eq. (2.1.1) versus x = 1/ z = T/θ. Two straight-line asymptotes are evident. When T << q, Cv is asymptotic to Cv 1 = (12π 4/5)rRx 3 = 1944rx3 J mol¯ 1 K¯ 1

(2.1.2)

When T << q, Cv is asymptotic to Cv2 = 3rR = 24.95r J mol¯1 K¯1

(2.1.3)

A family of curves that spans these two asymptotic limits is (2.1.4)

Cv=

The value n = 0.85 gives an excellent fit to the exact curve and is a useful approximation for numerical calculation. Eq. (2.1.2), known as the Debye T 3-law, is often written Cv1 = βT 3, where b = (12π 4/5)rR/θ3 = 1944r/θ3 J mol¯1 K¯ 4 . The only physical property of the solid that enters the Debye theory is the Debye temperature q. Table 2.1 shows its value for several materials of interest in applied superconductivity. Debye’s theory accounts for the contribution to the specific heat of the vibrations of the atomic lattice of which the solid is composed. (These vibrations are known conventionally as phonons, and Debye’s theory is said to give the phonon contribution to the specific heat.) If the solid is a metal, as are all the entries in Table 2.1, there is an additional contribution to the specific heat from the electrons. This additive contribution, which plays a significant role only at low temperatures, has the form Cve = γT

(2.1.5)

where g is the so-called Sommerfeld constant. Its value is given in Table 2.2 for the same materials that appear in Table 2.1. Table 2.1. The Debye Temperature q for Various Materials Material Ag Cu Al Fe

Debye temperature (K) 226 344 430 472

22

CHAPTER 2

Table 2.2. The Sommerfeld Constant g for Various Materials Sommerfeld constant (J mol¯ 1 K¯ 2)

Material

6.09 x 10¯ 4 7.44 x 10¯ 4 1.36 x 10¯ 3 5.02 x 10¯ 3

Ag Cu Al Fe

2.2. SPECIFIC HEATS OF TYPE-II SUPERCONDUCTORS The specific heat of a type-II superconductor in the normal state is also described by the Debye-Sommerfeld theory given in the previous section. In the superconducting state, it is given approximately by 2 (2.2.1) Cs(T,H) = Cn(T) + 3γT 3 /T co – γT+ γTH/Hc2(0) 2 )T 3 + γTH/Hc 2(0) = (β + 3γ/T co

(2.2.2)

where Cs(T,H) is the specific heat in the superconducting state at temperature T and applied magnetic field H, Cn(T) is the specific heat in the normal state at temperature T, Tco is the critical temperature at zero applied field, and Hc 2(0) is the (upper) critical field at zero temperature. The second line applies at low temperatures, when the Debye T 3-law, Cphonon = βT 3, is a satisfactory approximation. At the critical temperature Tc (at applied field H), where the phase change occurs, the specific heat undergoes a jump 2 Cs(Tc ,H) – Cn(Tc ) = γTc [3T c2 /T co – 1 + H/Hc 2(0)]

(2.2.3)

When there is no applied field, Cs(T co,H) – Cn(Tco ) = 2γTco

(2.2.4)

Such jumps in the specific heat (so-called l-anomalies) are characteristic of second-order phase changes in which there is no latent heat. Figure 2.2 shows a plot of measured values of C/T versus T 2 for the low-temperature superconductor Nb 44.6-wt-% Ti reported by Elrod, Miller, and Dresner (1982). The lines are best fits to Eq. (2.2.2) using the values γ = 0.145 J kg¯1 K¯2 , β = 2.3 x 10¯3 J kg¯1 K¯ 4 , and µ o Hc2(0) = 14 T. The fit is excellent and reinforces our belief that Eq. (2.2.2) is a description of superconductor specific heats at low temperatures adequate for use in the theory of superconductor stability. It is perhaps worth noting here that our goal in this chapter is to find, where possible, simple, widely applicable formulas that allow approximate computation of the physical properties we use to study the stability of superconductors. The Debye theory is based on an approximate phonon spectrum in which the density of phonon states varies as the square of the phonon frequency. At low

Material Properties

23

Figure 2.2. Measured values of C/T versus T 2 for Nb 44.6-wt-% Ti (Elrod et al., 1982). (Redrawn from an original appearing in Elrod et al. (1982) with permission of Plenum Publishing Corp., New York.)

temperatures, only low-frequency phonons are excited, and at this end of the phonon spectrum, the quadratic approximation is accurate. At higher temperature, such as might typify the new high-temperature superconductors, the fit to Eq. (2.2.1) might not be as good. Indeed, deficiencies in the Debye theory due to its simplified phonon spectrum have been known for some time—long ago, Born and von Karman (Reed and Clark, 1983) sought to improve the Debye theory by invoking a more realistic phonon spectrum, but only at the cost of increased computational complexity. Occasionally, the difficulties with the Debye theory have been circumvented in an ad hoc way by letting the Debye temperature q vary with T, but such a “cure” is only applicable when the specific heat is already known experimentally over a wide range of temperatures. This is exactly the situation with recent measurements of the specific heat of YBa2Cu3O7 recently published by Pavese and Malishev (1994): see Fig. 2.3. Shown in Fig. 2.4 are plots of the specific heat calculated using Eqs. (2.1.1) and (2.2.1) for five values of the Debye temperature together with seven points taken from Fig. 2.3. At low temperatures q ~ 375 K, whereas at high temperatures q ~ 525 K. Choosing the smaller value leads to a maximum error in the neighborhood of 100 K of about 30%. We shall not abandon Eqs. (2.1.1) and (2.2.1) for high-temperature superconductors, but rather accept the facts of the last paragraph as a caution on the limitation of their accuracy. The next four optional sections present a discussion of the thermodynamics of superconductors that culminates in the derivation of Eq. (2.2.1).

24

CHAPTER 2

Figure 2.3. Measured values of C/T versus T for YBa2Cu3O7 (Pavese and Malishev, 1994). (Redrawn from an original appearing in Pavese and Malishev (1994) with permission of Plenum Publishing Corp., New York.)

Figure 2.4. Comparison of the measurements of (Pavese and Malishev, 1994) for YBa2Cu3O7 with the Debye theory. g = 0.04 J/kg·K2, r = 13, MCLWT = 0.6662 kg/mol, Tco = 93 K.

Material Properties

25

2.3. FIRST LAW OF THERMODYNAMICS FOR A MAGNETIZABLE BODY Presented below is a phenomenological theory of the specific heat of the superconducting state. In this theory, the specific heat is deduced by thermodynamic arguments from the measured value of another, more accessible quantity, the magnetization M of the superconductor in the presence of an applied field H. The state of a magnetizable body is described by Maxwell’s equations (2.3.1) x H=J B = µ o (H+ M)

(2.3.2) (2.3.3)

where E is the electric field, B is the magnetic induction, H is the applied magnetic field, created by J, the current density produced by the external power supply, and M is the magnetization of the body. Because measurements of the magnetization are carried out slowly, the displacement current density has been omitted from the right-hand side of Eq. (2.3.2). In MKS units, M has the dimensions of A/m = Am2/m3 or the dimensions of dipole moment per unit volume. In a conceptual sense, M can be thought of as an increase in the total field inside the body brought about when the applied field orients the body’s atomic dipoles. To write the first law of thermodynamics for a magnetizable body, we need to know the increment of work the body does when its magnetic state changes. Change in the magnetic state, i.e., change in H and M, causes change in the induction B, which in turn creates the electric field E. This field applies a force Ee to the charge carriers produced by the external power supply that create the external field H. This force (the back emf) is opposed by an equal and opposite force –Ee produced by the external power supply. If the velocity of these charge carriers is v, –eE· v is the rate at which the external power supply does work on them. If their density is n, –neE·v = –E·J is the rate at which the external power supply does work on the charge carriers in a unit volume. The quantity d(vol) is thus the rate at which the external power supply does work when the state (H,M) is changing. To relate the quantity E·J to the state variables H and M, we scalar multiply Eq. (2.3.1) by H and Eq. (2.3.2) by E and subtract: H.∇xE−E.∇xH+H.∂B/∂t =-E.J

(2.3.4)

CHAPTER 2

26

Now the first two terms in Eq. (2.3.4) together equal (E x H), so that if we integrate Eq. (2.3.4) over a large volume V whose surface S surrounds the body and the external power supply, we find (2.3.5)

As the volume V grows large, the magnitude of the product E x H falls faster than the surface S increases, and the surface integral vanishes in the limit. Thus (2.3.6)

(2.3.7)

The passage from Eq. (2.3.6) to Eq. (2.3.7) has been achieved by substituting the right-hand side of Eq. (2.3.3) for B. The second volume integral only extends over the volume of the body, for only there is M different from zero. The left-hand side of Eq. (2.3.6) is the rate at which the external power source is supplying power. The first term on the right-hand side of Eq. (2.3.7) is the rate of increase of the energy of the magnetic field and the second term is the rate of increase of the energy of the body, i.e., the rate as which the external power supply does work on the body. For thermodynamic purposes, we need the rate at which work is done by the body during a change of state; it is (2.3.8)

In the simplest experiment to measure M, the external field H is applied parallel to the axis of a long wire so that both H and M are parallel and uniform over the body. Then the rate of work done by the body is –µ oH d(MV )/dt

(2.3.9)

where V is the volume of the body. Thus, finally, the work done by the body in a change of state is (2.3.10)

Material Properties

27

2.4. GlBBS FREE ENERGY OF A MAGNETIZABLE BODY The two laws of thermodynamics may be written together for the magnetizable body as (2.4.1) where T is the temperature, s the entropy of the body per unit mass, u its internal energy per unit mass, and δ its density. The most convenient independent variables to use in the thermodynamic discussion are the applied field H and the temperature T since they are under our control. Accordingly, we introduce the function (2.4.2) which is the Gibbs free energy per unit mass of the body. Then (2.4.3) so that (2.4.4) Finally, the specific heat at constant applied field CH is given by (2.4.5) According to Eq. (2.4.4), (2.4.6)

where the subscript s on g indicates the superconducting state. When the pair of values (T,H) lies on the phase boundary between the mixed and the normal states, the Gibbs free energy of the normal and superconducting states are equal (remember, the transition is second-order). Thus (2.4.7) where Hc2 is the upper critical field at temperature T (see Fig. 1.2). If we subtract Eq. (2.4.7) from Eq. (2.4.6), we find

28

CHAPTER 2

(2.4.8) It is important to note here that because superconductors are diamagnetic (i.e., exclude the magnetic field), M is oppositely directed to H. Thus gs(T,H) < gn(T) when H < Hc2 as required.

2.5. SPEClFlC HEAT AT ZERO FlELD At zero applied field, Eq. (2.4.8) takes the form (2.5.1) Now the measured curves of M versus H look like the sketch in Fig. 2.5. Each curve corresponds to a particular fixed temperature. As H increases from zero, M starts out equal to –H : this is the Meissner state, which persists until H = Hc1. Then M begins slowly to rise until it reaches zero at H = Hc2. The curves at different temperatures resemble one another; we make the assumption now that they are strictly similar geometrically to one another. Then since both the height Hc1 and 2 the base Hc2 of the M-H curve scale as (1 – T 2/T co ), where Tc o is the critical temperature at zero field, (2.5.2)

Figure 2.5. A sketch of the magnetization M versus the applied magnetic field H. Each curve corresponds to a particular fixed temperature. In the theory it is assumed that these curves are strictly similar geometrically to one another, and the sketch is drawn consistently with this assumption.

Material Properties

29

For convenience, we denote the integral on the right-hand side of Eq. (2.5.2) as –a, where a > 0. Then gs(T,0) = gn(T) – (µ 0 /δ )a(1 – T 2 /T 2co ) 2

(2.5.3)

so that the specific heats in the normal and superconducting states are related by (2.5.4) Now it is experimentally known that Cs(T,0) approaches zero with decreasing 2 temperature faster than linearly, so the linear term -4µ o /d)aT/T co must cancel the 2 Sommerfeld term γT in Cn(T). Therefore, 4(µo /δ)a/T co = γ, and Eq. (2.5.4) becomes Cs(T,0) = Cn(T) + 3γT 3/T 2co – γT= (β + 3γ /T 2co )T 3

(2.5.5)

the last equality applying when Cn(T) can be represented as γT+βT 3 (i.e., at low temperatures).

2.6. CONTRIBUTION OF THE MAGNETIC FIELD TO THE SPEClFlC HEAT To determine the effect of the magnetic field on the specific heat we must calculate

(2.6.1) Now we make the further assumption that the shallow curves in the M-H diagram of Fig. 2.5 are in fact parallel straight lines. The slope of these parallel lines is Hc1(0)/[Hc2(0)–Hc1(0)] ~ Hc1 (0) /Hc2(0). Thus, M = [Hc1(0)/Hc2(0)][H – Hc2(T)] 2 = [Hc1(0)/Hc2(0)][H–Hc2(0)(1–T 2/T co )]

(2.6.2)

Furthermore, (2.6.3)

so that

30

CHAPTER 2

M = 2a[H – Hc2(0)(1– T 2/T 2co )]/[ Hc2(0)] 2

(2.6.4)

Inserting Eq. (2.6.4) into Eq. (2.6.1) we find (2.6.5) so that finally Cs(T,H) = Cn(T) + 3γT 3/T 2co – γT+ γTH/Hc2(0)

(2.6.6)

2 = (β + 3γ/T co )T 3 + γTH/Hc2(0)

(2.6.7)

2.7. MATRIX RESISTIVITY; BLOCH–GRÜNEISEN FORMULA Electrical resistance is caused by the scattering of electrons as they move through a wire under the action of a potential difference. The electrons can be scattered by solid-state defects, by impurities, and by the phonons themselves. The phonon contribution to the resistivity rr is given by a formula similar in appearance to the Debye formula and called the Bloch–Grüneisen formula (Reed and Clark, 1983):

Figure 2.6. A log-log plot of the quantify y, which is proportional to the phonon resistivity, versus the dimensionless temperature T/θ according to the theory of Bloch and Grüneisen (Redrawn from an original appearing in Dresner (1993, “Stability”) with permission of Butterworth-Heinemann, Oxford, England.)

31

Material Properties

Table 2.3. The Ice-Point (273 K) Resistivities of Several Metals Material cu Ag A1

Resistivity(µΩ -cm) 1.55 1.48 2.43

where K is a constant yet to be determined. Again the quantity in the brackets has been tabulated by Abramowitz and Stegun (1965). The quantity y plotted against x = 1/z = T/θ as calculated using Eq. (2.7.1) is shown in Fig. 2.6. Again there are two asymptotes, namely,

T << θ: y1 = 124.4x 5

(2.7.2)

T >>θ:

(2.7.3)

y 2 = x/4

The approximation (2.7.4) fits the exact curve quite well when n = 0.65. The constant K in Eq. (2.7.1) is usually fitted to the room-temperature (300 K) or the ice-point (273 K) resistivity. Typical values of the latter for several matrix materials are shown in Table 2.3. In addition to scattering by phonons, the conduction electrons are scattered as they migrate by impurities and solid-state defects. Whereas the phonon contribution to the resistivity ρρ falls strongly with decreasing temperature, the contribution of the impurities and defects, called the residual resistivity and denoted by rr , is independent of temperature. According to Matthiesen’s rule, the total resistivity is the sum of the residual and phonon contributions; above about 30 K the phonon resistivity dominates. It is worth noting here that annealing (such as might occur in a heat treatment) decreases the residual resistivity whereas cold-working (such as might occur in drawing or extrusion) increases it.

2.8. MAGNETORESlSTIVITY When a metal carries current in a magnetic field, its electrical resistivity is larger than when there is no field. This is because the Lorentz force on the moving

32

CHAPTER 2

conduction electrons changes their trajectories. The additional resistivity is called the magnetoresistivity and is denoted by ρm. The magnetoresistivity depends on the direction of the magnetic field relative to the current encountering the resistance. In the high-field region of solenoids, the magnetic field is perpendicular to the current in the conductor (transverse magnetic field). In transverse magnetic fields, the magnetoresistivity is related to the applied magnetic field B by Köhler’s rule, namely, ρm/ρ = F(BρRT /ρ)

(2.8.1)

where ρRT is the room-temperature resistivity and F is an empirically determined function (which of course must vanish when its argument vanishes). For many metals, among them copper and silver, the function F is nearly a straight line (see Figs. 2.7 and 2.8). When F is linear, the transverse magnetoresistivity is directly proportional to B. The proportionality constants for copper and silver are, respectively, 4.5 x 10¯11 ohm-m/T and 3.0 x 10¯11 ohm-m/T. The transverse magnetoresistivity for A1 is more complicated and does not follow Köhler’s rule; instead it rises at low fields, but saturates quickly at relatively modest fields. According to Reed and Clark (1983), the longitudinal magnetoresistance is lower than the transverse, usually by a factor of two or more. Furthermore, it saturates at high enough fields.

Figure 2.7. A Köhler plot of the transverse magneto-resistance of copper (Reed and Clark, 1983). (Redrawn from an original provided courtesy of the American Society for Metals.)

Material Properties

33

Figure 2.8. A Köhler plot of the transverse magneto-resistance of silver (Iwasaet al., 1993). (Redrawn from an original appearing in Iwasa et al. (1993) with permission of Butterworth-Heineman, Oxford, England.)

At low temperatures, the transverse magnetoresistivity may be comparable with or even exceed the residual resistivity and so must be accounted for in studies of superconductor stability.

2.9. THE WIEDEMANN–FRANZ LAW Because both heat and electric charge are transported in metals by the electrons, there is a relation between the thermal and electrical conductivities. This relation, called the Wiedemann–Franz law, is most often written kρ =LoT

(2.9.1)

where k is the thermal conductivity, r the electrical resistivity, Lo the Lorenz constant, equal to 2.45 x 10 8 V2K-2 , and T is the temperature. The Wiedemann– Franz law is an approximation based on a simplified picture of the electronic structure of metals, and its accuracy depends both on the temperature T and the purity of the metal. For ordinary commercial metals, which are relatively impure,

34

CHAPTER 2

the law is reasonably accurate both at low temperatures < O.1θ and at high temperatures > q. Between these two extremes, the effective value of the Lorenz constant Lo falls somewhat, the fall being greater the greater the punty of the material. There is some question about whether the Wiedemann–Franz law holds in a magnetic field. We know that the resistivity is augmented by the field (magnetoresistance). Does the thermal conductivity fall in a magnetic field? According to Fevrier and Morize (1973), in copper it does and in such a way that the effective Lorenz constant Lo increases with increasing transverse field in their two samples by 4% and 13% up to 5 T (roughly 1–2% per tesla). Arenz, Clark, and Lawless (1982), also studying copper in fields up to 12 T, find an increase in the effective value of Lo with increasing transverse magnetic field, in their case by about 7% per tesla. In spite of these drawbacks, the Wiedemann–Franz law is widely used in studies of superconductor stability because of its simple form.

3 Flux jumping

3.1. THE CRITICAL-STATE MODEL As noted earlier, as the temperature or the magnetic field crosses the upper boundary in Fig. 1.2 there is a change of phase accompanied by a sudden change in some physical properties, e.g., resistivity and, as we saw in the last chapter, specific heat. But as the current density exceeds the critical current density and the fluxoid lattice is torn loose by the Lorentz force from its pinned attachment to the defects in the solid, no true phase change occurs,1 although the resistivity of type-II superconductors undergoes a sharp (but continuous) change (see Fig. 3.1). The sharpness of this change allows us to describe the behavior of type-II superconductors in terms of the so-called critical-state model (Bean, 1962; Anderson and Kim, 1964), in which the superconductor either carries a current density Jc , the critical current density, or no current at all. To see how such a situation comes about, let us consider the parallel circuit shown in Fig. 3.2, in which one conductor is copper and the other is a superconductor, for example, NbTi. Such a circuit in fact represents a multifilamentary superconductor in which many fine filaments of NbTi run longitudinally through a matrix of copper. Let the critical current in the superconductor (at the ambient field and temperature) be Ic and let I be the total current being driven through both branches. (The usual definition of Ic (or Jc) is the current (or current density) at which the superconductor resistivity reaches a specified value, typically 10-13 ohm-m.) If I < Ic , all the current flows in the superconductor. When I > Ic , the current divides according to the resistances of the two branches. But what is the resistance of the superconducting branch? Because of the steepness of the resistivity curve in Fig. 3.1, no matter what value the resistivity has (as long as it is not zero), the current in the superconductor is very close to Ic . Therefore, the current I – Ic must flow through the copper. The voltage drop across the copper is then V = Rcu (I – Ic). Since the copper and the superconductor are electrically in parallel, the superconductor 35

36

CHAPTER 3

Figure 3.1. A sketch of the sharp but continuous increase with current of the resistivity of type-II superconductors.

also experiences the voltage drop V, and its resistance consequently is V/Ic . If we raise the current I, the voltage V = Rcu (I – Ic) goes up and so does the resistance of the superconductor. This requires an increase in the current in the superconductor, but because of the steepness of the curve in Fig. 3.1, the required increase in the superconductor current is very small. For practical purposes, then, the superconductor can be considered always to be carrying the current Ic when it is resistive. The italicized statements above give the essence of the critical-state model, which brings with it enormous simplifications in computation, and which, it can be fairly said, underlies the entire field of applied superconductivity.

3.2. THE THREE-PART CURVE OF JOULE POWER The parallel circuit in Fig. 3.2 generates resistive (Joule) power when I > Ic and some of the current is carried in the copper. Since the voltage V spans both

Figure 3.2. A parallel circuit in which one conductor is copper and the other is a superconductor.

Flux Jumping

37

Figure 3.3. (a) A plot of critical current Ic versus temperature T showing a linear declineof Ic with T. (b) The three-part powergeneration curve that is the basis of most studies of superconductor stability.

branches of the parallel circuit, the entire current falls through that voltage drop. Then the Joule power Q being produced is IV = IRcu (I – Ic). Now empirical observations of many superconductors show that the critical current (or current density) falls linearly with increasing superconductor temperature T. Shown in Fig. 3.3a is a plot of Ic versus T showing a linear decline of Ic with T that reaches zero when T = Tc (where the superconductor becomes normal). Shown also is the current level I and the temperature Tcs at which Ic = I. This temperature is the largest at which the superconductor can still carry all of the impressed current I. At higher temperatures, some of the current I spills over into the copper. This is called current sharing and Tcs is called the current-sharing threshold temperature. Clearly, for T < Tcs , Q = 0 (see Fig. 3.3b). When T > Tc , Ic = 0, and Q has the constant value I 2R cu , all of the current being in the copper (since the normal-state resistivity of NbTi is much greater than that of copper). Between Tcs and Tc , Q = IRcu (I – Ic) is taken to vary linearly with T because Ic itself varies linearly with T. The resulting three-part curve of Fig. 3.3b, made up of three straight-line segments, plays a fundamental role in the science of applied superconductivity and is the basis of most studies of superconductor stability.

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CHAPTER 3

3.3. CHARGING OF A SUPERCONDUCTOR: CRITICAL-STATE MODEL When any conductor is exposed to a changing magnetic field, currents are induced in it: this is Faraday’s law of induction. These currents are called eddy currents or shielding currents. The latter term is a consequence of Lenz’s law that tells us that the induced currents flow in such a direction as to oppose the change in the external magnetic field. If the conductor is an ordinary conductor, e.g., copper, the induced shielding currents encounter resistance. If the external field change stops, the shielding currents decay as a consequence of the resistance. If the conductor is a type-II superconductor in which no pinning is present, the shielding currents interact via the Lorentz force with the fluxoids, causing them to move. The motion of the fluxoids creates an electric field that opposes the flow of the shielding currents, so that when the external field change stops, the shielding currents decay. (The opposing electric field induced by the motion of the fluxoids is often represented as though the shielding currents were flowing through a resistance, which is named the flux-flow resistance.) When the shielding currents have decayed and the magnetic field has become uniform throughout the sample (in the form of a uniform fluxoid lattice), a state of thermodynamic equilibrium is reached. When strong pinning is present, it restrains the motion of the fluxoid lattice, prevents the shielding currents from decaying, and greatly delays the attainment of thermodynamic equilibrium. In this case, the pattern of shielding currents is again determined by the critical-state model. If the shielding current density locally exceeds the critical current density, pinning fails and flux-flow resistance appears. But as soon as the current density decays to the critical value, pinning again becomes effective and the flux-flow resistance disappears. Furthermore, at any place in the sample where the magnetic field is changing and an electric field exists, the shielding current rises without hindrance (owing to the lack of resistance in the superconductor) until it reaches the critical value. So the current density in the sample always has the critical value, though at different points it may flow in different directions. If there are places where the magnetic field is completely shielded, the current density there is zero. But the only finite value the current density can have is the critical value. The application of the critical-state model to the interior of a superconductor can be illustrated by the following example. Let us consider a slab of superconductor of thickness 2d that is being charged with current (Fig. 3.4). Let I be the z-directed current per unit length of slab extending in the x-direction. If we apply the right-hand rule to the current, we see that the magnetic field created as the current increases is in the x-direction on the left-hand side of the slab and in the –x-direction on the right-hand side of the slab. If we apply the left-hand rule to the rising magnetic flux through the rectangle ABCD, we see that the induced voltage

Flux Jumping

39

Figure 3.4. A sketch illustrating the development of layers of critical current density in a slab of superconductor being charged with current.

around the rectangle opposes the flow of the current I in the center of the slab and promotes it near the edges of the slab. These relations, which apply to any conductor, are the origin of the skin effect in ordinary conductors. In a superconducting slab, they force the current density to adopt the critical value Jc in two layers of thickness a = 1/2Jc adjacent to each edge of the slab. According to the z-component of Maxwell’s equation x B = µ o J (3.3.1) and according to the x-component of Maxwell’s equation (3.3.2) The values of J, B, ∂B/∂t, and E are as shown in Fig. 3.5. The source of the electric field E is the external power supply, which must maintain it as long as the current I is increasing. The penetration depth, a, increases as I increases until at full penetration (a = d), the current I reaches the critical value 2Jc d.

3.4. CHARGING OF A SUPERCONDUCTOR: POWER-LAW RESlSTlVlTY The foregoing example has been chosen because more elaborate computations are possible for this example that enable us to understand in greater detail the nature of the critical-state model. The critical-state model as used above is based on the

40

CHAPTER 3

Figure 3.5. Sketches showing the values of J, B, ∂B/∂t, and E in the slab as it is charged with current.

assumption of an infinitely sharp resistive transition at J = Jc. Suppose, instead, the resistive transition is such that the resistivity r varies as a high power n of the current density. As mentioned earlier, the critical current density Jc is defined as that current density at which the resistivity reaches some fiducial value ρc , typically 10¯13 ohm-m. So our assumption of a power-law resistivity can be written ρ /ρc = (J/Jc )n

(3.4.1)

When J > Jc , r >> ρc and when J < Jc , r << ρc , the transition through r = ρc at J = Jc being very sharp. Now we can combine Eqs. (3.3.1), (3.3.2), (3.4.1), and Ohm’s law E = ρJ to find the following partial differential equation for the penetration in from the edges of the slab of the z-directed current density J :

Flux Jumping

41

(3.4.2) If we imagine the current to be instantaneously raised to its final, constant value I, the solution of Eq. (3.4.2) we are looking for is subject to the conservation condition (3.4.3)

Equations (3.4.2) and (3.4.3) have a solution2 in which the current density penetrates a distance a from each edge given by a = A(ρc I nt/ µ oJ cn )1/(n+2)

(3.4.4)

where A is a numerical constant that depends on the value of n: A = [ 2(n+ 1)(n+2) /n] 1/(n+2) [ Γ((3n+2) /2n) /π1/2Γ (( n+ 1) /n)]n/(n+2) (3.4.5 ) At any time t, the profile of J is J/Jc = [nA 2/2(n +1)( n+2)]1/n[(1/J c)2(µ o /ρc t )]1/(n+2) (1–y 2/a2)1/ n

(3.4.6)

A typical value for Jc at 4.2 K and zero field for NbTi (ρc = 10-13 ohm-m) is 8 x 109 A/m2 (Larbalestier et al., 1986). For n we choose 50. (Measured values of n for conductors composed of NbTi filaments in a copper matrix are in the range 20–40. Some workers (Warnes and Larbalestier, 1986) have attributed these values to periodic variation in the diameter of the filaments, called sausaging, that arises during processing. Then the measured values of n would be a lower limit to any value intrinsic to the NbTi itself. For the latter we have chosen 50 purely for the purposes of illustration.) According to Eq. (3.4.5), A = 0.5683. Suppose now that the slab is 2 mm thick (d = 1 mm) and is charged to 50% of its critical current, i.e., I= 8.00 x 106 A/m. The extreme critical-state model tells us that the layers of critical current at each edge should be 0.5 mm thick, whereas Eqs. (3.4.4) and Eqs. (3.4.6) give the results in Table 3.1 for the penetration depth a and the ratio J(0,t)/Jc. For these quantities, the results of the critical-state model are in fair agreement. Furthermore, according to Eq. (3.4.6), the current density at five-eighths of the penetration distance a is only 1% less that at the edges of the slab, and at a depth of 96% of a, the decrement in current density is only 5%, so that the current distribution in the layers at each edge is rather uniform, as expected from the critical-state model. Finally, according to the entries in the fourth column, the resistivity in the current layers falls as the current spreads out, dropping far below the fiducial value ρc = 10-13 ohm-m used to define the critical current as time goes on.

42

CHAPTER 3

Table 3.1. Penetration Depth a, Current RatioJ(0,t)/Jc and Resistivity ρ (0,t) Time (s)

a (mm)

J(0,t)/Jc

16 ms 1s 1000 s 1 month 2.3 x 106 years

0.500 0.541 0.618 0.719 1.000

1.012 0.935 0.819 0.704 0.506

ρ(0,t) (ohm-m) 1.85 x 10 -13 3.47 x 10-15 4.53 x 10 -18 2.33 x 10-21 1.63 x 10 -28

This numerical illustration shows that the critical-state model is a good, but hardly perfect, representation of the behavior of type-II superconductors. But in spite of any small drawbacks it might have, it is almost universally used to describe the behavior of superconductors because it enables us to see what is going on at a glance, so to speak.

3.5. FLUX JUMPlNG DEFINED After the current ramp stops, the current and field distributions according to the critical-state model are as shown in Fig. 3.5. When the current is steady, ∂B/∂t and E are zero. Steady states are not always stable against small perturbations. A frequently used example of stable and unstable steady states is provided by a weight on the end of a rigid rod pinned at the end opposite the weight (i.e., a rigid pendulum). The steady states are (1) the weight hanging straight down vertically and (2) the weight standing straight up vertically. State (1) is stable against small pushes—the oscillatory motion of the pendulum is damped by friction in the pinned fulcrum so that eventually the weight hangs straight down again and state (1) is restored. But if state (2) is given a slight push, the pendulum falls over and never again returns to state (2), but rather approaches state (1). State (2) is unstable against small perturbations. Because there are always thermal fluctuations in every system, unstable steady states cannot persist in the laboratory. Now as it happens, the steady state described by Fig. 3.5 is not always stable against small perturbations. Consider, for example, what would happen if the temperature of the sample were to be raised slightly by an amount dT. The critical current density Jc would decrease slightly, and then, since the total current I per unit length remains constant, the sheath of critical current density at the edges of the slab would have to broaden (dashed lines in Fig. 3.6). According to Eq. (3.3.1), the slope of the magnetic field is –µ o Jc. This slope decreases slightly when the temperature rises so that the magnetic field distribution also broadens, as shown by the dashed line in Fig. 3.6b. The changing magnetic

43

Flux Jumping

Figure 3.6. Sketches showing the change in the values of J, B, ∂B/∂t, and E in the slab caused by a slight temperature increase. The values before the temperature increase, established by charging the slab with current, are shown as solid lines, the values during the temperature increase are shown as dotted lines.

field creates an electric field (Figs. 3.6c and 3.6d). The electric field vector, being parallel to the current density Jc , does work on the charge carriers that is dissipated in the sample as heat. This heat raises the temperature yet more and suggests the possibility of a thermal runaway. Such a thermal runaway is called a flux jump (Wipf, 1967; Swartz and Bean, 1968) and is undesirable because it may quench the superconductor entirely. We can calculate the secondary temperature rise as follows. According to Eq. (3.3.1), in the left-hand current sheath B = mo Jc(a – y) = (m oI/2) – m o Jc y

(3.5.1)

where the origin (y = 0) has been placed on the left-hand edge of the slab, and a, the penetration distance, is given by a = I/2Jc. Then, ∂B/∂t = –µ o(dJc /dt)y so that

(3.5.2)

44

CHAPTER 3

E=µ o (dJc /dt)(y2–a2)/2

(3.5.3)

The power dissipation P per unit face area in the left-hand current sheath is then a

P = JcE dy = –µ o Jc(dJc/dt)a3/3

(3.5.4)

0

Now P dt is the heat dQ produced per unit face area by the rise in the temperature dT. Assuming for the moment that this heat is not conducted out of the current layer (adiabatic assumption), it causes a secondary temperature rise dT´ = dQ/Sa, where S = δ Cp is the volumetric heat capacity [J m¯3 K-1 ] of the superconductor. (Here δ is the density of the superconductor [kg m- 3] and Cp is its specific heat [J kg-1 K-1].) Thus dT´/dT = µo Jc (–dJc /dT)a2/3S

(3.5.5)

If dT´/dT >1, the critical current sheath continues to penetrate the slab and the current distribution of Fig. 3.5a is unstable to small perturbations. If dT/dT < 1, the penetration slows down and is assumed eventually to stop. The current distribution of Fig. 3.5a is then loosely called stable, although strictly speaking under the adiabatic conditions assumed there is no return to the original state. Stable here simply means that there is no flux jump. When Jc varies linearly with T, as is commonly the case, –dJc /dT=Jc /(Tc –T )

(3.5.6)

so that Eq. (3.5.5) becomes dT´/dT =µo J 2c a 2/3S(Tc – T)

(3.5.7)

Since a < d, if d is small enough, both sides of Eq. (3.5.7) are < 1 and the slab is stable against flux jumps. According to Larbalestier et al. (1986), for NbTi, Jc (4.2 K, 0 T) ~ 8 x 109A/m2. 2 Elrod et al. (1982) give b + 3γ/T co = 7.5 x 10-3 J/kg-K4, while Wilson (1983) gives δ = 6200 kg/m3, so that S(4.2 K, 0 T) = 3450 J/m3-K. The critical temperature Tco is 9.1 K. If d < 25 µ m, the two sides of Eq. (3.5.7) < 1. Thus thin enough slabs are stable against flux jumps when they are being charged with current. This is one of the main reasons for subdividing the superconductor into fine filaments in commercial conductors.

Flux Jumping

45

3.6. VALIDITY OF THE ADIABATIC ASSUMPTION Eq. (3.5.7) is based on the assumption that the heat generated by the motion of the flux stays in place during the flux jump. We called this the adiabatic assumption. In order to test its validity (cf. Akachi et al., 1981; Lubell and Wipf, 1966; Gandolfo et al., 1968), we need to determine how long a flux jump takes and how long it takes for the heat to diffuse out of the expanding current sheath. To determine the time scale for a flux jump, let us consider again Eq. (3.4.2). For an ordinary ohmic medium (n = 0), Eq. (3.4.2) becomes (3.6.1) which has the form of the ordinary diffusion equation with a diffusivity = ρ /µ o , where r is the constant resistivity of the medium. The units of diffusivity are m2/s, and so it connects the distance the current has diffused with the time elapsed. The thermal diffusivity is = k/S, where k is the thermal conductivity. If the expansion of the flux is fast compared with the diffusion of heat out of the current sheath, and the adiabatic assumption is valid. If the situation is reversed and the diffusion of heat out of the current sheath is much faster than the progress of the flux jump. In this latter extreme, the material outside the current sheath contributes to the heat capacity that determines the secondary temperature rise dT´. For a material obeying the Wiedemann–Franz law, (3.6.2) where Tb is the ambient (bath) temperature. If then and If then and << >> << >> << For NbTi at 4.2 K, = 4.87 x 10-3 m2/s, which corresponds to a resistivity ρmean = 6.12 x 10-9 ohm-m. This value is almost exactly two orders of magnitude smaller than the normal-state resistivity of NbTi. So if after the initial temperature rise dT, the current sheath were to develop its full normal resistance, would be >> and the adiabatic assumption would be valid. If the initial temperature rise dT is small enough, however, the current sheath may not develop its full normal resistance and the adiabatic assumption may not be valid. Consider, for example, the conductor of Section 3.4. If the initial perturbing temperature rise dT occurs 1 s after the conductor is charged, the resistivity of the current sheath (at the front face, where it is largest) is 3.47 x 10-15 ohm-m. To = the raise this resistivity to 6.12 x 10-9 ohm-m, the value at which ratio J/Jc must increase by a factor of 1.333 (n = 50). Since J remains constant, Jc must decrease by the same factor. If Jc vanes linearly with T, the temperature rise dT is related to the ratio JCafter/JCbefore as follows (see Fig. 3.7):

CHAPTER 3

46

Figure 3.7. A sketch justifying Eq. (3.6.3) relating the change in temperature and the critical currents before and after the temperature increase.

dT=(Tco –T)(1 –JCafter /JCbefore)

(3.6.3)

If Tco = 9.1 K and T = 4.2 K, dT = 1.22 K. This means that if the initial perturbation is much less than 1.22 K, << and the adiabatic assumption is invalid. In such a case, regions of the conductor outside the current sheaths also contribute heat capacity (and possibly cooling, if the exterior surface of the conductor is bathed in a cryogen), so that the right-hand side of Eq. (3.5.7) is an upper bound to the ratio dT´/dT. On the other hand, if the initial perturbation is much greater than 1.22 K, >> >> and the adiabatic assumption is valid. In any case, the adiabatic assumption is conservative, which is to say that if it implies that a slab is stable against flux jumping, adding heat conduction to the theory will strengthen this conclusion.

3.7. STABILITY AGAINST AN EXTERNAL MAGNETIC FIELD Another situation in which flux jumping occurs is that of a slab exposed to a rising external magnetic field Bo directed parallel to its faces. Again the field components obey Eqs. (3.3.1) and (3.3.2). As Bo increases, critical shielding currents are induced in the slab that seek to prevent the magnetic flux from entering it (Fig. 3.8). Again the dashed lines show what happens after a small rise dT in temperature. In the left-hand current sheath, B = Bo – µ o Jc y, 0 < y < Bo /µ o Jc

(3.7.1)

Flux Jumping

47

Figure 3.8. Sketches showing the change in the values of J, B, and E in the slab caused by a slight temperature increase. The values before the temperature increase, established by raising the external magnetic field, are shown as solid lines, the values during the temperature increase are shown as dotted lines. The case shown is the case of incomplete penetration.

so that again during the perturbation dT, E is given by Eq. (3.5.3) and P by Eq. (3.5.4). Now dT´/dT is given by Eq. (3.5.7) with a replaced by B o / µ o Jc : dT´/dT=(B o2 /3µ o)/S(Tc –T )

(3.7.2)

Continuing the illustrative example worked at the end of Section 3.5, we find Bo = 0.252 T when dT´/dT = 1. Thus as we raise the external field we expect flux jumps to be possible once it reaches 0.252 T. The foregoing analysis is based on the slab’s being wide enough that the invading flux does not reach its center, i.e., that d > Bo /µo Jc . This condition is called the case of incomplete penetration. If we substitute Bo = µ o Jc d into Eq. (3.7.2), we obtain Eq. (3.5.7) with a = d. Thus in the example we are pursuing, if d > 25 µ m, then flux penetration is incomplete up to the flux-jumping threshold.

48

CHAPTER 3

Figure 3.9. Sketches showing the change in the values of J, B, and E in the slab caused by a slight temperature increase. The values before the temperature increase, established by raising the external magnetic field, are shown as solid lines, the values during the temperature increase are shown as dotted lines. The case shown is the case of full penetration.

Suppose d is smaller than the limit just calculated; then the field components are as shown in Fig. 3.9. This condition is called the case of full penetration. Again in the left-hand sheath, B is given by Eq. (3.7.1), but now for 0 < y < d. The electric field E is now given by Eq. (3.5.3) with a replaced by d, and similarly P is given by Eq. (3.5.4) with a replaced by d. Finally, dT´/dT is given by Eq. (3.5.7) with a replaced by d. The quantities µ o Jcd and B* = [3µ oS (Tc – T)]½ have the dimensions of magnetic induction B. We can therefore summarize the results of this section by means of a graph in which µo Jc d is the abscissa and Bo is the ordinate (see Fig. 3.10). The region above the line OP of slope 1 through the origin corresponds to full penetration, the region below it to incomplete penetration. From Eq. (3.7.2) we can see that in the case of incomplete penetration, the stippled region of Fig. 3.10 is stable against flux jumps. From Eq. (3.5.7), with a being replaced by d, we see that in the case of full penetration, the hatched region is stable against flux jumps.

Flux Jumping

49

Figure 3.10. A graph summarizing the stability of a slab against flux jumping caused by application of an external magnetic field. The region above the line OP of slope 1 corresponds to full penetration, the region below the line corresponds to incomplete penetration. The shaded regions are stable against flux jumps.

In almost all applications, we are interested in external fields Bo greatly in excess of the field B* , and so we must keep µ o Jc d < B* . Thus, as mentioned earlier, the superconductor in modern, commercial conductors is divided into fine filaments whose diameters are well below the limit imposed by the condition µo Jc d < B* = [3µ o S (Tc – T)]1/ 2 .

3.8. TWISTED FILAMENTS Suppose now we consider a composite slab of multifilamentary superconductor. When this slab is exposed to a rising external magnetic field, critical shielding currents are induced in the filaments near the surface that shield the interior from the invading flux. The situation is quite similar to that just analyzed in Section 3.7 for a slab of pure superconductor, with two small differences. First, the effective critical current is λ Jc ,where λ is the volume fraction of superconductor and Jc is the critical current density of the superconductor itself. Second, the resistivity controlling the diffusion of magnetic flux through the composite is that of the matrix (typically copper). Since ρcu at 4.2 K is << ρmean, for copper, and the adiabatic assumption is invalid. If the composite slab is uncooled (the worst case) this simply means that the volumetric heat capacity S is not determined just by that of the material in the critical current sheaths, but must be augmented by the heat capacity of the material in the central parts of the slab as well. But these alterations in Jc and S do not change the order of magnitude of the maximum stable thickness 2d, which is again tens of µ m. Such thin composite conductors are undesirable.

50

CHAPTER 3

Figure 3.11. A schematic diagram of two twisted filaments in a composite slab.

As we shall see next, if the filaments are twisted within the conductor, the shielding currents decay and the external magnetic flux penetrates uniformly into the composite slab. Then no shielding current sheaths form and no flux jumps occur. This opens the possibility of composite conductors whose thickness is much greater than that just estimated. Shown in Fig. 3.11 is a schematic drawing of two twisted filaments in the composite slab. The arrows show the direction of the emf induced by the rising external field in the loop shown in the figure. By translational symmetry (which means that any loop is like any other to either side of it) conditions at point B must be the same as those at point A. As one traces the loop, one crosses the matrix at points B and A in the same direction; hence, since the superconductor supports no voltage drop, the voltage difference VB = VA = appears in the matrix. This means that induced currents necessarily flow partly in the matrix and so must decay when the external field stops rising. Since the transient shielding currents flowing in one loop do not cross into adjacent loops, we can determine their distribution by considering just the piece of composite slab conductor between the point B and the point A as though the rest of the conductor had been sheared off (Fig. 3.12). If the field created by the induced currents is neglected compared to the applied field Bo (we shall see later that this approximation is conservative), then the voltage around path ABCD in Fig. 3.12 is Since by symmetry, the y-axis is the locus E = 0, and since the superconducting filaments support no voltage, it follows from Faraday’s law of induction that E = – The induced current density is then J =– This current density flows in the matrix perpendicular to the filaments. By conservation of charge, the current dz = must flow through the plane z = 0 per unit length of the slab in the x-direction. This current must not exceed the critical value λJc d or the filaments will become resistive. Thus the pitch width is limited to

51

Flux Jumping

Figure 3.12. A schematic diagram of one pitch width of a composite superconductor showing the many filaments of superconductor embedded in the matrix.

p<

(3.8.1)

For NbTi at high fields, say 8 T, Jc = 120 kA/cm2. If we take l = 0.4, d = 0.5 mm (1-mm-thick slab), p = 5 x 10¯10 ohm-m (typical for copper at 4.2 K and 8 T), and = 0.1 T/s (typical for small magnets), we find p < 98 mm. This estimate is conservative for the following reason. The field created by the induced currents opposes the applied field in direction (Lenz’s law). Thus the actual field change we used to calculate the induced currents is an overestimate and thus overestimates those currents. Hence the pitch width limit given in Eq. (3.8.1) is smaller than it has to be and is thus conservative.

3.9. SELF-FIELD STABILITY Whereas the external field has the same direction at every point in the slab, the self-field created by the transport current (i.e., the current being driven through the conductor by an external power supply) changes sign in the slab. Indeed, by symmetry, the self-field has equal magnitudes and opposite directions at points that are images under reflection in the midplane of the slab. As a consequence, the self-field creates no net flux linkage with the loops shown in Fig. 3.12 and thus induces no shielding currents. As far as the self-field is concerned, the filaments behave as though they were untwisted. Then, harking back to Fig. 3.10, we see that slabs for which µoλJcd > B* = [3µ 0 S(Tc – T)]1/2 are stable against flux jumps induced by the self-field Bsf only if Bsf < B*. Now for a slab, the self-field at the edge of the slab is µ o Jd, where J is the transport current density in the slab. Hence, self-field stability is assured if J < B* /µ o d. The same result follows from Eq. (3.5.7), which describes the flux-jump stability of a conductor being charged with current, with the slight change that now Jc is replaced by λJc . Since a = Jd/λJc , we again find for self-field stability that

52

CHAPTER3

J < B* /m 0 d. If we take B * = 0.25 T in accordance with the estimates in Section 3.7, and consider slabs 1 mm thick, we find B* /m od = 40 kA/cm2. This limiting current density for self-field stability is well above the operating current densities in many, though not all, magnets.

3.10. A FINAL WORD ON FLUX JUMPING As a final word, it is worth remarking that flux-jump and self-field stability are not practical problems any more because modern, commercial conductors are manufactured to be well within the required limits.3 For example, prototype NbTi/Cu strands for the ill-fated Superconducting SuperCollider (SSC) described by Kallsen et al. (1991) have 4.8-µ m or 6.5-µ m filaments of NbTi in a copper matrix. The diameters of the wires themselves are 0.67 mm and 0.91 mm. No twist pitch was reported, but typically pitch widths are between 5 and 10 diameters; thus for these SSC strands, p is in the range 3–9 mm, which is amply small. For these strands, λJc ~ 120 kA/cm2 at 5 T and 4.2 K, so that when they operate at more than about one-half of their critical current, J exceeds the limit B* /µo d. But the strands are helium cooled, and since >> for copper, the helium cooling is sufficient to suppress flux jumps. Notes to Chapter 3 statement may be somewhat overdone, especially in the case of the high-temperature superconductors, where the irreversibility line, which may be well separated from the phase boundary, is said by some to separate a vortex glass phase from a vortex liquid phase. This issue is at present hotly debated and the interested reader is referred to Bishop et al., 1992; Bishop et al., 1993; and Huse et al., 1992. 2This solution is one of a type called similarity solutions. In this book such solutions arise in several places: the study underway, the flow of helium induced by heating in cable-in-conduit conductors (Chapter 8), the transfer of heat in turbulent superfluid helium (Chapter 9), and the determination of quench energies of high-temperature superconductors (Chapter 6). Detailed information on the calculation of similarity solutions can be foundin Dresner (1983). For the sake of completeness here, a short appendix on similarity solutions has been included at the end of the main text. 3 This is not really the last word by any means for the following reason. It has perhaps occurred to the reader when he examined Eq. (3.7.1) that as the external field Bo changes with time and the flux moves in and out of the filaments of superconductor, heat is dissipated in the filaments by the induced electric field E acting on the shielding currents. In fact this is so, and in the presence of a changing external field such dissipation, called hysteresis loss, is present in the superconducting filaments. Furthermore, there is additional dissipation caused by the induced currents that flow partly through the matrix as discussed in Section 3.8. This dissipation, because it arises from currents that cross the matrix from one filament to another, is called coupling loss. Collectively, these losses are known as AC-losses. In applications with alternating magnetic fields, these losses impose a load on the refrigeration system. The hysteresis losses, it turns out, are minimized by making the filaments as fine as possible, and the coupling losses are minimized by making the pitch width as short as possible. The subject of AC-losses is worthy of a book in itself, but it is outside the scope of the present work, which is devoted to stability problems, and will be pursued no further here. The interested reader should consult the relevant chapter of Wilson’s book (1983). 1This

4 Boiling Heat Transfer and Cryostability

4.1. FUNDAMENTALS OF BOILING HEAT TRANSFER As mentioned in Sections 1.6 and 1.7, one strategy for overcoming the innate instability of superconductors is to immerse them in a boiling liquid coolant (helium for the conventional, metallic, low-temperature superconductors and nitrogen for the newer, ceramic, high-temperature superconductors). To judge the effectiveness of this strategy, we need to know how much heat is transferred from a heated surface to a boiling liquid, and it is this subject we turn to next. Knowledge of boiling heat transfer is usually expressed in terms of a diagram of heat flux q (units, W/m2) plotted against superheat ∆T(units, K). The superheat ∆ T is the temperature difference between the heated surface and the bulk liquid far from the heated surface. Such diagrams have a characteristic appearance shown schematically in the plot of Fig. 4.1. In spite of the fact that q is the ordinate, it is the independent variable in the simplest kind of boiling heat transfer experiments. Fig. 4.2 shows a sketch of a thin horizontal sample supplied with an electric heater and immersed in a pool of boiling (saturated) helium. Data are taken by energizing the heater to supply a fixed power, waiting for thermal equilibrium to be established, and then measuring the temperature of the sample with an embedded thermometer. Such an experiment, in which the experimenter fixes the heat flux, is said to be under flux control. The coolant is saturated at the surface of the pool, but owing to the gravitational head, it is slightly subcooled at the depth of the sample. In other words, owing to the slight hydraulic pressure increase at the sample location, the saturation temperature there is slightly higher than the pool temperature. Hence, for very small heat fluxes (corresponding to very small superheats ∆T ) no phase change occurs at the heated surface. Heat is removed from the sample by convection, i.e., by flow driven 53

54

CHAPTER 4

Figure 4.1. The boiling heat flux q plotted against superheat ∆ T (the temperature difference between the heated surface and the bulk liquid).

by the buoyant force on the warm helium adjacent to the heated surface. In this regime, q is roughly proportional to ∆ T. As the heat flux increases, change of phase becomes possible and bubbles appear. These bubbles nucleate on the surface of the sample, grow in size, break away, and rise into the bulk of the liquid, where they either collapse again or reach the free surface and burst. This process is completely familiar to anyone who has ever boiled an egg or made a cup of tea. In this regime, known as the regime of nucleate boiling, q is roughly proportional to a power of ∆ T between 2 and 3. Eventually a heat flux is reached at which the bubbles are sufficiently numerous and grow fast enough to coalesce, blanketing the heated surface with a continuous film of vapor. Heat transfer through this vapor film is not as efficient as heat transfer when liquid contacts the heated surface, and so the superheat ∆ T

Figure 4.2. Sketch of a thin horizontal sample supplied with an electric heater and immersed in a pool of boiling helium.

Boiling Heat Transfer and Cryostability

55

jumps sharply to the higher value at point Q (remember, the flux q is being held fixed). The coalescence of the bubbles into a film and the attendant jump in D T is called burnout or the boiling crisis. The heat flux at the point P is called by several names, viz., the maximum nucleate boiling heat flux, the burnout heat flux, or the first critical heat flux. If we now reduce the heater power, we do not return at once to the nucleate boiling regime but instead continue in the regime QR of film boiling. In this regime, q is again roughly proportional to D T. Ultimately, as q is reduced, we reach a point at which the rate of vaporization at the liquid-vapor interface is not sufficient to hold the liquid away from the heated surface. Fingers of liquid penetrate and destroy the vapor film, and there is a sudden return to nucleate boiling. This restoration of nucleate boiling, which is accompanied by a sharp drop in the superheat ∆ T, is called recovery. The heat flux at point R is called by several names, viz., the minimum film boiling heat flux, the recovery heat flux, or the second critical heat flux. The description of boiling heat transfer just given corresponds to flux control. Temperature control is also possible although more difficult to arrange. Conceptually, it requires a feedback link between the thermometer and the heater to control the heater power so as to maintain a fixed superheat ∆ T. Again as ∆ T increases from small values, we begin by traversing the curve OSP. But now there can be no sharp change of ∆T at the burnout point P. Instead the heat flux decreases along the dashed curve PR, approaching the recovery point R. This regime, which can only be realized under temperature control, is called the regime of transition boiling. When a surface in transition boiling is examined visually, one sees that the surface is covered by patches of bubbles interspersed by patches of vapor film. These patches form and re-form continually, the bubbles here coalescing momentarily while the film there collapses. A fluctuating equilibrium is maintained in which the average fraction of the surface covered by vapor film increases as the superheat increases.

4.2. ADDITIONAL FACTORS AFFECTING BOILING HEAT TRANSFER The description just given of boiling heat transfer is schematic. The actual position of the q-∆ T curve in any specific case depends on a host of factors, among which the most important are: 1. The identity and thermodynamic properties of the coolant (densities, thermal conductivities, specific heats of the liquid and the vapor, the surface tension, the latent heat of vaporization, the saturation temperature). 2. The ambient (saturation) pressure, since all of the above properties vary with position along the saturation line (see Fig. 1.7).

56

CHAPTER 4

3. The orientation and shape of the surface. For example, in the case of a flat surface, heat transfer is quite different for a surface facing downwards than for a surface facing upwards. In the case of a horizontal wire, heat transfer depends on the wire diameter. 4. The condition of the surface. The shape of the heat transfer curve depends on the surface roughness, the presence of chemical impurities, and the presence of coatings applied to the surface. 5. Gravity. Since buoyancy is a principal driving force, boiling heat transfer is different in strong centrifugal fields from what it is at 1 g. Similarly, there are differences at zero g. 6. Channels. If the boiling coolant resides in a thin channel rather than in an open pool, the heat transfer curve depends on the shape, dimensions, orientation, and length of the channel. Especially important is the possible accumulation of vapor in the upper reaches of the channel, which may seriously diminish heat transfer there. All these factors are of great importance in determining the stability of superconductors cooled with boiling helium (called pool cooling, even when the helium is in thin channels between adjacent superconductors of a magnet winding). The task in this book, however, is to analyze the stability of the superconductor, assuming its heat transfer characteristics are known. We treat boiling heat transfer much as we treat the solid-state aspect of superconductivity: we assume its results are available to us and try to decide how to use them to design magnets. The literature of boiling heat transfer is immense. Two excellent references to boiling heat transfer to cryogens that the reader may consult are the article of Brentari and Smith (1965) and Chapter 6 of Sciver’s book (1986).

4.3. CRYOSTABILITY Stekly, with his coworkers (Kantrowitz, 1965; Stekly and Zar, 1965; Stekly et al., 1966), and Laverick and Lobell (1965) were the first to build superconducting magnets that recovered the superconducting state after a normalizing perturbation. Both groups of workers used boiling helium as the coolant, and both arranged for cooling to exceed Joule heating by adding copper to the conductor until the Joule heating was sufficiently reduced. We can understand how this is done by means of Fig. 4.3, which compares the steady-state boiling heat flux with the three-part curve of Joule power (cf. Fig. 3.3b). When the conductor is in the fully normal state, the Joule power produced per unit surface area of the conductor in contact with the helium Q = ρcu J2A/fP, where J is the overall current density in the conductor (transport current I divided by the total cross-sectional area A), f is the volume fraction of copper, P is the cooled perimeter of the conductor (i.e., the perimeter in contact with the helium coolant), and ρcu is the resistivity of the copper matrix. If

Boiling Heat Transfer and Cryostability

57

Figure 4.3. A comparison of the steady-state boiling heat flux with the three-part curve of Joule power from Fig. 3.3 when the superheatat recovery is greater than Tc – Tb.

Q is less than qr , the recovery heat flux, then the heating curve (curve b) lies below the cooling curve (curve a) at every temperature. In such a case, the heat balance at every point along the conductor is negative, so that the temperature everywhere ultimately returns to the saturation temperature of the helium pool. (This temperature is most often called the bath temperature and symbolized by Tb .) As we can see from the expression given above for Q, if we increase A by adding copper, we can eventually reduce Q below qr . Stability achieved in this way is called cryogenic stability, cryostability, or unconditional stability. Figure 4.3 has been drawn assuming the superheat at recovery (point R) is greater than Tc – T b. If it is not (cf. Fig. 4.4), Stekly’s criterion becomes Q =ρcu J 2A/fP < h(Tc – Tb ), where h is the slope of the film boiling part of the boiling curve. This last equation can be written [ρ cu(λJc)2A/fPh(Tc –Tb )](J /λJc)2 < 1

(4.3.1)

where, as in the last chapter, λ is the volume fraction of superconductor and Jc is the critical current density of the superconductor alone. The first factor in Eq. (4.3.1) is often eponymously called the Stekly number and symbolized by α. The ratio J/λJc , which gives the ratio of the transport current to the critical current (at Tb ), is usually symbolized by i. As mentioned in Section 1.6, one disadvantage of cryostability is the comparatively low current density J = I/A in the conductor (typically ~3 kA/cm2 at 8 T and ~5 kA/cm2 at 5 T for NbTi/Cu conductors). Low current density means large size, large weight, and high cost for magnets and, by extension, may mean large size and high cost for the rest of the apparatus in which the magnets serve. On the other

58

CHAPTER 4

Figure 4.4. A comparison of the steady-state boiling heat flux with the three-part curve of Joule power from Fig. 3.3 when the superheat at recovery is less than Tc – Tb.

hand, cryostability is attractive because it is unconditional—we do not need to know what thermal perturbations the magnet will be exposed to in order to know that it will recover. The range of thermal perturbations occurring in superconducting magnets is very poorly known. Being able to rely unconditionally on the continuous operation of a magnet without knowing what perturbations it will suffer is a great advantage. This idea is so appealing that three of the six magnets of IEA Large Coil Task (Beard et al., 1988) were cryostable pool boilers.

4.4. COLD-END RECOVERY Stekly and Laverick’s way of cryostabilizing penalizes the current density more than necessary, and Maddock, James, and Norris (1969) showed that uncon-

Figure 4.5. In an unconditionally stable superconductor, all parts of the conductor driven normal by a thermal perturbation recover simultaneously.

Boiling Heat Transfer and Cryostability

59

Figure 4.6. In a Maddock-stable superconductor, the edges of the normal zone propagate inward and the center is the last point to recover.

ditional stability can be preserved at somewhat higher current densities than the criterion Q = ρcu J2A/fP < qr allows. In magnets obeying this last criterion, all parts of the conductor driven normal by a thermal perturbation (called a normal zone) recover simultaneously, the temperature falling at every point in the normal zone (see Fig. 4.5). In Maddock-stable conductors, the edges of the normal zone propagate inwards and the center is the last point to disappear (see Fig. 4.6). This mode of recovery, which we analyze in detail below, is called cold-end recovery. The inward velocity v of the edges of the normal zone depends on the transport current I. As I increases, v gets smaller until at a certain current Im, v becomes zero. When I > Im, the edges of the normal zone propagate outwards, i.e., the normal zone grows instead of shrinking. In this case, of course, the conductor quenches. These relationships are graphically illustrated in Fig. 4.7. The current Im, corresponding to v = 0 that separates the region of cold-end recovery from the region of quenching is called the minimum propagating current. Maddock, James, and Norris have given a simple, elegant, and practical method of finding the minimum propagating current. They begin with the one-dimensional heat balance equation for a long superconductor in a helium bath: S(∂T/∂t)=∂/∂z[k(∂T/∂z )]+ QP/A –qP/A

(4.4.1)

where S is the heat capacity of the conductor per unit volume, T is the local temperature of the conductor, t is the time, z is the distance along the conductor, k is the thermal conductivity of the conductor, Q is the Joule power per unit cooled surface area, P is the cooled perimeter, A is the cross-sectional area of the conductor, and q is the heat flux being transferred to the helium. The quantities Q and q are the same as those plotted in Fig. 4.3. We look for a solution of Eq. (4.4.1) that looks like the curves in Fig. 4.6. The uniform central temperature is determined by Q = q, since near the center of the normal zone ∂T/∂t and ∂T/∂z are both zero. This means that the current I must be large enough for the heating and cooling curves of Fig. 4.3 to intersect (see Fig. 4.8).

CHAPTER 4

60

Figure 4.7. The normal zone propagation velocity v plotted versus current I.

8

8

Of the two possible intersections, only intersection 2 is stable against small perturbations. For if the steady state 2 is perturbed by a slight increase in temperature, the cooling curve exceeds the heating curve, and the temperature is restored to T2. Similarly, if steady state 2 is perturbed by a slight decrease in temperature, heating exceeds cooling, and the temperature is again restored to T 2. At T 1 the situation is just the reverse. A slight positive perturbation will cause the temperature to rise until it reaches T2; a slight negative perturbation will cause the temperature to fall to Tb. So if the initial normal zone does not disappear immediately, its central temperature quickly approaches T2. The left-hand edge of the normal zone has the form of a traveling wave T(z + vt), where T(– )= Tb and T( )= T2. If we substitute this into Eq. (4.4.1), it becomes

Boiling Heat Transfer and Cryostability

61

Figure 4.8. The case in which the steady-state boiling heat flux intersects the three-part curve of Joule power. The equality of the two stippled areas determines the minimum propagating current.

–vS(dT/dx) + d/dx[k(dT/dx)] + QP/A – qP/A = 0

(4.4.2)

where x is an abbreviation for z + vt. The minimum propagating current Im corresponds to v = 0. When v = 0, we can write Eq. (4.4.2) as d/dx[k(dT/dx)] + QP/A – qP/A = 0

(4.4.3)

Now we multiply both sides of Eq. (4.4.3) by kdT = k(dT/dx)dx and integrate from x = –∞ to x = ∞: (4.4.4)

The first term in Eq. (4.4.4) vanishes because far to the left and far to the right of the edge of the normal zone the temperature profile becomes flat, i.e., at x = ± ∞, dT/dx = 0. So the minimum propagating current is determined by the integral condition (4.4.5)

It has become customary to ignore the temperature variation of the thermal conductivity k and to replace it with a suitable constant value. In that case, the

62

CHAPTER 4

meaning of Eq. (4.4.5) is that the two stippled areas in Fig. 4.8 must be equal. In other words, the minimum propagating current is chosen to raise the curve Q until the two areas are equal. This is the famous equal-areas theorem of Maddock, James, and Norris. A rigorous approach requires that the temperature dependence of k be taken into account. As Maddock, James, and Norris point out, at low temperatures the Wiedemann-Franz law tells us that k varies directly as T. In that case, the equal-areas theorem holds when Q and q are plotted against T 2 instead of T. It is important to note that in applying the equal-areas theorem the transition boiling part of the boiling curve is taken into account. This is because the temperature of the conductor vanes continuously through the normal zone so that the heat transfer is under temperature control.

4.5. lMPROVlNG BOILING HEAT TRANSFER A typical value for Q for a fully normal cryostable conductor of the Stekly type is 0.15 W/cm2. If cold-end recovery is allowed, this flux (abbreviation: Qn) can perhaps be doubled to 0.30 W/cm2, corresponding to an increase in current density of about 40%. These values refer to conductors with relatively uncomplicated bare copper surfaces. Many workers have tried to improve boiling heat transfer (and thus current density) by roughening the conductor surface, coating it, chemically treating it, or some combination thereof. The literature contains many reports of such attempts. We mention only two in order to give the reader an idea of what can be done. Butler et al. (1970) coated heat transfer surfaces with thin layers of materials of low thermal conductivity and found that while the burnout point (point P in Fig. 4.1) moved to higher temperatures and lower fluxes, the recovery point R moved upwards markedly. In one example workedby Wilson (1983, p. 104), a 7-µ m coating ofcellulose paint raised the equal-area value of Qn from 0.31 W/cm2 to 0.48 W/cm2. Nishi et al. (1981) and Ogata and Nakayama (1982) compared heat transfer from a chemically oxidized, roughened surface called Thermo-Excel-C with that from a smooth copper surface. The Thermo-Excel-C surface, produced by making two families of parallel cuts in different directions, has a rasplike appearance. After it has been machined, it is chemically oxidized with an unspecified alkali. Figure 4.9, redrawn from the paper of Nishi et al., shows a marked improvement in heat transfer caused by the mechanical treatment of the surface and a further marked improvement caused by the subsequent chemical treatment. All in all, qr, the recovery heat flux, is raised roughly fourfold to a value slightly greater than 0.80 W/cm2. Other factors affecting boiling heat transfer are the shape of the conductor surface, the orientation of the conductor with respect to the direction of gravity, the thickness and orientation of the channels between adjacent conductors, the accu-

Boiling Heat Transfer and Cryostability

63

Figure 4.9. A diagram showing a marked improvement in heat transfer caused by scoring the surface and and an additional improvement caused by subsequent chemical treatment of the scored surface (Nishi et al., 1981). (Redrawn from an original appearing in Nishi et al. (1981) with permission of the IEEE; ©IEEE 1981.)

mulation of vapor in the channels, and vapor-induced convection of liquid helium. As with attempts to improve heat transfer by surface treatment, the papers reporting on these other factors are legion, and I shall cite only a couple in the discussion below. Walstrom (1982) studied heat transfer from a conductor with a complex shape, the conductor of the GE/ORNL coil of the IEA Large Coil Task (Beard et al., 1988), shown in Fig. 4.10. The figure also shows some of his results. Two things are noteworthy: first, there is no dip near the recovery point R, and, second, the heat transfer depends strongly on the orientation of the conductor with respect to the direction of gravity. The absence of the dip at R appears to be caused by the surface’s being made up of elements pointing in many directions. It has long been known that decreasing the thickness of the helium channels degrades boiling heat transfer (Wilson, 1983, pp. 105–6). Nishi et al. (1983) recently found such degradation at channel thicknesses of 1 mm or less. They also studied vapor-induced convection and vapor accumulation near the tops of long, vertical channels, and found them to play countervailing roles, with either able to dominate the other, depending on channel length and thickness and the rate of vapor production. As Christensen and Peck (1982) have pointed out, the deleterious effects of vapor accumulation can be avoided by inclining the channels so as to lead vapor diagonally away from vertical conductors that may have gone normal.

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Figure 4.10. Heat transfer from the conductor of the GE/ORNL coil of the IEA Large Coil Task (Walstrom, 1982). The angle is the angle from the horizontal. (Redrawn from an original appearing in Walstrom (1982) with permission of Plenum Publishing Corp., New York.)

Much ingenuity has been exerted in improving boiling heat transfer, and while such improvement has enhanced the performance of cryostable magnets, it has not changed our basic view of cryostability. Accordingly, I close this very brief discussion of boiling heat transfer and continue with the discussion of the problems of stability.

4.6. MINIMUM PROPAGATING ZONES (I) For large, expensive magnets like those of the international Large Coil Task, the possibility of failure is often deemed unacceptable, and the magnets are designed to be cryostable. Even when magnets are Maddock-stable, their overall current density is still low, but the attendant penalty of large size and high cost is borne in return for the certainty of stable operation.

Boiling Heat Transfer and Cryostability

65

8

When the magnets are smaller and the investment lower, designers are often willing to increase the current density beyond the Maddock limit in the hope that if the perturbations are small enough, the magnet will not quench in operation. How this is possible can be seen in principle by recalling our earlier discussion of Fig. 4.8. Suppose now that the right-hand stippled area is larger than the left-hand stippled area because we have raised Qn. Then we expect from Fig. 4.7 that an initial normal zone whose uniform central temperature approaches T2 will propagate, i.e., grow larger, leading to a quench. If the initial normal zone is very long (so that conditions at the center are not affected by heat conduction at the ends of the zone) and if the initial temperature is > T 1, then the central temperature will approach T 2, and such a zone will propagate. On the other hand, if the initial temperature is < T 1, the central temperature will fall to Tb and the normal zone will disappear (recovery). Thus for long normal zones, the initial temperature T 1 marks the bifurcation between quench and recovery. Long normal zones do not really fit the facts because thermal perturbations are caused most often by local slipping of the conductor under the action of the Lorentz force. What we should like to do is to characterize in some simple way those local perturbations that lead to quenches and those that lead to recovery. This ambition unfortunately exceeds our capabilities, for what it means is that we must determine which initial conditions T(z,0) asymptotically approach propagating solutions of the partial differential Eq. (4.4.1) and which asymptotically approach Tb . This involves solving the partial differential equation is some general way for an arbitrary initial condition T(z,0), and I do not know how to do this. A less ambitious but more fruitful approach has been suggested by Martinelli and Wipf (1972) and later used by Wilson and Iwasa (1978). These authors surmised the following: In addition to the uniform steady states T = Tb , T = T 1, and T = T 2, Eq. (4.4.1) has a nonuniform, localized steady state T(z), i.e., a steady state obeying the boundary conditions T( _+ ) = Tb (see Fig. 4.11). This steady state, called the minimum propagating zone (abbreviation: MPZ), is unstable against small perturbations just like the uniform solution T = T 1. Initial conditions T(z,0) that are > T MPZ (z) quench, while initial conditions T(z,0) < TMPZ (z) recover. Of particular interest to magnet designers is the formation energy of the MPZ, EMPZ =A dz S(T´) dT´, which is usually taken as an estimate of the minimum quench energy, i.e., the heat instantaneously deposited at a point that just causes a quench. Proving that (1) the MPZ is unstable and (2) that it separates quenching initial conditions from those that recover is not easy. The mathematical details, however, are instructive and have been included as Appendix B. If we introduce s = k(dT/dz ) as a new dependent variable and T as a new independent variable, the time-independent form of Eq. (4.4.1), i.e., with ∂T/∂t = 0, becomes the first-order ordinary differential equation

66

CHAPTER 4

8

Figure 4.11. Sketch of the minimum propagating zone, a nonuniform, localized steady state obeying the boundary conditions T( +_ ) = Tb.

s(ds/dT) + k(Q – q)(P/A) = 0

(4.6.1)

Now the solution TMPZ (z) that is sketched in Fig. 4.11 has s = 0 at z = 0, where where T = Tb. If we integrate Eq. (4.6.1) over T from T = Tmax, and s = 0 at z = Tb to Tmax, we find the result Tmax

(4.6.2)

k(Q – q) dT= 0 Tb

that determines Tmax (remember, Q, q, and k are functions of T only). Since Q(T) is now larger than the Maddock equal-area value, Tmax < T 2 . In principle, Eq. (4.6.1) is solvable; it is, to use an old phrase, immediately reducible to quadratures (i.e., to integrations). But if we use the three-part curve of Fig. 3.3b for the Joule power Q per unit cooled surface, the three-part curve of Fig. 4.1 for q, the heat flux through the cooled surface, and a temperature-dependent thermal conductivity k, the results are very complicated and make understanding of the MPZ difficult. What we need to do at this juncture is to simplify Q, q, and k so as to achieve simple results that enable us to see at a glance what is going on. We can do this by (1) using Newton’s law of cooling q = h(T– Tb ), where h is a temperature-independent heat transfer coefficient (units: W m-2 K-1), and (2) assuming k is a temperature-independent constant. We need to calculate the explicit form of Q as a function of T. When T >Tc , Q = Q n = ρ c u J 2 A/fP. When T b < T< T c s , Q=0. In between, when Tcs < T < Tc , Q = Qn(T – Tcs )/(Tc – Tcs ). If we refer to the sketch in Fig. 4.12, we can see that (Tc – Tcs )/(Tc – Tb ) = I/Ic = i, where Ic is the critical current (at the bath temperature Tb ). Thus Tcs =Tb +(1–i)(Tc–Tb ) and

(4.6.3)

Boiling Heat Transfer and Cryostability

67

Figure 4.12. Sketch to clarify the derivation of Eq. (4.6.3) for the current sharing threshold temperature Tcs.

= ρcu J 2 /f

Tc < T

QP/A = (ρcu J 2/f )(T – Tcs )/(Tc – Tcs ) Tcs < T < Tc

(4.6.4)

Tb < T < Tcs

=0

If we introduce the symbol τ = T – Tb for the temperature rise, Eq. (4.6.1) becomes s(ds/dτ) + k[(ρcu J 2/f )g(τ)– hPτ/A ] = 0

(4.6.5)

where τc < τ

=1

g(τ) = [τ +τ c (i – 1)]/iτ c (1 – i)τc < τ < τc 0 < τ < (1 – i) τc

=0

(4.6.6)

If the volumetric heat capacity S is also taken to be independent of temperature, the MPZ formation energy can be written 0

τ max τ dz = 2SA k τ(dz/kdτ ) dτ

–

0

EMPZ = 2SA 8

τmax = 2SAk

(τ/s) dτ 0

(4.6.7)

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4.7. MINIMUM PROPAGATlNG ZONES (II) Now we introduce special units in which k = S = hP/A =τ c = 1. The dimensions of these quantities are, respectively, PL-1Θ-1, PTL-3Θ-1, PL-3Θ-1 and Q, where P is power, L is length, T is time, and Q is temperature. The special unit of PL-3 is then (hP/A)τc so that the quantity ρcu J2 /f has the numerical value (ρcuJ2/f)/(hp/A)τc = αi 2. Here α is the Stekly number. Then, in special units, Eq. (4.6.5) becomes s(ds/dτ) + αi2g(τ) – τ = 0

(4.7.1)

where =1

1<τ

g(τ )= (τ + i – 1)/ i (1 – i ) < τ < 1 =0

(4.7.2)

0 < τ< ( 1 – i )

Even the problem represented by Eqs. (4.7.1) and (4.7.2) is difficult to solve. But for small MPZ’s for which τmax < 1, an analytic solution is attainable with relative ease. In the first place, if τ max< 1, then (see Fig. 4.13): τ max = (1 – i >(α i)1/2/[(αi)1/2 – 1]

(4.7.3)

The requirement that τmax < 1 thus translates into the requirement that αi 3 > 1. Direct integration of Eq. (4.7.1) shows that

(4.7.4)

Figure 4.13. An auxiliary sketch to aid in the determination of τ max .

Boiling Heat Transfer and Cryostability

69

Figure 4.14. A plot of the dimensionless formationenergy ε of the MPZ plotted against i = I/Ic with the Stekly number α as parameter.

These partial solutions vanish at τ = 0 and τ = τmax and join continuously at τ = 1 – i, as they should. A straightforward integration (Elrod et al., 1981) shows that τ max e=

dt 0

= [αi (1– i)/(αi – 1)][1 + (π/2 + arcsin [(αi)-1/2])/(αi – 1)1/2]

(4.7.5)

when αi3 > 1. The quantity ε is a measure of the formation energy of the MPZ. Shown in Fig. 4.14 is e plotted against i with the Stekly a number as a parameter. Also shown is the location of the minimum propagating value of i, iMP, for each a, calculated from the equal-area requirement αi2 = 2 – i (see Fig. 4.15). At i= iMP, ε has a vertical asymptote. We see from Fig. 4.14 that as i barely exceeds iMP, ε falls precipitously. Thus little is to be gained from exceeding iMP slightly. If the decision is made to operate the conductor beyond iMP, one should use values well beyond iMP . Otherwise, one might as well use values slightly less than iMP and reap the benefits of cryostability. A special design problem that sometimes arises in practice is to choose the copper-to-superconductor ratio once the size and shape of the conductor have been fixed (Elrod et al., 1981). With all else fixed except the volume fraction f of copper, the Stekly number a varies as (1– f )2/f and i varies as 1/(1 –f ).

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Figure 4.15. A sketch to aid in the determination of the minimum propagating (equal-area) value of i.

Shown in Fig. 4.16 are contours of e in the (i,a)-plane. Shown also are the curves C1: αi 2 = 1, the boundary of unconditional stability and C2: αi 2 = 2 – i. Now the (i,a) locus of constant αi2f cuts across the contours of ε so that by appropriate choice off we can try to maximize ε. If f c is the largest allowable volume fraction of f (i.e., that f that reduces the critical current Ic to the operating current I and thus makes i = 1) and αχ is the corresponding value of α, then the (i,a) locus of constant αi 2f has the following equation in the (i, α)-plane:

Figure 4.16. Contours of ε in the (i,α) plane. Shown also are the curves C1: αi 2 = 1, the boundary of unconditional stability, and C2: αi 2 = 2 – i, the boundary of cold-end recovery.

Boiling Heat Transfer and Cryostability

a = αc fc /[i(i – 1 +fc )]

71

(4.7.6)

The curve C3 in Fig. 4.16 is the locus (4.7.6) for αc = 5 and fc = 0.8 (Cu/SC = 4). The arrow on curve C3 points in the direction of decreasing f, i,e., decreasing volume fraction of copper. As we proceed in the direction of the arrow from i = 1, there is at first a rapid increase in e. But eventually the locus (4.7.6) becomes nearly parallel to the contours of e, and further decrease in f brings little gain in stability. In the example under discussion, e = 0.75 at i = 0.6 (Cu/SC = 2) whereas at i = 0.95 (Cu/SC = 3.75), e = 0.10. Thus there is a clear preference for the lower copper-tosuperconductor ratio. Trade-off studies similar to this one have also been done by Wipf (1978).

4.8. THE FORMATION ENERGY OF THE MINIMUM PROPAGATING ZONE Next we estimate the order of magnitude of the MPZ formation energy. To do this we must return from special units to ordinary units. According to Eq. (4.6.7), in special units e = E/2A. Now E/2A has the units PTL-2 and e is dimensionless. Hence in ordinary units we must have E/2A = e S(Tc – Tb )(kA /hP)1/2

(4.8.1)

Typical parameter values for a NbTi wire in an 8-T field might be S = 3000 J m-3 K-1, Tc – T b = 1.4 K, D (wire diameter) = 0.8 mm, h = 1000 W m-2 K-1, and k = 200 W m-1 K-1 (ρ cu= 5 x 10-10 ohm-m). Then E/ε = 2.67 x 10-5 J. Hence we expect the MPZ energies to lie in the range of a few to a few tens of µ J. Such energies are very small. The traditional illustration is a 1-gram weight falling through a distance of 1 mm; it gains 9.8 µ J. But such an illustration tells us very little about how to design a superconducting magnet. Instead, we must study the energy released when the conductor slips under the action of the Lorentz force. Let us consider a magnet with a ventilated winding, i.e., a winding in which spaces are left between conductors to allow the infiltration of liquid helium. Such magnets are typically wound with spiral-wrapped insulation separating adjacent conductors, and the conductors are partly held in place by the frictional force between them and the spacers. If a conductor slips at the point of contact with a spacer, the Lorentz force does work, which eventually is converted into heat. To understand the stability of the magnet, we must compare this work with the MPZ energy calculated above. A conceptual model that will serve as the basis for our calculations is sketched in Fig. 4.17. The conductor is taken to be square with side a. It is shown supported by several spacers. The arrows show the direction of the Lorentz force per unit length w = JBa2, assumed to be parallel to the surface of contact between the

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CHAPTER 4

Figure 4.17. A conceptual model for calculating the heat released when the conductor slips at its point of contact with a spacer.

conductor and the spacers (worst case). If the conductor slips suddenly at the center spacer, it suffers an average deflection (pinned boundary condition at adjacent spacers; again a worst case) d = wb4/( 120YM )

(4.8.2)

where b is the unsupported length of the conductor and YM is its flexural rigidity. Here Y is the Young’s modulus of the conductor (139 GPa for copper at cryogenic temperatures) and M is the geometric moment of inertia of the conductor around the neutral axis, namely, a4/12. The energy E´ released when the conductor slips is wbδ so that E´ = (JB)2b5/10Y

(4.8.3)

Let us continue our example by taking (Jc)NbTi = 120 kA/cm2 and Cu/SC = 1.5 ( f = 0.6). Then the Stekly number a = 27.4. If we choose i = 0.6, then e = 0.623 and E = 16.6 µ J. Now J = Jc(1- f )i = 28.8 kA/cm2. Then it follows from Eq. (4.8.3) that b = 5.34 mm. So in this example we must support the conductor at intervals less than about 2.5 mm to insure stability against wire slippage. While these numbers vary somewhat from example to example, their impact is clear: designs must avoid long unsupported spans of conductor.

4.9. THE MAXIMUM ALLOWABLE RESISTIVE FAULT

8

A design problem that can be studied by a slight extension of the foregoing theory is to determine the maximum allowable resistive fault that can exist in a conductor without quenching it. If we model the fault as a steady power source w (units:W) at z = 0, then we must add a term (w/A )δ(z) to the right-hand side of Eq. (4.4.1); here δ(z) is the Dirac delta function. Now we look for steady, local solutions of Eq. (4.4.1), i.e., time-independent solutions T (z ) forwhich T( _+ )= Tb. We expect two such solutions: one is the analog of the uniform state T = Tb that would exist in the absence of the fault, and the second is the MPZ. The first is stable against small perturbations; the second is not.

73

Boiling Heat Transfer and Cryostability

The presence of the delta function source (w/A )δ(z) causes a discontinuity at z = 0 in the slope dT/dz of the temperature. If we integrate the time-independent form of Eq. (4.4.1) from z = 0– to z = 0+, we find (4.9.1) so that T(z) has a cusp at z = 0. Since T(–z) = T(z), it follows from Eq. (4.9.1) that (4.9.2)

8

Now we again introduce the variable s = k(dT/dz ) in the time-independent form of Eq. (4.4.1), but now s obeys the boundary conditions s(Tb) = 0 (z = ) and s(Tmax ) = –w/2A (z = 0+). Now if we integrate Eq. (4.6.1) over T from T b to Tmax we find the result (4.9.3)

Figure 4.18 shows a sketch of the general behavior of k(Q – q) to be expected from the behavior depicted in Fig. 4.8. Because we are beyond the Maddock stability limit, the area of lobe 2 is greater than the area of lobe 1. There are at most three possible values for Tmax, shown at T1, T 2, and T3. The value T3 can be ruled out thus: If we integrate Eq. (4.6.1) from Tb to T, we find (4.9.4)

Figure 4.18. A sketch showing three possible valuesT1, T2, and T3 of Tmax that fulfill Eq. (4.9.3).

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CHAPTER 4

Now since the solutions T(z) we seek vary from Tmax to Tb continuously, if Tmax were equal to T3, there would be temperatures just to the left of point P for which the integral in Eq. (4.9.4) would be positive, contrary to the requirement of Eq. (4.9.4). As mentioned earlier, the lower solution, for which Tmax = T 1, is the local analog of the uniform, stable state T = Tb and the upper solution, for which Tmax = T2, is the analog of the MPZ. If as before, we use Newton’s law of cooling for q and assume k to be a temperature-independent constant, Eq. (4.9.3) becomes (4.9.5)

In special units, the left-hand side of Eq. (4.9.5) becomes (4.9.6)

The largest value of w for which the lower solution exists is that for which the left-hand side of Eq. (4.9.6) equals the entire area of lobe 1, i.e., for which τmax = τ1 = the temperature rise at point P1, namely, (αi)( 1 – i)/(αi – 1) (see Fig. 4.19). The area of lobe 1 in special units is –τ1(1 – i)/2 = – (α i)(1– i)2/2(αi – 1) so that in ordinary units

Figure 4.19. An auxiliary sketch to aid in the calculation of w max .

(4.9.7)

Boiling Heat Transfer and Cryostability

75

wmax = (ai)1/2(ai – 1)–1/2(1 – i)·2(kA/hP )1/2 hP(Tc – Tb )

(4.9.8)

Now we continue the numerical example begun at the start of Section 4.8. The second factor in Eq. (4.9.8) is then 44.5 mW. If we take J = 28.8 kA/cm2, as was done following Eq. (4.8.3), then I = 145 A. With a = 27.4 and i = 0.6, the first factor in Eq. (4.9.8) is 0.413 so that the maximum fault resistance is 0.875 mW. This calculation can thus serve to determine the maximum allowable joint resistance. Since good joints typically have resistances of the order of nW, we do not expect them to have any appreciable effect on stability. This last conclusion can be quantified by comparing the difference De in the formation energies of the upper and lower states with the formation energy of the MPZ in the absence of a resistive fault. This calculation has been carried out by the author (Dresner, 1982). The exact result for De as a function of a, i, and w is rather complicated, but the upshot of the calculations is that De varies almost linearly with w for fixed a and i. Thus we may use the rule of thumb De/De w=0 = 1 – w/w max

(4.9.9)

Clearly, then, in the numerical example given after Eq. (4.9.8), a nW joint will have virtually no effect on the quench energy.

4.10. STABILITY OF PARTLY COVERED CONDUCTORS Another application of the formation energy of the MPZ as a measure of stability is to the stability of partly covered conductors. Neighboring conductors in a pool-cooled magnet are separated by insulation that partly covers their surfaces and prevents full contact with the liquid helium. In studying the stability of such conductors, the custom was to take the partial occlusion of the surface into account simply by reducing the cooled perimeter. But Meuris and Mailfert (1981) eschewed this simple approximation and studied the effect on stability of point-to-point variation in cooling. Their elegant study gave most surprising results, described below. Working in the idealized case of constant properties, Meuris and Mailfert considered a long conductor having a single uncooled region of half-length L To simplify their calculations, they ignored current sharing and took for the function g(t) =1

1
g(t)

(4.10.1) =0

t<1

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Figure 4.20. The four steady states found by Meuris and Mailfert (1981). (Redrawn from an original appearing in Meuris and Mailfert (1981) with permission of the IEEE; © IEEE 1981.)

What they found was that besides the pure superconducting state t = 0, there are four other steady states having cold ends allowed by the heat balance equation. Meuris and Mailfert label these four states according to their sizes as shown in Fig. 4.20, redrawn from their paper. States 2 and 4 are stable, states 1 and 3, unstable. These states do not all occur together but rather occur in various combinations

Figure 4.2 1 . The (αi2,xs) plane divided intoregions in which different steady states occur(Meuris and Mailfert, 1981). (Redrawn from an original appearing in Meuris and Mailfert (1981) with permission of the IEEE; © IEEE 1981.)

Boiling Heat Transfer and Cryostability

77

according to the values of αi 2 and the dimensionless half-length of the uncooled region xs = (Ph/kA )1/2 Ι. Figure 4.21, also redrawn from the paper of Meuris and Mailfert (1981), shows the (α i 2 , xs)-plane divided into regions according to the steady solutions that occur there. When the choice (4.10.1) is made for g(t), the equal-area criterion of Maddock, James, and Norris (1969) is simply αi 2 = 2. When the conductor is metastable, i.e., when α i 2 > 2, the (αi2,xs)-plane is divided into two regions, C and D, distinguished by whether or not xs > 1/αi 2. In each of these two regions, one nonzero steady state occurs; these steady states are unstable. In each region, two outcomes are possible following a perturbation: quench or recovery. As usual, Meuris and Mailfert propose the formation energy of the unstable steady state as a measure of conductor stability. When the conductor is stable (αi 2 < 2), the situation is slightly more complex. In the cross-hatched region A, the conductor always recovers the superconducting state. In the white regions B, E, and F, two steady states occur, one stable, the other unstable. The unstable state is always smaller than the stable state (see Fig. 4.20). Again two outcomes are possible following a perturbation: recovery of the superconducting state or transition to a steady normal zone centered on the uncooled region and described by the stable steady state. Meuris and Mailfert again propose the formation energy of the intermediate unstable steady state as a quantitative criterion of stability. Figure 4.22 shows the dimensionless formation energy of the unstable state as a function of the dimensionless uncooled length xs for five values of αi 2. (N.B.: Meuris and Mailfert’s dimensionless formation energy ec is twice that defined in Eq. (4.8.1).) The stable normal zones predicted by Meuris and Mailfert have been observed (Claudet et al., 1979). Their existence represents a kind of loss of stability which, though far less severe than a quench, is nonetheless discomfiting.

4.11. TRANSIENT HEAT TRANSFER I was careful everywhere in the foregoing sections to refer to the formation energy of intermediate unstable steady states only as a figure of merit or a measure or a quantitative criterion of conductor stability. I think it cannot be emphasized too strongly that the MPZ formation energy represents an artificial standard of stability useful only because it allows us to make quantitative decisions about conductor design. Always lurking in the background is the unproven assumption that the conductor with the larger MPZ formation energy has the larger minimum quench energy, too. By its very definition as a steady state, the MPZ must be calculated using a steady-state heat transfer coefficient. But in reality, the heat transfer coefficient greatly exceeds the steady-state heat transfer coefficient for a brief interval immediately following a pulsed heat addition to the conductor. As we shall see below,

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Figure 4.22. The dimensionless formation energy of the unstable state as a function of the dimensionless uncooled length xs for various αi 2 (Meuris and Mailfert, 1981). (Redrawn from an original appearing in Meuris and Mailfert (1981) with permission of the IEEE; © IEEE 1981.)

this high but transient heat transfer allows a certain limited deposition of heat in the conductor without creating a normal zone at all. The phenomenology of transient heat transfer in saturated helium has been elucidated by Tsukamoto and Kobayashi (1975), Schmidt (1978; 1981) and Steward (1978). Although they used different techniques of experimentation, in essence they did the same thing, namely, suddenly energize a heater of low thermal inertia in contact with a saturated helium bath and measure its temperature as a function of time. What they observed is summarized in Fig. 4.23, redrawn from Steward’s paper. Right at the start, the heater temperature rises suddenly by a few tenths of a Kelvin. This modest, early temperature rise corresponds to heat transfer limited by the so-called Kapitza resistance. Kapitza observed that there is a temperature discontinuity at the interface between a solid surface and liquid helium related by the following equation to the heat flux q being transferred from the solid to the helium: q = a(T ns –THne )

(4.11.1)

where Ts is the solid temperature, T H e is the liquid helium temperature at the interface, and a and n are constants characteristic of the surface. Eq. (4.11.1) implies

Boiling Heat Tranfer and Crystability 79

Figure 4.23. Steward’s curves of transient heattransferfrom a vertical surface to boiling liquid helium at atmospheric pressure (Steward, 1978). (Redrawn from an original appearing in Steward (1978) with kind permission from Elsevier ScienceLtd.,The Boulevard, Langford Lane, Kidlington, OX5 IGB, UK.)

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that there is an interfacial resistance to heat flow since, in point of fact, at a discontinuity in temperature the temperature gradient is infinite. A theory of the Kapitza discontinuity based on considerations of the phonon spectrum in the solid and the liquid predicts n = 4; experimental values lie mainly in the range 3 +_ 0.5 (Sciver, 1986). The experimental values of the coefficient a are scattered; a suitable value for purposes of rough estimation is a = 40 mW cm-2 K-3 when n = 3 (Sciver, 1986). As time progresses, the heater temperature gradually increases, approaching at long enough times proportionality to the square root of the elapsed time. In this regime, which is clearly visible for the smaller heat fluxes in Fig. 4.23, the rate-determining heat transfer process is transient conduction through the heated layer of liquid helium adjacent to the heater surface. After a time, the heat being transferred to the helium causes bubbles to appear, and the dominant mode of heat transfer becomes nucleate boiling. For small heat fluxes, this nucleate boiling may persist indefinitely. For larger fluxes, it ends when there is a transition to film boiling accompanied by a large increase in the heater temperature. This transition is called takeoff. The time tf to takeoff as a function of the constant heat flux q has been measured by several workers and the measurements compiled by Schmidt (1982) (see Fig. suggested to Schmidt the following elementary 4.24). The near constancy of qt1/2 f theory of transient heat transfer in saturated helium. According to ordinary diffusion theory, when the flux entering a half-space is suddenly clamped at a constant value q, the temperature rise ∆ T at the front face is given by ∆ T= (4q2t/ pkS )1/2

(4.11.2)

where t is the elapsed time since the heater was energized, k is the thermal conductivity of helium and S is its volumetric heat capacity. The heat introduced per unit face area during the time t is qt. If this heat is absorbed in a thin layer of

Figure 4.24. The time to takeoff as a function of the constant heat flux according to Schmidt (1982). (Redrawn from an original provided courtesy of the International Institute of Refrigeration.)

81

Boiling Heat Transfer and Cryostability

helium of thickness d in which the temperature rise is uniform at the value given in Eq. (4.11.2), then S ∆ T d = qt, so that d = (πkt/4S )1/2

(4.11.3)

Schmidt postulates that takeoff occurs when the heat transmitted qt equals the latent heat Lδ of the heated layer, for at that time he supposes that there is enough heat present to vaporize the entire heated layer. Thus qt f1/2 = (πk/4S)1/2L = 39 mW cm-2 s1/2

(4.11.4)

(k/S = 2.84 x 10-8 m2/s and L = 2.59 J/cm3 according to Arp and McCarty (1989). The curve in Fig. 4.24 corresponds to a value of qt f1/2 = 51 mW cm-2 s1/2 ; its rather good agreement with the experimental facts supports Schmidt’s picture of what is going on. (N.B.: Schmidt (1978; 1982) first used a factor of 2, later a factor of π/2 in Eq. (4.11.4) instead of the factor (π/4)1/2.) Armed with Schmidt’s formula (4.11.4), we can attack the following problem. If an intense disturbance lasts for a short time ∆ t, what is the maximum heat it can produce without causing a transition to film boiling? The maximum allowable heat input per unit area H is clearly that which just makes tf = ∆ t. Now since q = H/∆ t, we find from Eq. (4.11.4) that H = (πk/4S )1 / 2L(∆ t)1/2

(4.11.5)

Eq. (4.11.5) is based on neglect of any Joule heat produced during transient heat transfer. This is a satisfactory assumption for short enough times ∆ t because H varies as (∆ t)1/2 whereas the Joule heat varies as ∆ t. Table 4.1 shows for several values of ∆ t the heat H calculated from Eq. (4.11.5) and the maximum Joule heat per unit surface area (ρcu J2A/fP)∆ t for the numerical example of Section 4.8. In this example, for the times given, the neglect of the Joule heat is roughly justified. The time ∆ t can be estimated for the conceptual model of conductor slippage introduced in Section 4.8. If the conductor slips freely, its motion is restrained only by its inertia. The time to reach mechanical equilibrium is of the order

Table 4.1. Comparison of the Allowable Heat H with the Joule Heat ∆t (s)

H (J/m2)

10–6 10–5 10–4

0.510 1.61 5.10

(ρcu J 2A /fP)∆ t (J/m2) 0.0138 0.138 1.38

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(b2/a)(δ/Y )1/2, where d is the density of the conductor (8960 kg/m3 for copper, Southwell, 1969). This time turns out to be about 9 µ s for b = 5 mm and a = (π/4)1/2D = 0.71 mm. The total Joule heat H(A/P)b produced in a 5-mm span is then about 1.5 µ J. This heat can be added to the MPZ energy 16.6 µ J, since the latter value is predicated on the film-boiling heat transfer coefficient. In this case, transient heat transfer slightly improves the stability of the conductor; its neglect is thus slightly conservative.

5 Normal Zone Propagation

5.1. EXACT CALCULATION OF THE PROPAGATION VELOCITY When the current lies beyond the minimum propagating current (see Fig. 4.7), local normal zones whose formation energy exceeds the minimum quench energy grow. Their central temperature approaches the temperature T2 of point 2 in Fig. 4.8 and their edges propagate outwards. The situation is much the same as shown in Fig. 4.6b, except that now t4 is the earliest time and t1 the latest. In most cases, the propagation is uniform, i.e., the velocity is constant and the temperature profile at the edges of the normal zone does not change its shape with time. The propagation velocity is comparatively easy to measure. One method is to measure the voltage produced by the normal zone as a function of time. In well-cooled magnets in which the central temperature T2 is less than about 20 K, the resistivity of the matrix is nearly independent of temperature, and the rate of increase of the normal-zone voltage is proportional to the propagation velocity. In uncooled magnets, the central temperature of an expanding normal zone keeps on rising as does the resistivity, and the contribution to the voltage from the center of the normal zone becomes disproportionately large. In such a case, voltage measurement is not a reliable way to measure the propagation velocity. A second way, free from this defect, is to time the flight of the normal–superconducting front bet ween two voltage taps. Owing to the complexity of boiling heat transfer, the propagation velocity is measured in order to determine the minimum propagating current. A typical series of measurements provides points on the curve of propagation velocity versus transport current (Fig. 4.7). The minimum propagating current is then determined by interpolation. The part of the curve so measured can be used to extract information about heat transfer during normal zone growth or shrinkage. Several attempts of this kind have been made; in order to discuss them, we must first calculate the 83

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propagation velocity in order to determine its dependence on the various parameters of the conductor. The basic equation from which we start is Eq. (4.4.2), which describes the left-hand edge of a normal zone, i.e., the edge that propagates from right to left. Again, to obtain a solvable problem, we use Newton’s law of cooling and assume k and S to be independent of temperature. We again introduce the variable s = k(dT/dz) and finally employ the special units of Section 4.7. The result is the first-order ordinary differential equation s(ds/dτ) – vs + ai2g(τ) – τ = 0

(5.1.1)

the solution of which must obey the boundary conditions s(0) = 0 and s (ai 2) = 0 (since τ2 = ai 2). Altov et al. (1973) were the first to solve the problem just formulated using finite differences. Later, the author gave an analytical solution to the problem (1979, “Analytic Solution”). The results of these exact calculations (in the special units of Section 4.7) is the functional dependence of v on a and i: v = F (a,i). Remembering that (r cuJ 2/f )/(hP t c /A ) = ai 2, we can write v = [(rcu J 2/f )/(hP tc /A)] 1/2[F(a,i)/a1/2i]

(5.1.2)

which becomes in ordinary units v = (k/S )(hP/kA)1/2[(rcu J 2/f )/(hP tc /A)]1/2[F (a,i)/a1/2i] = (2λJc/S)(ρcuk/f τ c)1/2 · [F(α,i)/2α1/2]

(5.1.3)

Shown in Fig. 5.1 is the second factor in Eq. (5.1.3) plotted versus i with a as a parameter. It approaches 1 when i approaches 1 for all a. When a →∞,i.e., when h →0 (an uncooled superconductor), F (a,i)/2a1/2 → ci 1 / 2 /2, where c is related to i by the equation (Dresner, 1980, “Propagation”): [c/(4 – c2)1/2]arctan[c(4 – c2)1/2/(2 – c2)] + 2 In c = ln[i/(1 – i)]

(5.1.4)

The curve ci1/2/2 is the uppermost curve shown in Fig. 5.1. It alone extends down to i = 0. All the other curves cross the axis v = 0 at values of i satisfying the equal-area requirement ai 2 = 2 – i. Between this value of i and the Stekly value, which satisfies ai 2 = 1, initial normal zones collapse by cold-end recovery, the flanks of the zone moving inwards with the (negative) velocity v. As i approaches the Stekly value from above, v →– ∞.Below the Stekly value of i, all points of the initial normal zone recover simultaneously, and the notion of propagation is inapplicable. (N.B.: In evaluating the left-hand side of Eq. (5.1.4), use the principal value of the arctan lying in the interval (0,p).)

Normal Zone Propagation

85

Figure 5.1. The second factor in Eq. (5.1.3) plotted versus i: it gives the dependence of the propagation velocity on the current.

The first factor on the right-hand side of Eq. (5.1.3) (callit v *)can be simplified for a matrix material obeying the Wiedemann–Franz law, Eq. (2.9.1). Since the thermal conductivity of the matrix is typically >> than that of the superconductor, k/f= kcu and the first factor in Eq. (5.1.3) becomes v * = (2λ Jc /S)[(LoTb /(Tc – Tb )]1 / 2

(5.1.5)

where we have given the quantities ρcu and k cu , which have been assumed independent of temperature, the values they have at the ambient temperature Tb. Using the figures given in the example of Section 4.8, we find v* = 86.8 m/s; since in this example a = 27.4 and i = 0.6, we see from Fig. 5.1. that F(α,i)/2α1/2 = 0.32, so that v = 27.8 m/s. This single example gives some idea of the order of magnitude of the propagation velocity. But it can vary widely depending on the circumstances. In the new high-temperature superconductors, it can be as little as a few mm/s, whereas in the cabled conductor of the Superconducting SuperCollider 17-m dipoles, it was as much as several hundred m/s.

5.2. APPROXIMATE CALCULATION OF THE PROPAGATION VELOCITY The implicit relation between c and i in Eq. (5.1.4) is inconvenient when one wishes to undertake repetitive calculations even on an electronic computer. The analytic solution of reference (Dresner, 1979, “Analytic solution”) for F (a,i) involves an even more complicated relation than that of Eq. (5.1.4) and greater inconvenience. A simple, explicit, and quite accurate approximate result for F (a,i) can be arrived at by replacing in Eq. (5.1.1) the three-part curve (4.7.2) for g(τ) by the two-part curve

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τ < 1 – i/2

= 0 g(τ)

(5.2.1) τ > 1 – i/2

=1

This replacement has the desirable property that the differences in area of the right-hand and left-hand stippled regions in Fig. 4.15 are the same for the three-part g(τ) of Eq. (4.7.2) as for the two-part g(τ) of Eq. (5.2.1), namely, αi2(αi2 – 2 + i)/2. Thus the minimum propagating (v = 0) value of i is the same in both cases. When τ < 1 – i/2 and g = 0, s = l+t

(5.2.2)

where l+ is the positive root of the quadratic equation l2 – v l – 1 = 0

(5.2.3)

(Remember, we are calculating the velocity of the left-hand edge of a normal zone for which s = k(dT/dz ) > 0.) When t > 1 – i/2 and g = 1, s = m+(ai 2 – t)

(5.2.4)

where m+ is the positive root of the quadratic equation m2 + v m – 1 = 0

(5.2.5)

The partial solutions (5.2.2) and (5.2.4) satisfy the boundary conditions s(0) = 0 and s(αi 2) = 0, as they should. They must be equal at t = 1 – i/2, and this condition gives the result v = (C – 1)/C1/2 where C = [αi 2 – (1 – i/2)]/( 1 – i/2)

(5.2.6)

Since this result obtains in special units, F(α,i) = (C – 1)/C1/2

(5.2.7)

Figure 5.2 shows a plot of the approximate value of F(α,i)/2α1/2 plotted versus i with the same values of as a parameter as used in Fig. 5.1. The agreement is excellent over most of the range of i, with serious deviations only near i = 1. According to the author (Dresner, 1979, “Analytic solution”), considerable improvement in agreement can be achieved by applying the empirical factor 1 + 0.561a–1.45 to the result calculated from Eq. (5.2.6).

Normal Zone Propagation

87

Figure 5.2. The approximate value of the second factor in Eq. (5.1.3) plotted versus i [cf. Eqs. (5.2.6) and (5.2.7)].

5.3. COMPARISON WITH EXPERIMENTS OF IWASA AND APGAR The curves in Fig. 5.1 can be compared with experimental propagation velocities in the following way. The intersections of the curves with the i-axis are the dimensionless minimum propagating currents corresponding to the various values of the Stekly number a. Experimental values of the minimum propagating current can thus be used to determine a. Then we can use the value of a so determined to calculate the propagation velocity as a function of i using Eq. (5.1.3) and the data in Fig. 5.1 or Eq. (5.2.7). Straightforward as this procedure seems, it fails to reproduce the experimentally measured propagation velocities. The experimental and calculated curves have the same intercept on the i-axis, of course, but the theoretical curve is usually several times steeper than the experimental curve. This difficulty is not eliminated by use of a three-part boiling curve and temperature-dependent thermophysical properties, as extensive numerical calculations have shown (Dresner, 1976). I gave a hint at the resolution of this difficulty, when I became convinced “that steady-state heat transfer coefficients are inadequate to describe the growth of normal zones” (Dresner, 1976). Accordingly, I introduced ad hoc corrections for transient heat transfer that raised the heat transfer coefficient at the propagating wave front but allowed it to approach the steady-state value far behind. Agreement was much improved, and I concluded this work by recommending “a velocity-dependent correction that increases with velocity.” Later, I used a velocity-dependent correction that raised heat transfer for expanding zones and lowered it for collapsing zones (Miller et al., 1977). Such a correction slowed both expansion and collapse; thus it decreased the slope of the curve of v versus i in the neighborhood of the minimum propagating current. This idea proved right: experiments by Iwasa and Apgar (1978) identified the growth and collapse of the vapor film blanketing the conductor surface as the source

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of the velocity-dependent heat transfer. When the normal zone is expanding, the vapor film forms at the advancing wave front. The latent heat of formation is absorbed from the metal, slowing the rate of advance of the front. When the normal zone is contracting, the vapor film condenses at the retreating wave front. The latent heat of the film is then released to the metal, slowing the rate of retreat of the front. Iwasa and Apgar measured the instantaneous temperature of a helium-cooled copper plate as a function of heat flux. They expressed the heat flux as a sum of two terms, qs (t), the steady-state term, and a(τ)(dT/dt), a term proportional to the time rate of change of the temperature. From their measurements, they obtained the temperature-dependent coefficient a (τ). By fitting curves of propagation velocity versus transport current measured by Miller et al. (1977), I was also able to obtain the coefficient a (τ) (Dresner, 1979, “Transient heat transfer”). The agreement between these latter values and those directly measured by Iwasa and Apgar was excellent. Tsukamoto and Miyagi (1979) and Nick, Krauth, and Ries (1979) performed virtually identical analyses, using Iwasa and Apgar’s transient term a (τ )(dT/dt) to fit measured velocities of propagation. The particular form chosen by Iwasa and Apgar to represent their transient correction, namely, a(τ)(dT/dt), makes correcting the propagation velocity relatively simple. If one adds this term to q in the fundamental Eq. (4.4.1), one finds that the term a(τ)(dT/dt) can be combined with the left-hand side and has the sole effect of increasing the volumetric heat capacity S by Pa/A. One can therefore use the formulas previously derived, e.g., Eq. (5.1.3), merely by increasing S.

5.4. EFFECT OF TRANSIENT HEAT TRANSFER The formation and decay of the vapor film is a relatively slow process and the work of Iwasa and Apgar applies only for relatively long transits of the front past a fixed point. In the experiments of Miller et al. (1977), which could be fitted well using the Iwasa–Apgar correction, this transit time was of the order of 10 ms. Judging from Steward’s data in Fig. 4.23, we expect the heat transfer coefficient that controls the motion of the front to be that of film boiling. It is worth noting that the Iwasa–Apgar correction affects both the positive (propagation) and negative (cold-end recovery) velocity. When the time of transit is much shorter than 10 ms, we expect the heat transfer coefficient that controls the motion of the front to be higher than that of film boiling because of the transient heat transfer processes discussed in Section 4.11. In the experiments of Funaki et al. (1985), the transit time was of the order of a few tenths of a ms. If we fit Eq. (5.1.3) to the data of Funaki et al. by suitably choosing the heat transfer coefficient, we find that it lies in the range 0.5–1.5 W cm-2 K-1 when v is in the range 5–20 m/s. This is larger than the film boiling heat transfer coefficient by roughly a factor of 10 and is typical of the lower left-hand corner of Steward’s diagram, Fig. 4.23.

Normal Zone Propagation

Author

89

Table 5.1. Summary of the Agreement of Various Theories with the Data of Funaki et al. Reference Assumed mechanism a Agreement

Dresner Lvovsky and Lutset Nick, Krauth, and Ries Funaki et al.

Constant adjustable parameters S, h D+S N+ S + I D +N

Good with suitably chosen parameters Poor Good Good

(1979, “Analytic Solution”) (1982) (1979) (1985)

aD = transient conduction heat transfer;

N = metastable nucleation; S = steady-state film boiling; I = Iwasa–Apgar correction.

Funaki et al. and the authors they quote (Lvovsky and Lutset (1982); Nick et al. (1979); Dresner (1979), “Analytic solution”) undertook rather laborious computations, some numerical and some analytic. The bases of all these computations were not the same. Table 5.1 summarizes the agreement of the various theories with the experimental data presented by Funaki et al. in their Fig. 4 (redrawn here as Fig. 5.3). Probably the simplest thing to do in practical circumstances is to use Eq. (5.1.3)

Figure 5.3. A comparison of propagation data of Funaki et al. (1985) with the theories of various authors (cf. Table 5.4). (Redrawn from an original appearing in Funaki et al. (1985) with permission of Butterworth -Heineman, Oxford, England.)

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with F given by Eq. (5.2.7), choose S and h in accordance with Steward’s diagram, Fig. 4.23, and use Iwasa–Apgar’s correction to S, if applicable.

5.5. TRAVELING NORMAL ZONES (I) In the simple theory of Sections 5.1 and 5.2, both the Joule heat source Q and the heat flux q being transferred to the helium are functions only of the metal temperature T. In Sections 5.3 and 5.4 we discussed the complexity introduced when the heat flux q at a point depends not only on the temperature there but also on the time elapsed since the normal-superconducting front passed the point. In certain other circumstances described below, it is the Joule heat source Q that depends not only on the temperature but also on the time elapsed since the normal-superconducting front passed. These circumstances arise in very large composite conductors in which the matrix and the superconductor are sharply segregated. One conductor, designed for use in very large magnets proposed for superconducting magnetic energy storage (SMES) is shown in Fig. 5.4 (Huang and Eyssa, 1991). In this conductor, the superconductor is confined to the circumference of a large cylinder of aluminum. Other similar conductors have been proposed for use in fusion magnets (Mito et al., 1991) and in space applications (Huang, Eyssa, and Hilal, 1989). Typically, such conductors operate with currents in the range 30–100 kA, are of the order of 2.5–5 cm in diameter, and consist of large blocks of high-purity aluminum in which much smaller superconductors are embedded. The segregation of the aluminum matrix and the superconductor has the following deleterious effect on the stability of the conductor. When the superconductor is first normalized, the current enters the matrix but is confined to the vicinity of the superconductor. Thereafter, the current diffuses throughout the matrix, tending toward a state of uniform current density. In this uniform state, the Joule power is much lower than at the start. The relaxation time of current redistribution is typically some tens to hundreds of milliseconds. Thus the excess Joule heat (over the uniform state) appears as a short pulse immediately following normalization. If the conductor is not cryostable, this short heat pulse diminishes the external energy it takes to create a propagating normal zone. Whereas for small (and therefore flexible) conductors, conductor motion is local, for the very large (and therefore stiff) conductors we are considering here, conductor motion may be spread out over many diameters. Then the quench energy (defined in Section 4.6 as the heat instantaneously deposited at a point that just causes a quench) is no longer useful as a measure of stability. Instead, we use the stability margin, defined as the uniform heat density instantaneously deposited in a long length of conductor that just causes a quench.1 The pulse of excess Joule heat produced during current redistribution reduces the stability margin compared to

Normal Zone Propagation

91

Figure 5.4. A large, high-current conductor with segregated matrix and superconductor for superconducting magnetic energy storage (SMES) (Huang and Eyssa, 1991). (Redrawn from an original appearing in Huang and Elyssa (1991) with permission of the IEEE; © IEEE 1991.)

what it would be if the superconductor were homogeneously distributed over the entire matrix. In addition to affecting the stability margin, the excess Joule heat affects the velocity of propagation of normal zones; since it is released at the head of the propagating wave, it increases the propagation velocity. But an even more remarkable thing happens: a special kind of propagation takes place in conductors that would be unconditionally cryostable if the superconductor were distributed homogeneously throughout the matrix. When a long normal zone is created in such a conductor, the edges propagate outward continually, driven by the release of excess Joule heat from the newly normalized conductor. In the center, the current distribution eventually becomes uniform, after which local cooling exceeds local heating and the superconductor recovers. All that then remains are two vestigial normal zones at the ends that continue to move outwards. The normal zones, called traveling normal zones (TNZ), are finite in extent and move without change of shape at a constant velocity away from the site of the original disturbance. Boom and his coworkers (Christianson and Boom, 1984) were the first to recognize the adverse influence of current redistribution on the stability of large

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SMES conductors. Christianson (1986) and Devred and Meuris (1985) studied the increase in the propagation velocity caused by current redistribution, but neither group of authors suggested the existence of traveling normal zones. To my knowledge, the first hint came from Luongo, Loyd, and Chang (1989); but, ironically, after a long and essentially correct discussion, they concluded their conductor would not recover behind the outward moving fronts. But like their predecessors, they did realize that the excess Joule heat produced during current redistribution would increase the propagation velocity and decrease the minimum propagating current. As far as I know, the first authors to state explicitly that traveling normal zones were possible are Huang et al. (1990; 1991). My own contribution to this subject (1990) was formulation of a simplified model, discussed below, which could be treated analytically. This model predicts a current threshold below which the conductor is cryostable and above which TNZs occur. At this threshold, the propagation velocity jumps to a finite value rather than rising smoothly from zero. When the current becomes high enough, recovery far behind the fronts no longer occurs and instead of TNZs, there is a single expanding zone. Lately Mints and his coworkers have presented an analytical treatment of TNZs based on equivalent circuits (Kupferman et al., 1991). TNZs have been seen experimentally (Pfotenhauer et al., 1991) in a SMES proof-of-principle experiment. In addition to verifying the existence of TNZs, the authors saw the expected jump to a finite velocity at the threshold current.

5.6. TRAVELING NORMAL ZONES(II) To account for the excess Joule heat released during current redistribution, we must add to the left-hand side of Eq. (4.4.2) the term wU(t+z/v ), where w is the excess Joule heat density (i.e., the excess Joule heat released when a unit volume of conductor is normalized) and U(t) dt is the fraction of this heat released during the time interval dt beginning a time t after the superconductor becomes normal. The function U(t) is thus normalized so that U(t)dt = 1; for t < 0, of course, U = 0. Finally, note that the function U has the physical dimensions of reciprocal time. We again (1) assume that k and S are independent of temperature, (2) assume that Newton’s law of cooling applies, (3) employ the special units of Section 4.7, and (4) use Eq. (4.10.1) for g(t). Then Eq. (4.4.2) becomes d 2t/dx 2 – vd t/dx + ai 2 – t + wU(x/v ) = 0,

x>0

(5.6. la)

d 2t/dx 2 – vd t/dx – t = 0,

x <0

(5.6.1b)

where the origin of x is the point on the traveling-wave profile at which the superconductor first goes normal, namely, the point at which τ = 1.

Normal Zone Propagation

93

The solution we seek must obey the boundary conditions: (5.6.2a)

τ(+ ) = 0 (TNZ)

(5.6.2b)

8

τ(– ) = 0 8

τ(0) = 1

(5.6.2c)

as well as the condition dτ/dx continuous at x = 0

(5.6.2d)

When x < 0, τ = exp(λ +x), where λ+ is again the positive root of Eq. (5.2.3). Therefore, τ(0) = 1 and dτ/dx = λ+. These last two conditions plus Eq. (5.6.2b) in general overdetermine the solution to Eq. (5.6.1a), but a solution is possible for certain particular values of v, one of which is the value we seek. If we introduce the Laplace transform c(p) of t(x) defined by (5.6.3) the transform of Eq. (5.6. la) can then be solved for c to give c = [p +l+ – v – ai 2/ p – wvu (vp)]/[p 2 – vp – 1]

(5.6.4)

where (5.6.5) is the Laplace transform of U. The singularities of the transform (5.6.4) are a pole at p = 0 and poles at p = λ+ and p = λ_, the roots of Eq. (5.2.3). Now the residue at the pole p = λ+ must vanish if τ is not to rise exponentially with distance behind the normal-superconducting front. Thus when p = λ+, the first bracketed term in Eq. (5.6.4) must vanish. Then we find w = [l+(v 2 +4)1/2 – ai 2 ]/[v l + u (v l + )]

(5.6.6)

5.7. TRAVELING NORMAL ZONES (III) According to the initial-value and final-value theorems (Spiegel, 1965)

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CHAPTER 5

(5.7.1)

lim pu(p) = U( ) = 0

(5.7.2)

8

lim pu(p) = U(0+)

8

the last equality following from the integrability of U. Thus w becomes infinite in the limits v → 0 and v → . If ai 2 < 1 (the condition for unconditional cryostability when g(t) is given by Eq. (4.10.1)), so that the numerator of Eq. (5.6.6) is positive for all v, and w > 0 for all v, then w must have a minimum for some value of v. Shown in Fig. 5.5 is a sketch of w versus v showing the minimum. For any value of w above the minimum, two values of v are possible. The arguments given so far do not determine which of these occur in the laboratory. But because the velocity at point P´´ decreases with increasing w, we are inclined to think the traveling wave corresponding to P´´ does not occur in the laboratory.The reason for this nonoccurrence is that this traveling wave is unstable against small perturbations. Even if once created, the thermal fluctuations that are always present would immediately destroy it. On the other hand, the traveling wave that corresponds to the intersection P´ is stable and does occur in the laboratory. Of course, these explanatory remarks in no way constitute a proof. We see then that as w increases, a point suddenly occurs at which propagation of TNZs is possible and the velocity jumps suddenly to a finite value. This behavior has been observed in the laboratory (Pfotenhauer et al., 1991). Furthermore, right at the threshold dv/dw = ; a hint of this behavior exists in the experimental data, but the spacing of the experimental points is wide and the point is still largely moot. 8

Figure 5.5. An auxiliary sketch of the dependence of w, the excess Joule heat density, on the propagation velocity v.

Normal Zone Propagation

95

8

In strict point of fact, Eqs. (5.6. la and 5.6.1b) do not represent the situation we are trying to model as we can see in the following way. If we apply the final-value theorem (5.7.2) to Eq. (5.6.4) we see that t( ) = ai 2 rather than zero as the boundary condition (5.6.2b) requires. We presume that the temperature distribution far behind the front does not greatly affect the motion of the front. To improve the model for TNZs, we would need a third equation like Eq. (5.6.1 b) that would apply for x > X, where X is the value of x at which τ becomes 1 again. It has probably not escaped the reader’s notice that the model as it now stands is correct for ai 2 > 2. For, since ai 2 = 2 is the condition for cold-end recovery based on Eq. (4.10.1), there would be no recovery far behind the front. But then the argument that w has a minimum as a function of v no longer applies since the numerator of Eq. (5.6.6) is not always positive. When ai 2 > 2, w approaches as v → but approaches – as v → 0. Of course, negative values of w are unphysical. When w = 0, it follows from Eq. (5.6.5) that 8

8

8

v = (ai 2 – 2)/(ai 2 – 1)1/2

(5.7.3)

This is the same result as that of Eq. (5.2.6) except that the transition temperature occurs at t = 1 instead of t = 1 – i/2, so that now C = ai 2 – 1. When w > 0, the value of v one calculates from Eq. (5.6.6) is always greater than that given by Eq. (5.7.3). It is not possible to go further with general arguments, so next we turn to the calculation of w and U(t), following the method of Dresner, 1991, “Excess heat”.

5.8. THE EXCESS JOULE HEAT DUE TO CURRENT REDlSTRlBUTION The equations that govern the diffusion of current in a conducting medium are the two Maxwell equations D x B = µ0 J (5.8.1) D

x E+

=0

(5.8.2)

Ohm’s law

E= r J

(5.8.3)

and the equation of conservation of current D ·J=0

(5.8.4)

The resistivity r of the matrix is a constant independent of J. From these equations it follows at once that

CHAPTER 5

96

(5.8.5) Let the axes be so chosen that the z-axis points along the conductor in the direction of flow of the transport current I. Let W denote the cross sectional area of the conductor in the (x,y )-plane and let W o denote the area of W in which the transport current is initially confined. At t = 0, imagine this confinement to be abrogated. The transport current then diffuses transversely, tending toward a state of uniform current density. During this redistribution, the current density J (now the z-component of J) obeys the diffusion equation (5.8.6) We wish to solve Eq. (5.8.6) under the boundary and initial conditions J(r,0) = I/Ω o in W o , 0 elsewhere in W

(5.8.7)

J(r ,8) = I /W everywhere in W

(5.8.8) (5.8.9)

where dw = dx dy and r is the two-dimensional radius vector (x,y ). It proves useful to expand J in terms of certain eigenfunctions fk of the Helmholtz equation in W, namely, those defined by the equations D2 fk +a k2 f k = 0 in W D n · fk = 0

on S, the boundary of W

(5.8.10) (5.8.11)

where n is the outward normal to S. Thus we set (5.8.12) which satisfies Eq. (5.8.6). Equation (5.8.12) also satisfies condition (5.8.8). It follows by well-known arguments (Courant and Hilbert, 1953) that (5.8.13)

(5.8.14)

Normal Zone Propagation

97

With the help of Eq. (5.8.13) we can see that Eq. (5.8.12) satisfies condition (5.8.9). We choose the coefficients to satisfy condition (5.8.7), which can be written 8

(l /W o )g(W o) = l /W +

SA f

k k

(5.8.15)

k=1

whereg(W o) is the characteristic function of W o, that is, the function that is 1 inside W o and 0 outside. Using Eqs. (5.8.13) and (5.8.14) we find Ak =

(5.8.16)

The Joule power expended per unit length of conductor is (5.8.17) W

k=1

8

The first term on the right in Eq. (5.8.17) is the Joule power produced when the current density is uniform throughout W. The second term is the excess Joule power. When integrated over t from 0 to , the second term gives the excess Joule heat per unit length of conductor. Dividing the result by W gives w, the excess Joule heat density:

k =1 (5.8.18)

k=1 The summation on the right-hand side of the last equation is a geometric factor, which has been calculated by the author (1991, “Excess heat”) for a variety of situations. We consider here only the case in which W is a circle of radius R and Wo is an infinitely thin annulus at its circumference. 5.9.

THE SPECIAL CASE OF A CYLINDRICAL CONDUCTOR

In a cylinder of radius R, for current distributions that depend only on the radius (azimuthal symmetry) fk =Jo (gkr/R )

(5.9.1)

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CHAPTER 5

ak = gk /R

(5.9.2)

where Jo is the Bessel function of the first kind of order zero and gk is the kth root of J1, the Bessel function of the first kind of order one. Then [fk ,fk ]= pR2J 02 (gk )

(5.9.3)

If W o is the annulus R1 < r < R2, then [fk ,g]= (2pR2/gk )[(R2/R)J 1(gk R2/R) – (R1/R)J 1(gk R1/R)] If we set and R 2 = R and R 1 = R( 1 – e), we find,2 in the limit as e

(5.9.4)

0,

8

w = (m 0 J 2/2p 2R2)

S (1/g k2) = mo I 2/16p2R2

(5.9.5)

k =1

The exact value of the sum in the middle term of Eq. (5.9.5) is 1/8, as can be shown by a method of Euler’s (Dresner, 1991, “Excess heat”).

5.10. COMPARISON WITH EXPERIMENT OF PFOTENHAUER ET AL. According to Eq. (5.8.17), the function U(t) is a sum of decaying exponentials with relaxation times t k = mo /2rak2 = m o R 2/2rg k2 . If we ignore all but the lowest mode, then U(t) = exp(–t/t1)/t1 where t1 = mo R 2/2rg 21 and g1 = 3.83171. In the special units of Section 4.7, t1 = mo R2hP /2rg 12SA . When U(t) has this value, u(p) = (1 + pt1)–1 and Eq. (5.6.6) becomes (still in special units) w = [l+(v 2+4)1/2– ai 2][1/v l+ +t1]

(5.10.1)

Let us use this formula to analyze the experiment reported by Pfotenhauer et al. (1991). Their conductor consisted of a 2.54-cm-diameter rod of high-purity aluminum in the circumference of which eight 2.8-mm-diameter NbTi/Cu conductors were embedded. Their experiment was carried out at Tb = 2.5 K in a self-field of about 1.25 T (1 ~ 50–60 kA). At that field, the critical temperature Tc of NbTi is about 8.5 K, so that Tc – Tb = 6.0 K. Between the temperatures Tc and Tb , the specific heat of aluminum varies between 0.1 and 0.8 J kg-1 K-1, which is a substantial variation. We use the mean value of 0.45 J kg-1 K-1; multiplying by the density of aluminum (2700 kg/m3), we then find S = 1200 J m-3 K-1 and S(Tc – Tb) = 7200 J m-3. In their article, Pfotenhauer et al. do not specify the residual resistivity of their aluminum but merely state that it is high-purity aluminum. We guess a residual resistivity ratio rRT/r of 500 and find (since rRT = 2.4 mW-cm)that r = 4.80 x 10-11 W- m. The Wiedemann–Franz law then gives k = 2810 W m-1 K-1 at the average

Normal Zone Propagation

99

8

temperature (Tc + Tb )/2 = 5.5 K. The normal-state heat flux with a uniformly distributed, 50-kA current, Qn = rI2/AP , is then 2970 W/m2. The relaxation time t1 = 0.144 s. According to Eq. (5.9.5), when I = 50 kA, the excess Joule heat density w = 1.23 x 105 J/m3. Then the average excess Joule heat flux during the time t1, wA/Pt1, = wR/2t1, is 5420 W/m2. Thus the total heat flux during the first 0.144 s is roughly 8390 W/m2. Reference to Steward’s diagram, Fig. 4.23, then shows that the heat transfer coefficient should be that of film boiling. Accordingly, we assume h = 1000 W m-2 K-1. The special unit of time, SA/hP, is 7.62 x 10-3 s; the special unit of length, (kA/hP)1/2, is 0.134 m; the special unit of velocity (khP/A)1/2/S is then 17.5 m/s; and the special unit of energy density, S(Tc – T b), is 7200 J/m3. Then, in special units, w = 17.1 (I=50kA)and t1 = 18.9.Finally,ai 2 = 1 when I = 71.1 kA(unconditionalstability). Eq. (5.10.1) can easily be evaluated with a programmable hand-held calculator. Remembering that both w and ai 2 scale as I2, we find after a little trial and error (improved between steps by interpolation) that when I = 67.3 kA, the value of w is equal to the minimum value obtained from Eq. (5.10.1). Thus the threshold for the creation of TNZs is 67.3 kA. At this current, ai 2 = 0.896 and the velocity v in special units is 0.217 or in ordinary units 3.8 m/s. At I = 70 kA, ai 2 = 0.970 and v = 8.3 m/s, and at I = 75 kA, ai 2 = 1.11 and v = 12.9 m/s. Comparison with the experiment of Pfotenhauer et al. shows only very rough agreement. Their measured threshold was about 55 kA; slightly beyond that their propagation velocity reached values around 20 m/s. The small value of v at the threshold that we have calculated is not so worrisome in view of our expectation that dv/dw = , especially since the experimental points v, plotted versus I, show a downward concavity just beyond the threshold. But well away from the threshold, our calculated velocities are still low by about a factor of 2. Perhaps the most important conclusion that can be gleaned from the calculations follows from the comparison of the threshold current for the appearance of TNZs (67.3 kA) with the current for unconditional stability (71.1 kA). There is for this conductor a loss of only 5% in threshold current, which seems a rather slight penalty to pay to avoid the cost of distributing the superconductor uniformly throughout the matrix. Notes to Chapter 5 It follows from the discussion of Fig. 4.8 in the third paragraph of Section4.6, that the stability margin of a small, pool-cooled conductor that is not cryostable is the volumetric enthalpy of the conductor between Tb and T1. 2 The result, Eq. (5.9.5), can teach us a valuable lesson of an unexpected sort: The result w ~ mo I2/R2 can easily be obtained from dimensional analysis, the only relevant variables being w itself, mo , I, R, and the matrix resistivity r. It is a common (though in fact indefensible) practice to assume the dimensionless coefficient in such dimensional formulas to be close to 1. In the case at hand, that constant, 1/16p2, equals 6.333 x 10–3, so that assuming it to be close to 1 makes an error of more than two orders of magnitude. 1

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6 Uncooled Conductors

6.1. THE BIFURCATION ENERGY Strictly speaking, the title of this chapter is a misnomer because all superconductors must be cooled in order to function. But in certain kinds of magnets, most prominently those potted in epoxy, the rate of heat transfer from the conductor to the coolant is slow enough that the issue of quench or recovery following a thermal perturbation is decided as though there were no cooling at all. The mechanism of recovery is longitudinal conduction of heat along the matrix away from the normal zone. In common parlance such magnets are called uncooled. With the advent of the high-temperature superconductors, particularly the BSCCO conductors, it became possible to operate magnets in the 20–40 K range. Such magnets can be cooled with helium gas. It may happen then that the heat capacity of the gas is small compared with that of the conductor and the magnets may be treated as uncooled in determining their stability. Conductively cooled magnets of the kind recently proposed by Laskaris (1990; Herd et al., 1993) can also be treated as uncooled in determining stability. As in Chapter 4, we shall focus on the quench energy as the measure of stability. But now, when there is no cooling, there is no minimum propagating zone like that pictured in Fig. 4.11. For when q = 0, it is impossible to fulfill Eq. (4.6.2) since kQ > 0. Thus although some initial conditions (corresponding to large heat depositions) lead to quenches and others (corresponding to small heat depositions) lead to recovery, there is no steady-state solution that separates them. We must therefore turn to the time-dependent heat balance Eq. (4.4.1) without the term qP/A, namely: + QP/A

(6.1.1)

to decide whether a given initial condition leads to recovery or to a quench. The initial condition of greatest interest to us is that corresponding to an instantaneous pulse of energy E per unit area of the plane z = 0 at time t = 0. (The 101

102

CHAPTER 6

pulse energy must be expressed as an energy per unit area because Eq. (6.1.1) is a one-dimensional heat balance equation. In two dimensions, the pulse energy would be expressed as an energy per unit length, and only in three dimensions would it be expressed simply as an energy.) If the pulse is large, we expect a quench; if it is small, we expect recovery. What interests is the value of the pulse energy that divides the two regimes. Again we take QP/A to be given by Eqs. (4.6.3) and (4.6.4); again we let t = T – Tb be the temperature rise; again we assume k and S are independent of temperature; and again we introduce special units in which k = S = t c = 1. Now since h does not appear in the partial differential Eq. (6.1.1), we can complete our system of special units by taking QnP/A = ρcuJ 2/f = 1. The dimensions of these quantities are PL -1 Q-1 , PTL-3 Q -1 , Q and PL-3, where, as before, P is power, L is length, T is time, and is Q temperature. Then Eq. (6.1.1) becomes (6.1.2) where g(t) is given by Eq. (4.7.2). When t is small, the last term in Eq. (6.1.2) can be neglected and then t = E(4πt)–1/2exp(–z 2/4t)

(6.1.3)

We take this form for the initial condition at very short times. The evolution of the initial condition is controlled by Eq. (6.1.2) and its ultimate fate (quench or recovery) thus depends on the two parameters E and i (which enters through g(t)). Thus the bifurcation value of E, Ebif = G(i), where G is a dimensionless function yet to be determined. In ordinary units this equation becomes 2 -1/2 G(i ) Eb i f = Sk 1/2 t 3/2 c (rcu J /f )

(6.1.4)

since E has the dimensions of an energy per unit area (PTL-2).

6.2. GROUP ANALYSIS OF THE BIFURCATION ENERGY As shown in Dresner (1985), we can go a step further in determining the function G(i) if we assume that when E = Ebif, the temperature rise t of the normal zone drops below t = 1 before much Joule heat is produced. This assumption replaces Eq. (4.7.2) for g(τ) by = (t + i – 1)/ i

1–i
g(t)

(6.2.1) = 0 0
Uncooled Conductors

103

This functional dependence of g on t is a member of the family of dependences = bt – a a /b < t g(t)

(6.2.2) 0 < t < a/b

=0

When g(t) is given by Eq. (6.2.2), Eqs. (6.1.2) and (6.1.3) are invariant to the group of transformations z´ = lz t´=l2t 8

, – <w< 8

0
8

t´ = lwt

(6.2.3)

b´ =l-2b a´ = lw–2 a E´= lw+1E What this means is that if we use Eqs. (6.2.3) to replace z, t, t, b, a, and E in Eqs. (6.1.2) and (6.1.3) by their primed counterparts, we obtain Eqs. (6.1.2) and (6.1.3) again. Thus if we have a solution τ(z, t, b, a, E) of Eq. (6.1.2) with the initial condition (6.1.3), then τ´(z´, r´, b´, a´, E´) is also a solution corresponding to the value E´ of the pulse energy. Solutions that quench clearly transform into solutions that quench and solutions that recover transform into solutions that recover. Now the bifurcation value of E can only depend on the values of a and b: Ebif = G(a,b). Since this relation must be valid for any consistent values of E, a, and b, it must be invariant to the transformations (6.2.3). Thus lw+1Ebif =G(lw–2 a,l-2b)

(6.2.4)

The only possible form for G that satisfies this functional equation is G = Cab –3/2

(6.2.5)

where C is an as yet undetermined constant (Cohen, 1931; Dresner, 1983). When a = (1 – i)/i and b = l/i, so that Eqs. (6.2.2) and (6.2.1) are the same,

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CHAPTER 6

G = C(1 – i)i1/2

(6.2.6)

Then Eq. (6.1.4) becomes 2 -1/2(1 – i)i 1/ 2 Ebif =CSk 1/2 t 3/2 c (rcu J /f )

(6.2.7)

in ordinary units. As already remarked in Note 2 of Chapter 5, it is dangerous to assume that a constant like C is close to 1. A more detailed but still approximate derivation of Eq. (6.2.7), given below in Section 6.3, yields the value C = p.

6.3. ESTIMATION OF THE UNDETERMINED CONSTANT Suppose we try an approximate solution to Eqs. (6.1.2), (6.1.3), and (6.2.1) of the form t(z,t )=H(t) F(z,t ); F(z,t )=(4pt )-1/2 exp(–z 2/4t)

(6.3.1)

where H(t) is an as yet to be determined function for which H(0) = E. If we substitute this trial solution into Eq. (6.1.2), we find = dH/dt) H´(t) F(z,t ) = g[H(t) F(z,t )] (H´ _

(6.3.2)

There is no solution to Eq. (6.3.2), which is the same thing as saying that Eq. (6.3.1) is not an exact solution of Eq. (6.1.2). But we expect Eq. (6.3.1) to be a good approximation, so we determine H(t) by requiring Eq. (6.3.2) to be satisfied at z = 0 (method of collocation): H´(t)(4pt )-1/2 =g[H(t)(4pt )-1/2]

(6.3.3)

Since we are only interested in times for which t > 1 – i, we take g(t) = (τ+ i – 1)/i in Eq. (6.3.3). Then we can solve for H(t) and find t(0,t )=(4pt -1/2 H(t ) =(4pt )-1/2et /i [E– ((1 –i)/i)(4p)1/2

t

(6.3.4) 0

If 8

– i)i1/2

E = ((1 0

then

(6.3.5)

Uncooled Conductors

105

(6.3.6)

8

If E is larger than the value given in Eq. (6.3.5), t(0,t ) diverges exponentially toward with time. If E is smaller than the value given in Eq. (6.3.5), t(0,t ) diverges exponentially toward – ) with time. But because we have taken g(t) = (τ + i – 1)/i, the expression (6.3.4) fails to apply once t falls below 1 – i. The value (6.3.5) of E is the bifurcation value; comparing Eq. (6.3.5) with Eq. (6.2.6), we see that C = p. This is, of course, an approximate value, but we expect it to be of the right order of magnitude. 8

6.4. SIZE OFTHE BIFURCATION ENERGY How big is the bifurcation energy? I estimate it here for the conductor of a small laboratory magnet I recently built using potted NbTi/Cu strands (diameter, 0.81 mm; f = 0.6). The magnet was designed to produce a field of 7 T at 4.2 K, which it did when the strand current density J was 16.5 kA/cm2 (corresponding to i = 0.23). The critical temperature of NbTi at 7 T is 6.0 K so that tc = 1.8 K. The resistivity of the copper, including magnetoresistivity, is about 5 x 10-10 Wm. Using an average temperature of 5.1 K, we find from the Wiedemann–Franz law that kcu =250Wm-1 K-1 andk=f kcu = 150Wm-1 K-1. The average volumetric heat capacity S of the conductor in the range 4.2–6.0 K is 3400 J m-3 K-1. Then we find from Eq. (6.2.7) that the bifurcation energy is 24.5 J/m2. Since the cross-sectional area of the strands is 5.15 x 10-7 m2, the total bifurcation energy EA is 12.6 mJ . The magnet in question trained twice (at 6.3 and 6.4 T) before reaching design field. These training quenches were presumably caused by local cracking of the epoxy potting at points of high stress (e.g., at unpurged gas bubbles in the epoxy). When the epoxy cracks and initiates a quench, it releases an amount of strain energy exceeding the bifurcation energy. The fact that there were only two training quenches suggests that when the epoxy cracks it does not always release an amount of energy exceeding the bifurcation energy. There is some evidence of another sort, coming from acoustic emission studies, that this is so. When epoxy cracks, it makes noise that can be detected. In these studies the noise was sometimes accompanied by a voltage across the magnet terminals, signifying the creation of a normal zone, and sometimes unaccompanied by voltage. The tentative conclusion is that epoxy cracking liberates energies of the order of a few to a few tens of mJ. Since this energy release is independent of temperature, training should not occur in potted magnets wound with high-temperature superconductors because for the latter the bifurcation energy is much larger than for the low-temperature superconductors (as we shall see later).

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6.5. THE EFFECT OF TRANSVERSE HEAT CONDUCTION ON THE BIFURCATION ENERGY So far we have treated the epoxy as though it were a perfect insulator of heat. In strict point of fact, this is not so, and transverse heat conduction through the epoxy to adjacent turns may need to be considered. To judge whether or not transverse conduction is important, we need to estimate how far heat can diffuse through the epoxy during the time in which the issue of quench or recovery is decided. The unit of time in our special system of units is Stc /(rcu J 2/f ), which equals 0.27 ms in the example of the last section. According to Reed and Clark (1983), for epoxies k ~0.1 W m-1 K-1 and Cp ~1 J kg-1 K-1. If we take the density of epoxy to be around 1000 kg/m3, we find the thermal diffusivity k/S of epoxy to be ~10-4 m2/s. In a unit of time, then, heat can penetrate about 0.16 mm into the epoxy. If the separation of adjacent turns is greater than 0.16 mm, transverse conduction of heat from one strand to its neighbor may be neglected. In the example of Section 6.4 the strands are close packed. The epoxy-filled interstices have a volume fraction of 1 – 0.0931 and the average thickness of epoxy between turns is only about 0.04 mm. In this case, it seems likely that transverse conduction cannot be neglected. When similar estimates show that transverse heat conduction cannot be neglected, Eq. (6.1.1) may be replaced by the following three-dimensional heat balance equation for a homogeneous, anisotropic medium: (6.5.1) where kt is the thermal conductivity in the transverse direction. If we introduce and y´ = yk 1/2/k 1/2 the new transverse coordinates x' = xk 1/2/k 1/2 t , Eq. (6.5.1) t becomes (6.5.2) where r= (x´ 2 + y´ 2 + z 2)1/2 is the radial coordinate in (x´,y´,z) space. In special units, this becomes (6.5.3) The initial condition we need to fulfill is (6.5.4) which is the instantaneous point-source solution to Eq. (6.5.3) when the last term is neglected. Note now that E1 has the units of energy. When g(t) is given by Eq. (6.2.2), Eqs. (6.5.3) and (6.5.4) are invariant to the transformation

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107

r´= λr t´ = λ2t <w<

8

8

0
t = l wt

(6.5.5)

b´ = l -2b a´= lw–2 a E1´ = lw+3E1 Then G = Cab-5/2 so that when a = (1 – i)/i and b = l/i, G = C( 1 – i)i3/2 . We have to be careful at this juncture about the meaning of E1. According to Eq. (6.5.4) tdx dy dz = (k/kt )E

(6.5.6)

where E is the pulse energy we are seeking. In ordinary units, then,

E = CSk 1/2k t τc5/2(rcu J 2/f )-3/2(1 – i)i 3/ 2

(6.5.7)

A calculation exactly like that described in Section 6.3 yields the result C = 6p2 (Dresner, 1985).

6.6. BIFURCATION ENERGIES OF HIGH-TEMPERATURE SUPERCONDUCTORS The work of the foregoing sections has been based on the assumption that the quantities k, rcu , and S could be taken independent of temperature. This assumption is satisfactory for the old, low-temperature superconductors for which the bath and critical temperatures are close together. But it is not satisfactory for the new, high-temperature superconductors. A Ag/BSCCO composite operated at 30 K in fields of a few tesla, for example, may have a critical temperature of 70 or 80 K. Over the range 30–80 K, the physical properties k, rcu , and S vary strongly with temperature, and we cannot assume their constancy. But we can salvage the group-invariance method of Section 6.2 if we fit the temperature variations of the physical properties with power laws. Accordingly, we take S=Sc (T/Tc )m

(6.6.1)

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r Ag = r Age (T/T c )n

(6.6.2)

Furthermore, we assume that the thermal conductivity of the silver and its resistivity obey the Wiedemann–Franz law. (It is worth remembering here the results of Chapter 2, according to which 0 < m < 3 and 1 < n < 5.) These assumed temperature variations make Eq. (6.1.1) nonlinear, and an immediate consequence of this is that we cannot usefully introduce the temperature rise t = T – Tb as a new dependent variable. To attain a solvable problem, we therefore assume that the bath temperature Tb is zero. This is a drastic assumption, but as we shall see later, it probably does not vitiate the utility of the results. If we now introduce special units in which Sc = kc = T c = r Age J 2/f = 1 (dimensions PTL-3Θ-1, PL-1Θ-1, Q, and PL-3, respectively), then Eq. (6.1.1) becomes (6.6.3) where g(T) is given (literally) by Eq. (4.7.2). The nonlinear Eq. (6.6.3) can be dealt with in the same group-theoretic way as we dealt with the linear Eq. (6.1.2) in Section (6.2), but the calculations are more complicated (Dresner, 1994, “Quench energies”). We quote only the results here. The bifurcation energy per unit cross-sectional area Ebif , defined by (6.6.4)

is given by -1 -1/2 (1– i)m-n+3 / 2 Ebif = CScT c2 L1/2 o f (ρAge Jc ) i

(6.6.5)

where Lo is the Lorenz constant and C is a dimensionless constant that depends on m and n. Detailed calculations (Dresner, 1994, “Quench energies”) show that Eq. (6.6.5) is only valid if m + 3/2 > n so that the exponent of 1 – i in Eq. (6.6.5) is always positive. An approximate calculation similar to that in Section 6.3 shows that when m = n, C is close to 4.5. Let us use this formula to estimate the quench energy of a small experimental coil described by Schwenterly et al. (1993). The conductor was a 2.5 mm x 0.2 mm multifilamentary Ag/BSCCO tape with a silver-to-BSCCO volume ratio of 6. At 5.5 T, the extrapolated critical temperature was ~65 K; at Tb = 20 K, the critical current was ~2.5 A. Figure 6.1 shows a plot of the parameter cluster Sc T 2c / r c , as a function of temperature for silver for residual resistance ratios of 100, 500, and 2500. There is

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109

Figure 6.1. A plot of the parameter cluster ST2/ρ as a function of temperature for silver for residual resistance ratios of 100, 500, and 2500. (Redrawn from an original appearing in Dresner (1993, “Stability”) with permission of Butterworth-Heinemann, Oxford, England.)

a knee at about 20 K, beyond which the parameter cluster depends only slightly on temperature and almost not at all on the residual resistance ratio. At 65 K, we use the value 3 x 1018 A2 m-4 s K. Since the critical current density in the tape is 5 x 106 Am-2, the first factor in Eq. (6.6.5) is 362 J/mm2. When multiplied by the area of the tape, 0.5 mm2, this gives 181 J, which is an enormous perturbation energy. The value just calculated needs to be multiplied by the second factor in Eq. (6.6.5). Reasonable values for m and n are m = 2 and n = 3 (see Table 6.1). When i = 0.9, the second factor in Eq. (6.6.5) is 1/3, and the quench energy is roughly 60 J, which is still very large.

Table 6.1. Approximate Values of m and n over Various Temperature Ranges for Ag/BSCCO Conductors Temperature(K) 10–20 20–65 65–130 130–200 200

m 3 2 0.67 0.33 0

n < 5, depends on RRR 3 1.5 1.2 1

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The rather low current density of ~0.5 kA/cm2 of this coil is responsible for its very high stability. Recently, Moore (1992) argued that critical current densities in the range 100–1000 kA/cm2 are necessary for practical application to transmission lines, motors and generators, and magnetic energy storage. Such high current densities would reduce the quench energy by about three orders of magnitude to, say, tens of mJ. Such energies are still roughly three orders of magnitude greater than the energy released by epoxy cracking that we estimated in Sections 6.4 and 6.5. So as mentioned there, potted magnets wound with Ag/BSCCO superconductor should be immune to training. The preceding work depends on three assumptions, namely, that the temperature variations of S and rAg could be described by power laws, that the thermal conductivity and resistivity of silver obeyed the Wiedemann–Franz law, and that we could take Tb = 0. The first two assumptions are relatively benign, but the last requires some discussion. It had to be made because without it the group-invariance method and the other detailed calculations of Dresner (1994, “Quench energies”) would not have been possible. It can be justified because of the rapid drop of the specific heat at low temperatures. Perhaps the simplest way of seeing this is to employ an argument given in the same reference. Consider the extreme limiting case in which the volumetric heat capacity S falls discontinuously to zero at some finite temperature Tb. Then, as soon as any heat is introduced to the medium, its temperature jumps to Tb. Thus a nascent normal zone behaves as though it were growing in a medium whose initial temperature were Tb , not zero. If we invert this argument, we can say that if the ambient temperature were actually Tb , the normal zone would grow as though the ambient temperature were zero. For these reasons, we expect that if the volumetric heat capacity S falls rapidly with temperature, the quench energy can be adequately approximated by assuming Tb = 0.

6.7. PROPAGATION VELOCITIES OF UNCOOLED SUPERCONDUCTORS The absence of cooling actually complicated the calculation of the quench energy by forcing us to consider a time-dependent problem where formerly we had only to consider a time-independent one. The calculation of the normal zone propagation velocity is similarly complicated because in the absence of cooling the central temperature no longer approaches a constant value (T 2 of Fig. 4.8; cf. the first paragraph of Section 5.1). When k, rcu , and S can be taken as independent of temperature, we already know the propagation velocity—it is given by Eqs. (5.1.3) and (5.1.4). As noted above, this assumption of temperature independence is satisfactory for the classical low-temperature superconductors, but not for the newer high-temperature superconductors. To calculate the propagation velocity, we again need to look for traveling wave solutions T(z + vt) of the heat balance equation, now Eq. (6.1.1). As

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111

8

in Section 4.4, T(– ) = Tb (we no longer need to take Tb = 0 as we did in the determination of the quench energy); but T( ) can no longer be set equal to T 2 (indeed, there is no T2). The determination of the temperature profile far behind the propagating front is, as noted above, a complication that does not exist when cooling is present. To learn how to deal with this complication (Dresner, 1994, “On the connection”), let us first consider the case in which k, rcu , and S can be taken as independent of temperature, even though we already know the answer to this case. In addition, in order not to obscure the principle involved, we simplify the computations by ignoring current sharing; thus we take for QP/A the two-part curve 8

=r cu J 2/f, Tc < T QP/A

(6.7.1) = 0,

T b < T < Tc

Now let us introduce special units in which k = S = Tb = r cu J 2/f = 1 (dimensions: PL -1 Q-1,PTL-3Q-1,Q,PL-3).Eq. (6.1.1)becomes + g(T)

(6.7.2)

where = 1,

Tc < T

= 0,

1 < T < Tc

g(T) (6.7.3)

For a traveling-wave solution that propagates from right to left, T(z + vt), Eq. (6.7.2) becomes d2T/dz2 – v(dT/dx) + g( T) = 0

(6.7.4)

where x = z + vt. We again introduce s = dT/dx as a new dependent variable in order to reduce Eq. (6.7.4) to a first-order differential equation: s(ds/dT) – vs + g(T) = 0

(6.7.5)

Now the advantage of a first-order differential equation is that it may be analyzed graphically by studying its direction field. (The direction field of a first-order differential equation dy/dx = F(x,y ) is the diagram one obtains if one draws at points (x,y ) short hatch marks having the slope dy/dx calculated from F(x,y ). The field of hatch marks then gives at a glance the course of the integral

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Figure 6.2. The first quadrant of the direction field of Eq. (6.7.5). (Redrawn from an original appearing in Dresner ( 1994, “On the Connection”) with permissionof Butterworth-Heinemann, Oxford, England.)

8

curves.) The easiest way to do this is to sketch the curves on which ds/dT = 0 or ds/dT = , for these curves divide the (T,s)-plane into regions in any one of which ds/dT has only one algebraic sign. Figure 6.2 shows the first quadrant of the direction field of Eq. (6.7.5) (N.B.: for a wave traveling from right to left, s = dT/dx > 0, and, of course, T > 0). In region II (T > Tc ), where g = 1, ds/dT = 0 on the horizontal line s = 1/v and ds/dT = along the T-axis for T > Tc. In region I, where g = 0, ds/dT= v. Since s(1) = 0 [T(– ) = Tb , therefore s = (dT/dx ) x = - = 0], the solution we want in region I is 8

8 8

s = v(T– 1)

(6.7.6)

In order to determine v uniquely, we must join the solution (6.7.6) at T = Tc to an integral curve in region II. To determine which integral curve, we consider how

Figure 6.3. Profiles of temperatureT versus z at various times after the initial normal zone is established that show the evolution of traveling waves. (Redrawn from an original appearing in Dresner (1994, “On the Connection”) with permission of Butterworth-Heineman, Oxford, England.)

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113

Figure 6.4. Profiles of s = dT/dx versus x that correspond to the profiles in Figure 6.3. (Redrawn from an original appearing in Dresner(1994, “On the Connection”) withpermissionof Butterworth-Heinemann, Oxford, England.)

the initial normal zone approaches a traveling wave solution as time progresses. Figure 6.3 shows profiles of T versus z for various times after the initial normal zone is established. Figure 6.4 shows the corresponding profiles of s versus T corresponding to the left halves of the profiles in Fig. 6.3. These profiles resemble the integral curves in the (s,T)-plane that lie below the horizontal lines = 1/v in Fig. 6.2. As the maximum temperature of the normal zone becomes larger and larger, the value of s at T = Tc approaches 1/v more and more closely. Thus we must match Eq. (6.7.6) to this value at T = Tc . Thus v = (Tc – 1)–1/2

(6.7.7)

in special units or v =(J/S)(k r cu /f)1/2(Tc – Tb )–1/2

(6.7.8)

in ordinary units. This result agrees with Eq. (5.1.3) in the limit i 0 as it should; for in this limit, the three-part curve (4.6.4) for QP/A becomes the two-part curve (6.7.1). (N.B.: As i 0, ci1 / 2 i)

6.8. PROPAGATION WITH TEMPERATURE-DEPENDENT MATERIAL PROPERTIES Now let us considerwhat happens when k cu, r cu, and S all vary with temperature and kcu, and rcu ,obey theWiedemann–Franz law.Then atraveling wave solution T(z + vt) of Eq. (6.1.1) obeys the ordinary differential equation d/dx[k(dT/dx )] – vS (dT/dx ) + (rcuJ2/f)g(T) = 0

(6.8.1)

If we now take s = k(dT/dx ) and multiply Eq. (6.8.1) by k, we obtain the first-order differential equation

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Figure 6.5. A sketch of the direction field of Eq. (6.8.3). (Redrawn from an original appearing in Dresner (1994, “On the Connection”) with permission of Butterworth-Heineman, Oxford, England.)

s(ds/dT) – vSs + (krcu J2/f)g(T) = 0

(6.8.2)

Now since k = fkcu , k rcuJ 2/f = LoTJ2. If we introduce special units in which Sb = Tb = Lo J2 = 1 (dimensions: PTL-3 Q-1, Q and P2L-4 Q-2), Eq. (6.8.2) becomes s(dS/dT ) – v ss + Tg(T) = 0

(6.8.3)

where s(T) = S(T)/Sb . Finally, we ignore current sharing and take g(T) to be given by Eq. (6.7.3). Figure 6.5 shows a sketch of the direction field of Eq. (6.8.3). In region II, where g(T) = 1, the locus of zero slope is the curve s = T/v s

(6.8.4)

At low temperatures, where s ~ T 3, s falls as T-2; at high temperatures, where s approaches a constant, s rises as T. Hence, s has a minimum as shown in the diagram. The locus of infinite slope is, as before, the portion of the T-axis to the right of Tc. Fig. 6.5 shows a separatrix that lies between the subfamily of integral curves that intersect the locus of zero slope and the subfamily of integral curves that do not. As in Section 6.7, we expect that the s-T profile of the actual normal zone approaches the separatrix as time goes on. Now we do not know the equation of the separatrix, but we guess that it intersects the vertical line T = Tc near the intersection of that line with the locus of zero slope, i.e., at s = Tc /v s(T c). In region I, where g = 0, the solution of Eq. (6.8.3) for which s( 1) = 0 is (6.8.5)

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115

Since for this solution, s must take the value Tc /v s(Tc ) at T = Tc , it follows that (6.8.6) or, in ordinary units, (6.8.7) From the manner in which this last result has been derived we expect it to overestimate v. This result bears a striking resemblance to an earlier result of Whetstone and Roos (1965). These authors neglected the term k(d2T/dx 2) in Eq. (6.8.1) when T > Tc , whereas in the present work it is the entire term d/dx[k(dT/dx )] that is neglected when T > Tc.

6.9. THE EFFECT OF CURRENT SHARING ON THE PROPA GATION VELOClTY The neglect of current sharing always leads to an underestimate of the propagation velocity (Dresner, 1994, “On the connection”). This is easy enough to understand because current sharing adds additional Joule heating below Tc to the source term given in Eq. (6.7.3). I have calculated the velocity v as a function of (Tc – Tcs)/(Tc – Tb ) numerically for several values of Tc/Tb and several values of the exponent m in the assumed expression s = (T/Tb)m. Results are shown in Figs. 6.6, 6.7, and 6.8 for m = 1, 2, and 3, respectively. In each figure, five curves are plotted of v/v0, where v0 is the propagation velocity in the absence of current sharing, i.e., when Tcs = Tc. These curves correspond to values of the ratio Tc/Tb = 3, 2,1.5, 1.2, and 1.1, reading from top to bottom. Note that when Tc/Tb and m are large, and Tcs is near T b (e.g., the right-hand edge of Fig. 6.8), the correction to v due to current sharing can be quite large. The physical reason for this strong increase in v is that under the circumstances just mentioned, the volumetric heat capacity S is quite small at the point on the leading edge of the front at which Joule heating begins.

6.10. AN INTERESTING COUNTEREXAMPLE It has become deeply ingrained in the minds of many workers in the field of applied superconductivity that traveling-wave solutions always exist, so it may come as a surprise to some to learn that such solutions do not always exist. The following counterexample is given in (Dresner, 1994, “On the connection”).

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Figure 6.6. The ratio v/v0 of the propagation velocity with current sharing (v) to the propagation velocity without current sharing (v0) for m = 1 and severalvalues of Tc /Tb.

Figure 6.7. The ratio v/v0 of the propagation velocity with current sharing (v) to the propagation velocity without current sharing (v0 ) for m = 2 and several values of Tc /Tb .

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117

Figure 6.8. The ratio v/v0 of the propagation velocity with current sharing (v) to the propagation velocity without current sharing (v0 ) for m = 3 and several values of Tc /Tb.

Suppose (1) S is a constant independent of temperature, (2) kcu and rcu obey the Wiedemann–Franz law, and (3) current sharing is neglected (g(T) given by Eq. (6.7.3)). Then s = 1 in Eq. (6.8.3), the direction field of which is given in Fig. 6.9. The dashed straight lines represent the two special solutions s = m +_ T of Eq. (6.8.3) when T > Tc and g(T) = 1. Here the slopes m+_ are the roots of the quadratic equation

Figure 6.9. The direction field of Eq. (6.8.3) when S is independent of temperature, i.e., when s(T)= 1. (Redrawn from an original appearing in Dresner (1994, “On the Connection”) with permission of Butterworth-Heinemann, Oxford, England.)

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m2 – vm + 1 = 0

(6.10.1)

These roots are real only when v ≥ 2. It is easy to prove that m+ ≥ m_ > 1/v, so Fig. 6.9 is correctly drawn when v ≥ 2. When T < Tc , s = v(T – 1). Since s is continuous at T = Tc, we must have v(Tc – 1) = m_Tc = [v – (v 2 – 4)1/2]Tc /2

(6.10.2)

which can be rewritten 2(Tc – 1)/Tc = 1 – (v 2 – 4)1/2/v

(6.10.3)

Figure 6.10 shows sketches of the two sides of Eq. (6.10.3). We see at once that when v ≥ 2, the right-hand side is ≥ 1. But then from the other sketch we see that 1 ≥ Tc ≥ 2. If Tc > 2, and we assume v ≥ 2, we are led by the reasoning just given to the conclusion that Tc ≥ 2, a contradiction. Hence if Tc > 2, the only remaining possibility is that v < 2. But then m+ and m_ are not real, and the direction field of Eq. (6.8.3) looks like Fig. 6.11. This direction field fails to supply a condition like Eq. (6.10.3), and so when Tc > 2, no traveling wave is determined.

Figure 6.10. Plots of the two sides of Eq. (6.10.3). (Redrawn from an original appearing in Dresner (1994, “On the Connection”) with permission of Butterworth-Heineman, Oxford, England.)

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119

Figure 6.11. The direction field of Eq. (6.8.3) when S is independent of temperature,i.e., when s(T) = 1, and Tc > 2. (Redrawn from an original appearing in Dresner (1994, “On the Connection”) with permission of Butterworth-Heinemann,Oxford, England.)

6.11. THE APPROACH TO A TRAVELING WAVE It has also become deeply ingrained in the minds of many workers in the field of applied superconductivity that if a traveling-wave solution exists, it represents the asymptotic limit to which nonrecovering normal zones tend. This turns out not to be generally true, the situation being considerably more complicated. To deal with it, we use the following comparison theorem (which can be proved using the methods described in Sections B.1. and B.2.): If T1(z,t) and T 2(z,t) are two solutions of Eq. (6.1.1) that obey the same boundary and initial conditions (T 1(a,t) = T 2(a,t), T 1(b,t) = T 2(b,t), and T 1(z,0) = T 2(z,0), a ≥ z ≥ b) but belong to two different source terms Q1(T) ≥ Q2(T),then T 1(z,t) ≥ T 2(z,t) for all t > 0 and a ≥ z ≥ b. Now let us continue the example of Section 6.10 but add the further assumption that k cu is a constant independent of temperature and that rcu = LoT/kcu . The Wiedemann–Franz law is thus still obeyed, and the traveling-wave solutions are still those of Section 6.10. Now using the special units of Section 6.8, the time-dependent heat balance Eq. (6.1.1) becomes (6.11.1) If we add to our list of special units k = 1, then k cu = 1/f, and r cuJ 2/f = LoTJ2, which in special units equals T. Thus Eq. (6.11.1) becomes (6.11.2) If we take Q1(T) = T and Q2(T) = Tg(T), then Q1(T) of the time-dependent equation

Q2(T). Therefore, solutions

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(6.11.3) give an upper bound to the solutions of Eq. (6.11.2) that obey the same boundary and initial conditions. Equation (6.11.3) is linear and has the infinite-medium point-source kernel (4pt)–1/2exp(–z 2/4t + t)

(6.11.4)

If the initial value of the temperature rise T – 1 (call it DT(z,0)) is > 0 only on a finite interval, say 0 ≥ z ≥ c, then for z < 0, (6.11.5a)

(6.11.5b) The right-hand side of Eq. (6.11.5b) is then an upper bound to T2 – 1, where T2 is the solution of Eq. (6.11.2). The locus of a constant value K of the right-hand side of Eq. (6.11.5b) is given by (6.11.6) As t grows large, this locus asymptotically becomes the locus z 2 = 4t2 or z = –2t (remember, we are considering the leftward propagating edge of the normal zone). Let us switch to a frame of reference traveling to the left with a velocity –2 by introducing the new variable z´ = z + 2t. Then in the z´-frame, (6.11.7) for z´ < 2t (i.e, z < 0). Shown in Fig. 6.12 is a sketch of the right-hand side of Eq. (6.11.7) in the z´-frame. For l z´l << 4t, the profile is close to ez´ , whereas for z´ < –4t, the profile drops more rapidly than a simple exponential. The arrows show the direction of migration of the profile and the various boundaries as time advances. Shown also as a dashed curve is the profile of the traveling-wave solution of Section 6.10 corresponding to a velocity v > 2. A contradiction is evident in this diagram. Since the traveling-wave solution is given by s = v(T – 1) when T < Tc ,

121

Uncooled Conductors

Figure 6.12. A sketch of the right-hand side of Eq. (6.11.7).

T– 1 ~ exp[v(z + vt)] = exp[vz´ + (v – 2)t]

(6.11.8)

8

8

Thus at any fixed instant of time, the traveling-wave solution falls exponentially as z´ – , whereas T1 falls as a Gaussian function as z´ – . Thus T1 cannot be larger than T everywhere. Therefore, an initial condition that is confined to a finite interval cannot asymptotically mature into a traveling wave. If this contradiction were all the trouble created by the analysis we have just completed, it would not be so serious because the departure of the true solution from the traveling-wave solution only occurs in the remote wings of the propagating front where the temperature rise is very small. Furthermore, this situation gets better as time goes on. But unfortunately there is more. The curve marked “RHS (6.11.7)” moves down but not laterally as time increases, whereas the dashed traveling wave moves to the left. Clearly the traveling wave must “collide” with the vertically descending curve and thus must gradually alter. The consequent change propagates in from the ends. The traveling-wave solution is then, at best, a kind of intermediate asymptotic state, which, though it may be established fairly quickly, cannot endure forever. In spite of these somewhat arcane objections of principle, propagation experiments are usually analyzed by means of the traveling-wave theory.

6.12. THE EFFECT OF HEAT TRANSFER TO THE POTTING ON THE PROPAGATION VELOCITY The assumption that a potted conductor is uncooled is extreme,1 and in point of fact, some of the heat capacity of the epoxy potting may be available to cool the superconductor and thus slow the expansion of the normal zone (Dresner, 1980, “Propagation”). If we augment the special units of Section 6.8 with the additional requirement that kb = 1, the special unit of time is kb Sb /Lo J 2. The thermal diffusivity of the epoxy is kep /Sep ~ 10–4 m2/s (cf. Section 6.5). Thus heat penetrates a distance

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d ~ [(kep /Sep)(kb Sb /Lo J2)1/2 during the passage of the propagating front. Therefore the effective volumetric heat capacity for a potted wire of diameter D is (6.12.1) The reciprocal of the bracketed term is the correction to the propagation velocity. If we evaluate this bracket for the example conductor described in Section 6.4 using the epoxy properties given in Section 6.5, we find the bracket equals 1.40, so that the propagation velocity should be reduced by a factor of 0.713.

6.13. THE ADIABATIC HOT-SPOT FORMULA In potted magnets wound with low-temperature superconductors (typically NbTi) the propagation velocity is often large enough (1–10 m/s) to drive the whole magnet normal before very much of the energy stored in the magnet can be converted to Joule heat. When this happens, the temperature rise of the quenched magnet can be calculated by equating the stored energy E to the enthalpy increase M where M is the mass of the magnet and cp is its average specific heat. The temperature rises are often quite modest (< 100 K) and do not injure the magnet. Such a magnet is said to be self-protecting. Nonrecovering normal zones in potted magnets not only spread longitudinally along the conductor but also spread transversely through the epoxy potting to adjacent conductors (cf. Section 6.5). The transverse propagation velocity is less than the longitudinal propagation velocity by a factor (kt /k)1/2, where kt is the thermal conductivity in the transverse direction and k is the thermal conductivity in the longitudinal direction. While transverse propagation is much slower than longitudinal propagation (usually it is only several percent as large), it is often the most efficient means by which the normal volume of the magnet increases. In magnets cooled with boiling helium, normal zones may arise that fail to propagate beyond a certain distance. For example, if the liquid level drops and uncovers part of the winding, that part may become normal, whereas the part still immersed may remain superconducting. Such a situation cannot be allowed to endure, for the eventual overheating of the uncovered conductor may destroy it. How much time have we got to recognize this threat and turn off the magnet? To answer this question in detail is difficult, but a simple conservative approach is to assume the nonrecovering normal zone is uncooled either by contact with a cryogen or by longitudinal heat conduction. Then the heat balance Eq. (6.1.1) becomes S(dT/dt) = QP/A = r cu J 2/f = rJ 2

(6.13.1)

where r = rcu /f is the effective longitudinal resistivity of the conductor. Then

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Uncooled Conductors

(6.13.2)

8

Eq. (6.13.2) is called the adiabatic hot-spot formula. It has the useful characteristic that the material properties of the conductor (S, r, and T) appear on one side of the equations while the electrical property J(t) appears on the other. The right-hand side is called the adiabatic hot-spot function of the conductor. It can be calculated without too much difficulty using the formulas for the specific heat and the resistivity given in Chapter 2. Once an upper limit to the hot-spot temperature has been decided on, the value of J2 dt is determined. The integral is composed of two parts. The first part, J2 t1, is contributed by the time interval 0 < t < t1 during which the nascent normal zone is undetected. At t1, the voltage across the normal zone is large enough to be detected, and the terminals of the quenching magnet are then connected to a dump resistance, R. Thereafter, the current in the magnet falls exponentially with time with a relaxation time t2 = L/R,where L is the inductance of the magnet. The integral on the left-hand side of Eq. (6.13.2), with upper limit , then equals J2(t1 + t2 /2), where J now represents the value of the current density when the quench began. The use of a dump resistance to dissipate most of the stored energy of the magnet outside of the dewar is an example of active protection. The voltage V across the normal zone can be determined by a method devised by Iwasa and Sinclair (1980). For a traveling wave T(z + vt), Eq. (6.1.1) becomes vS (dT/dz ) = d/dz[k(dT/dz )] + (r cu J2/f)g(T)

(6.13.3)

8

If we integrate Eq. (6.13.3) from z = – to z = 0, the symmetry center of the expanding normal zone, we find (6.13.4)

where Tmax is the maximum temperature at the center. The integral in the middle term is the voltage difference across half the normal zone. Aprocedure for determining the hot-spot temperature could be this: (1) Choose a voltage V that can be detected reliably. (2) Calculate Tmax from Eq. (6.13.4), using Eq. (6.8.7), say, to determine v. (3) Use Tmax to calculate t1 from Eq. (6.13.2). (4) Choose a value of R, the dump resistance, and calculate t2. The value of R is usually constrained by the maximum voltage IR across the magnet, which must not exceed the breakdown strength of the insulation. (5) Calculate the hot-spot temperature from the adiabatic hot-spot function, which is equated to J 2(t 1 + t2 /2).

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When the high-temperature superconductors are used, the propagation velocity is much lower than when the low-temperature superconductors are used. The reason for this is the strong increase in the volumetric heat capacity S with temperature. For silver, the volumetric heat capacity at 77 K is three orders of magnitude as great as that at 4.2 K. The propagation velocity is then correspondingly smaller, perhaps only 1–10 mm/s. Some workers have felt that such slow propagation would worsen the problem of quench protection. In the words of Oberly et al. (1992), “The propagation velocity of the quench is of the order of 100 times slower at 20 K than at 4.2 K due to greatly increased heat capacity and greatly decreased thermal conductivity. This very slow propagation velocity can result in damage to the superconducting coil as the large stored energy is absorbed in a small region of the coil.” Certainly magnets wound with high-temperature superconductors and operated at high temperatures cannot be self-protecting. But the problem of active protection is not worsened as one might infer from the quotation. For although the velocity v is greatly reduced (in rough inverse proportion to S), the countervailing increase in the integrand on the left-hand side of Eq. (6.13.4) largely offsets the effect of this decrease in v, and the voltage V remains detectable at quite modest temperature rises (~100 K) (Dresner, 1993, “Stability”; 1994, “On the connection”). The question of active protection of magnets wound with high-temperature superconductors has been discussed here as though it were conceivable that such magnets occasionally quenched. The great increase in stability noted in Section 6.6 may change our view of the protection problem, but it is still too early to tell what the situation will be when we have high-temperature superconductors good enough to wind large, high-field, high-temperature magnets.

6.14. THERMAL STRESSES DURING A QUENCH The thermal stresses that arise during a quench are generally the most influential factor in determining the maximum allowable temperature rise at the hot spot. In magnets wound with NbTi, this temperature rise can be quite large because NbTi is very strain tolerant. Even strains of several percent do not significantly degrade its critical current density. This is not so for Nb3Sn, which can stand at most oneor two-tenths of a percent strain before there is a noticeable degradation in the critical current. This is because of the frangibility of Nb3Sn, a quality that is shared by the ceramic oxide high-temperature superconductors. Ochiai et al. (1991) have suggested the following mechanism by which thermal strain may damage silver-clad ceramic oxide superconductors. When the conductor is cooled down after reaction, the silver contracts more than the ceramic oxide, putting the silver in tension and the ceramic in compression. The silver, which is strongly annealed by the high-temperature of reaction and therefore very soft, reaches its elastic limit in the early stages of cooldown, after which its tensile stress

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125

Figure 6.13. An idealized stress-strain curve for silver. (Redrawn from an original appearing in Dresner (1993, “Stability”) with permission of Butterworth-Heinemann, Oxford, England.)

remains fixed at its yield stress sy during the rest of the cooldown. Fig. 6.13 shows an idealized stress-strain curve for silver that will be used for the remainder of this discussion. Once the stress in the silver has saturated at the yield stress, the conductor perseveres in a state in which the silver suffers a tensile stress sy and the high-temperature ceramic oxide a compressive stress –f 1 sy /f 2, where f1, is the volume fraction of silver and f2 the volume fraction of the ceramic oxide. If a normal zone forms, the silver expands thermally more than the ceramic oxide; with enough heating, the ceramic oxide can go into tension and the silver into compression. However, the maximum compressive stress that the silver can support is – sy , so that the ceramic oxide suffers at most a tensile stress f1sy /f 2 . In the case studied by Ochiai et al., the yield stress of the annealed silver was 13 MPa, whereas the ultimate tensile strength of their ceramic oxide (BSCCO) was about 60 MPa. Their tapes were 70 vol.-% silver and 30 vol.-% BSCCO, so that their f1sy /f 2 was 30.3 MPa. The thermal stresses accompanying a quench of their conductor should not harm it. Handling, winding, and previous quenches can all work harden the silver. If more than one quench is anticipated, we may limit the hot-spot temperature so that the compressive stress in the silver never reaches sy and the silver never leaves the elastic range. Then successive quenches do not work-harden it. Figure 6.14 depicts schematically the state of stress in a Ag/BSCCO conductor after cooldown. To simplify our analysis, let us determine strain based on material displacements from this configuration; the stress-strain curve for the silver is then that shown in Fig. 6.15. We take the stress-strain curve of the BSCCO to be entirely elastic:

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Figure 6.14. The state of stress in a Ag/BSSCO conductor after cooldown. (Redrawn from an original appearing in Dresner (1993, “Stability”) with permission of Butterworth-Heinemann, Oxford, England.)

s2 = E2h 2– f 1s y /f 2

(6.14.1)

where henceforth s represents stress, h strain, E elastic modulus, subscript 1 Ag, and subscript 2 BSCCO. The force-free condition for the conductor is f 1 s 1 +f 2 s 2 =0

(6.14.2)

Figure 6.15. The stress-strain curve of silver when strain is reckoned on the basis of displacements from the configuration of Fig. 6.14. (Redrawn from an original appearing in Dresner (1993, “Stability”) with permission of Butterworth-Heinemann, Oxford, England.)

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127

while the condition that the silver and the BSCCO have the same length is h1 + a1DT =h2 +a2DT

(6.14.3)

where a1,2 are the linear coefficients of thermal expansion of silver and BSCCO, respectively, and DT is the temperature rise. On eliminating s2 and h2 from Eq. (6.14.3), using Eqs. (6.14.1) and (6.14.2) yields h 1+ ( f1 /f 2E 2)(s1 – sy ) + (a1 –a2)DT= 0

(6.14.4)

The values of h1 and s1 are determined by the intersection of the stress-strain curve of Fig. 6.15 with the straight line represented by Eq. (6.14.4.). If this intersection is the point (–2 s y /E1, – sy ), i.e., if this point lies on the line (6.14.4), the temperature rise DT is then the largest possible that just avoids further cold working of the silver. Ochiai et al. (1991) give the following values for the parameters: E1 = 80 GPa, E2 = 54 GPa, a1 = 2 x 10–5 K–1, a2 = 8 x 10–6 K–1. With these values, DT = 121 K. If the intersection point is (–sy /E1, 0), the BSCCO will never go into tension at all. Then DT will be half as large as formerly, namely, 60 K. In at least one example worked in Dresner (1994, “On the connection”), at the instant the temperature rise in a normal zone in a Ag/BSCCO tape reached 100 K, the normal zone voltage was about 64 mV, which, though small, is detectable. Furthermore, this voltage was based on estimation of the propagation velocity using Eq. (6.8.7), which neglects the helpful effects of current sharing. Notes to Chapter 6 1Nonetheless, propagation in completely insulated conductors has been examined and with some interesting results. While studying the propagation of normal fronts in a vacuum-insulated composite conductor, Bartlett, Carlson, and Overton (1979) observed that the velocity of propagation was somewhat larger (~10%) when the electrons flowed in the direction of propagation than when they flowed in the opposite direction. Gurevich and Mints (1981) suggested that the Thomson effect was responsible. Their idea was that heat would be released when the electrons flowed down a temperature gradient, speeding up propagation, whereas heat would be taken up when the electrons flowed up the gradient, slowing down propagation. They were on the right track, but the thermoelectric effect on which they focused their attention gave an effect about an order of magnitude smaller than observed, and, as Clem and Bartlett showed (1983), in the wrong direction. Clem and Bartlett explained the observed asymmetry using the Peltier effect—the heat absorbed or rejected when current passes from one material to another (copper to superconductor). This small effect is not likely to have any practical influence on magnet design, but the elegance and ingenuity of the explanation must certainly command our attention.

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7 Internally Cooled Superconductors

7.1. STABILITY MARGINS AND INDUCED FLOW As mentioned in Section 1.7, magnets have been built that are cooled with supercritical helium (Tb ~ 4 K, pressure ~ 1MPa; point P3 in Fig. 1.7). Since the helium pressure is greater than atmospheric pressure, the helium must be tightly confined, and the simplest way to confine it is in the interior of a hollow conductor. Three of the magnets of IEA Large Coil Task (Beard et al., 1988) were wound with such internally cooled superconductors. Fig. 7.1 shows sketches of their conductors. The Westinghouse conductor is a cable-in-conduit conductor, a type that has already been described briefly at the end of Section 1.7. The Swiss conductor is a solder-filled cable penetrated by a central, helium-filled tube. The Euratom conductor is a coarse, flat cable in a stainless steel box. The rationales of stability of the three magnets are quite different from one another. The Euratom and Swiss magnets were designed with “the intention to reduce the possible heat release as much as possible by using a rigid conductor and a monolithic winding pack” (Young et al., 1982). The Westinghouse design, on the other hand, foresaw the possibility of strand motion but sought to compensate for it through “a very high cooled surface [that] can be obtained by subdividing the conductor in[to] many individual strands [and that] provides the ability to rapidly remove heat input by sudden energy release and/or Joule heating from the conductor” (Young et al., 1982).1 Internally cooled conductors are not cryostable because the helium inventory available for recovery is limited. The issue of quench or recovery is decided in tens of milliseconds, whereas the residence time of the helium in the conductor is minutes or even tens of minutes. Consequently, the replacement of warm helium by fresh cold helium (that could take place in a large bath) may not occur fast enough to promote recovery. Therefore, if a thermal perturbation is strong enough, it may raise the helium temperature beyond the current sharing threshold. The Joule 129

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Figure 7.1 . Sketches of the internally cooled conductors of the IEA Large Coil Task (Beard et al., 1988). a) Westinghouse Conductor. Features Nb3Sn transposed compacted cable in a JBK-75 stainless steel sheath, 17.6 kA at 8 T, 20.8 x 20.8 mm. The compaction ratio provides mechanical support for internal radial loads while still permittingaxial slippage during winding, minimizing strain. b) Swiss Conductor. Features NbTi compact high strength solder filled around a central cooling tube, 13 kA at 8 T, 18.5 x 18.5 mm. Low mechanical hysteresis could be demonstrated in low temperature test. Key factors in its choice are low leak risk and quench pressure. c) Euratom Conductor. Features roebel cabled around a kapton insulated stainless steel core, then enclosed in a stainless steel sheath, 11 kA at 8 T, 40 x 10 mm. The strands are fixed mechanically by soft soldering onto the CrNi core with high resistance solder. (Redrawn from an original appearing in Dresner (1984, “Superconductor”) with permission of ButterworthHeinemann, Oxford, England.)

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heating of the conductor cannot be extinguished and a quench ensues. Internally cooled superconductors, especially cable-in-conduit conductors, are thus not truly cryostable but rather metastable. The item of central interest regarding a metastable system is how great a perturbation it takes to produce instability. Hoenig, Iwasa, and Montgomery (Proceedings, 1975) proposed the use of the stability margin (introduced in Section 5.5) as a figure of merit, and it has become the generally accepted standard. As a reminder to the reader, I state the definition again: The stability margin is the uniform heat density instantaneously deposited in a long length of conductor that just causes a quench. Most workers express the stability margin in J/cm3 of conductor, i.e., J/cm3 of metal excluding interstitial helium. This standard has been chosen not so much because uniform perturbations are expected but rather because it provides a simple basis for the comparison of different conductors. Hoenig et al. (1975, Proceedings) pointed out that the helium in a hollow conductor like the Swiss conductor in Fig. 7.1 would have to flow very fast in order to provide adequate interfacial heat transfer. Only high-speed turbulent flow could provide a large enough heat transfer coefficient to result in a high stability margin. High-speed turbulent flow requires a large pressure drop and can dissipate substantial pumping power at cryogenic temperatures. This dissipation creates very large room-temperature refrigeration loads. To reduce the pressure drop and the pumping power while preserving stability, Hoenig and Montgomery proposed using conductors with very large cooled surfaces, which could be obtained by subdividing the superconductor into many fine strands. The germinal work of Hoenig and his coworkers spawned a profusion of studies, most of them numerical, of how best to reduce pumping power while preserving stability margin. (Dresner, 1980, “Stability” reviews these early studies.) Most presupposed that the helium flow was imposed by external pumps and that the imposed flow remained unaffected by the heat produced in the normal zone. This early work has become obsolete because, as it turns out, the heat produced in the normal zone does profoundly affect the helium flow. The first hint of this came from an experiment of Iwasa, Hoenig, and Montgomery (1977) in which they measured the stability margin as a function of imposed flow rate. To their (and everyone else’s) surprise, the stability margin was nearly independent of the helium flow rate right down to zero flow! Hoenig and his coworkers (1977; 1979; 1979) as well as others (Miller et al., 1979 and 1980, “Stability”; Lue et al., 1980) confirmed this observation in additional experiments. This meant that vigorous pumping would not be needed to ensure adequate stability. Iwasa, Hoenig, and Montgomery ascribed recovery in stagnant helium to transient conduction of heat in the supercritical helium. But a drawback of this explanation soon arose out of an experiment of Lue, Miller, and Dresner (1978), who were studying vapor locking in a triplet of superconducting strands sheathed in a stainless steel tube and cooled with stagnant, saturated helium. When the ends

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of the stainless steel tube were open, the stability margin was high, but when the ends of the tube were closed, the stability margin fell more than tenfold. Such behavior is incomprehensible if good recovery in stagnant helium is fostered by transient conduction. Instead, it suggests that the good recovery is caused by transient flow induced in the helium by its thermal expansion as it is heated. Plugging the ends of the tube suppresses such flow and strongly diminishes the stability margin. Strong induced flows had already been suggested by others: Arp (1979) and Krauth (1980) reported calculated induced flows, the Reynolds number of which reached instantaneous values ~ 10 5. In a cable-in-conduit conductor with a hydraulic diameter ~ 1 mm, a Reynolds number ~ 10 5 corresponds to flow velocities of several meters per second. Such high flow across the heat transfer surface substantially increases the heat transfer through the surface. The calculations of Arp and Krauth were carried out numerically. In keeping with the spirit of this book, which is to obtain simple, widely applicable analytic approximations wherever possible, we follow here an approach of the author’s (1979, “Heating-induced flow”) based on applying Riemann’s method of characteristics to the linearized equations of compressible flow.

7.2. THE ONE-DIMENSIONAL EQUATION OF COMPRESSIBLE FLOW The full-blown one-dimensional equations of compressible flow in a tube (the continuity, momentum (Euler’s), and energy equations) are three coupled time- and space-dependent partial differential equations. They must be supplemented by the equation of state of the fluid, and together these four equations present a formidable obstacle to analytic solution. However, analytic solutions are attainable in two limiting cases described below. When the helium in a cable-in-conduit conductor is subjected to a pressure gradient, its motion is restrained by inertia and by friction with the walls and the cable. In the earliest stages of motion, friction with the walls can be neglected. Then the motion can be treated by the same techniques used to analyze motion in a shock tube. In the late stages of the motion, when the disturbance has spread over many tube diameters, the chief restraining force on the fluid is friction, and the inertia of the fluid may be neglected. In both of these limiting cases, analytic solutions are attainable. The one-dimensional equations of compressible flow in a tube are the following (Taub, 1967): dp/dt + r (∂v/∂z) = 0 ρ(dv/dt) = –∂p/∂z – rF

(mass balance)

(7.2.1)

(momentum balance)

(7.2.2)

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133

r d(u + v 2/2)dt = –∂(pv)/∂z + rq

(energy balance)

(7.2.3)

where r is the fluid density, t is the time, z measures distance along the tube, v is the flow velocity, p is the pressure, F is the frictional force per unit mass of fluid, u is the internal energy per unit mass of fluid, and q is the external (Joule) power input into a unit mass of fluid. The total derivatives dv/dt in Eq. (7.2.2) and d(u + v 2/2)/dt in Eq. (7.2.3) are so-called hydrodynamic derivatives, defined by d( )/ = ∂( )/¶t + v∂( )/∂z. The frictional force F = 2fv 2/D, where f is the Fanning friction factor and D = 4AHe /P is the hydraulic diameter. Here AHe is the area of helium in the cable cross section and P is the perimeter wetted by helium. Although the frictional force F appears in Eq. (7.2.2), the momentum balance equation, it does not appear in Eq. (7.2.3), the energy balance equation. The reason for this is that while the frictional force on the fluid decelerates it, i.e., destroys momentum, it does not destroy energy, but merely converts kinetic energy of flow to an equal amount of heat. If we multiply Eq. (7.2.2) by v and subtract it from Eq. (7.2.3), we obtain r(du/dt ) = –p(∂v/∂z) + rFv + rq

(7.2.4)

Now Eq. (7.2.1) can be written dp/dt = – r(∂v /∂z )

(7.2.5)

with the help of which Eq. (7.2.4) can be recast as du/dt + p(dw/dt) = Fv+ q

(7.2.6)

where w = 1/ r is the specific volume of the fluid. (The word specific means per unit mass .) Now, du + p dw = T ds, where T is the temperature and s the specific entropy of the fluid (this is the second law of thermodynamics). Thus Eq. (7.2.6) becomes T(ds/dt) = Fv + q

(7.2.7)

which says that the increase in entropy is caused by (1) the conversion of kinetic energy into heat (dissipation term Fv) and (2) heat input into the fluid (term q).2 Now we use Eq. (7.2.7) and the thermodynamic identity dr = dp/c2 – (brT/cp ) ds

(7.2.8)

where c is the sonic speed, b is the volume coefficient of thermal expansion, r(∂w/∂T)p , and cp is the specific heat, to eliminate dp/dt from Eq. (7.2.1). The result is dp/dt + rc 2 (∂v/∂z) = (b ρ c 2/cp )(Fv + q)

(7.2.9)

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Equations (7.2.9) and (7.2.2) comprise two partial differential equations in p and v for the determination of the pressure and flow velocity of the fluid. In strict point of fact they are not enough. As time progresses and p and v change, s also changes (according to Eq. (7.2.7)). The change in r must then be calculated using Eq. (7.2.8), and finally the change in T must be calculated from the equation of state. But in the problems we shall treat approximately, Eqs. (7.2.9) and (7.2.2) will be enough. (N.B.: Since p enters Eqs. (7.2.9) and (7.2.2) only in derivatives, we henceforth interpret p as the pressure rise.)

7.3. INDUCED FLOW IN A LONG HYDRAULIC PATH In the very earliest stages of heating-induced flow, the frictional retarding force can be neglected (shock-tube approximation). Accordingly, we set F = 0 in Eqs. (7.2.2) and (7.2.9). Next we linearize them by dropping second-order terms: r (∂v/∂t) + ∂p/∂z = 0

(7.3.1)

rc 2 (∂v/∂z) + ∂p/∂t = brc 2q/cp

(7.3.2)

If we now ignore changes in r, c, b, and cp , we can introduce special units in which r = c = brc 2q/c p = 1 (dimensions: ML-3, LT–1, and ML-1T-3, where M is mass, L is length, and T is time). Then Eqs. (7.3.1) and (7.3.2) become ∂v/∂t + ∂p/∂z = 0

(7.3.3)

∂v/∂z + ∂p/∂t = 1

(7.3.4)

It is these equations that we treat by Riemann’s method of characteristics. If we add and subtract Eqs. (7.3.3) and (7.3.4) we get

3

∂(v + p)/∂t + ∂(v + p)/∂z = 1

(7.3.5)

∂(v – p)/∂t – ∂(v – p)/∂z = –1

(7.3.6)

These characteristic equations can be interpreted as saying that The quantity v ± p increases by ± dt as we traverse a segment of a (linear) characteristic dz = ± dt. In the absence of heat source q, The quantity v ± p is constant along a linear characteristic dz = ± dt.

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135

Figure 7.2. The wave diagram for a uniformly heated zone of length b in the interior of a very long pipe.

These statements can be used to find the velocity v and the pressure rise p as a function of position z and elapsed time t by geometric means using a figure called (in gas dynamics) a wave diagram. The abscissa in a wave diagram is z and the ordinate is t. Fig. 7.2 is the wave diagram for a uniformly heated zone of length b in the interior of a very long pipe (so long, in fact, that conditions at the ends of the pipe (open, closed, restricted) do not matter). The characteristics dz = ± dt are lines of slope ± 1; those with positive slope are called positive characteristics and those with negative slope are called negative characteristics. According to the statements above, we know how the quantity R+ = v + p varies along a positive characteristic and how the quantity R_ = v – p varies along a negative characteristic. Following these quantities, called Riemann invariants in gas dynamics,4 allows us to find v and p at any point. Consider, for example, the point Q in the interior of the heated zone for any time t > b. The positive and negative characteristics through Q intersect the z-axis (t = 0) at points outside the heated zone. Since v = p = 0 when t = 0, we see by considering the positive characteristic SQ through Q that v Q + pQ = tTA + AP = b/2 + PQ

(7.3.7)

From the negative characteristic we find v Q – pQ = –tBQ = –BR = –QR

(7.3.8)

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If we add Eq. (7.3.7) to Eq. (7.3.8) we find 2v Q = b/2 + PQ – QR = 2PQ

(7.3.9)

v Q = PQ = z Q

(7.3.10)

or

If we subtract Eq. (7.3.8) from Eq. (7.3.7), we find 2pQ = b/2 + PQ + QR = b

(7.3.11)

p Q = b/2

(7.3.12)

so that

Thus when t > b, the pressure rise in the heated zone is uniform and constant at b/2 (in special units!) and the flow velocity rises linearly from zero at the center to b/2 at the edges. The region of the wave diagram for which 0 < t < b and –b/2 < z < b/2 is shown divided into four congruent triangles numbered 1 through 4. Application of the method just used shows that in triangle 1, v = 0 and p = t. In triangle 2, v = (t + z – b/2)/2 and p = (t – z + b/2)/2. In triangle 3, the pressure rise p is the same as in triangle 2, but the velocity has its sign reversed. In triangle 4, v = z and p = b/2. The maximum flow velocity of b/2 is attained at the edge of the heated zone at and after the time t = b. In ordinary units this is vmax =bqb/2cp

(7.3.13)

In the Westinghouse coil of the Large Coil Task (Beard et al., 1988), when the conductor is fully normal, q = 8.42 x 104 W/kgHe in the high-field region (8 T). At 3.9 K and 1.2 MPa, which is close to the operating condition of the coil, b = 0.043 K-1 and cp = 2690 J kg-1 K-1. Then if a 2-m-long section of the conductor becomes normal, the maximum induced velocity is 1.35 m/s. This is nine times as great as the ambient flow rate of 15 cm/s. According to the Dittus-Boelter equation (Bird et al., 1960) the heat transfer coefficient varies as the 0.8-power of the flow velocity and so is raised by a factor of 5.8 by this induced flow. This factor is an underestimate because it does not include in q the transfer to the helium of the heat pulse that initially drove the conductor normal.

7.4. INDUCED FLOW N THE EXPERIMENTS OF LUE ET AL. The foregoing analysis is satisfactory for a long initial normal zone created in the interior of a long hydraulic path. It does not conform to the details of an

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137

experiment undertaken by Lue et al. (1980) in which the transient pressure rise was also measured. The details of their experiment are these: A soldered, twisted triple of 1-mm-diameter wires in a 3.25-m-long, 2.41-mm-ID steel sheath with open ends; void fraction, 44%; Cu/NbTi = 4.5; ambient pressure, 0.5 MPa; zero imposed flow; ambient field, 7 T; r cu (including magneto-resistivity), 4.7 x 10–10 W -m; critical current, 400 A; transport current, 300 A; heater power, 80 W; duration of heat pulse, 10 ms. The Joule power when the conductor is normal is 71.3 W; then q = 165 W/gHe when the heater is on and the conductor is normal. At the operating conditions of 0.5 MPa and 4.2 K, b = 0.0843 K-1, cp = 3560 J kg-1 K-1, r =141 kg m–3, and c = 240 m/s. The open ends of the heated zone change the wave diagram, and we must recompute the formulas for the velocity and the pressure rise. Fig. 7.3 is the new wave diagram. At z = ±b/2, the open ends of the heated zone, the pressure rise p = 0 and R_ = R+. At z = 0, the center of reflection symmetry, v = 0. Using these results

Figure 7.3. The wave diagram for a uniformly heated zone of length b with open ends.

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we can easily show that the pressure rise and velocity are periodic with period 2b (remember, we are still using special units in which c = 1). For, R +(D) = R +(C) + GC = R _(C) + GC = [R _(B) – LC] + GC = [R +(B) – LC] + GC = [R +(A) +BE] – LC + GC = R +(A)

(7.4.1)

because GC + BE = LC (LC = b, GC + BE + LC = 2b). A similar argument shows that R _(D) = R _(A). Since v = (R + + R _)/2 and p = (R + – R _)/2 , the periodicity of the Riemann invariants implies the periodicity of v and p. Fig. 7.4 shows the point z = zA at two successive instants a time b apart. Using the same rules of calculation as before, we find R _(D) = R _(B) – BC = R +(B) – BC = R +(A) + HB – BC = R +(A) – 2zA

(7.4.2)

since HB – BC = HB – (HC – HB) = 2HB – HC = 2AH – GH = –2OA = –2z A . Similarly, R +(D)=R_ (A) –2z A

(7.4.3)

Adding and subtracting these last two equations, we find

Figure 7.4. The wave diagram for a uniformly heated zone of length b with open ends showing a point A at two successive instants of time an interval b apart.

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Figure 7.5. The wave diagram for a uniformly heated zone of length b with open ends showing two points A and E symmetrical with respect to the quarter-point at two times an interval b/2 apart.

vD = vA – 2zA or vA´ = 2zA – vA

(7.4.4a)

pD = – pA or pA´ = – pA

(7.4.4b)

Thus the pressure reverses sign every half-cycle. Furthermore, to get the velocity profile at any time (for z > 0), subtract the velocity profile at a time b earlier from the line through the symmetry center having slope 2. Fig. 7.5 shows two points A and E symmetrical with respect to the quarter-point at times b/2 apart. Using a procedure similar to that above we find that vA = pE + b/2 – zE and pA = – vE + zE

(7.4.5)

These equations mean that to get the velocity profile at any time add to the pressure profile at a time b/2 earlier the line of slope –1 through the end of the heated zone and then reflect the resulting curve around the quarter-point. To get the pressure profile at any time subtract the velocity profile at a time b/2 earlier from the line of slope 1 through the origin and then reflect about the quarter point. In the triangle in Fig. 7.3 corresponding to triangle 1 in Fig. 7.2, v = 0 and p = t, whereas in triangle 2, v = t + z – b/2 and p = b/2 – z. Then using the rules just derived, we can show how the velocity and pressure profiles develop with time. Fig. 7.6 shows these profiles at intervals b/2 over a full cycle of duration 2b. The maximum velocity vmax = b and the maximum pressure pmax = b/2 in special units. Note that these maxima occur at different times. In ordinary units,

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Figure 7.6. The profiles of velocity and pressure in a uniformly heated zone of length b with open ends at time intervals of b/2 over a full cycle of duration 2b. (Redrawn from an original appearing in Dresner (1979, “Heating-induced flow”) with permission of Butterworth-Heinemann, Oxford, England.)

vmax = βqb/cp

(7.4.6)

Pmax = ρcvmax /2

(7.4.7)

Using the figures given at the beginning of this section for the experiment of Lue et al., we find vmax = 12.7 m/s and pmax = 0.2 14 MPa. This calculated value of Pmax compares favorably with the maximum pressure rise of 0.15 MPa measured by Lue et al. However, the measured maximum occurred roughly 20 ms after the heater was energized rather than at the expected b/2c = 6.8 ms. In Dresner, 1979, “Heating-induced flow,” these discrepancies were attributed to the presence of a l-m-long, 1.9-mm-ID tee connecting the cold pressure transducer to the stainless steel tube that encases the conductor. The open area of the tee connector (2.8 mm2) is slightly larger than the open area of the conductor (2.0 mm2), and the volume of the connector is 43% of the void volume of the conductor. When the heater is energized, the helium at the center of the conductor expands into the unheated tee, reducing the maximum pressure rise below what is predicted by Eq. (7.4.7). Furthermore, the diversion of helium into the tee may delay the attainment of the maximum pressure rise. These effects vitiate a detailed comparison of the theory with the experiment of Lue et al., and only the order of magnitude agreement is noteworthy.

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7.5. MULTIPLE STABILITY The discovery of recovery in stagnant helium was a pleasant surprise because it obviated the need for strenuous pumping. The coupling between heat transfer to helium and its state of flow that is the cause of recovery in stagnant helium has another effect that was, at the time of its discovery, even more surprising. As mentioned in Section 7.1, the item of central interest is how great a perturbation it takes to produce a quench. Using the experimental setup described in the first paragraph of Section 7.4, Lue et al. (1980) observed whether the Cu/NbTi triplet quenched or recovered in response to various heat pulses. They carried out their study by fixing the transport current and the background magnetic field and then, in successive shots, gradually raising the pulse heat. What they expected to observe was recovery at small pulse heats and a transition to quenching at larger pulse heats. The largest pulse heat density that still allowed recovery would then be taken as the stability margin. In some of their experiments, they did see the expected sequence recovery, quench. But in other experiments, they saw the double sequence recovery, quench, recovery, quench, i.e., they saw recovery following small heat pulses, quench following larger heat pulses, recovery following yet larger heat pulses, and finally quench following the largest heat pulses. Typical experimental results are shown in Figs. 7.7 and 7.8. In both of these figures, the cross-hatched area corresponds to quench and the clear area corresponds to recovery. In the course of their experiments, Lue et al. explored the dependence of this multivalued stability margin on transport current and externally imposed flow. The qualitative results of their experiments are summarized in Fig. 7.9, which is a schematic representation of the stability margin DH as a function of transport current I and imposed flow velocity v. Points below the surface correspond to recovery; points above the surface correspond to quench. The surface is folded so that certain slices parallel to the (I,∆H)-plane result in a 2-shaped curve like that of Fig. 7.7, while other slices parallel to the (v,∆ H)-plane lead to the configuration of Fig. 7.8. In my 1980 review of this subject (“Stability”), I wrote, “This strange folded surface cries for an explanation . . .” Now to formulate a complete explanation of stability in cable-in-conduit conductors, one must take into account the mutual coupling of the flow-dependent heat transfer and the heating-induced flow. This difficult problem can only be dealt with numerically and has, indeed, attracted the attention of many numerical analysts. In a qualitative analysis designed to avoid this difficulty, Lue et al. (1980) artificially broke the coupling between heat transfer and induced flow by considering two half-problems in which they (1) studied the stability margin imagining the heat transfer coefficient to be externally imposed, and (2) studied the induced flow and associated heat transfer coefficient assuming the heating rate of the helium to be externally imposed. In other words, in problem

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Figure 7.7. Typical measurements (Lue et al., 1980) of the stability margin versus transport current. The cross-hatched area corresponds to quench and the clear area to recovery. (Redrawn from an original appearing in Dresner (1984, “Superconductor”) with permission of Butterworth-Heinemann, Oxford, England.)

8

(1) they studied DH(h) and in problem (2) they studied h(DH), assuming a fixed initial heat pulse duration. The qualitative nature of the results in problem (1) are sketched in Fig. 7.10. When h is very large, recovery is very rapid, and the total Joule heat produced during recovery is small compared to the initial heat pulse (consider the limiting case when h = ). Then (7.5.1)

where AHe is the cross-sectional area of the helium in the cable, Aco is the cross-sectional area of the conductor (metal), S is the volumetric heat capacity of the helium, Tb is the ambient temperature, and Tcs is the current-sharing threshold temperature. The integral in Eq. (7.5.1) is the volumetric enthalpy difference of the

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Figure 7.8. Typical measurements (Lue et al., 1980) of the stability margin versus externally imposed mass flow. The cross-hatched area corresponds to quench and the clear area to recovery. (Redrawn from an original appearing in Dresner (1984, “Superconductor”) with permission of Butterworth-Heinemann, Oxford, England.)

helium between Tb and Tcs . Only the enthalpy of the helium is considered in this expression because at low temperatures it greatly exceeds that of the metal. It only takes a heat density equal to the volumetric enthalpy difference of the metal between Tb and Tcs to warm the metal to the point where the superconductor

Figure 7.9. A schematic representationof the stabilitymargin DH as a function of transport current I and imposed flow velocity v. Points below the surface correspond to recovery, points above the surface to quench. (Redrawn from an original appearing in Dresner (1984. “Superconductor”) with permission of ButterworthHeinemann, Oxford, England.)

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Figure 7. 10. A schematic representation of the stability margin as a function of an externally imposed heat transfer coefficient. (Redrawn from an original appearing in Lue et al. (1980) with permission of the Journal of Applied Physics of the American Institute of Physics.)

is fully normal. As just mentioned, this heat density is very small compared to the asymptote (7.5.1). Thus, when a point having this heat density for its ordinate is plotted in Fig. 7.10 it appears to be on the h-axis. If the conductor is to recover, the heat transfer coefficient h must at least be large enough to remove the Joule power in the fully normal state, i.e., it must fulfill the equation I 2rcu /Acu =hP(T c – Tb )

(7.5.2)

where I is the transport current, rcu is the resistivity of copper, Acu is the cross-sectional area of the copper, and P is the wetted perimeterof the cable. Thus the foot of the DH(h)-curve in Fig. 7.10 lies at the point (h,0), where h is given by Eq. (7.5.2). In problem (2), the pulse duration is held fixed so that q ~ DH. Now from the results of Section 7.3, the induced flow velocity v is proportional to q, and the heat transfer coefficient associated with it is proportional to v 0.8 ~ DH0.8 (curve OAB in Fig. 7.11). Now when DH is very small, both q and the induced flow velocity v are very small. But when q is small, the interval of transient heat transfer is long' and transient heat transfer may endure for the entire heat pulse. Then h will be large (segment CD in Fig. 7.11). As DH (and q) increase, the time to takeoff decreases and so does the effective heat transfer coefficient (arc DA in Fig. 7.11). Eventually, for large DH, the curve of h(DH) joins the curve OAB. The effective heat transfer coefficient h as a function of the pulse heat DH for a fixed pulse duration then looks like curve CDAB in Fig. 7.11.

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Figure 7.11. A schematic representation of the heat transfer coefficient arising from the induced flow as a function of the size of an externally imposed heat pulse, assuming the same duration for all pulses. (Redrawn from an original appearing in Lue et al. (1980) with permission of the Journal of Applied Physics of the American Institute of Physics.)

Now we combine the results of the two half-problems by plotting the curves in Figs. 7.10 and 7.11 on one plot, Fig. 7.12; this necessitates rotating Fig. 7.11 around its 45° diagonal. The two curves are shown intersecting at three points, P, Q, and R. This is one of two possibilities; the other is that the minimum of the h(DH)-curve lies to the right of the DH(h)-curve and the intersections P and Q do not occur. The configuration of curves shown in Fig. 7.12 corresponds to a multivalued stability margin. Suppose, for example, we introduce a heat pulse lying on the DH-axis between O and P (fine line 1). The heat transfer coefficient that the pulse

Figure 7.12. The curves of Figs. 7.10 and 7.11 plotted on the same axes. (Redrawn from an original appearing in Lue et al. (1980) with permission of the Journal of Applied Physics of the American Institute of Physics.)

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induces lies on the curve h(DH) at point M. The heat transfer coefficient that the pulse requires for recovery lies on the curve DH(h) at point N. Since hM > h N, the conductor recovers. So the segment OP of the DH-axis corresponds to recovery. By identical reasoning, so does the segment RQ. The segment QP, on the other hand, corresponds to quenching, for along fine line 2, the induced hK < the required h L. Finally, points lying above R on the ∆Η-axis correspond to quenching. Here, then, is the double sequence recovery, quench, recovery, quench. When the minimum of the η(∆Η)-curve lies to the right of the ∆Η(η)-curve and only the intersection R occurs, we obtain the single sequence recovery, quench. The qualitative argument just given serves three purposes. First, it strips away the mystery of the folded surface in Fig. 7.9 by showing how the interplay of a few simple phenomena can lead to behavior that at first sight seems incomprehensible. Second, it reveals what ingredients are essential to include in a numerical stability program if one hopes to reproduce multivalued stability. These ingredients are transient heat transfer, takeoff, and augmentation of turbulent heat transfer by heating-induced flow. Third, it provides a basis for estimating the location of the point B in Fig. 7.9.

7.6. THE LIMITING CURRENT In Fig. 7.9, we call the value of DH on the upper sheet AFCD the upper stability margin and the value on the lower sheet BKEF the lower stability margin. The experimental data in Fig. 7.7 show that the upper stability margin can be many times larger than the lower stability margin. Since we are ignorant of the perturbation spectrum, it is prudent to assume that the effective stability margin in the region of multivalued stability is the lower stability margin. So for practical purposes, there is a sharp drop in stability as the current increases past the current IB at point B. This current has been given the name limiting current and the symbol Ilim. Operating below the limiting current guarantees the high upper stability margin. When we operate beyond the limiting current in the region of multivalued stability, there are perturbations less than the upper stability margin but larger than the lower stability margin that can quench the magnet. Now magnets have been built that operate successfully beyond the limiting current (Lue and Miller, 1982). In such magnets, the thermal perturbations happily did not exceed the lower stability margin. But there have been other magnets, which, though they worked well below the limiting current, quenched above it (Painter et al., 1992). Whether one should operate below the limiting current or not is a subjective decision up to the individual designer. The size of the limiting current, on the other hand, is a purely technical question. We cannot determine the limiting current from the elementary considerations discussed so far, but we can use them to obtain a scaling rule that expresses the dependence of the limiting current on various parameters of the conductor (Dresner,

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1981, “Parametric study”). The condition determining the limiting current is the confluence of the intersections P and Q in Fig. 7.1 2. If the DH(h)-curve rises sharply near its foot, the heat transfer coefficient at this confluence is close to that at the foot. Hence, Ilim and h satisfy Eq. (7.5.2). To estimate h at the confluence of P and Q, we proceed as follows. The heat flux qJ into the helium depends on the takeoff time t according to the relation qJ = Ct –1/ 2

(7.6.1)

where C is a constant that depends on the thermodynamic state of the ambient helium. The lower stability margin (intersection P in Fig. 7.12) is characterized by the condition that the duration of the normalizing pulse just equals the time to takeoff. In other words, at point P, the normalizing heat pulse is just finished being drained away by transient heat conduction into the helium at the moment of takeoff. Thus t in Eq. (7.6.1) is the pulse duration. A heat flux qJ will induce a velocity v of the order of v ~ b(qJP/rHe AHe)b/CpHe

(7.6.2)

since qJ P/ rHe AHe = q, the power density in the helium (here rHe is the density of the helium). Combining Eqs. (7.6.1) and (7.6.2) and ignoring thermodynamic quantities and numerical constants, we find v ~ bt –1/2D–1

(7.6.3)

where D = 4AHe/P is the hydraulic diameter of the helium space. The upper branch of the curve h(DH) in Fig. 7.12 represents steady heat transfer in turbulent helium and can be taken to be described by a heat transfer correlation of the form Nu ~ (Re)m (Pr)n

(7.6.4)

where Nu is the Nusselt number, Re the Reynolds number, and Pr the Prandtl number (Bird et al., 1960). Again ignoring thermodynamic and numerical constants, we find that Eq. (7.6.4) says h ~ v mD m–1

(7.6.5)

If we combine Eqs. (7.6.5) and (7.6.3) we find h ~b m t –m /2D –1 Inserting Eq. (7.6.6) into Eq. (7.5.2), we find, after some manipulation

(7.6.6)

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Jlim ~ [f(1 – fco ) fco ]1/2 ρcu–1/2D–1(Tc – Tb )1/2 b m/2t –m/4

(7.6.7.)

where Jlim = Ilim /A, A is the cross-sectional area of the cable space, f is the volume fraction of copper in the strands, fco is the volume fraction of the strands in the cable space, and, as before, rcu is the electrical resistivity of copper. In the classical Dittus–Boelter correlation m = 4/5, and this value appeared to agree well with the first experimental data on limiting currents reported by Lue and Miller (1981, “Parametric study”; also Dresner, 1981, “Parametric study”). But later experiments (Lue and Miller, 1981, “Heated length”) were better fitted by the value m = 4/15. Arp (1979) has pointed out that Kawamura (1976; 1977) showed that the Nusselt number can vary by almost an order of magnitude from its steady-state value when the Reynolds numbers changes rapidly. Perhaps this may make the different reported values of m seem less troubling; in any case, there is little more we can say about the value of m. The proportionality of Jlim to r–cu1 / 2 (Tc – Tb )1/2 has been tested by Lue and Miller (1981, “Parametric study”; also Dresner, 1981, “Parametric study”) by changing the magnetic field to which the sample was exposed, and quite good agreement was found. There is a proportionality constant missing from Eq. (7.6.7) that Miller (1985) has determined from a study of the experimental data. In SI units, the constant is close to 1 for 4-K helium. It varies linearly with pressure from 1.2 at 0.3 MPa to 0.95 at 0.6 MPa. The scaling rule (7.6.7) does not depend on the critical current Ic , and it may happen that the value of Ilim > Ic. What this means is simply that we cannot operate the conductor in the regime of multivalued stability because it becomes resistive before we reach that regime. Such a conductor has single-valued stability for any attainable current, and its stability margin always equals the upper stability margin. An example is the Westinghouse conductor of the Large Coil Task (Beard et al., 1988): the large value of Tc at 8 T forNb 3Sn pushes Ilim well beyond Ic . Interestingly, a lower magnetic field may restore multivalued stability because with decreasing field Ilim increases as (T c – Tb )1 / 2 whereas Ic increases faster, as Tc – Tb .

7.7. DISCUSSION OF THE ISOBARIC ASSUMPTION Eq. 7.5.1. applies to the special limiting case in which all the enthalpy of the helium between Tb and T cs is devoted to absorbing the initial normalizing pulse. In general, this available enthalpy is split between the heat of the initial normalizing pulse and the Joule heat produced during recovery, as we see now by considering the process of recovery a little more closely. Just after the initial pulse, the conductor is hot, with a temperature above the current sharing threshold, while the helium is still cold (at temperature Tb). Then the conductor and the helium exchange heat, the conductor temperature aways being greater than or equal to that of the helium. As

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long as the conductor temperature exceeds the current sharing threshold Tcs, the conductor continues to produce Joule power. If the conductor and the helium equilibrate their temperatures at a temperature < Tcs , the conductor recovers, for as its temperature falls below Tcs, it stops producing Joule power. On the other hand, if the helium temperature reaches Tcs first, the conductor never stops producing Joule power and so quenches. In the limiting case in which the initial normalizing pulse is the largest consistent with recovery, i.e., in which it equals the stability margin, the conductor and helium temperatures equilibrate exactly at Tcs . Thus the helium has been heated from Tb to Tcs by the combined heat of the initial normalizing pulse, DH, and the Joule power produced during the temperature equilibration. Without solving the time-dependent problem of heat exchange between the conductor and the helium, it is not possible to say how much Joule heat is produced during recovery. But when the heat transfer coefficient is very large, making the process of heat exchange very rapid, we expect little Joule heat to be produced during recovery. This is the situation that corresponds to the upper stability margin, where strong induced flow greatly augments the heat transfer coefficient. Thus we expect the limiting value of DH given in Eq. (7.5.1) to approximate the upper stability margin and to be larger than it. Eq. (7.5.1) is based on the tacit assumption that recovery is an isobaric process, for only at constant pressure is the heat absorbed by the helium equal to its increase in enthalpy. Now we have already seen in Sections 7.3 and 7.4 that heating-induced flow is accompanied by a pressure rise, so recovery cannot be a strict isobaric process. In the stability experiments of Lue et al. (1980) on single triplets and in those of Miller et al. (1980, “Stability”) on a one-third scale Westinghouse Large Coil Task conductor, the stability margin DH sometimes exceeded the right-hand side of Eq. (7.5.1) by as much as a factor of 2. This seems to me to clear evidence that recovery cannot be treated as isobaric and that the pressure transient associated with heating-induced flow has an important effect on the cooling capacity of the helium. This point was first made by Wilson (1977); Lue et al. (1980) showed that pressure excursions of several atmospheres could account for the discrepancies noted between DH and the right-hand side of Eq. (7.5.1). We can only argue approximately because we do not know the exact thermodynamic trajectory of the heated zone. Qualitatively, it must look in the pressurespecific volume plane like the curve shown in Fig. 7.13. (N.B.: p represents pressure rise.) Point A is the initial state of the helium; the hatch marks label the part of the curve over which Joule power is produced. According to the earlier discussion, when the normalizing heat pulse equals the stability margin, the helium temperature just reaches Tcs when the Joule heating stops (point B). Thereafter, the helium expands isentropically back to ambient pressure (arc BC). It can be shown straightforwardly that the heat absorbed by a unit mass of helium on arc AB exceeds the specific enthalpy difference – Since dq = – w dp,

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Figure 7.13. Thermodynamic path in the pressure-specific volume plane.

(7.7.1) Now since w on the upper branch of curve ABC, where dp < 0, is > w on the lower branch, where dp > 0, w dp < 0. It follows immediately then that q > – Now, in strict point of fact, the enthalpy difference – is different from the right-hand side of Eq. (7.5.1). For the temperature at point C is less than Tcs because in the isentropic expansion BC, the temperature falls. The enthalpy difference – can be calculated as follows: (7.7.2)

and Now since

Then

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(7.7.3) Then, subtracting the second of Eqs. (7.7.2) from the first, we find

= T(∂s/∂p)T(–pB)= bTwp B

(7.7.4)

by making use of the Maxwell relation (∂s/∂p)T = – (∂w/∂T)p = –bw. Thus the error in using Eq. (7.5.1) for the stability margin is bTpB (AHe /Aco). If we use pmax in place of pB , then for the conditions of the experiment of Lue et al. quoted in Section 7.4, pB = 0.214 MPa, b = 0.0843 K-1, Tb = 4.2 K, and AH e /Aco = 0.786. Then bTpB (AHe /A co ) = 5.96 x 104 J /m3 = 59.6 mJ/cm3. Typically, stability margins are several hundred mJ/cm3 (cf. Fig. 7.7), so the fractional error incurred by using Tcs as the upper limit in the integral in Eq. (7.5.1) is not large. The thermodynamic arguments given so far are based on the assumption that a single pair of thermodynamic variables describes the entire normal zone, and from the work of Sections 7.3 and 7.4, we know this is not so. At any single instant, the thermodynamic states of different fluid elements may differ from one another. In view of all these caveats, it would appear that the estimate of Eq. (7.5.1) has no rigorous interpretation. Nonetheless, it is useful as a figure of merit, for when it is high the upper stability margin is high, and when it is low, the upper stability margin is low. In addition, it is easy to calculate, a property that has great utility in survey calculations.

7.8. THE LOWER STABILITY MARGIN As mentioned in Section 7.6, the lower stability margin is characterized by the condition that the duration of the normalizing pulse just equals the time t to takeoff. If we ignore the Joule power compared to the heat flux qJ = DHAco /Pt and substitute the last expression into Eq. (7.6.1), we obtain DH = 4Ct 1 / 2/D w

(7.8.1)

where Dw is the wire (strand) diameter. The right-hand side of Eq. (7.8.1) should be an upper bound to DH because of the neglect of Joule power generation. Lue and Miller (1981, “Heated length”) measured the lower stability margin of a triplet of 1-mm strands soldered around a central heater wire. The ambient helium pressure was 0.5 MPa and the ambient temperature was 4.2 K, for which conditions, I have calculated that C = 26 mW cm–2 s1/2 (Dresner, 1984, “Superconductor stability”). For pulse times of 10, 50, and 100 ms, we calculate from Eq. (7.8.1) that DH = 87, 194, and 274 mJ/cm3, respectively. (The results from Eq. (7.8.1) have been reduced by a factor of 5/6 to account for the fact that in a soldered

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triplet only 5/6 of the surface is wetted.) The corresponding experimental results are 60, 100, and 140 mJ/cm3 and are slightly lower than the upper bound calculated from Eq. (7.8.1). In Dresner (1984, “Superconductor stability”), an estimate is presented of the Joule power required to reduce the calculated DH to the measured value. It is given as 36%, 25%, and 18%, respectively, of the fully normal heat flux, indicating that the conductor appears only to have been driven into the current sharing range by the initial pulse. Notes to Chapter 7 1 The idea of increasing the surface by subdividing a fixed volume is ancient. To my knowledge, the first suggestion in this direction regarding superconductors came from Chester (1987), who wrote, “Clearly, excellent thermal contact is desirable between the superconductor and the thermal ballast . . . this is achieved by subdivision of the superconductor to present greater interfacial area.” The present era of development of cable-in-conduit conductors began in 1975 with a reminder from Hoenig, Iwasa, and Montgomery (IEEE Transactions) of the advantages of subdividing the superconductor by cabling. A review of this earlier work can be found in Dresner, 1980, “Stability.” 2 If the reader has any doubts about the correctness of leaving the frictional force out of the energy equation, the following observation should settle them. The power dissipated by the frictional force per unit volume is –rFv. If this term were included on the right-hand side of Eq. (7.2.3), the term Fv would not appear on the right-hand side of Eq. (7.2.7). Then the dissipation of kinetic energy into heat would produce no entropy, an obvious contradiction. 3 Riemann’s method was invented to deal with hyperbolic partial differential equations, the classical example of which are the equations of compressible flow. The interested reader can find detailed discussions of the method in the books of Courant and Hilbert (1953), or Courant and Friedrichs (1948). 4 In the simplest problems of gas dynamics the Riemann invariants do not change along the characteristics and this accounts for their names. We continue to use this name here for convenience even though the presence of a heat source causes the “invariants” to change along the characteristics. 5 Strong heat transfer promoted by transient heat conduction occurs early in supercritical helium as well as in boiling helium. Whereas in boiling heat transfer the transient phase comes to a close (takeoff) when the surface is blanketed by vapor, in supercritical heat transfer the transient phase comes to a close when the temperature of the surface reaches the so-called pseudo-critical line where the density of the helium drops sharply. Takeoff in supercriticalhelium is thus caused by the blanketing of the heat transfer surface by low-density helium, a process sometimes called pseudo-boiling. The time to takeoff in supercritical helium varies as the reciprocal square of the heat flux just as in boiling helium and the two times are not terribly different in magnitude for the same heat flux.

8 Hydrodynamic Phenomena

8.1. NEGLECT OF FLUID INERTIA When a potted magnet quenches, the main protection issue is the spreading of the normal zone and the final hot-spot temperature. When a magnet cooled by a boiling cryogen quenches, another protection issue arises: the pressure rise in the dewar caused by the vaporization of the boiling coolant. The resolution of this issue is to provide a suitably large pressure-relief tube and does not usually influence the design of the conductor. When an internally cooled conductor quenches, especially a cable-in-conduit conductor, almost all of the hydraulic resistance to the expulsion of the helium comes from the conductor itself. (Consider, for example, the Westinghouse conductor (Fig. 7.1), for which the hydraulic diameter D is 0.4 mm and the hydraulic path length L is 120 m; it has an L/D ratio of 3 x 105!) For such conductors, the issues of protection cannot be divorced from the design of the conductor, and so we consider them here. These issues, all interconnected, are the rise in internal pressure, the thermal expulsion of helium from the ends of the conductor, the propagation of normal zones, and the possibility of hydrodynamic quench detection. The rise in internal pressure in a quenching cable-in-conduit conductor such as the Westinghouse conductor is a relatively slow process compared with the process of recovery, the former lasting for seconds, the latter only for milliseconds. In seconds, the disturbance created by the expanding helium in the normal zone spreads over many diameters. As already noted in Section 7.2, when a disturbance has spread over many tube diameters, the chief restraining force on the fluid is friction, and the inertia of the fluid may be neglected. Therefore, in treating the late-stage hydrodynamic problems of protection, we shall set to zero the left-hand side of Eq. (7.2.2), which is the inertial term. Then Eq. (7.2.2) simply becomes ∂p/ ∂z = –rF

(8.1.1) 153

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Using Eq. (8.1.1) for the derivative ∂p/∂z, Eq. (7.2.9) becomes ∂p/∂t + rc 2(∂v/ ∂z) = (brc 2/cp )[q + Fv(1+ cp /bc 2)]

(8.12)

Consulting the tables of Arp and McCarty (1989), we find that the quantity cp /bc 2 is always close to 1.

8.2. MAXIMUM QUENCH PRESSURE To bound the internal pressure that the conductor might suffer, let us assume an entire hydraulic path has gone normal all at once. We presume the ends of the hydraulic path intrude into large plenums held at constant pressure, so that there is no pressure rise at the ends of the hydraulic path. According to Eq. (8.1.1), the pressure rise is largest at the central element, where it equals p o = (2f/D) rv 2 dz 0

(8.2.1)

where = L/2. We now assume that the velocity has a linear profile, being zero at the center and having a maximum vm at the open ends. Then Eq. (8.2.1) becomes po = (2 f/D )rvm2

(8.2.2)

For the central fluid element, for which v = 0, Eq. (8.1.2) becomes ∂p o / ∂t + rc 2(vm / ) = brc 2q/cp

(8.2.3)

or using Eq. (8.2.2) to eliminate vm in favor of po, ∂po/∂t+ (ρc2/ )(3poD/2ρ ƒ)1/2 =βρc2q/cp

(8.2.4)

When po achieves its maximum pm , ∂po/∂t = 0, and then it follows from Eq. (8.2.4) that pm = (2f /3)(b2r/c p2 )(q2 /D) = (2f /3)(b2/rc 2p )(Q2 /D)

(8.2.5)

where Q = rq is the Joule power density in the helium. In Miller et al. (1980, “Pressure”), it is shown that the thermodynamic group b2/rc 2p depends only weakly on density and can be approximated roughly by 0.45p–1.8 in SI units, where p denotes absolute pressure, rather than pressure rise. Substituting this correlation into Eq. (8.2.5), we obtain the approximate formula pm= 0.65f 0.36(Q /D)0.36(1– pa /pm) –0.36 (SI units)

(8.2.6)

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where pm is the maximum absolute pressure and pa is the ambient pressure. The last term varies slowly with the ratio pm /pa when this ratio is large compared to 1. For 3 < pm /pa < 20, the last term in Eq. (8.2.6) varies between 1.16 and 1.02; here we use the mean value of 1.09. Measurements of the friction factors of typical cable-in-conduit conductors (Lue et al., 1979; Daugherty and Sciver, 1991) indicate that ƒ~ 0.02 and is determined largely by the roughness of the cable. With these simplifications, Eq. (8.2.6) becomes pm = 0.17(Q

/D)0.36

(SI units)

(8.2.7)

At different times in the past, the constant in Eq. (8.2.7) has been given different values, depending on the state of knowledge about the friction factor f at the time. In the original work in which the results (8.2.6) and (8.2.7) were derived (Miller et al., 1980, “Pressure”), the constant was 0.10. In a somewhat later work (Lue et al., 1982), it was raised to 0.14, and now we prefer the value 0.17. Fig. 8.1 shows experimental points collected by Lue et al. (1982) compared with Eq. (8.2.7) using values of the constant of 0.10 and 0.14. The agreement is good, and Eq. (8.2.7) can be used confidently for design purposes. The formula (8.2.7) has occasionally been misapplied by using it when the ratio pm/pa is close to 1. A single example will suffice to show what kind of errors are possible. Suppose ƒ= 0.02, Q /D = 1019 W2 m–4, and pa = 1 MPa. Eq. (8.2.7) gives pm = 1.176 MPa, whereas Eq. (8.2.6) gives pm = 1.578 MPa. The calculated

Figure 8. 1. Experimental data from Lue et al. (1982) compared with Eq. (8.2.7) using values of the constant of 0.10 and 0.14. (Redrawn from an original appearing in the Proceedings of the Ninth Cryogenic Engineering Conference, K. Yasukochi (ed.), Kobe, Japan, May 11–14, 1982, pp. 814–818, by permission of the publishers, Butterworth-Heinemann Ltd. ©)

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pressure rises differ by a factor of 3.3! If, on the other hand, Q /D = 1022 W2 m–4, Eq. (8.2.7) gives pm = 14.14 MPa and Eq. (8.2.6) gives pm = 13.59 MPa. The absolute pressures differ by only 4.0% and the pressure rises by only 4.4%.

8.3. THERMAL EXPULSION Thermal expulsion of the helium from the open ends of a fully normal hydraulic path has also been studied experimentally (Lue et al., 1982), but the emphasis was on times somewhat shorter than that required for a linear velocity profile to develop (cf. Fig. 8.2). When the elapsed time is short, the pressure relief waves penetrating inward from each open end have not yet reached the center. Until they do, the expanding helium near each end behaves as though the hydraulic half-length were infinite. This means we must solve the partial differential Eqs. (8.1.1) and (8.1.2) subject to the boundary and initial conditions p(0,t) = 0, v(∞,t) = 0, p(z,0) = 0, v(z,0) = 0

(8.3.1)

Here z = 0 is taken to be the open end of the conductor with positive z pointing inwards, and t = 0 is taken to be the instant at which the conductor becomes normal.

Figure 8.2. The thermal expulsion velocity from the open ends of a fully nomal test conductor as a function of time (Lue and Miller, 1982). (Redrawn from an original appearing in the Proceedings of the Ninth Cryogenic Engineering Conference, K. Yasukochi (ed.), Kobe, Japan, May 11–14, 1982, pp. 814–818, by permission of the publishers, Butterworth-Heinemann Ltd. ©)

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Before we make a direct attack on this problem, it behooves us to inquire if Eqs. (8.1.1) and (8.1.2) and the conditions (8.3.1) are invariant to stretching groups similar to that used in Section 6.2. Such a group is z´ = lz v´ = l–1/2v 0
(8.3.2)

t´ = l3/2t q´ = l–3/2q with the other quantities(p, b, r, cp, c) being unchanged. For a fixed initial thermodynamic state, the expulsion velocity v(0,t) can only depend on q and t, i.e., v(0,t) = (q,t). This relation must be invariant to the transformations (8.3.1), i.e., v´(0,t´) = (q´, t´ ) or (8.3.3) From Eq. (8.3.3) it follows that (8.3.4) where is an as yet undetermined function of the single variable qt. This means that if we plot the data in Fig. 8.2 using vt1/3 as ordinate and I2t ~ qt as abscissa, all the curves in Fig. 8.2 should, and do, collapse to a single curve (cf. Fig. 8.3). The points in Fig. 8.3 correspond to points on the curves in Fig. 8.2 for which t < 1 s. For times t ~ 2 s or longer, the disturbance has reached the center of the conductor and the expanding helium no longer behaves as though the hydraulic half-length were infinite. The function (qt) has been calculated by the method of similarity solutions (Dresner, 1981, “Thermal expulsion”), discussed in Appendix A. The result of this calculation is vt 1/3 = 0.952 (bc/cp )2/3(D/ƒ)1/3(qt )2/3

(8.3.5)

It is important to note here that this explicit form has been attained only at the cost of (1) assuming the thermodynamic properties of the helium remain constant, and (2) ignoring frictional heating. In fact, r diminishes as helium is expelled from the conductor, so that on both counts Eq. (8.3.5) should underestimate v. As we can see from Fig. 8.3, it does so. The overall slope of the experimental points matches the theoretical slope of 2/3 fairly well, but the similarity solution is low by roughly a factor of 2.

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Figure 8.3. The data (for t < 1 s) of Fig. 8.2 plotted using the similarity variables vt 1/3 and I 2t. Shown, too, is the similarity Eq. (8.3.4). o ) 500 A; ) 760 A; ∆ )1010 A; ) 1250 A. t = 0.2, 0.4, 0.6, and 1.0 s. (Redrawn from an original appearing in the Proceedings of the Ninth Cryogenic Engineering Conference, K. Yasukochi (ed.), Kobe, Japan, May 11–14,1982, pp. 814–818, by permission of the publishers, Butterworth-Heinemann Ltd. ©)

8.4. EXPULSION INTO AN UNHEATED PART OF THE CONDUCTOR A problem closely related to thermal expulsion from an open end is thermal expulsion into an unheated part of the conductor. Imagine a very long tube, the left half of which is producing Joule heat while the right half is not. The heated helium on the left expands, intruding into the right half. We seek the velocity at which the hot helium crosses the boundary, i.e., the velocity of efflux of the helium from the left side into the right. With the same provisos as before, namely, constant thermodynamic properties and elapsed time short enough that there is no influence from the ends, we find the same result as that given in Eq. (8.3.4) except that the numerical constant 0.952 is replaced by 0.600. This result can be used to estimate the initial rate of expansion of a newly created normal zone, assuming, as we shall, that the normal zone is propagated by the expansion of the hot helium. For early times, then, the normal-superconducting front should move with a velocity proportional to the cube root of the elapsed time. Luongo et al. (1989, “Thermal hydraulic simulation”) report that Eq. (8.3.5) with the modified numerical coefficient agrees well with their numerical simulations.

8.5. SHORT INITIAL NORMAL ZONES The problems discussed so far are all based on long initial zones: in Sections 8.2 and 8.3 an entire hydraulic path was assumed to go normal at the outset; in

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Section 8.4, an infinite pipe was envisioned. We need to know in addition what happens when the initial normal zone is short. Now the problem is a two-region problem and is substantially more complicated than the problems just dealt with. Again we can simplify the problem by artificially breaking the problem into two one-region half-problems. The first half-problem concerns the flow induced in the cold helium by the expanding hot helium in the initial normal zone. To break the coupling we assume that we know how this hot-cold boundary moves. The expanding hot-cold boundary then acts like a piston that accelerates the cold helium, which, for the long times we are interested in (seconds, not milliseconds) is restrained by friction.

8.6. THE PISTON PROBLEM If the disturbance created by this “piston” has not yet reached the open end of the hydraulic path, we can study the motion induced by the piston as though it were taking place in an infinite tube. Thus we must solve Eqs. (8.1.1) and (8.1.2) with the boundary and initial conditions v(z,0) = p(z,0) = v(∞ ,t) = p(∞ ,t) = 0

(8.6.la)

v(Z,t) = dZ/dt

(8.6.1b)

where Z(t) is the location of the piston, assumed known. Indeed, in what follows we take Z(t) = Xtn, where X is a known constant of proportionality. Thus we take the displacement of the piston to be proportional to a power n of the elapsed time. Furthermore, in the cold helium q = 0. To make the calculation possible, we restrict ourselves to small velocities v, so that the frictional v 3-term on the right-hand side of Eq. (8.1.2) can be neglected. (Later, we shall mention the conditions under which this neglect is permissible, i.e., the conditions under which the solution we shall obtain is self-consistent.) Finally, we treat the thermodynamic properties of the helium as constants independent of temperature and pressure. If we introduce special units in which r = c = D/4f = 1 (respective dimensions: ML–3, LT–1, L), Eqs. (8.1.1) and (8.1.2) become ∂p/∂z = –v 2/2

(8.6.2)

∂p/∂t + ∂v/∂z = 0

(8.6.3)

from which it follows by elimination of p that v(∂v/∂t) = ∂2 v/ ∂z2

(8.6.4)

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It will prove convenient in what follows to transform to a frame of reference moving with the piston: z = z – Z(t). Then Eq. (8.6.4) becomes v[∂v/∂t – (dZ/dt )(∂v /∂z)] = ∂2v /∂z2

(8.6.5)

We shall assume that the second term in the bracket on the left-hand side is small compared with the first—later, we shall discuss this assumption, too. If we drop the second term, we again obtain Eq. (8.6.4) written in terms of z and t, namely, v(∂v/∂t) = ∂ 2v/∂ζ2

(8.6.6)

but now the last boundary condition in Eq. (8.6.1b) becomes v(0,t) = dZ/dt

(8.6.1c)

Equation (8.6.6) is well suited to treatment by the method of similarity solutions, which is described in great generality in Appendix A. Because this method plays such an important role in this book, I shall use it here to solve Eq. (8.6.6) subject to the boundary and initial conditions (8.6.1a) and (8.6.1c). The details of application to a concrete problem may serve as a steppingstone toward full understanding of Appendix A. We start by exploring the invariance of Eq. (8.6.6) to stretching groups like those used in Sections 6.2 and 8.3. Let us try the group z´ = lz v´ = la v

0< l<∞

(8.6.7)

t´ = λδt where a and d are constants yet to be determined. If we substitute Eqs. (8.6.7) into Eq. (8.6.6), we find that the resulting equations will be precisely Eq. (8.6.6) written in terms of the primed variables if, and only if, α− δ = –2

(8.6.8)

Thus not both of the constants a and d may be chosen at will. The values of a and d are, as we shall see subsequently, determined by the boundary and initial conditions. But for the time being let us consider them arbitrary, save that they obey Eq. (8.6.8). Now suppose we look for solutions of the form v = t a/dy([z /t 1/d )

(8.6.9)

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where y is an as yet undetermined function of the single variable x = z/t 1/d . It is easy to see that Eq. (8.6.9) remains unchanged if we substitute for v, z, and t in terms of v´, z´ , and t´ from Eq. (8.6.7). It is this form that is called a similarity solution. Since the as yet unknown function y(x) is a function of one variable only, if we substitute Eq. (8.6.9) into the partial differential Eq. (8.6.6), an ordinary differential equation will result. The calculation is simple and yields δ(d2y/dx 2) + xy (dy/dx ) – ay 2 = 0

(8.6.10)

When z = 8 or when t = 0, v = 0, according to Eq. (8.6.1a). The first of these conditions requires that y(∞)= 0; this satisfies the second condition as well.1 Finally, since v(0,t) = nXt n–1, we must have a/d = n – 1 and y(0) = nX. If a/d = n – 1, it follows from Eq. (8.6.8) that a = 2(n – 1)/(2 – n) and δ = 2/(2 – n). If Eq. (8.6.9) is to represent a disturbance that spreads out with time, d must be > 0, i.e., n must be < 2.

8.7. THE PISTON FORMULA The stage is now set for the solution of the ordinary differential Eq. (8.6.10). But before we plunge ahead, it pays to imagine what we would do with the solution y(x) if we already had it. By integrating Eq. (8.6.2) from z = 0 to z = ∞ (N.B.: ∂p/ ∂ z = ∂p/∂z), we find (8.7.1) Now Eq. (8.6.10) is itself invariant to a stretching group, namely, y´ = m–2 y 0<m<∞

(8.7.2)

x´ = mx (This is no coincidence and the existence and form of this group could have been predicted from the outset; cf. Appendix A.) It follows then that (8.7.3) Since y(0) = nX can be given any value for a fixed y´(0) by varying m, the right-hand side of Eq. (8.7.3) must be a constant independent of y(0) = nX; call it C. Then from Eq. (8.7.1),

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p(0,t) =C(nX )3/2t 3n/2–1= (n3/2C)Z 3/2 t –1

(8.7.4)

Although C is independent of y(0) = nX, it still depends on n because a and d, both of which depend on n, appear in the differential Eq. (8.6.10). The quantity n3/2C, which we henceforth abbreviate as A, varies almost linearly with n and is well approximated by A = 0.473n + 0.208

(8.7.5)

as numerical solution of Eq. (8.6.10) for several values of n shows (Dresner, 1991, “Theory”). In ordinary units, Eq. (8.7.4) becomes p(0,t) = A ρc 2 (4fZ/D ) 3/2(D/4fct)

(8.7.6)

Except for the numerical value of A, Eq. (8.7.6) has been derived entirely on the basis of group-theoretic arguments. The two self-consistency conditions, namely that the frictional term in Eq. (8.1.2) be negligible, and that the second term in the bracket in Eq. (8.5.5) be negligible, have been investigated in Dresner, 1991, “Theory,” where it is shown that n must be > 2/3 and p(0,t) < < rc 2 for Eq. (8.7.6) to be valid. For supercritical helium at low temperatures rc 2 ~ 10 MPa.

8.8. SLUG FLOW Equation (8.7.6) gives the pressure rise at the piston caused by a displacement Z of the piston in a time t. Strictly speaking, Eq. (8.7.6) applies to cases in which Z(t) = Xt n, i.e., in which the displacement Z is proportional to a power n of the elapsed time t, and for 2/3 < n < 2. Furthermore, it only applies for pressure rises p(0,t) << rc 2, but since rc 2 ~ 10 MPa for low-temperature supercritical helium, this last condition is not too serious a restriction. The numerical constant A, which depends on the exponent n, is given by Eq. (8.7.5). Eq. (8.7.6) applies only as long as the disturbance created by the piston has not yet reached the open end of the hydraulic path. Long after the disturbance has reached the open end of the hydraulic path, we take the helium to be in slug flow (which implies the case n = 1). Slug flow means v = X everywhere. Then, from Eq. (8.1.1), it follows at once by integrating that p(0,t) = (2f Dz/D) rX 2

(8.8.1)

where Dz, is the distance from the instantaneous position of the piston to the open end of the tube. When n ≠ 1, we generalize2 Eq. (8.8.1) to p(0,t) = (2f Dz/D) r(dZ/dt) 2

(8.8.2)

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When faired together, Eqs. (8.7.6) and (8.8.2) represent the pressure rise the piston creates by pushing the cold helium through the conductor and thus constitute the needed solution to the first half-problem.

8.9. THE PROPAGATION VELOCITY Normal zones expand because heat is transferred from the resistive part of the conductor to the part that is still superconducting. In magnets cooled with boiling helium or potted magnets, the heat is transferred by conduction through the copper matrix, but in cable-in-conduit conductors, it is largely transferred by the expansion of the hot helium. Then, the velocity of normal zone propagation equals the velocity of expansion of the hot–cold front. This last notion implies that the normal zone engulfs no new helium. Therefore, the heated helium consists only of the atoms originally present in the initial normal zone. The picture is thus of a bubble of hot helium expanding against the cold helium that confines it on either side. We treat the hot helium as a perfect gas having a uniform absolute pressure p and a uniform temperature T. If 2Li is the initial length of the normal zone and 2Lf is its length at a time t, the equation of state of the hot helium is pLf = ri Li RT

(8.9.1)

Eq. (8.9.1) represents the solution to the second half-problem, namely, the state of the hot helium assuming its extent Lf and temperature T are known as functions of the elapsed time t. The two half-problems can now be combined by noting that the pressure p given in Eq. (8.9.1) must equal the pressure p(0,t) + pa , i.e., that the pressure must be continuous across the hot–cold front. If p(0,t) is given by Eq. (8.7.6), if we note that Z = Lf – Li, and if we set h = Lf /L i , we obtain f

h[A (ric 2/p a )(4fLi /D )3/2(D/4fct )(h – 1)3/2 + 1] = ρiRT/pa

(8.9.2)

Here we have used the fact that the density of the cold helium is the same as the initial helium density ρi in the normal zone. If we know T as a function of t (we discuss the function T(t) directly below), we can solve Eq. (8.9.2) to find η(t) and thus the trajectory of the hot–cold front. This must be done numerically, but if the Newton–Raphson method is used, it can easily be done on a programmable pocket calculator, If p(0,t) is given by Eq. (8.8.2) and we set dZ/dt = (Lf – Li )/t, we find h[(2f ri Dz/Dpa )(L i / t )2(h – 1)2 + 1]= ri RT/pa

(8.9.3)

To utilize Eqs. (8.9.2) and (8.9.3), we need the temperature T as a function of time. A simple, reasonable estimate is the adiabatic hot-spot temperature (cf. Eq. (6.13.2)). The volumetric heat capacity S in Eq. (6.13.2) is that of the copper and

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the superconductor together. In the present context, we must also include in S a prorated contribution from the conduit because on the long time scale we are considering (seconds) it may come close to thermal equilibrium with the cable. The temperature calculated from the hot-spot formula is an overestimate for several reasons: (1) we have neglected the heat capacity of the helium; (2) we have ignored the work done by the expanding hot helium on the cold helium confining it; (3) different parts of the normal zone have been resistive for different lengths of time. The temperature we calculate from Eq. (6.13.2) is for the central, hottest point. As the normal zone grows, the mass of metal it engulfs becomes larger and larger while, as noted at the beginning of this section, the mass of helium in the normal zone remains constant. Furthermore, the specific heat of the metal rapidly exceeds that of the helium as the temperature rises. Dresner (1989) presents as an illustration a normal zone in the Westinghouse cable-in-conduit conductor that is initially 2 m long; 3 s after its creation it has expanded to 13 m and its temperature has reached 60 K. By that time, 90% of the Joule heat produced has gone into the metal, 6% having gone into the helium, and 4% into work done by the expanding normal zone on the adjacent cold helium. As noted in Dresner (1989), “The lion’s share of the Joule heat produced in the normal zone goes to the metal.” Regarding the fact that different parts of the normal zone have been resistive for different lengths of time, little more can be said than that use of the hot-spot temperature should be better for smaller expansion ratios h than for larger. Figure 8.4 shows the temperature, pressure, expansion (hot–cold) velocity, and the expulsion velocity3 as functions of time for the Westinghouse conductor for Li = 0.2, 1.0, and 5.0 m. The hot–cold velocity has been calculated using the piston Eq. (8.9.2) for short times and the slug Eq. (8.9.3) for long times and the results joined graphically (cf. Fig. 8.4c). The time of transition can be estimated from the time at which the expulsion velocity approaches the hot–cold velocity calculated from the slug-flow Eq. (8.9.3). The propagation (hot–cold) velocity and pressure both increase as the length of the initial normal zone increases. For short initial zones the pressure is substantially less than that calculated from Eq. (8.2.7). But unless we can be sure of the maximum length of an initial normal zone, the safest design is a conductor that can withstand the maximum pressure given by Eq. (8.2.7). Recently, Shaji and Friedberg (1994I,1994II) have published a comprehensive study of the propagation of normal zones in cable-in-conduit conductors.

8.10. THERMAL HYDRAULIC QUENCHBACK As a normal zone expands it compresses and displaces the cold helium on either side of it. The cold helium is heated by this compression and by the friction accompanying the displacement. Fluid elements quite far from the normal zone can

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Figure 8.4. The temperature, pressure, hot–cold front expansion velocity, and expulsion velocity as functions of time for the Westinghouse conductor of Fig. 7.1 (Dresner, 1989). (Redrawn from an original appearing in Dresner (1989) with permission of the IEEE; © IEEE 1989).

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be heated in this way. If the temperature of these fluid elements should rise to the current-sharing threshold Tcs , the strands that they wet become resistive. Then, rather rapidly, quite long segments of the conductor become normal, giving the appearance of a sudden, sharp increase in the propagation velocity. This acceleration of the normal-superconducting front has been dubbed thermal hydraulic quenchback (THQ). Its existence was first discovered by Luongo et al. (1989, “Thermal hydraulic simulation”) in the course of numerical simulations of thermal expulsion from a cable-in-conduit conductor. The partial theory given below for the time of onset of THQ is due to the author (1991, “Theory”). In contrast to the situation before, after the onset of THQ the normal zone engulfs more and more new helium, causing the entire hydraulic path to become normal very rapidly. This adds additional justification for using the quench pressure Eq. (8.2.7) in design. The first fluid element to reach Tcs is the one next to the hot–cold “piston” because it has the largest velocity and highest pressure of all. In a time dt, the frictional work dW done on a fluid element of length dz next to the piston is dW = ( f/ 2)rV 2 Pdz Vdt

(8.10.1)

where henceforth we shall use the abbreviation V for dZ/dt, the velocity of the piston. Here the first factor is the shear stress exerted on the fluid by the wetted surface, P is the perimeter of the wetted surface, and Vdt is the displacement of the fluid element. If we divide dW by rAHedz , the mass of the fluid element, and integrate over time, we find (8.10.2)

where w is the specific frictional work. The temperature rise caused by this work is w/cp. If we add to this temperature rise that caused by compression, we find the overall temperature rise to be (8.10.3) where p(0,t) is given by the piston Eq. (8.7.6). When n = 1 we can write Eq. (8.10.3) as cp DT/c 2 = (1/2)(4f V 3t/Dc 2) + A (∂T/∂p)v rcp (4fV 3t/Dc 2 )1/2

(8.10.4)

which is a quadratic equation for the dimensionless unknown (4fV 3t/Dc 2)1/2 in which the coefficients are also dimensionless. Let us estimate the size of these coefficients. Typical values (4.2 K, 1 MPa) of the various parameters are r = 150 kg m-3, cp = 3000 J kg-1 K-1, c = 280 m/s, (∂T/∂p)v = 2 K/MPa, and A = 0.681 ( n

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= 1). Then the coefficient A(∂T/∂p)v c p = 0.613; with DT = Tcs – Tb = 1 K, the coefficient cp DT/c 2 = 0.0383. With coefficients of this order, at the positive root of Eq. (8.10.4), the first term on the right is much smaller than the second. Dropping it, we have as the condition for the onset of THQ (4fV 3t/Dc2) 1/2 = [(∂p/∂T) v DT ] /Arc 2

(8.10.5)

This result means that with the given parameters, the onset of THQ is caused by compression, not frictional dissipation. When n ≠ 1, Eq. (8.10.5) should be replaced by (4f Z/D)3/2 (D/4fct) = [(∂p/ ∂T )v DT ] /A r c 2

(8.10.6)

If we add to the assumed values above, the parameters f= 0.02 and D = 0.4 mm, we then find V 3t = 1.53 m3/s2. For velocities much greater than 1 m/s (cf. Fig. 8.4), THQ begins almost at once, supporting in such a case the use of the conservative quench pressure Eq. (8.2.7). The left-hand side of Eq. (8.10.6) varies as t(3n/2–1). In the parts of Fig. 8.4 referring to times less than about 2 s, the velocity is decreasing with time, which means n < 1. For n = 3/4, for example, the exponent 3n/2 – 1 = 1/8. Then, the onset time t varies as (DT)1/(3n/2–1) = (DT)8 .A small change in DT can then be reflected in quite a large change in the onset time t. In an experiment of finite duration, when t is long, THQ is simply not observed, so that the strong dependence of t on DT can appear as a threshold. Such threshold behavior has been observed in the THQ experiments of Lue et al. (1993; 1994). Recently, Shaji and Friedberg (1995) have presented a detailed study of thermal hydraulic quenchback.

8.11. HYDRODYNAMIC QUENCH DETECTION The hydrodynamic disturbances caused by an expanding normal zone can be used to detect an incipient quench by monitoring the efflux of helium from the ends of a hydraulic path (Dresner, 1988). Since we wish to detect a normal zone as early as possible, we are interested in situations in which the velocity of efflux is much less than the piston velocity. The velocity of efflux veff =t a/d y (Dz/t 1/d), where as before Dz is the distance from the instantaneous position of the piston to the open end of the tube. If the velocity of efflux is to be very much less than the piston velocity, y (Dz/t 1/d) must be < < y(0). Now the asymptotic form of y(x) fo r large x is 6/x 2. That y(x) = 6/x2 satisfies Eq. (8.6.10) is easily verified; for a proof that it is the asymptotic form we seek, cf. Appendix A. Substituting this form of y(x) into Eq. (8.6.9) and using Eq. (8.6.8), we find veff = 6t/(D z )2 in special units, or

(8.11.1)

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veff =[3c 2Dt]/[2f (Dz)2]

(8.11.2)

in ordinary units. Using the data given in the previous section for the Westinghouse conductor (c = 280 m/s, D = 0.4 mm, f = 0.02) and taking Dz = 60 m, the half-lengt h of a hydraulic path, we find veff = 33 cm/s at t = 0.5 s. This is roughly twice the ambient flow velocity of 15 cm/s and should be detectable without great trouble. It should be noted that no signal can arrive at the outlet faster than a sound wave, so that no change in the velocity of efflux can be observed until a time t = Dz/c has elapsed. In the foregoing example, this time is 0.21 s.

8.12. RATIONAL DESIGN OF CABLE-IN-CONDUIT CONDUCTORS The author has evolved a design procedure for cable-in-conduit conductors based on the stability and protection considerations discussed in this and the last chapter (Dresner, 1991, “Rational design”). The goals of this design procedure are the two composition variables f and fco, the volume fraction of copper in the strands and the volume fraction of strands in the cable space, respectively. The procedure is best explained by an example (taken from Dresner, 1991, “Rational design”). Suppose we wish to design a 4.0-K, 8-T, high-current-density, NbTi/Cu cable-in-conduit conductor for a fusion magnet. We assume the strand diameter to be 0.7 mm, the residual resistance ratio of the copper to be 100, and take the ambient helium pressure to be 5 atm. Since we want high current density, we aim for 20 kA/cm2 over the cable space. We know already that high current density entails high quench pressure, so we choose a hydraulic path length of only 20 m and decide to limit the quench pressure, if possible, to 50 MPa or less. Figure 8.5 displays a portion of the (fco f)-plane containing a particular triangular region. The region is bounded on the left by a curve that separates compositions (to its left) for which the stability margin is multivalued from compositions (to its right) for which the stability margin is single-valued and equal to the upper stability margin. This curve has been calculated using the scaling law, Eq. (7.6.7).4 The triangular region is bounded on the right by a curve that separates compositions (to its left) for which the maximum quench pressure is < 50 MPa from compositions (to its right) for which the maximum quench pressure is > 50 MPa. This curve has been calculated with Eq. (8.2.7). The triangular region is bounded at the top by a curve above which there is not enough superconductor to carry the current density in the superconducting state. Within the triangle are contours of quench pressure (atm), stability margin (mJ/cm3), and fraction of critical current (dimensionless). We see from the diagram that we can meet the proposed design constraints with f ~0.7 (cu/sc ~ 2.3) and fco ~0.8 (void fraction ~20%), at which point the current density is about two-thirds of critical, the stability margin is ~70 mJ/cm3, and the maximum quench pressure is ~350 atm = 35 MPa.

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169

Figure 8.5. The portion of the (fco f ) plane that corresponds to cable-in-conduit conductors that have single-valued stability, a limited quench pressure, and are able to carry a specified transport current. (Redrawn from an original published in Superconducting Materials and Magnets, International Atomic Energy Agency 1991.)

8.13. PERFORATED JACKETS: MODIFIED HYDRODYNAMIC EQUATIONS The hydraulic path length of 20 m is rather short, and if we hope to maintain a high current density over the cable space, we cannot increase it very much without sending the quench pressure out of bounds. One way to modify the design of the conductor to reduce the quench pressure is to perforate the jacket. Figure 8.6 shows a conceptual arrangement for doing this. The inner cylindrical jacket surrounds the cable and the supercritical helium (region A). Region B, between the two jackets, is space available for pressure relief. The inner jacket is perforated by small holes to bleed helium from region A into region B during a quench. The following two questions immediately come to mind: (1) How big must the holes be and what must their spacing be to provide substantial pressure relief ? (2) Do the holes seriously affect the heating-induced flow and so reduce the stability margin? To answer these questions we need to modify the hydrodynamic equations (7.2.1–3) to account for the presence of a distributed sink of strength -m [dimensions: ML–3T –1], which is how we shall describe the loss of helium from region A by percolation through the wall (Dresner, 1993, “Reducing”). To make this modification we shall have to derive the hydrodynamic equations from first principles by means of mass, momentum, and energy balances over the fixed control volume lying between z and z + dz in space A.

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Figure 8.6. A conceptual arrangement of a double-jacketed cable-in-conduit conductor. The inner jacket is perforated by small holes that bleed helium from region A to region B. (Redrawn from an original appearing in Dresner (1993, “Reducing”) with permission of Butterworth-Heinemann, Oxford, England.)

The mass balance is (8.13.1) which becomes in the limit dz → 0, ∂r/∂t + ∂(rv )/∂z = – m

(8.13.2)

This is the same as Eq. (7.2.1) except for the appearance of the sink term –m on the right-hand side. The momentum balance over the control volume is – mAH evdz – (f /2)ρv 2Pdz

(8.13.3)

The fifth term on the right-hand side is based on the view that the helium leaving space A carries its momentum with it. This seems plausible because as the helium enters the perforations its forward motion is arrested by collision with the walls and its momentum thereby annihilated. The last term on the right-hand side is the frictional retarding force exerted by the cable and the walls of the main channel. As dz → 0, Eq. (8.13.3) becomes ∂(rv )/∂t + ∂(rv 2)/∂t = –∂p/∂ z – mv – rF

(8.13.4)

where, as in Sections 7.2ff., F=2fv 2/D. If we multiply Eq. (8.13.2) by v and subtract it from Eq. (8.13.4.), we obtain

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171

r(dv/dt ) = –∂p/∂z – rF

(8.13.5)

which is the same as Eq. (7.2.2). (Cf. Section 7.2 for the definition of the total (hydrodynamic) derivative, d( ) /dt.) The energy balance over the control volume is

– meAHe dz– p(m/r)A He dz +rqA He dz

(8.13.6)

where e = u + v 2/2 is the sum of the specific internal energy u and the specific kinetic energy v 2/2. The fifth term on the right-hand side is the loss of energy accompanying the helium percolating through the walls. The idea here is that the kinetic energy of the helium departing space A is converted to internal energy by collision with the walls of the perforations and then that energy and the original internal energy are swept out of space A. The sixth term is the pressure work done by the helium remaining in space A in pushing the departing fluid through the perforations. The form of this term is based on the tacit assumptions that the helium percolating into space B does not significantly raise its pressure and that the pressure in space A is much larger than that in space B. As explained in Section 7.2, the frictional force F does not appear in the energy balance. In the limit as dz → 0 and after subtraction of e times Eq. (8.13.2) , Eq. (8.13.6) becomes r(de/dt) = –∂(pv)/∂z – p (m/r) + rq

(8.13.7)

Except for the additional term –p (m/r) on the right-hand side, Eq. (8.13.7) is the same as Eq. (7.2.3). Now we follow the procedure that led to the entropy Eq. (7.2.7), namely, multiplying the momentum Eq. (8.13.5) by v and subtracting from the energy Eq. (8.13.7); we obtain thereby r(du/dt) = – p(∂v/∂z) – p(m/r) + rq + rFv

(8.13.8)

Now we use the mass balance Eq. (8.13.2) to eliminate ∂v/∂z, obtaining precisely Eq. (7.2.6), namely, du/dt + p(dw/dt ) = q + Fv

(8.13.9)

It follows then, as before, by using the second law of thermodynamics, that T(ds/dt) = q + Fv

(8.13.10)

Now if we eliminate r from the mass balance Eq. (8.13.2) using the thermodynamic identity (7.2.8), we obtain, with the help of Eq. (8.13.10)

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dp/dt + rc 2(∂v /∂z) = (βrc 2/cp )(q + Fv) – mc2

(8.13.11)

which is the same as Eq. (7.2.9) except for the extra term –mc2 on the right-hand side.

8.14. PERFORATED JACKETS: REDUCTION OF THE QUENCH PRESSURE Equations (8.13.5) and (8.3.11), like their counterparts Eqs. (7.2.2) and (7.2.9), are partial differential equations that serve to determine p and v. If we set the inertial term, r(dv/dt), to zero in Eq. (8.13.5), as we did at the beginning of this chapter in its counterpart Eq. (7.2.2), these two equations become ∂p/ ∂z = –rF

(8.14.1)

∂p/ ∂t + rc 2 (∂v/∂z) = (brc 2/cp )[q + Fv (1 + cp /bc 2)] – mc 2

(8.14.2)

and

which are identical with Eqs. (8.1.1) and (8.1.2) except for the last term on the right-hand side of Eq. (8.14.2). If we now follow the steps in Section 8.2 that led there to Eq. (8.2.4), we find instead (8.14.3) To go any further, we need to relate the sink strength m to the diameter D´ and the spacing of the perforations. The perforations are short channels of length equal to the thickness of the inner conduit. The helium is presumed to traverse these channels in turbulent flow. If we let s be the porosity of the inner conduit (the ratio of perforation area to total surface area), then m = g(rp)1/2/D1 where g = 4s (D´ /2f´ )1/2

(8.14.4)

where f´ is the Fanning friction factor in the perforations, and D1 = 4AHe /PA, where PA is theperimeter of space A. Eq. (8.14.4) is based on the tacitassumptions, already noted in the last section in connection with Eq. (8.13.6), that the helium percolating into space B does not significantly raise its pressure and that the pressure in space A is much larger than that in space B. If we substitute Eq. (8.14.4) into Eq. (8.14.3), we can rewrite the latter as

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173

Comparing Eq. (8.14.5) with Eq. (8.2.4) we see that the maximum value of po , pm, is reduced by the factor [1+ g(2

3

f /3D 21 )1 / 2]–2

(8.14.6)

compared with the result we would calculate from Eq. (8.2.4) for an unperforated inner conduit. As a numerical example, let us again consider the Westinghouse conductor = 60 m, D = 0.4 mm, f = 0.02, AHe = 88 mm2, P A = 83 mm). Then 6.33 x 105. If the correction factor (8.14.6) is chosen equal to, say, 0.25, then g must have the value 1.58 x 10–6. The jacket thickness is 1.75 mm. Let us take D´ = 0.1 mm for the purpose of illustration and assume f´ = 0.005 (since the perforations are likely to be fairly smooth inside). Then the porosity s = 1.65 x 10–7. The required spacing d of the holes is given by d = (Ahole /AHe )(D1/4 σ)

(8.14.7)

Then d = 0.573 m. Thus 100-µm holes spaced every 57 cm in the Westinghouse conductor should reduce the maximum quench pressure, when an entire hydraulic path goes normal, by a factor of 4.

8. 15. PERFORATED JACKETS: EFFECT ON THE STABILITY MARGIN The next question is whether the perforations adversely affect the heating-induced flow that is responsible for the high upper stability margin. If we treat Eqs. (8.13.5) and (8.13.11) as we treated Eqs. (7.2.2) and (7.2.9) in Section 7.3, they become r(∂v/∂t) + ∂p/ ∂z = 0

(8.15.1)

rc 2(∂v /∂z) + ∂p/∂t = brc 2q/cp – mc 2

(8.15.2)

We can preserve the entire work of Sections 7.3 and 7.4 by replacing q by the effective value q[1 – mc p /(brq)]

(8.15.3)

For the Westinghouse conductor (b = 0.043 K–1, r = 156 kg/m3, cp = 2690 J kg–1 K–1, q = 84 kW/kgHe), the quantity brq/cp = 210 kg m–3 s–1. According to Eq. (8.14.4), when g = 1.58 x 10–6, m = 2.08 kg m–3 s–1 for p = 0.2 MPa, which is a typical pressure rise accompanying heating-induced flow. According to Eq. (8.15.3), the effective value of q to use in the determination of the heating-induced

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flow is less by only 1% than the actual value of q. This means that the perforations in the numerical illustration of the last section have virtually no effect on heatinginduced flow, or by extension, on the upper stability margin. If the porosity of the inner jacket is greatly increased over what has been used in the foregoing illustration, it is likely that the perforations will affect the heatinginduced flow and the stability margin. Just what this effect will be is difficult to say. Imagine, for example, that the cable were disposed as a cylindrical annulus surrounding a large, inner, helium-filled plenum and held in place by a coiled spring. Then there would be almost no hydraulic resistance between the interstitial helium in the cable (region A) and the central plenum (region B). There would likely be strong heating-induced flow transverse to the conductor axis. Luongo et al. (1994, “Helium”) have suggested that such transverse flow might blow the helium out of the cable interstices and into the central plenum very quickly, and there is experimental evidence supporting their contention (Luongo et al., 1994, “Quench”). One presumes the effect on stability of such helium depletion of the cable would be deleterious. Notes to Chapter 8 1It is shown in Appendix A that y ~ x –2 ~ t 2/d z-2 when x > > 1, i.e., when either z > > 1 or t < < 1 or both. But then v ~ t (a+2)/d z–2 = t z–2 , which approaches 0 as t approaches 0. 2The condition for this generalization is that the instantaneous velocity V= dZ/dt should change little during the time it takes sound waves to cross the length _ Dz of the column of moving fluid, i.e., that IdV/dtl(Dz/c) << V. Since V = nXt n –1, this becomes t >> ln-1l(Dz/c). 3 The expulsion velocity has been calculated for early times, where it is small compared with the piston velocity by means of an asymptotic solution to Eq. (8.6.4) valid far from the piston. For a more detailed discussion, see Section 8.11. 4For the sake of completeness, let it be noted that the initial heat pulse was taken to be 10 m long and of 1 ms duration. The exponent m was chosen as 4/15, and the critical current density of pure NbTi at 4.2 K and 8 T was taken to be 1.34 kA/mm2 (from data of Larbalestier, 1986).

9 Cooling with Superfluid Helium

9.1. THE SUPERFLUID DIFFUSION EQUATION As mentioned in the second paragraph of Section 1.7 (which can now be reread with profit), below 2.2 K helium enters a second liquid phase. This phase, called superfluid helium or He-II, has properties that are quite different from those of ordinary liquid helium (called He-I). The nature of these properties has been a subject of intense interest ever since He-II was discovered in Kammerlingh-Onnes’ laboratory in Leiden. Three excellent and extensive books describing superfluid helium are London’s Supefluids (1954), Wilks’s The Properties of Liquid and Solid Helium (1967), and Sciver’s Helium Cryogenics (1986). For the uninitiated reader who wants a brief overview, I recommend a quick read of Chapter 4 of D. K. C. MacDonald’s Near Zero (1961). The most striking properties of superfluid helium are its complete lack of viscosity and its very high thermal conductivity. It is the high thermal conductivity that makes superfluid helium attractive as a coolant. When we use the words “thermal conductivity,” we employ a parlance that is appropriate to Fourier’s law of heat conduction, which says that the heat flux is proportional to the temperature gradient. As it happens, superfluid helium does not obey Fourier’s law, as was first demonstrated by Keesom and Saris (1940). Their experiments showed that instead the heat flux q in superfluid helium is proportional to the cube root of the temperature gradient, i.e., that D (9.1.1) q = – K ( T )1/3 In strict point of fact, this proportionality does not extend to vanishingly small temperature gradients and heat fluxes, but does apply for heat fluxes greater than roughly 0.1 W/cm2. The state of the superfluid helium when the cube-root law (9.1.1) holds is said to be turbulent, but this superfluid turbulence, composed of a tangle of quantized fluid vortices, is different in structure from ordinary turbulence. 175

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Although it may be something of a misnomer, I shall call the transport of heat in turbulent He-II superfluid heat conduction. Likewise, I shall call the coefficient K the superfluid thermal conductivity. When stagnant, subcooled He-II at atmospheric pressure (point P2 in Fig. 1.7) is used as the cryogen in a superconducting magnet, cooling takes place by superfluid heat conduction. Because Eq. (9.1.1) is different from Fourier’s classical law, we cannot use the wealth of known solutions to ordinary heat conduction problems to help us analyze the stability of superconducting magnets. Instead, we must start again at the beginning. The ordinary diffusion equation for heat is derived by substituting Fourier’s expression for the heat flux q into the heat balance (continuity) equation S (∂T/∂t) +

D

·q=0

(9.1.2)

where S is the volumetric heat capacity of He-II. If instead we substitute Eq. (9.1.1) for q into Eq. (9.1.2), we find D (9.1.3) S(∂T/∂t) = D · [K( T) 1/3] I shall refer to Eq. (9.1.3) as the superfluid diffusion equation. It is this equation that governs the temperature distribution in turbulent He-II. Because it is strongly nonlinear (owing to the cube root), we cannot solve it with any of the classical methods designed for linear equations, such as separation of variables, expansion in series, or Laplace transforms. It can be treated by the method of similarity solutions, however, and this method will afford us the solutions we shall need.1

9.2. SUPERCONDUCTOR STABILITY: THE METHOD OF SEYFERT ET AL. Let us consider the following generic problem of superconductor stability: a superconductor is cooled by contact with a closed channel of length L filled with He-II (cf. Fig. 9.1). The thermodynamic state is that denoted by point P2 in Fig. 1.7. Suppose the superconductor is driven normal by a sudden, uniform heat pulse density DH after which it produces a steady Joule heat flux qJ. If DH is small enough, the He-II cools the superconductor well enough to overcome the Joule power and the superconductor recovers. If DH is too large, the He-II cannot cool the superconductor well enough to overcome the Joule power and the superconductor quenches. We want the value of DH that separates these two alternatives, i.e., we want the stability margin. It will prove convenient in what follows to calculate the related quantity E = (DH)V/A, where V is the volume of superconductor and A its area of contact with He-II. The quantity E is thus the bifurcation energy per unit wetted surface. In order to determine E, we need to calculate the rate of transfer of heat from the superconductor to the helium channel. In its full generality, this is a two-region problem, one

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177

Figure 9.1. A closed, He-II-filled channel of length L in contact at one end with a superconductor (Dresner, 1987). (Redrawn from an original appearing in Dresner (1987) with permission of the IEEE; ©IEEE 1987.)

region being the superconductor, the other the helium. But owing to the peculiar properties of He-II, the problem can be reduced to a one-region problem, as Seyfert, Lafferranderie, and Claudet (1982) have shown. If the temperature of the superconductor is high enough, the helium contacting it has a temperature higher than the He-I/He-II transition temperature. Thus a layer of liquid He-I (and possibly a vapor layer) forms adjacent to the superconductor, separating it from the He-II filling the channel. The downstream boundary of this He-I layer has the temperature of the line LL´ in Fig. 1.7 (approximately T l, the lambda temperature of point L). In most practical situations, the heat flux down the channel (and thus across the He-I layer) is a few tens of kW/m2. The temperature difference between the superconductor and T l is a few Kelvins. Since the thermal conductivity of He-I is ~10–2 W m–1 K–1, the thickness of the He-I layer is ~10–6 m. Thus the He-I/He-II interface is always very close to the surface of the superconductor. Furthermore, owing to the extreme thinness of the layer, it has an extremely small heat capacity (~0.4 J m–2 K–1) and so its width can respond to changes of the heat flux in times of the order of tens of microseconds. Looking back from the He-II-filled tube toward the superconductor, one therefore sees a surface whose temperature is clamped at T l, and whose locations is, for practical purposes, that of the superconductor surface. This crucially important observation was first made by Seyfert, Lafferranderie, and Claudet (1982), who summarized the situation succinctly as follows: “At the onset of burnout [transition from He-II to He-I at the surface of the heat bath], formation of the thermal barrier starts. The He-II near the heated surface experiences a phase transition. A He-II/He-I interface appears which has its temperature locked at T l. . . . We assumed that this barrier has a negligible thickness and that it only affected heat transport in He-II by the condition of a constant temperature, i.e., T = T l, at the hot end of the channels in our test section.” Accordingly, the temperature distribution in the channel is given by the solution of the one-region problem comprising Eq. (9.1.3), in one dimension, S(∂T/∂t) = ∂/ ∂z[K( ∂T/∂z)1 / 3]

(9.2.1)

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and the boundary and initial conditions T(0,t) = T l (∂T/∂z)z=L = 0

(9.2.2)

T(z,0) = T b

9.3. SIMILARITY SOLUTION IN A LONG CHANNEL The problem posed in Eqs. (9.2.1) and (9.2.2) has no simple solution as written; but when L → ∞ and S and K are treated as constants, it has a similarity solution (Dresner, 1984, “Transient”), namely, (T– T b )/(T l – Tb ) = 1 – x(x 2 + a2)–1/2

(9.3.1)

where x = z(T l – Tb)1 / 2 (S/Kt)3/4 and a2 =

(9.3.2)

The reader can verify by substitution that Eqs. (9.3.1) and (9.3.2) do indeed satisfy the partial differential Eq. (9.2.1) and the boundary and initial conditions (9.2.2). The manner in which Eq. (9.3.1) is obtained in the first place is explained in Appendix A. According to Eq. (9.3.1), the instantaneous heat flux into the superfluid channel q(0,t) is given by (9.3.3) This flux represents the cooling capacity of an infinite channel filled with superfluid helium. Seyfert et al. (1982) have shown how this result can be used to calculate the stability margin E based on the balance of areas shown in Fig. 9.2. In this figure, the ordinate is the heat flux from the conductor into the helium and the abscissa is the time elapsed since the beginning of the heat pulse. The stepped curve depicts the power production in the superconductor. The initial heat pulse E, which has a duration t1, is the first part of the stepped curve. After the time t1 elapses, the superconductor is assumed to be normal and to be producing a steady Joule heat flux qJ (post-heating). The smooth curve labeled “similarity solution” is the heat flux q(0,t) given by Eq. (9.3.3). At time t2, it crosses the level qJ of the post-heating flux. If the helium has not withdrawn all of the heat produced or deposited in the superconductor by the time t2, the superconductor will not have cooled enough to recover the super-

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179

Figure 9.2. A sketch depicting the balance-of-areas argument of Seyfert et al. (1982). (Redrawn from an original appearing in Seyfert et al. (1982) with permission of Plenum Publishing Corp., New York.)

conducting state. Then the heat flux qJ will persist beyond t2, where it exceeds q(0,t), and the conductor will quench. It is clear from this argument that the largest value of E that still permits recovery is attained when all the heat produced or deposited up to time t2 just equals the heat withdrawn by the superfluid helium. This means that areas A and B in Fig. 9.2 must be equal, i.e., that q(0,t)dt = E + qJ (t2 – t1)

(9.3.4)

0 where t2 = (k/q j ) 4 . If in Eq. (9.3.4) t2 >> t1, we find E = qJt2/3. Combining these last two results with Eq. (9.3.3), we obtain E = K 3S(T l – Tb ) 2/4qJ3

(9.3.5)

This result applies to long (strictly speaking, infinite) channels. If the channel is short enough, the temperature profile in it may be taken as uniform and the maximum amount of heat that can be withdrawn from the superconductor without causing a phase transition in the helium is

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(9.3.6)

Emax =

where is the enthalpy per unit volume of the helium.2 If we define a fiducial heat flux q = KS 1/3(T l–Tb )2/3/(4E max )1/3

(9.3.7)

*

we can combine Eqs. (9.3.5) and (9.3.6) into qJ << q

1,

*

E/Emax =

(9.3.8) (q /qJ ) 3, *

q J >> q

*

9.4. THE TEMPERATURE DEPENDENCE OF THE PROPERTIES OF HE-II The simple estimates (9.3.8), which can be faired together graphically, are based on the assumption that the material properties K and S can be treated as constants independent of temperature. In fact, they are anything but constants. According to Sciver (1986), useful fits to the temperature dependences of K and S are K= 1.04x 105 t5.7(1 –t5.7)Wm–5/3 K–1/3

(9.4.1)

where t = T/Tλ, and S= 1.32 x 106 t5.6 J m–3 K–1

(9.4.2)

The temperature dependences described by these correlations are strong, and so it is difficult to say what effective values to use in Eq. (9.3.7). An analysis of some experimental data in the range 1.8 K < T < 2.1 K led the author to the following empirical rule (Dresner, 1987, “Arapid”): Calculate K and S from Eqs. (9.4.1) and (9.4.2) and correct the value of KS1/3 so obtained by the factor 1.3(T l – Tb )0.6. The method just described has been used by Peck and Michels (1989) to design a 200-kA cable-in-conduit conductor cooled with He-II for use in superconducting magnetic energy storage (SMES). The problem posed in Eqs. (9.2.1) and (9.2.2) has been solved numerically by Seyfert et al. (1982), who used the power 1/3.4 instead of 1/3 in Eq. (9.1.3).

9.5. THE KAPITZA LIMIT The high-flux limit (second line of Eq. (9.3.8)) cannot be valid for arbitrarily high post-heating fluxes qJ. For if it is large enough, the temperature jump at the

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181

Figure 9.3. Sketches showing the relation of the Kapitza flux qK and the Joule power qJ in the case of nonrecovery (a) and in two limiting cases (b,c) (Dresner, 1987a). (Redrawn from an original appearing in Dresner (1987) with permission of the IEEE; ©IEEE 1987.)

helium-superconductor interface induced by the Kapitza resistance (cf. Eq. (4.11.1)) is large enough to keep the superconductor temperature above the current sharing threshold. When this happens, the curve of Kapitza flux (Eq.(4.11.1)) and the curve of Joule heat flux qJ produced by the superconductor (cf. Fig. 3.3) must intersect as shown in Fig. 9.3a. Just as in the discussion in Section 4.4 of Fig. 4.8, the intersection 1 represents an unstable steady state and the intersection 2 a stable steady state. If the metal temperature Tm is driven above T1 by the initial heat pulse, it settles down at T2, and recovery of the superconducting state is impossible. Two limiting cases are possible in each of which the curve of Joule heat flux touches the Kapitza cooling curve in one place only. In the case of Fig. 9.3b, n ) qJmax =qK (Tc) =a(T cn – T He

(9.5.1)

For Joule heat fluxes < qJmax, the Kapitza resistance will not prevent recovery. The critical temperature Tc of NbTi at 8 T is 5.6 K. If we take T He = 1.8 K, and use the values of a and n suggested in Section 4.4 for rough estimates (a = 40 mW cm–2 K–n, n = 3) we find qJmax = 6.79 W/cm 2. In the case of Fig. 9.3c, the condition of tangency at T = To is qJmax/(Tc – Tcs ) =qK (To )/(To – Tcs ) = (dqK/dT)T=To

(9.5.2)

n If T on > > T He , it follows from the last equality in Eq. (9.5.2) that

To = nTcs/(n – 1)

(9.5.3)

For the case of tangency (Fig. 9.3c) to apply, To must be less than the critical temperature Tc. If it is, then qJmax = qK (To )(Tc –Tcs )/(To –Tcs )

(9.5.4)

The critical temperature of NbTi at 3 T is 7.8 K. If we imagine the superconductor to be carrying two-thirds of its critical current, Tcs = 3.8 K (cf. Eq. (4.6.3)). Finally, then, To = 5.7 K. Then qK (To ) = 7.41 W/cm2 and from Eq. (9.5.4), qJmax = 15.6

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W/cm2. The numerical examples show that except for very high heat fluxes, the Kapitza resistance does not prevent recovery.

9.6. THE TWO-DIMENSIONAL CHANNEL Pfotenhauer and Sciver (1986) have studied the stability margin in the two-dimensional channel shown schematically in Fig. 9.4. We should like to know how the stability margin E varies as a function of the Joule power per unit heated surface qs. When qs is small, the transverse temperature distribution (in the x-direction) is nearly uniform, and the channel behaves like a one-dimensional channel of length L in the y-direction subjected to a Joule heat flux qJ = (w/d)qs. The stability margin such a channel is shown in Fig. 9.5 spanning the asymptotes E = Eo = – 2 and E = Cq J–3, where C = K 3S(T l – T b ) /4.

Figure 9.4. The two-dimensional channel studied by Pfotenhauer and Sciver (1986). (Redrawn from an original appearing in Dresner (1987) with permission of the IEEE; ©IEEE 1987.)

Cooling with Superfluid Helium

183

Figure 9.5. A sketch of the stability margin E of the two-dimensional channel of Figure 9.4 as a function of qJ (Dresner, 1987). (Redrawn from an original appearing in Dresner (1987) with permission of the IEEE; ©IEEE 1987.)

When qs is large and d << w, the two-dimensional channel behaves like a one-dimensional channel of length d and Joule heat flux qs = (d/w)qJ. The stability margin of such a channel can be plotted in Fig. 9.5 spanning the asymptotes E = (d/L)Eo and E = Cq s–3 = C(w/d)3 qJ–3 . If these two curves are faired together, we get the right-hand heavy curve that represents how the stability margin varies with qJ when d << w. If d >>w, thetwo-dimensional channel will behavelikea one-dimensional channel –3 = C(w/d)3q –3 , which only for very large q s when E approaches the asymptote C qs J –3 now lies to the left of the asymptote Cq J . We expect E to depart from the asymptote Cq J–3 when it is of the order of (d/L)Eo . The left-hand heavy curve represents how the stability margin varies with qJ when d >> w. Finally, we must add the Kapitza limits (9.5.1) or (9.5.4) to Fig. 9.5 if they apply. Figure 9.6 shows the experimental points of Pfotenhauer and Sciver. In their paper (1986), they noted that the Joule power did not remain constant during the course of an experiment. They defined qJ in such a way that the experimental points lie slightly to the right of the theoretical curve defined by the limits in Eq. 9.3.8. In Fig. 9.6, I have normalized qJ so that the asymptote passes through the cluster of

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Figure 9.6. The experimental values of the stability margin E reported by Pfotenhauer and van Sciver (1986). E0 = 1.43 J/cm2, w = 1.65 mm, L = 50 mm. d1 = 2.5 mm, o) d 1 = 0.7 mm. (Redrawn from an original appearing in Dresner (1987) with permission of the IEEE; ©IEEE 1987.)

points near E/Eo = 0.1. Also shown in the figure are the asymptotes C(w/d)3 q J–3 for both sets of points and the corresponding values of (d/L)Eo, shown as horizontal line segments. Both sets of points behave as expected from the preceding argument. Notes to Chapter 9 keep a narrow focus in this chapter by concentrating on superconductor stability. There are other technological problems that have been studied for ordinary (Fourier-law) fluids that must be reconsidered for He-II. One such problem area is the design of heat exchangers, including such details as heat transfer from tube banks to a He-II bath. Another is bubble growth in He-II and the related problem of cavitation in pumps. A third is the motion of He-II through transfer lines in the presence of a heat leak. A fourth is the motion of He-II through porous plugs, including the design of phase separators and thermomechanical pumps. The literature concerned with these technological applications of superfluid He is quite extensive, and a useful starting point for the interested reader is the proceedings of the biennial Space Cryogenics Conferences published as special issues of Cryogenics. 2Eq. (9.3.6) is based on the tacit assumption that the conductor quenches if the He-II in the channel is all converted to He-I. This assumption results in an underestimate of the stability margin since the He-I also has some cooling capacity. 1I

10 Miscellaneous Problems

10.1. AN UNCOOLED SEGMENT OF A HIGH-TEMPERATURE SUPERCONDUCTOR When a magnet is cooled with a liquid cryogen, the liquid level may inadvertently fall and expose a segment of the conductor. If the uncooled segment becomes normal, it cannot recover the superconducting state because it is uncooled, but its temperature may not rise indefinitely because its ends, which reenter the liquid cryogen, are heat-sunk at the saturation temperature of the cryogen. This is generally the case when the cryogen is liquid helium. At helium temperature, the resistivity of the matrix, which determines the heating rate when the superconductor is normal, is nearly independent of temperature. Then the central temperature of the uncooled normal segment can rise until the temperature gradient is large enough to balance the production of Joule heat by conduction out the heat-sunk ends. When the cryogen is liquid nitrogen, the situation is otherwise because at nitrogen temperature both the resistivity and the thermal conductivity of the matrix vary strongly with temperature. As the central temperature rises so does the Joule power, and to find out what happens in the competition between heating and cooling we must undertake a detailed calculation (Dresner, 1994, “Stability”). The basic heat balance equation we use is Eq. (6.1.1) and we look for steady-state solutions for which T(±a) = Tb , where z = ±a denotes the location of the heat-sunk ends of the uncooled segment. We take the matrix thermal conductivity kcu to be related to the matrix resistivity rcu by the Wiedemann–Franz law, Eq. (2.9.1). Finally, we ignore current sharing, i.e., we assume that the central temperature rise is large enough that the uncooled segment is fully normal over virtually its entire length. As in earlier chapters, we introduce the auxiliary variable s = k(dT/dz) and find that in the steady state s(ds/dT) + kQP/A = 0

(10.1.1) 185

186

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Now, as noted in Section 4.3, Q = rcuJ 2A/fP. Since k = fkcu and kcurcu = LoT, Eq. (10.1.1) can be written s(ds/dT) + Lo J 2T = 0

(10.1.2)

Integrating Eq. (10.1.2) we have at once s2 = Lo J2 (T 2max –T 2 )

(10.1.3)

where Tmax is the maximum temperature, which occurs at z = 0. From Eq. (10.1.3) and the definition of s it follows that 2 2 –1/2 L 1/2 o Jdz = –(L o T dT /ρ)(T max –T )

(10.1.4)

where r = rcu/f is the effective resistivity of the normal conductor. Integration from z = 0 to z = a yields (10.1.5)

10.2. THE CRITICAL LENGTH The functional dependence of Tmax on a is by no means transparent, and to clarify it we consider cases in which r is given by a power law in T: r = rb (T/Tb )n

(10.2.1)

To simplify the evaluation of the integral in Eq. (10.1.5) we introduce two new variables, y and q, defined by y = Tmax /Tb

(10.2.2)

T = Tmax cosq

(10.2.3)

and

Then (10.2.4) The integral on the right-hand side can be evaluated easily for n = 0, 1, 2, 3, 4, and 5 (we go no higher, because n = 5 is the maximum exponent expected from the Bloch–Grüneisen law; cf. Fig. 2.6). The results are

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187

2 1/2 Jarb /L1/2 o Tb = (y – 1)

(n = 0)

= arccos( 1/y)

(n = 1)

= In[y + (y 2 – 1)1/2]/y

(n = 2)

= (y 2–1)1/2/y 2 = {(y 2

1/2 + In[y

–1)

(10.2.5)

(n = 3) 2

1/2

+ (y – 1)

= (y 2 + 2)(y 2 – 1)1/2/3y 4

]/y}/2y

2

(n = 4) (n = 5)

These results are plotted in Fig. 10.1. The following conclusions are evident from these calculations: 1. For all these values of n ≥ 1, a solution for y =Tmax /Tb is possible only when the dimensionless half-length of the uncooled segment Jarb /L 1/2 o T b is less than some critical value. For larger values of this dimensionless half-length there is no solution, i.e., no steady state. Physically, this means that as the central temperature rises, the increase in Joule heating outstrips the increase in conductive cooling and the central temperature runs away. 2. For n = 2, 3, 4, and 5, there are two values of y = Tmax /Tb for each value of the dimensionless half-length Jarb /L1/2 o T b less than the critical value. For one of these (the upper one), Tmax decreases as a increases, while for the

Figure 10.1. The dimensionless peak temperature y = Tmax /Tb versus the dimensionless uncooled length Jarb /L 1/2 o T b for n = 1(1)5. (Redrawn from an original appearing in Dresner (1994, “Stability”) with permission of Plenum Publishing Corp., New York.)

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other (the lower one), Tmax increases as a increases. It is shown in Section B.6 that the upper state is unstable against small perturbations and so it cannot occur in practice. By way of example, let us evaluate the critical length for a ceramic-silver superconductor cooled by liquid nitrogen. In the temperature range 77–200 K, the resistivity of silver varies closely as the 3/2 power of the temperature, so we take n = 1.5. The limiting value of Jarb /L1/2 o Tb is then 0.845. The resistivity rb of silver at Tb = 77 K is 2.9 x 10 –9 ohm-m. If we take the current density in the silver in the normal state to be 5 x 107 A/m2, we find a = 7.0 cm, a comparatively short distance. In a recent EPRI review (1992), Moore has suggested that for high-field applications such as motors, generators, and energy storage, current densities an order of magnitude greater than that given above may be needed. In that case the limiting uncooled half-length would be only 7.0 mm. A cryogen vapor bubble trapped in a tight winding could easily create an uncooled segment this long. The saving grace in this case is the rather large formation energy of the limiting steady state, which has been calculated in Dresner (1994, “Stability”) to be 2.2 J/mm2.

10.3. VAPOR-COOLED LEADS Soon after the discovery of the high-temperature superconductors, it was suggested that they could be used to make current leads whose heat leak into a helium-filled dewar is much less than that of conventional copper leads. To analyze the thermal behavior of such leads, let us consider the conceptual setup shown in Fig. 10.2. There, a current lead extends from a bath of liquid nitrogen at saturation temperature Tc to a bath of liquid helium at saturation temperature T b. The current lead is shown penetrated by a central hole through which passes the helium vapor formed inside the dewar. Such a current lead is called a vapor-cooled lead because the upward vapor flow opposes any downward heat flow through the lead. When the vapor-cooled lead is made of copper, as is the usual practice, the heat that enters the dewar through it and vaporizes liquid helium comes from two sources: (1) Joule power is created in the copper lead by the passage of the current, and (2) heat is conducted down the copper body of the lead from the hot end to the cold end. If we make the cross-section of the lead smaller to reduce conduction, we increase its resistance and therefore we increase the Joule power. If we make the cross-section larger to decrease the resistance and the Joule power, we increase conduction. Clearly, there is an optimum at which the heat leak into the dewar is minimized. Calculations by Lock (1969) show this minimum to lie close to 1 W/kA, the exact value depending on the temperature Tc (which is usually 300K rather than 77 K) and the residual resistivity of the copper.

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189

Figure 10.2. A schematic drawing of a vapor-cooled lead operating between a helium bath and a nitrogen bath.

If we make the lead out of a high-temperature superconductor and keep Tc = 77 K, the lead is superconducting over its entire length and thus produces no Joule power. If we make it long enough, we can make the conduction heat leak through it as low as we please and thus improve its performance beyond that of a copper lead. As it happens, if the superconducting lead is too long, it may become unstable and assume a temperature distribution in which part of its length is normal. This partly normal state only occurs when the length L of the lead exceeds some critical length. Leads shorter than the critical length can only exist in the fully superconducting state and are said to be cryostable. The heat leak for the longest possible cryostable lead is smaller than the heat leak of the optimized copper lead by only a modest factor, roughly 4 in the illustrative calculations described below. In these calculations, when the lead is longer than the critical length, two partly normal states, PN1 and PN2, appear in addition to the fully superconducting state, FS (cf. Fig. 10.3). As the length L becomes greater, the peak temperature of PN1 decreases while that of PN2 increases. We surmise, therefore, that PN2 is stable and PN1 unstable. The state PN1 thus appears to act as a bifurcation state, and the difference between its formation energy and that of FS can be used as a measure of the stability of state FS. Noncryostable leads can be operated in practice just as noncryostable magnets can, so the large anticipated reduction in heat leak they promise should be realizable.

