Stirling And Pulse-Tube Cryo-Coolers: Inside Story, The

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; and, 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. The 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; and, 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:
Preface xiii; Notation xvii; Chapter 1 Background and scope; 1.1 Introduction 1.1; 1.2 Stirling types 1.4; 1.3 The basic pulse-tube 1.6; 1.4 The thermo-acoustic cooler 1.10; 1.5 Scope 1.10; 1.5.1 Scope from linear wave theory 1.11; 1.5.2 Scope from the methods of non-linear gas dynamics 1.12; 1.5.3 Scope from extension of Rott's thermoacoustics 1.13; 1.5.4 Scope from incorporation of regenerator theory 1.13; 1.5.5 Scope from taking account of 'real' gas behaviour 1.14; 1.5.6 Scope from similarity and scaling 1.18; 1.5.7 Scope from optimization 1.18; 1.5.8 Scope from continuously graded regenerator packing 1.20; 1.5.9 Scope from re-acquisition of regenerator heat transfer and flow correlations 1.21; Chapter 2 Ideal reference cycles; 2.1 Introduction 2.1; 2.2 Stirling cycle - equivalence of volume variations 2.2; 2.3 In search of an ideal cycle for the Gifford pulse-tube 2.7; 2.3.1 Kittel's ideal COP 2.9; 2.3.2 In the footsteps of Gustav Schmidt 2.10; 2.3.3 Specimen ideal gas processes 2.18; 2.4 Coefficient of performance of ideal Gifford cycle 2.22; 2.5 Deductions for first-principles pulse-tube design 2.26; Chapter 3 Ideal Stirling cycle - real gas; 3.1 Background 3.1; 3.2 Role of the ideal cycle in the present study 3.1; 3.3 Basic reference cycle 3.2; 3.3.1 Reference cycle with ideal gas 3.2; 3.3.2 'Real' gas 3.5; 3.4 Reformulation - the complete ideal cycle 3.7; 3.4.1 Ideal gas 3.7; 3.4.2 'Real' gas 3.8; 3.5 Heat quantities 3.9; 3.5.1 Basic thermodynamic relationships 3.9; 3.5.2 Engineering thermodynamics form 3.12; 3.5.3 Application to the ideal gas 3.13; 3.5.4 Application to the 'real' gas 3.14; 3.6 Computed results 3.15; 3.7 Implications for practical design 3.16; Chapter 4 Isothermal Stirling cycle with van der Waals gas; 4.1 A criterion for moving forward 4.1; 4.2 The 'isothermal' cycle generalized 4.1; 4.2.1 Simplifying assumptions 4.1; 4.2.2 'Integral' formulation adapted to van der Waals gas 4.2; 4.2.3 Equation of state in terms of simulation variables 4.6; 4.3 Simulated gas processes 4.9; 4.4 Implications for practical cooler design - update 4.11; 4.5 Standard solution of cubic equation 4.11; Chapter 5 A first model of electro-magnetic dynamics; 5.1 Context 5.1; 5.2 Mechanical equations of motion 5.2; 5.3 Discretization and normalization 5.4; 5.4 The electro-magnetic circuit 5.5; 5.4.1 Instantaneous solenoid force 5.5; 5.4.2 Determination of operating point 5.7; 5.5 Gas process model 5.9; 5.5.1 Energy equation for variable-volume spaces 5.9; 5.5.2 Gas law 5.11; 5.5.3 Mass conservation 5.12; 5.5.4 Evaluation of working-space NTU 5.12; 5.6 Regenerator pressure drop 5.13; 5.6.1 Distributed pressure drop 5.14; 5.6.2 Pressure drop based on mean flowrate 5.15; 5.7 Regenerator transient thermal response 5.16; 5.8 Preparation for solution 5.17; 5.9 Specimen simulated performance 5.19; 5.10 Deductions from computed performance under rated operating conditions 5.25; 5.11 Real gas effects 5.27; 5.12 Implications for practical cooler design - update 5.29; Chapter 6 Towards a cook-book method of thermodynamic design; 6.1 Background 6.1; 6.2 The inevitability of scaling 6.1; 6.3 Scaling principles revisited 6.2; 6.4 Improvements in or relating to regenerator scaling 6.3; 6.5 Similarity of working-space NTU 6.5; 6.6 Scaling and experiment 6.10; 6.7 Scaling in practice 6.12; 6.8 Some realities 6.12; 6.9 Similarity and the Stirling prime mover 6.18; 6.10 Extension to the regenerative cryo-cooler 6.21; 6.11 Insights from unconventional test procedures 6.23; 6.12 Zen and the art of scaling 6.26; Chapter 7 The Gifford low-frequency pulse-tube; 7.1 Background 7.1; 7.2 Equivalent pulse-tube 7.2; 7.3 Particle trajectories 7.2; 7.4 Integration grid 7.5; 7.5 Temperature solutions 7.8; 7.6 Specimen temperature solutions 7.11; 7.7 Conclusions 7.13; Chapter 8 Classic regenerator problem - real gas; 8.1 Introduction 8.1; 