Effects of Acoustic and Fluid Dynamic Interactions in Resonators: Applications in

Thermoacoustic Refrigeration

A Thesis

Submitted to the Faculty

of

Drexel University

by

Dion Savio Antao

in partial fulfillment of the

requirements for the degree

of

Doctor of Philosophy

June 2013

ii

© Copyright 2013

Dion Savio Antao. All Rights Reserved

iii

If you can trust yourself when all men doubt you, But make allowance for their doubting too;

If you can meet with triumph and disaster And treat those two imposters just the same;

Or watch the things you gave your life to broken, And stoop and build 'em up with wornout tools;

Rudyard Kipling, If, ln. 2, 6 and 8

Omni autem cui multum datum est multum quaeretur ab eo et cui

commendaverunt multum plus petent ab eo

LVKE 12:48

iv

Dedications

I would like to first dedicate this dissertation to my role models, my biggest critics and my most fervent supporters, my mum and dad. Next I would like to dedicate this dissertation to my sister; may it motivate her to achieve her own goals and ambitions in life and reach for the stars. Finally, I would like to dedicate this dissertation to the people of India; the taxpayers whose contributions helped provide for my education (from kindergarten to university) and that of so many others.

v

Acknowledgements

At this point, it is important to thank the people and recognise and acknowledge the contributions that made this dissertation possible. First I must thank my dad and mum for their constant love and support and for teaching me (in both words and deeds) that hard work and perseverance are the most important keys to success. I would also like to thank my sister Noelle for her love and support and Vera and Uday Kunte for welcoming me into their family and making their home my home away from home.

This dissertation would not be possible without the guidance of my adviser Prof.

Bakhtier Farouk. His constant drive to do great research and publish has always inspired

(and will always inspire) me to push myself a bit harder and a bit further. I also would like to thank my committee members (Prof. N. Cernansky, Prof. A. Lau, Prof. S. Bose,

Prof. A. Clyne, Prof. J. Tangorra and Prof. M. McCarthy) for their useful suggestions that helped improve my dissertation research.

I must acknowledge the NSF grant (CBET-0853959) and CFIC (Troy, NY) for supporting the research work reported in this dissertation. I am grateful for the financial support received from the department of Mechanical Engineering and Mechanics (MEM) in the form of teaching assistantships, the Freshman Design Education Fellowship and the Frederic O. Hess Assistantship. I am very grateful to the machinists in the Drexel

Machine Shop (Mark Shiber, Paul Velez, Earl Boling and Scott Eichmann) for their guidance which resulted in the two fabricated experimental setups reported in this dissertation. I would like to acknowledge the help and guidance of Dr. Ray Radebaugh

(National Institute of Standards and Technology (NIST), Boulder, CO) and Dr. Greg vi

Swift (Los Alamos National Laboratory (LANL), Albuquerque, NM), both of whose suggestions guided my initial experiments in cryogenics and thermoacoustics. I would also like to acknowledge Dr. Phil Spoor, Dr. John Corey (both Chart Industries) and Prof.

Bart Lipkens (Western New England University) for their help during various stages of this dissertation research. I would also like to thank the undergraduate students in the

MEM department who worked with me during my dissertation research, especially

Michael Loftus.

Finally, I am thankful to the MEM department and Drexel University for providing an excellent graduate education and an invaluable learning experience. I would like to thank my colleagues and friends in the MEM department and the

Combustion and Energy group, especially Nusair Hasan, Ersin Sayar and Viral Chhasatia for their camaraderie and always being there to discuss new ideas. vii

Table of Contents

List of Tables ...... xvi

List of Figures ...... xviii

Abstract ...... xxxii

Chapter 1 : Introduction ...... 1

1.1. Background ...... 1

1.1.1. Acoustics ...... 1

1.1.2. Thermoacoustics ...... 6

1.1.3. Cryogenics ...... 7

1.2. Dissertation statement ...... 10

1.3. Motivation for the dissertation research ...... 11

1.3.1. Development of accurate multi-dimensional PTR models ...... 11

1.3.2. Investigation of wave-shaping for cryocooler performance enhancement . 14

1.4. Objectives of the dissertation research ...... 16

1.5. Overview of the dissertation ...... 18

Chapter 2 : Background and Literature Review ...... 21

2.1. Acoustic resonators: consonant and dissonant systems ...... 21

2.1.1. Consonant cylindrical resonators ...... 21

2.1.1.1. Limitations of consonant cylindrical resonators ...... 22 viii

2.1.2. Dissonant wave-shaped resonators ...... 23

2.1.2.1. Experimental and numerical studies of dissonant wave-shaped

resonators ...... 24

2.2. Cryogenic refrigeration systems ...... 27

2.2.1. Dilution refrigeration ...... 28

2.2.2. Adiabatic demagnetization refrigeration (ADR) ...... 29

2.2.3. Sorption refrigeration ...... 30

2.2.4. Gas cycle refrigeration ...... 31

2.2.4.1. Recuperative cryocoolers ...... 33

2.2.4.1.1. Joule-Thomson cryocoolers ...... 34

2.2.4.1.2. Brayton cryocoolers ...... 35

2.2.4.2. Regenerative refrigerators ...... 37

2.2.4.2.1. Gifford McMahon and Stirling type cryocoolers...... 38

2.2.4.2.2. Pulse tube refrigerators/cryocoolers ...... 41

2.3. Modeling of PTR systems ...... 47

2.4. Acoustic streaming and streaming in PTRs ...... 49

2.5. Identification of research areas ...... 51

Chapter 3 : Numerical Studies of Transport Phenomena in Acoustic Resonators .. 55

3.1. Introduction ...... 55

3.2. Geometry of the consonant and dissonant resonators ...... 56 ix

3.3. Numerical model of the acoustic resonator ...... 58

3.3.1. Governing equations ...... 58

3.3.2. Numerical scheme ...... 61

3.4. Numerical model validation with past computational studies ...... 62

3.4.1. Initial conditions for the numerical model ...... 64

3.4.2. Boundary conditions for the numerical model ...... 64

3.5. Validation results: Temporal pressure variation ...... 65

3.6. Acoustic streaming in a cylinder resonator ...... 69

3.7. Numerical studies of cone shaped resonators: Results and discussion ...... 74

3.7.1. Initial and boundary conditions for the numerical model ...... 76

3.7.1.1. Initial conditions ...... 76

3.7.1.2. Boundary conditions ...... 77

4.7.2. Results: Un-steady processes in the dissonant cone resonator ...... 77

3.7.3. Results: Cycle-steady behavior in the dissonant cone resonator ...... 87

3.8. Non-linear phenomena in dissonant cone resonators ...... 91

3.9. Summary of numerical studies on acoustic resonators and brief conclusions .... 97

Chapter 4 : Experimental Studies of a Pulse Tube Refrigerator...... 100

4.1. Introduction ...... 100

4.2. Characterization of an orifice type PTR (Mk-I) ...... 101

4.2.1. Mk-I Orifice type PTR geometry and fabrication ...... 101 x

4.2.2 Experimental operating conditions ...... 105

4.2.3. Results: Characterization of the orifice type PTR ...... 110

4.2.3.1. Optimum frequency of operation ...... 111

4.2.3.2. Effect of mean operating pressure ...... 113

4.2.3.3. Effect of operating pressure amplitude ...... 115

4.2.3.4. Effect of flow coefficient of the orifice valve ...... 116

4.3. Investigation of the inertance effect on a PTR (Mk-II) ...... 119

4.3.1. Mk-II inertance type PTR geometry and fabrication ...... 120

4.3.2 Experimental operating conditions ...... 123

4.3.3. Results: Characterization of the orifice type PTR ...... 125

4.3.3.1. Effect of inertance tube geometry ...... 127

4.3.3.2. Effect of input electrical power ...... 132

4.4. Chapter summary and brief conclusions ...... 134

Chapter 5 : Numerical Investigation of Flow and Heat Transfer Processes in an In- line Pulse Tube Refrigerator ...... 137

5.1. Introduction ...... 137

5.2. Governing equations of PTR numerical model ...... 139

5.3. Code validation with past computational fluid dynamic studies ...... 142

5.4. Code validation with experimental studies ...... 143

5.4.1. Problem geometry ...... 144 xi

5.4.2. Numerical model conditions ...... 145

5.4.2.1. Boundary conditions ...... 146

5.4.2.2. Initial conditions ...... 147

5.4.2.3. Porous media conditions ...... 147

5.4.2.4. Numerical scheme ...... 148

5.4.3. Model validation: Simulation results and discussion ...... 149

5.4.3.1. Transient processes in the system ...... 149

5.4.3.2. Spatial temperature and velocity profiles ...... 151

5.4.3.3. Comparison of experimental and computational results ...... 153

5.4.3.4. Effects of component wall thickness ...... 155

5.4.4. Summary ...... 164

5.5. Effects of frequency on pulse tube refrigerator performance ...... 165

5.5.1. Problem geometry ...... 165

6.5.2. Numerical model conditions ...... 166

6.5.2.1. Boundary conditions ...... 166

5.5.2.2. Initial conditions ...... 167

5.5.2.3. Porous media conditions ...... 167

5.5.2.4. Numerical scheme ...... 168

5.5.3. Results and discussion ...... 169

5.5.3.1. Temporal processes in the system ...... 169 xii

5.5.3.2. Spatio-temporal temperature and velocity variations ...... 176

6.5.3.3. Steady-periodic multi-dimensional effects ...... 179

6.5.4. Summary ...... 185

5.6. Tapering the pulse tube for acoustic streaming suppression ...... 186

5.6.1. Suppression of streaming ...... 187

5.6.2. Problem geometry ...... 188

5.6.3. Boundary, initial and porous media conditions ...... 192

5.6.3.1. Boundary conditions ...... 192

5.6.3.2. Initial conditions ...... 193

5.6.3.3. Porous media conditions ...... 193

5.6.3.4. Numerical scheme ...... 193

5.6.4. Results and discussion ...... 194

5.6.4.1. Transient processes in the pulse tube refrigerator ...... 194

6.6.4.1.1. Standard OPTR (geometry- A) ...... 195

5.6.4.1.2. Variable diameter hot heat exchanger OPTR (geometry- B) .. 196

5.6.4.1.3. Gas temperature v/s solid temperature ...... 198

5.6.4.1.4. System performance ...... 199

5.6.4.2. Spatial temperature variation in the pulse tube ...... 201

5.6.4.3. Cycle-averaged multi-dimensional transport processes in the pulse

tube ...... 204 xiii

5.6.5. Summary ...... 210

5.7. Effects of the inertance tube on pulse tube refrigerator performance ...... 211

5.7.1. Problem geometry ...... 212

5.7.2. Boundary, initial and porous media conditions ...... 214

5.7.2.1. Boundary conditions ...... 214

5.7.2.2. Initial conditions ...... 214

5.7.2.3. Porous media conditions ...... 215

5.7.2.4. Numerical scheme ...... 216

5.7.3. Results and discussion ...... 216

5.7.3.1. Temporal transport processes in the cryocooler ...... 218

5.7.3.2. