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Stirling and Pulse-tube Cryo-coolers

Book Description

Modern technology calls increasingly for provision of cooling at cryogenic temperatures: super-conductivity research; imaging equipment for search-and-rescue; contemporary diagnostic medicine (MRI – magnetic resonance imaging); space exploration; advanced computer hardware; military defence systems. Where it is desirable to generate the cooling effect close to the point of heat removal, electrically powered Stirling and pulse-tube machines offer advantages over traditional, passive systems (Leidenfrost and Joule-Thomson).

Until now there has been no agreed approach to the thermodynamic design of either type. In particular, the choice of regenerator packing has remained a matter for time-consuming – and thus expensive – trial-and-error development. There has been no way of knowing whether an existing 'fully developed' unit is performing to the limit of its thermodynamic potential.

Stirling and Pulse-tube Cryo-coolers addresses these problems.

Features include:

  • An ideal cycle for the pulse-tube yielding heat, mass-flow and work;

  • Previously unseen phenomena of real gas behaviour;

  • Pictorial reliefs of pressure wave interactions;

  • Multiple wave reflections in graphic perspective

  • First solution of the 'regenerator problem ' by a full, unsteady gas dynamics treatment;

  • First ever depiction of pulse-tube boundary-layer events (heat conduction, 'streaming') driven by interacting left-and right-running pressure waves

  • First analysis of the graded regenerator and optimisation of gas path design;

  • Embryonic 'cook-book' method of ab initio cooler design based on dynamic similarity and thermodynamic scaling.

Stirling and Pulse-tube Cryo-coolers raises the threshold from which first-principles design of regenerative cryo-coolers may start. Those wishing to extend their study of the subject beyond the well-trodden, ideal gas/quasi-steady-state rationalisations will require this book.

Table of Contents

  1. Cover Page
  2. Title Page
  3. Copyright
  4. Contents
  5. Preface
  6. Notation
  7. Chapter 1: Background and scope
    1. 1.1 Introduction
    2. 1.2 Stirling types
    3. 1.3 The basic pulse-tube
    4. 1.4 The thermo-acoustic cooler
    5. 1.5 Scope
  8. Chapter 2: Ideal reference cycles
    1. 2.1 Introduction
    2. 2.2 Stirling cycle – equivalence of volume variations
    3. 2.3 In search of an ideal cycle for the Gifford pulse-tube
    4. 2.4 Coefficient of performance of ideal Gifford cycle
    5. 2.5 Deductions for first-principles pulse-tube design
  9. Chapter 3: Ideal Stirling cycle — real gas
    1. 3.1 Background
    2. 3.2 Role of the ideal cycle in the present study
    3. 3.3 Basic reference cycle
    4. 3.4 Reformulation – the complete ideal cycle
    5. 3.5 Heat quantities
    6. 3.6 Computed results
    7. 3.7 Implications for practical design
  10. Chapter 4: Isothermal Stirling cycle with van der Waals gas
    1. 4.1 A criterion for moving forward
    2. 4.2 The ‘isothermal’ cycle generalized
    3. 4.3 Simulated gas processes
    4. 4.4 Implications for practical cooler design - update
    5. 4.5 Standard solution of cubic equation
  11. Chapter 5: A first model of electro-magnetic dynamics
    1. 5.1 Context
    2. 5.2 Mechanical equations of motion
    3. 5.3 Discretization and normalization
    4. 5.4 The electro-magnetic circuit
    5. 5.5 Gas process model
    6. 5.6 Regenerator pressure drop
    7. 5.7 Regenerator transient thermal response
    8. 5.8 Preparation for solution
    9. 5.9 Specimen simulated performance
    10. 5.10 Deductions from computed performance under rated operating conditions
    11. 5.11 Real gas effects
    12. 5.12 Implications for practical cooler design - update
  12. Chapter 6: Towards a cook-book method of thermodynamic design
    1. 6.1 Background
    2. 6.2 The inevitability of scaling
    3. 6.3 Scaling principles revisited
    4. 6.4 Improvements in or relating to regenerator scaling
    5. 6.5 Similarity of working-space NTU
    6. 6.6 Scaling and experiment
    7. 6.7 Scaling in practice
    8. 6.8 Some realities
    9. 6.9 Similarity and the Stirling prime mover
    10. 6.10 Extension to the regenerative cryo-cooler
    11. 6.11 Insights from unconventional test procedures
    12. 6.12 Zen and the art of scaling
  13. Chapter 7: The Gifford low-frequency pulse-tube
    1. 7.1 Background
    2. 7.2 Equivalent pulse-tube
    3. 7.3 Particle trajectories
    4. 7.4 Integration grid
    5. 7.5 Temperature solutions
    6. 7.6 Specimen temperature solutions
    7. 7.7 Conclusions
  14. Chapter 8: Classic regenerator problem - real gas
    1. 8.1 Introduction
    2. 8.2 Fluid particle paths
    3. 8.3 Temperature solutions
    4. 8.4 Specimen temperature solutions
    5. 8.5 Temperature dependence of matrix material
  15. Chapter 9: The ultimate regenerator?
    1. 9.1 Context
    2. 9.2 Criteria for grading
    3. 9.3 Sample specification
    4. 9.4 Regenerator solutions revisited
    5. 9.5 Cyclic counterflow and graded hydraulic radius
    6. 9.6 ... And graded free-flow area, A ff
    7. 9.7 In conclusion
  16. Chapter 10: A question of streaming
    1. 10.1 Background
    2. 10.2 Acoustic theory revisited
    3. 10.3 Streaming
    4. 10.4 The boundary layer
    5. 10.5 Conservation equations of the boundary layer
    6. 10.6 ‘Acoustic’ streaming
    7. 10.7 Streaming and finite-particle displacement -a Lagrange formulation
    8. 10.8 The next step
  17. Chapter 11: Driving function for pulse-tube events – a gas dynamics option
    1. 11.1 Status quo
    2. 11.2 A role for unsteady gas dynamics
    3. 11.3 Temperature-determined gas dynamics
    4. 11.4 Implementation
    5. 11.5 Application to the cryo-cooler
    6. 11.6 Interim implications for design
    7. 11.7 The equations of temperature-determined gas dynamics
    8. 11.8 Extension to real gas behaviour
    9. 11.9 Approximate wave traverse times
    10. 11.10 Review
  18. Chapter 12: Bridging the gap
    1. 12.1 Non-linear versus linear – or both
    2. 12.2 Linear waves
    3. 12.3 The building blocks of linear wave algebra
    4. 12.4 Linear waves and the Method of Characteristics
    5. 12.5 Unrestricted number of wave reflection sites
    6. 12.6 Applicability to the pulse-tube
    7. 12.7 Further assumptions
  19. Chapter 13: A missing link
    1. 13.1 From Stirling to pulse-tube
    2. 13.2 Particle displacement under linear waves
    3. 13.3 Particle motion and the MoC
    4. 13.4 Integration grid for the pulse-tube regenerator
    5. 13.5 ‘Acoustic’ coordinates
    6. 13.6 Résumé
  20. Chapter 14: Polytropic gas dynamics – and other potential resources
    1. 14.1 Background
    2. 14.2 Shapiro's derivation
    3. 14.3 ‘Polytropic gas dynamics’
    4. 14.4 The Gifford pulse-tube – a non-conformist view
    5. 14.5 Closure
  21. Chapter 15: The pulse-tube cooler with ‘inertance duct’
    1. 15.1 Context
    2. 15.2 Linear wave mechanics and flow friction
    3. 15.3 Extension to arbitrary number of duct elements
    4. 15.4 A computational consideration
    5. 15.5 Uniform isothermal duct with friction
    6. 15.6 Extension to distributed temperature and unlimited number of duct elements
    7. 15.7 Heat exchange intensity (unlimited NTU)
    8. 15.8 Linear versus non-linear
    9. 15.9 Matters arising
    10. 15.10 Extension of ideal cycle to orifice and inertance line variants
  22. Chapter 16: Any other business
    1. 16.1 Status quo
    2. 16.2 A preliminary appraisal of the TRW 3503 pulse-tube
    3. 16.3 Example of data reduction
    4. 16.4 The Gifford pulse-tube – a reappraisal
    5. 16.5 In conclusion
  23. Appendix A: Conditions for equivalence of volume variations
    1. A.1 Kinematics
    2. A.2 Additional dead (unswept) volume
  24. Appendix B: The Ergun equation – and beyond
  25. Appendix C: Modified Newton–Raphson method
  26. References
  27. Name Index
  28. Subject Index