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Figure 10.3. A sketch showing the temperature profiles of the fully superconducting state FS and the two partly normal states PN1 and PN2.

10.4. THE HEAT BALANCE EQUATION FOR VAPOR-COOLED LEADS The thermal behavior of any vapor-cooled lead is governed by the heat balance Eq. (4.4.1). The vapor cooling term qP/A has a special form which we now calculate. Figure 10.4 shows the control volume between z and z + dz of the helium channel. The mass flow of helium up the channel is m [kg/s]. A heat balance over this control volume takes the form

Figure 10.4. The control volume between stations z and z + dz of the helium channel.

Miscellaneous Problems

191

qP = mCp(∂T/∂z)

(10.4.1)

where Cp is the specific heat of the helium and P is the perimeter of the helium channel. Eq. (10.4.1) is based on the assumption of thermal equilibrium between the helium vapor and the lead. In the actual construction of leads, the helium channel is in fact many fine channels in parallel having a very large ratio of exchange surface to volume. Thus the assumption of thermal equilibrium is quite a reasonable one. The mass flow m is related to the heat leak [kA (∂T/∂z)]z=0 into the liquid helium by [kA (∂T/∂z)]z=0 = mCL (10.4.2) where CL[J/kg] is the latent heat of helium. In steady state, Eq. (4.4.1) then becomes d/dz[k(dT/dz)] + QP/A – [k(dT/dz )]z=0(Cp/CL )(dT/dz ) = 0 (10.4.3) which must be solved subject to the boundary conditions T(0) = Tb and T(L) = Tc.

10.5. COPPER LEADS When the lead is made of copper, QP/A =ρJ 2. Again, we introduce the auxiliary variable s = k(dT/dz) and find as before s(ds/dT) + Lo J 2T– sb(Cp/CL)s = 0

(10.5.1)

where sb = s(T b) = [k(dT/dz)] z=0. Now we introduce special units in which Lo J 2 = Cp/CL = 1. The dimensions of these quantities are, respectively, P2L–4 Q–2 , and Q–1 , where P is power and Q is temperature. Then Eq. (10.5.1) becomes s(ds/dT) + T– sbs = 0

(10.5.2)

The direction field of Eq. (10.5.2) is plotted in Fig. 10.5. The smallest value of sb, which is the heat leak into the helium, occurs for the integral curve C for which s = 0 when T = Tc.1 Since Eq. (10.5.2) is homogeneous, its variables may be separated if we introduce u = s/T as a new dependent variable. Then we find dT/T= –[u/(u 2 – sbu + 1)] du

(10.5.3)

which gives upon integration from Tb to Tc (10.5.4)

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Figure 10.5. The direction field of Eq. (10.5.2).

Eq. (10.5.4) is a transcendental equation for the determination of sb given Tc and Tb. The integral in Eq. (10.5.4) is tabulated and equals ln(s2b /T b2 – s b2 /T b + 1)1/2 +sb(4–sb2 ) –1/2arctan[sb(4–s b2 )

1/2

2

/(2Tb –sb )]

(10.5.5)

where the arctan must be given its value in the interval (0,p). According to the tables of Arp and McCarty (1989), Cp = 5.2 J g–1 K–1 for helium at atmospheric pressure over most of the range from saturation (4.2 K) to room temperature. At the same pressure CL = 20.7 J g–1, so that the special unit of temperature is CL/Cp = 3.98 K. Thus Tb = 4.2 K = 1.055 in special units. If Tc = 77 K, then it follows from a little trial and error using Eqs. (10.5.4) and (10.5.5) that sb = 1.68 in special units. Now sb is the heat leak into the helium and has the units PL–2. The special unit of heat leak is then Lo1/2J(C L /C p ). Then, in ordinary units, Sb/J= 1.68Lo1/2 (C L /C p) = 1.05 W/kA

(10.5.6)

When Tc = 300 K, sb = 1.79 is special units, yielding a heat leak of 1.12 W/kA. Thus when we use copper leads, little is to be gained by an intermediate heat intercept at nitrogen temperature, as was pointed out long ago by Williams (1963). These conclusions hold not just for copper leads, but for leads made of any metal which obeys the Wiedemann–Franz law, Eq. (2.9.1). The above figures are very close to the more precise results of Lock (1969), who took into account the weak temperature dependence of the Lorenz ratio kr/T for copper. We can calculate the length L of the optimized copper lead from the formula (10.5.7)

Miscellaneous Problems

193

using the values of s = uT we obtain by integrating Eq. (10.5.3). This calculation is tedious, but if we use an idea of Williams (1963) and assume k is constant, Eq. (10.5.7) can be written with the help of Eq. (10.5.3) as (10.5.8)

The integral in Eq. (10.5.8) is likewise tabulated and equals 2(4–s2b )

–1/2

arctan[s b (4–s b2 )

1/2

2

/(2Tb –s b )]

(10.5.9)

where again the arctan must be given its value in the interval (0,p). When sb = 1.79 (Tc = 300K), L/k = 4.87 in special units. In ordinary units this last equality becomes LL1/2 o J/k = 4.87, which in turn gives the ratio L/A for the optimized lead once the total current has been chosen.

10.6. SUPERCONDUCTING LEADS When the lead is superconducting, Q = 0 in Eq. (10.4.3) and accordingly the second term (the one proportional to T is absent in Eqs. (10.5.1) and (10.5.2). The solution of Eq. (10.5.2) is then s = (T– Tb + 1)sb

(10.6.1)

(special units), which can be written as sb dz = k dT/(T– Tb + 1)

(10.6.2)

Thus, (10.6.3)

Again using Williams’s approximation of constant k, we carry out this integral to obtain sbL = k 1n(Tc – Tb + 1)

(10.6.4)

in special units, or sb L =k(CL /Cp )1n[(Cp/CL )(Tc – Tb + 1)]

(10.6.5)

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CHAPTER 10

in ordinary units. The heat leak sb thus varies inversely as the lead length L and so can be made as small as desired by making L long enough.

10.7. PARTLY NORMAL STATES As noted earlier, if L exceeds some critical length, partly normal states become possible. Then in Eq. (10.4.3), QP/A = 0, T < Tcr = rJ 2 , T >Tcr

(10.7.1a) (10.7.1b)

where Tcr is the critical temperature of the superconductor. For convenience, let us take Tcr = Tc, the saturation temperature of nitrogen. It will prove more convenient in what follows to work directly with Eq. (10.4.3) so we do not introduce the variable s. Again we assume that k is constant and take r = LoT/k in order to fulfill the Wiedemann–Franz law. Finally, we add k = 1 (dimensions PL–1 Q–1 ) to the definition of our special units. In special units, Eq. (10.4.3) then becomes d 2T/dz 2 – sb(dT/dz) = 0, 0 < z < w

(10.7.2a)

d2T/dz2 + T – sb(dT/dz) = 0, w < z < L

(10.7.2b)

where w is the length of the lower, superconducting segment of the lead. Eqs. (10.7.2a) and (10.7.2.b) must be solved subject to the boundary conditions (in special units) T(0) = Tb; T(L) = Tc; T(w) = Tc; (dT/dz )z=0 = sb

(10.7.3)

plus the condition of continuity of dT/dz at z = w. Now Eq. (10.7.2a) can easily be solved to give T = exp(sbz) + Tb – 1

(10.7.4)

which satisfies the first and fourth of the conditions (10.7.3). Then using the third of these conditions, we find w = ln(Tc – Tb + 1)/sb

(10.7.5)

(dT/dz)z=w =sb(Tc – Tb + 1)

(10.7.6)

so that

When w < z < L, the solution of Eq. (10.7.2b) is given by

Miscellaneous Problems

195

T = exp(µz )[Acos(vz ) + Bsin(vz )]

(10.7.7)

where k+ = m + iv and k_ = m – iv are the conjugate roots of the characteristic equation k2 – sbκ + 1 = 0

(10.7.8)

The second and third boundary conditions (10.7.3) then give Tc = exp( µw)[A cos(vw ) + Bsin(vw )]

(10.7.9a)

Tc = exp( µL)[A cos(vL ) + Bsin(vL )]

(1 0.7.9b)

while the condition of continuity of dT/dz at z = w becomes Sb(Tc – Tb + 1) = mTc + vexp( µw)[Bcos(vw ) – A sin(vw )]

(10.7.9c)

Now we proceed as follows: (1) Choose sb and calculate m and v from sb = 2m and m2 + v 2 = 1. (2) Find w from Eq. (10.7.5). (3) Solve Eqs. (10.7.9a) and (10.7.9c) for A and B. (4) Solve Eq. (10.7.9b) for L. These calculations are tedious but straightforward; the results are shown in Fig. 10.6 and discussed in Section 10.8. Similar calculations have been made by Matrone et al. (1989) and Hull (1989).

10.8. PARTLY NORMAL STATES: RESULTS AND DISCUSSION When L exceeds some critical length, partly normal states become possible. The results of illustrative calculations described in Section 10.7 are shown in Fig. 10.6. The abscissa is the dimensionless lead length b = LL1/2 o J/k and the ordinate is the dimensionless product of heat leak and lead length a = (sbL/k )(Cp/cL ). These calculations refer to a case for which Tcr = Tc = 78 K, T b = 4.2 K, and CL /Cp = 3.98 K, k is independent of temperature, and r = LoT/k in the normal state. The numbers alongside the points on the curve are values of the dimensionless peak temperature Tmax ; multiplication by C L /Cp converts them to Kelvins. When the dimensionless lead length b is < 4.989, the only steady state is the superconducting state. Now sb /J = L1/2 o (CL/Cp )(a/b)

(10.8.1)

so that for the longest cryostable superconducting vapor-cooled lead (b = 4.989), sb /J = 0.371 W/kA, which is only slightly better than an optimized vapor-cooled lead. If we use the difference in peak temperature of the lower, unstable partly normal state PN1 and the fully superconducting state FS as a measure of the stability of the latter, we see that increasing the length L of the lead, while reducing the heat leak (10.8.1), also sharply reduces the stability of the superconducting state.

196

CHAPTER 10

Figure 10.6. A plot of the dimensionless product a of heat leak and lead length versus the dimensionless lead length b for the states FS, PN1, and PN2. (Redrawn from an original appearing in Dresner (1991, “Superconductor”) with permission of Butterworth-Heinemann, Oxford, England.)

Furthermore, as the lead length increases, the peak temperature of the upper, stable partly normal state PN2 increases sharply. This means that if a long lead should transfer from the state FS to the state PN2 owing to some perturbation, it may overheat and be destroyed. Three factors mitigate the gloomy picture just presented. First, superconducting leads are often made of rods of high-temperature superconductors and are therefore relatively rigid. Second, they are located in a region where the magnetic field is weak so that they experience only a small Lorentz force. Third, owing to the rapid increase of specific heat with temperature, thermal runaway from state FS through state PN1 to state PN2 may be slow enough that we can intercede when a voltage first develops across the lead. Notes to Chapter 10 1The proof of this statement depends on the following theorem: If s and s are two solutions of Eq. 1 2 (10.5.2) and sb1 > sb2 , then s1 > s2 for all T. It is sufficient to prove it only when s1 and s2 are infinitesimally close. We subtract Eq. (10.5.2) written for s2 from Eq. (10.5.2) written for s1 and set h = s1 – s2; then we find dh/dT + P h = hb , where P = (ds/dT – sb) /s . This first-orderlinear differential equation for h has the solution

Miscellaneous Problems

197

Since exponentials are always positive, the algebraic sign of h and that of hb are always the same, as was to be proved.

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Appendix A The Method of Similarity Solutions

A. 1. PARTIAL DIFFERENTIAL EQUATIONS INVARIANT TO ONE-PARAMETER FAMILIES OF ONE PARAMETER STRETCHING GROUPS Roughly speaking, similarity solutions are special solutions of partial differential equations that can be obtained by solving a related ordinary differential equation. This makes them calculable with a great deal less work than other solutions of the partial differential equation, and that is the chief reason that they interest us. However, not every solution of the partial differential equation is a similarity solution. In fact, similarity solutions are a small subset of the totality of all solutions. So not every problem described by the partial differential equation leads to a similarity solution. But some do, and we profit from the method described here when we succeed in identifying a problem of technological interest that is described by a similarity solution. The partial differential equations that we have treated with the similarity method in this book are (1)

ct = (cn)zz

(Chapter 3: critique of the critical-state model)

(2)

cct = czz

(Chapter 8: hydrodynamic phenomena)

(3)

ct = (c1/3 z )z

(Chapter 9: cooling with superfluid helium)

(A.1.1)

Here we have used c for the dependent variable and z and t for the independent variables for the sake of a uniform notation. The common feature of these equations that allows us to find similarity solutions to them is their invariance to a family of groups of stretching transformations of the form 199

200

Appendix A

c´ =λac 0<λ <

8

t´ = λbt

(A.1.2)

z´ = λz Here l is the group parameter that labels individual transformations of a group and a and b are parameters that label groups of the family. The word “invariance” means that if we imagine the partial differential equations in Eq. (A.1.1) written in terms of the primed variables and then substitute the equations in Eqs. (A.1.2), the powers of l thereby introduced cancel, and we recover the original partial differential equation again, this time written in terms of the unprimed variables. In general, for the powers of l to cancel, the parameters a and b must obey a linear equation of constraint Ma + Nb = L

(A.1.3)

where the coefficients M, N, and L depend on the form of the partial differential equation. For the partial differential equations (1), (2), and (3) above, we find (M,N,L ) = (n–1,1,2), (1,–1,–2), and (2,–3,–4), respectively.

A.2. SIMILARITY SOLUTIONS If c(z,t) is a solution of the invariant partial differential equation, so is c´(z´,t´ ) since the partial differential equation in the primed variables is the same as the partial differential equation in the unprimed variables. In general, the transformed solution c´(z´,t´) is not the same as its pre-image c(z,t). But there may be some solutions that are carried into themselves under transformation by one group of the family (i.e., invariant to the transformations (A.1.2) for all l but for a particular choice of a and b). Solutions of the form c = t a/b y (z/t 1/b )

(A.2.1)

where y is a function of the single variable x = z/t1/ b, have this property. They are the similarity solutions. It can be shown (Cohen, 1931) that all invariant solutions must have the form (A.2.1). When we substitute Eq. (A.2.1) into the partial differential equation, we obtain an ordinary differential equation for the function y(x) since it is a function of only one variable. I call this ordinary differential equation the principal ordinary differential equation. The reader probably realizes that the form (A.2.1) restricts the boundary and initial conditions the similarity solutions fulfill. For example, c(0,t) = y(0)t a/b , a power law in t. Furthermore, if c(∞,t)= 0, then y(∞) = 0, so that then c(z,0) = 0 also.

201

The Method of Similarity Solutions

Different choices of a and b (that fulfill the constraint A.1.3) may lead to different boundary and initial conditions and thus define different problems, but the form A.2.1 is still rather limiting.

A.3. THE ASSOCIATED GROUP The form of the principal ordinary differential equation depends, naturally, on the form of the partial differential equation, but it can be proved (Dresner, 1983) that the principal ordinary differential equation, whatever its form, is invariant to the stretching group y´ =mL / M y 0<m<8

(A.3.1)

x´ = mx I call this group the associated group. The usefulness of the associated group rests on a theorem of Sophus Lie’s (Cohen, 1931; Dresner, 1983), according to which we can reduce the order of the principal ordinary differential equation by one if we use as a new independent variable an invariant of the group and as a new dependent variable a first differential invariant. An invariant is any function f (x,y) that is unchanged under the transformations A.3.1, i.e, any function f(x,y) for which f (x,y) = f (x´,y´) = f (mx,mL/M y). The function u = yx–L/M is such an invariant, and it can be shown that the most general invariant is a function of u. A first differential invariant is any function g(x,y,ý) (here – ý = dy/dx) such that g(x,y,ý) = g(x´,y´,ý´) = g(mx,mL / My , µ L / M 1ý). (N.B.: The L/M –1(dy/dx ) = mL/M –1 ý.) The transformations A.3.1 require that ý´ = dy´/dx´ = m function v = ýx–L/M+1 is a first differential invariant, and it can be shown that the most general first differential invariant is a function of u and v. For all three of the partial differential equations in Eq. (A.1.l), the principal differential equation for y(x) is of second order. Hence the use of u and v as new variables reduces the principal differential equation to first order. I call the first-order differential equation in u and v the associated differential equation. When the second-order principal differential equation is not readily solvable, the first-order associated differential equation may be studied geometrically by means of its direction field.

A.4. ASYMPTOTIC BEHAVlOR Before we turn to a concrete example of how this procedure works, certain additional generalities need to be described. There is a solution that is invariant not

202

Appendix A

just to one group of the family, but to all groups of the family, namely, c = Az L/Mt – N / M , where A is a constant that is determined by the structure of the partial differential equation. When the ratios L/M and N/M are both negative, this totally invariant solution obeys the partial boundary and initial conditions c(∞,t) = 0 and c(z,0) = 0. Often, technologically interesting solutions of the partial differential equation obey the same boundary and initial conditions. For those that do, the totally invariant solution, under additional conditions described below, gives their asymptotic behavior for large z (Dresner, 1993, “General Properties”). The additional conditions are these: The solutions of the partial differential equation must be ordered according to their boundary condition at z = 0. This means that if c1(0,t) ≥ c2(0,t) for t > 0, and if both c 1 and c2 obey the boundary and initial conditions c(∞,t) = 0 and c(z,0) = 0, then c1(z,t) ≥ c2(z,t) for all z and t. The ordering condition is fulfilled for the partial differential equations (l), (2), and (3) given above. For, these partial differential equations have the general form of conservation equations, namely, S(c)ct + qz = 0, where S > 0, and where ∂q/∂cz ≥ 0; it can be shown that solutions of such conservation equations for which c(∞,t) = 0 and c(z,0) = 0 are ordered according to their boundary condition c(0,t) (Dresner, 1993, “General Properties”); the argument is quite similar to that given in Section B.2 of Appendix B.

A.5. EXAMPLE: THE SUPERFLUID DlFFUSlON EQUATION To show how the procedure outlined above works let us consider partial — differential equation (3), for which M = 2, N = –3, L = –4, and A = 4/3√3. The principal ordinary differential equation is (A.5.1) bd(ý 1 / 3)/dx + xý – ay = 0 Choosing as an invariant and a first differential invariant p = u1/2 = xy1/2 and q – v 1/3 = xý 1/3, we find an associated differential equation dq/dp = 2p(bq – q3 + ap 2)/b(2p 2 + q 3)

(A.5.2)

Different choices of a and b correspond to different physical problems, i.e., to different boundary and initial conditions. Three problems of technological interest correspond to the following boundary and initial conditions: (1) c(0,t) = 1, c(∞,t) = 0 and c(z,0) = 0 (α= 0, b = 4/3); (2) cz (0,t) = –1, c(∞,t) = 0 and c(z,0) = 0 (a = 1, b = 2); (3) c dz = 1, c(∞,t) = 0 and c(z,0) = 0 (a = –1, b = 2/3). Since the partial differential equation describes heat transport, we can use the language of that subject to describe these problems: (1) is the problem of an initially uniform semi-infinite tube of He-II the temperature of whose front face is raised and clamped at t = 0; hence, it is called the clamped-temperature problem; (2) is the clamped-flux problem for the same semi-infinite tube; and (3) is the problem of a

203

The Method of Similarity Solutions

tube infinite in both directions subjected to a sudden heat pulse per unit area at the plane z = 0 at time t = 0; it is called the pulsed-source problem. Different values of a and b lead to different forms of the principal and associated ordinary differential Eqs. (A.5.1) and (A.5.2). For the clamped-temperature problem (1) and the pulsed-source problem (3), the principal differential equation is analytically solvable: —

(1)

c(z,t) = 1 –x/(x2 + a2)1/2, a 2 = 8/3√3, x = z/t 3/4

(3)

c(z,t) = t – 3/2 (4/3√ 3)/(x 4 + b4)1/2 ,

(A.5. 3a)

—

b = 2[G(1/4)]2/3(3p)1/2 = 2.854535, x = z/t 3/2

(A.5.3b)

Both of the solutions (A.5.3a) and (A.5.3b) are asymptotic to the totally invariant — solution c(z,t) = (4/3√ 3)z –2t3/2 when x >> 1, as expected. Solutions for different c dz than those given above can be obtained from the values of c(0,t) or solutions (A.5.3a) and (A.5.3b) by scaling c(z,t) with the group (A.1.2) or y(x) with the associated group (A.3.1). Scaling does not affect the asymptotic limit of the solutions since it is totally invariant to the family (A.1.2). For the clamped-flux problem (2), there is no simple solution to the principal differential equation, which must be solved numerically subject to the two-point boundary conditions ý(0) = –1 and y(∞) = 0. To avoid the labor of the shooting method, we turn for help to the associated differential equation. Corresponding to the solution y(x) we seek, there is a curve in the (p,q) plane which we now must identify. Shown in Fig. A.1 is the fourth quadrant of the direction field of (A.5.2) for a = 1 and b = 2. Only the fourth quadrant interests us since p > 0 and q < 0 (because y > 0 and ý < 0). The curves C1 and C2, the loci of zero and infinite slope dq/dp, respectively, divide the direction field into regions in which the slope dq/dp has one sign only. The intersections of these curves, the points O (0,0) and P (2/33/4, –2/31/2), are the singular points of (A.5.2). The totally invariant solution c = Az L/Mt –N/M corresponds to the solution y = L/M Ax of the principal differential equation, which is invariant to the associated group (A.3.1). For this solution, the invariant u = A and the first differential invariant v = (L/M)A. Thus the totally invariant solution maps into a single point in the (p,q) plane, namely, the point (A 1/2,[(L/M )A ]1/3), which is the singular point P. (That the totally invariant solution always maps into a singular point in the (u,v) plane follows from the fact that for the solution y = AxL/M of the principal differential equation, du = dv = 0 as x changes.) Thus the curve in the (p,q) plane that corresponds to the solution y(x) that we seek must pass through the singular point P. Furthermore, since P corresponds to the asymptotic behavior y ~ AxL/M of the solution y(x) that we seek, it corresponds to the limit x = ∞. When x = 0, on the other hand, p = q = 0, and the curve in the (p,q) plane must also pass through the origin O. Only the

204

Appendix A

Figure A.1. The fourth quadrant of the direction field of Eq. (A.5.2) when a = 1 and b = 2.

separatrix S does so. It is the curve in the (p,q) plane defined by the solution y(x) that we seek. Near the origin in the (p,q) plane, the integral curves behave linearly, i.e., p = –Bq. Substituting the definitions of p and q, we find that [y(0)]1/2 = –Bý1/3(0), which means that y(0) = B2 since ý(0) = –1. To find the value of B, we proceed as follows: Since the point P is a saddle point, two separatrices cross it. We can find their slopes — by applying L’Hospital’s rule to (A.5.2.) The negative slope m = –31/4(3 + √17)/6 = –1 .562422. Using this slope to get starting values p = pp – e, q = qp – me near P, we can integrate (6) numerically from P to O and find B = 0.912582. Now we have values of both y(0) (= B 2) and ý(0) (= –1), so we can integrate (A.5.1) in the forward direction. Here a slight problem arises because integrating (A.5.1) in the forward direction carries us along the separatrix S from O towards P. Because the integral curves in the (p,q) plane diverge away from O, integration in the direction from O to P is unstable: a small error (roundoff or truncation) throws us off the separatrix S and we eventually diverge to one side or the other. This instability is reflected in a corresponding instability as we attempt to integrate (A.5.1) in the forward direction. Nevertheless, as a practical matter, it is possible to advance to about x ~ 1 without undue errors; the computed behavior can then be joined to the known asymptotic behavior to achieve a reasonable estimate of y(x).

The Method of Similarity Solutions

205

Fortunately, a way exists to integrate (A.5.1) in the backward, stable direction. We proceed as follows: (1) we choose a point (p,q) on S close to P; (2) guess a (large) value of x, say x 1; (3) calculate y1 and ý1 from the chosen values of p and q ; and (4) use these values of as starting values for a backward, stable integration from x1 to 0. This procedure works for the following reason. Any image point of x1, y1, ý1, say mx 1, m-2 y1 , m–3ý1, has the same values of p and q as the point x 1, y 1, ý1 itself because p and q are functions of the group invariants u and v. Thus any value of x can be made to correspond to anyp and q on the separatrix. In general, the backward integration will not give the curve for which ý(0) has some specified value. But once the curve y(x) has been calculated, it can be scaled with the associated group to a curve with any desired ý(0).

A.6. INFORMATION OBTAINABLE BY GROUP ANALYSIS ALONE Since all the curves y(x) corresponding to different values of ý(0) are images of one another under the associated group (A.3.1), all have the same value B of –y1/2(0)/ý1/3(0) because this quantity is invariant to transformations of the associated group. (Note that the point x = 0 transforms into the point x´ = 0.) From this it immediately follows that c(0,t) = B 2ý2/3(0)t1/2. This formula gives the dependence of the temperature of the front face c(0,t) on the time t and the clamped flux –ý1/3(0), which are the only two parameters in the problem on which it can depend. We could have obtained this formula directly from knowledge of the associated group so that by group analysis alone we can obtain a formula for c(0,t) correct up to a single undetermined constant. To find the value of the constant, however, we must integrate the associated differential equation. The method outlined here does not depend on the partial differential equation being linear. On the other hand, it does depend on the partial differential equation being invariant to a one-parameter family of one-parameter stretching groups. This is a high degree of algebraic symmetry that is only found in the simplest equations. But many equations of technological interest have the high symmetry required and so can be dealt with by the method of this Appendix.

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Appendix B Stability of the MPZ

B.1. THE ORDERING THEOREM The task in this appendix is to show that the MPZ is unstable and that it separates quenching initial conditions from those that recover. To do this we shall study some general properties of the solutions of the time-dependent partial differential Eq. (4.4.1). To simplify the notation and bring it into conformity with the notation of Section A.4, let us set (B.1.1)

and then replace S/k by S(c) and QP/A – qP/A by Q(c). Then Eq. (4.4.1) takes the form S(c)ct = czz + Q(c)

(B.1.2)

The first result we prove is the following ordering theorem: If c1(z,t) and c2(z,t) are two solutions of the partial differential Eq. (B.1.2) for which c1(a,t) = c 2(a,t) and c1(b,t) = c2(b,t) but for which c1(z,0) ≥ c2(z,0), a ≥ z ≥ b, then c1(z,t) ≥ c2(z,t), a ≥ z ≥ b, for all t ≥ 0. In other words, if two solutions obey the same boundary conditions but one starts out bigger than the other, it always remains bigger.

B.2. PROOF OF THE ORDERING THEOREM The proof of the ordering theorem is a straightforward application of the methods of Protter and Weinberger (1967). Let us subtract Eq. (B.1.2) written for c2 from Eq. (B.1.2) written for c1. If we set h = c1 – c2, we find 207

Appendix B

208

Figure B.1. An auxiliary diagram to aid in the proof of the ordering theorem.

S(c)ht = hzz + [Q´(c) – S´(c)ct]h

(B.2.1)

In obtaining Eq. (B.2.1) we have used the law of the mean in the following two ways:

= (∂/∂t )[S(c)h], c 2 ≥ c ≥ c1

(B.2.2)

and Q(c1) – Q(c2) = Q´(c)η , c2 ≥ c ≥ c1

(B.2.3)

Owing to this usage, we do not know the exact values of S(c) and Q´(c) – S´(c)ct in Eq. (B.2.1), but all we need to know is that S(c) > 0 and Q´(c) – S´(c)ct is bounded. In terms of h, the boundary and initial conditions become h(a,t) = h(b,t) = 0 and h(z,0) >0, a ≥ z ≥ b.What we need to prove is that h(z,t) ≥ 0, a ≥ z ≥ b, for all t > 0. To do this we employ a trick of Protter and Weinberger (1967) to overcome the fact that we do not know the algebraic sign of Q´(c) – S´(c)ct. We introduce the auxiliary variable z = he–lt , which satisfies the following partial differential equation and boundary and initial conditions (cf. Fig. B.1): S(c)z t = z zz + [Q´ (c) – S´ (c)ct –lS(c)]z

(B.2.4)

z(a,t) = z(b,t) = 0, z(z,0) ≥ 0, a ≥ z ≥ b

(B.2.5)

Stability of the MPZ

209

If z has a minimum in the rectangle ABCD or on its boundary, that minimum must be ≥ 0. Now we prove that z cannot have a negative minimum at a point P in the interior of the rectangle ABCD in Fig. B.1. For at such a minimum, zt(P) = 0, z zz (P) ≥ 0, and z(P) < 0. If we choose l > max[Q´(c) – S´(c)ct]/min[S(c)]

(B.2.6)

we find that the right-hand side of Eq. (B.2.4) is > 0 while the left-hand side = 0, a contradiction. Furthermore, z cannot have a negative minimum at a point Q in the interior of line segment BC. For at such a minimum, z zz (Q) ≥ 0 and z(Q) < 0 so that zt (Q) > 0. Then yet smaller values of z would exist in rectangle ABCD just below point Q, again a contradiction. Thus the minimum value of z must lie on segments AB, AD, or DC of the boundary. The minimum value there is 0, so that z ≥ 0 in or on the boundary of rectangle ABCD, which is what we wished to prove.

B.3. APPLICATION OF THE ORDERING THEOREM TO THE MPZ The immediate impact of this theorem is that solutions whose initial state is everywhere (a = –∞, b = ∞) greater than the MPZ continue greater than the MPZ and solutions whose initial state is everywhere less than the MPZ continue less than the MPZ. Now the solutions of partial differential Eq. (B.1.2) can evolve in one of three ways: they can grow without bound in amplitude or extent (quench), or they can approach one of the two steady states c = 0 or cMPZ. If is cMPZ unstable against perturbation, it is commonly assumed that the initial conditions that occur in practice diverge away from cMPZ. The initial states < cMPZ and > 0 must recover (i.e., approach the steady state c = 0) whereas initial conditions > cMPZ must quench. In point of fact, however, all instability of a steady state means is that in every neighborhood of the steady state an initial condition exists whose corresponding solution diverges from the steady state. Stability, on the other hand, is a stronger statement. It means that there is some neighborhood of the steady state in which the solution corresponding to every initial condition approaches the steady state. In the case of an unstable steady state, there may well be some initial conditions that approach the steady state, but not all of them. As noted above, it is commonly assumed that the initial conditions that occur in practice diverge from the steady state.

8.4. LAGRANGIAN FORMULATION AND THE STABILITY OF STEADY STATES To prove the instability of the MPZ, we note that Eq. (B.1.2) can be written as an Euler–Lagrange equation

210

Appendix B

S(c)ct = ∂/∂z(∂L/∂cz) – ∂L/∂c

(B.4.1)

where (B.4.2) The action integral of this Lagrangian (B.4.3)

is a nonincreasing function of time (Dresner, 1982):

(B.4.4)

Here we have integrated the first term by parts; the integrated term vanishes since cz (±∞) = 0. The stationary values of A correspond to the steady states. For all nonsteady states, A decreases monotonically with time. When there are just two steady states, the one with the larger action is always unstable. Let the two steady states be c1(z) and c2(z) and let A1 > A2. Consider the one-parameter family of initial conditions c(z,0) = ac 1 + (1 – a)c 2, 0 ≥ a ≥ 1

(B.4.5)

Sketched in Fig. B.2 is the initial action as a function of a, which is shown as a continuous curve. This curve intersects the horizontal line corresponding to the intermediate value A* of the action in point P. As time goes on, all points on this curve move down except the fixed end points A1 and A2 that correspond, respectively, to the steady states c1 and c2. The point P can therefore only move to the right. Since P is bounded on the right by the line a = 1, it must approach a limiting value a* ≥ 1.

Stability of the MPZ

211

Figure B.2. A sketch of the initial action as a function of a when there are two steady states c1 and c2.

If a* < 1, the action of the solution corresponding to the initial condition B.4.5 having a = a* approaches A* as t → ∞ . But then this solution approaches a steady state that is neither c1 nor c2, contrary to the hypothesis that c1 and c2 are the only steady states. Therefore, a* = 1, which means that eventually the point P lies in every neighborhood of a = 1. Thus there are initial conditions arbitrarily close to c1 whose solutions do not approach c1. Thus c1 is unstable, as was to be proved.

B.5. ACTION INTEGRAL OF THE MPZ It remains to prove that the action of the steady state c = 0 is less than that of the MPZ. Now, if we examine Fig. 4.8 we see that Q(c) must have the general form shown in Fig. B.3. Using the same argument that led to Eq. (4.6.2) we find that (B.5.1)

Thus when c < cmax , (B.5.2)

so that LMPZ > 0. Thus AMPZ > 0. On the other hand, A (c = 0) = 0.

212

Appendix B

Figure B.3. The general form of Q(c).

B.6. STABILITY OF THE STEADY STATES OF AN UNCOOLED SEGMENT OF A SUPERCONDUCTOR The problem of Sections 10.1–10.2 can also be recast into the form (B.1.2), now with the boundary conditions c(±a,t) = 0 and the stipulation Q(c) > 0. The Lagrangian is again given by Eq. (B.4.2), but the action is now (B.6.1)

The proof that dA/dt ≥ 0 goes through as before since ct(±a,t) = 0. In Section 10.2, it was found that there are two steady states (when steady states exist at all), and to study their stability we must determine the value of the action A of each. In a manner similar to that of Eq. (B.4.4) we prove that dA/da < 0 for steady states:

= –cz2 (a)/2 < 0

(B.6.2)

since czz + Q(c) = 0 for a steady state and ca(a) = –cz (a) (cf. Fig. B.4).

(B.6.3)

213

Stability of the MPZ

Figure B.4. Geometric relations near the foot of the solution showing that ca(a) = –cz(a).

If we multiply Eq. (B.6.3) by cz and integrate from z = 0 to z = a, we find (B.6.4) Thus A decreases with increasing a and does so faster on the upper branch (larger cmax [or larger Tmax in the parlance of Sections 10.1–10.2]) than on the lower branch. If we go backwards from the single state of largest a (cf. Fig. 10.1), the action A increases on both branches but it increases faster on the upper branch than the lower. Thus for a given a, A is larger on the upper branch than on the lower. From the results at the end of Section B .4, we see that the upper state is unstable and the lower state stable.

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Index

Abrikosov, 5–6 Adiabatic hot-spot formula, 122–124, 163 Bifurcation, 65, 169 Bifurcation energy, 101–110, 176 Bloch–Gruneisen formula, 30–31 Boiling crisis, 55 Boiling film, 54, 79–81, 87–89 nucleate, 54, 79–80, 89 transition, 55 Burnout, 55 Cable-in-conduit conductors, 15, 129–132, 153, 164, 168, 174, 180 rational design of, 167–169 Characteristics, Riemann’s method of 134–140, 152 Compressible flow, equations of, 132–133, 169–172 Convection, 53–54 Copper-to-superconductor ratio, 69 Critical point, 13–14 Critical-state model, 35–38 Critical surface, 9, 16–17 Cryostability, 56–58 Cu/SC ratio: see Copper-to-superconductor ratio Current density, critical, 8, 11, 35 Current redistribution, 90, 95–98 Current critical, 35 eddy, 38 limiting, 146–148

Current ( cont .) minimum propagating, 59–60 shielding, 38, 46, 50 transport, 51 Debye T-cubed law, 21 Diffusivity magnetic, 45 thermal, 45 Dulong–Petit, 19 Epoxy, 12, 105–106, 121 Equal-areas theorem, 62 Essmann–Trauble, 5 Expulsion velocity, 156–158, 164–165 Field critical, 4, 9 lower critical, 5 upper critical, 5 Filaments, twisted, 49–52 Flow induced 132, 134–140, 146 slug, 162–163 Flux creep, 9 flow, 9 jump, 11, 42–52 lattice, 5–9, 35, 38 Fluxoid lattice: see Flux lattice Fluxoids, 5–9, 38 Formation energy, 65–72, 77–78 Free energy: see Gibbs free energy 223

224

Gibbs free energy, 27–28 Heat conduction, transverse, 106–7, 121–122 Heat flux burnout, 55 first critical, 55 maximum nucleate boiling, 55 minimum film boiling, 55 recovery, 55 second critical, 55 Heat transfer boiling, 53–56, 62–64 transient, 77–82, 87–90 Helium boiling, 13 supercritical, 14 superfluid, 13–14, 175–184 Helium-II: see Helium, superfluid Hot spot, 124: see also Adiabatic hot-spot formula Irreversibility curve, 9 Kammerlingh-Onnes, 1–4, 175 Kapitza resistance, 78–79, 180–182 Kohler’s rule, 32 Lambda line, 13 Lambda point, 13 Large Coil Task, 3, 58, 63–4, 136, 148 Lenz’s law, 38, 51 Lorenz constant, 33 Losses ac, 52 coupling, 52 hysteresis, 52 Maddock limit: see Recovery, cold-end; Equalareas theorem Magnetization, 25 Magnetoresistance: see Magnetoresistivity Magnetoresistivity, 31–33 copper, 32 silver, 33 Magnets cryostable, 13 metastable, 13 potted, 12–13 self-protecting, 122–124 Matrix, 10–11 Matthiesen’s rule, 31

INDEX

Meissner effect, 4 Minimum propagating zone, 64–75, 101, 215– 219 Mixed state, 5 MPZ: see Minimum propagating zone Niobium tin, 9, 11, 15, 124, 130, 148 Niobium titanium, 9, 11, 14, 16, 41, 44, 51–52, 71–72, 98, 105, 124, 130, 168 Normal state, 3 Normal zones, 59 propagating, 83–99, 110–122, 163–167 stable, 77 traveling, 90–99 voltage across, 123 Partly convered conductors, stability of, 75–77 Penetration depth, 39–42 Penetration full, 48 incomplete, 47 Phase diagram helium, 13–15 superconductors, 3, 5 Phonon spectrum, 20–23, 80 Phonons, 20–23, 30 Pinning, 8–9, 35 Piston problem, 159–162 Pool cooling, 56 Propagation velocity, 83–99, 110–122, 167 effect of current sharing on, 115–117 measurement of, 83 tranverse, 122 vacuum-insulated composites, 127 Protection, 123–124, 153 Quench detection, hydrodynamic, 167–168 Quench energy, 65, 77; see also Bifurcation energy Quench pressure of internally cooled superconductors, 153–156, 164–165, 168-169 reduction of, 172–173 Quench, 12 Recovery, 13 cold-end, 58–62 Resistance dump, 123 flux-flow: see Resistivity, flux-flow residual: see Resistivity, residual Resistive fault, maximum allowable, 72–75

INDEX

Resistivity flux-flow, 7–8, 38 ice-point, 31 power-law, 40, 108, 186 residual, 2 Riemann invariants, 135, I52 Saturation line, 13–14 Sausaging, 41 Silsbee, 9 Similarity solutions, 52, 157, 160–162, 178, 207–213 SMES (superconducting magnetic energy storage), 90, 110, 180 Sommerfeld constant, 21–22 Specific heat contribution of magnetic field to, 29–30 Debye theory of, 19–23 jumps in, 22 power-law, 108 superconductors, 22–30 superfluid helium, 180 zero-field, 28–29 Stability margin. 90–91, 131, 141–146, 151, 176–184 effect of perforated jacket on. 173–174 lower, 146, 151–152 in a two-dimensional superfluid channel, 182–184 upper, 146, 168–169 Stability cryogenic, 57 multiple, 141–152, 168–169 self-field, 51–52 unconditional, 57 Stable state, 42 Stekly number, 57 Superconductors ceramic: see Superconductors, high-temperature

225

Superconductors ( cont.) composite, 10 hard: see Superconductors, type-II high-temperature, 15–18, 52, 101, 105, 107– 110, 124 thermal stresses in, 124–127 uncooled segment of, 185–188,220 soft: see Superconductors, type-I type-I, 4 type-II, 4 Superheat, 53 Takeoff, 80, 146–147, 151–152 Temperature bath, 57 critical, 3–4 current-sharing threshold, 37 Debye, 20–21 Thermal conductivity, 33–34 superfluid, 176, 180 Thermal expulsion, 156–158 Thermal hydraulic quenchback, 164–167 Thermo-Excel-C, 62 THQ: see Thermal hydraulic quenchback TNZ: see Normal zones, traveling Tore-II Supra, 14 Training, 13 Traveling wave, 60, 110–121, 123 Twist pitch, 52 Unstable state, 42 Vapor, accumulation of in channels, 56, 62–63 Vapor-cooled leads, 188–196 Vaporization, latent heat of, 81, 192 VCL: see Vapor-cooled leads Wave diagram, 135–139 Wiedemann–Franz law, 33–34, 45, 85, 108, 113, 117, 119, 185, 192, 194