8.2 Fluid particle paths 8.1; 8.2.1 Mass of ideal gas contained between entry at TE and a plane at fractional distance x/Lr from entry 8.2; 8.2.2 Determination of fractional linear distance, [lambda] (= x/Lr), occupied by specified fraction, [nu] (= m[lambda]/Mr), of regenerator fluid content - ideal gas case 8.3; 8.2.3 Determination of fractional linear distance, [lambda] (= x/Lr), occupied by specified fraction, m[lambda]/Mr, of regenerator fluid content - any working fluid in which density, [rho], is a function of pressure, p, and temperature, Tg, viz. [rho] = [rho] (p, Tg) 8.5; 8.3 Temperature solutions 8.7; 8.3.1 Aspects of formulation common to ideal and real gas 8.7; 8.3.2 Enthalpy change - van der Waals gas 8.8; 8.4 Specimen temperature solutions 8.10; 8.5 Temperature dependence of matrix material 8.12; Chapter 9 The ultimate regenerator? 9.1 Context 9.1; 9.2 Criteria for grading 9.2; 9.3 Sample specification 9.5; 9.4 Regenerator solutions revisited 9.9; 9.5 Cyclic counterflow and graded hydraulic radius 9.12; 9.6... and graded free-flow area 9.15; 9.7 In conclusion 9.19; Chapter 10 A question of streaming; 10.1 Background 10.1; 10.2 Acoustic theory revisited 10.1; 10.2.1 Linear waves; duct of finite length; graduation of temperature 10.2; 10.2.2 Corresponding 'static' process 10.5; 10.3 Streaming 10.8; 10.4 The boundary layer 10.8; 10.5 Conservation equations of the boundary layer 10.10; 10.6 'Acoustic' streaming 10.15; 10.7 Streaming and finite-particle displacement - a Lagrange formulation 10.18; 10.7.1 Gas process model 10.18; 10.7.2 Mass conservation 10.18; 10.7.3 Momentum conservation 10.19; 10.7.4 Energy conservation 10.20; 10.7.5 Preparation for solution and specimen results 10.20; 10.7.6 Some reservations 10.23; 10.8 The next step 10.24; Chapter 11 Driving function for pulse-tube events - a gas dynamics option; 11.1 Status quo 11.1; 11.2 A role for unsteady gas dynamics 11.3; 11.3 Temperature-determined gas dynamics 11.5; 11.4 Implementation 11.5; 11.5 Application to the cryo-cooler 11.8; 11.6 Interim implications for design 11.15; 11.7 The equations of temperature-determined gas dynamics 11.15; 11.8 Extension to real gas behaviour 11.26; 11.9 Approximate wave traverse times 11.27; 11.10 Review 11.28; Chapter 12 Bridging the gap; 12.1 Non-linear versus linear - or both 12.1; 12.2 Linear waves 12.1; 12.2.1 Assumptions 12.2; 12.2.2 The wave equation 12.2; 12.3 The building blocks of linear wave algebra 12.4; 12.4 Linear waves and the Method of Characteristics 12.6; 12.5 Unrestricted number of wave reflection sites 12.16; 12.6 Applicability to the pulse-tube 12.21; 12.6.1 Optional transformation to 'pseudo-uniform' acoustic speed c0 12.21; 12.6.2 Geometry of equivalent one-dimensional duct in terms of that of the gauze 12.22; 12.6.3 Parameters of operation 12.22; 12.7 Further assumptions 12.23; Chapter 13 A missing link; 13.1 From Stirling to pulse-tube 13.1; 13.2 Particle displacement under linear waves 13.1; 13.3 Particle motion and the MoC 13.4; 13.4 Integration grid for the pulse-tube regenerator 13.10; 13.4.1 Redevelopment of the wave equation 13.10; 13.4.2 Construction of the integration grid 13.11; 13.5 Acoustic coordinates 13.14; 13.6 Resume 13.18; Chapter 14 Polytropic gas dynamics - and other potential resources; 14.1 Background 14.1; 14.2 Shapiro's derivation 14.1; 14.3 Polytropic gas dynamics 14.6; 14.3.1 Acoustic speed 14.6; 14.3.2 State plane relationships 14.7; 14.3.3 Evaluation of polytropic index 14.8; 14.4 The Gifford pulse-tube - a non-conformist view 14.10; 14.4.1 Equivalent piston motion 14.10; 14.4.2 Tentative application of the MoC 14.11; 14.4.3 Linear wave analysis 14.12; 14.5 Closure 14.15; Chapter 15 The pulse-tube cooler with 'inertance duct' 15.1 Context 15.1; 15.2 Linear wave mechanics and flow friction 15.2; 15.3 Extension to arbitrary number of duct elements 15.5; 15.4 A computational consideration 15.6; 15.5 Uniform isothermal duct with friction 15.7; 15.6 Extension to distributed temperature and unlimited number of duct elements 15.7; 15.7 Heat exchange intensity (unlimited NTU) 15.16; 15.8 Linear versus non-linear 15.20; 15.9 Matters arising 15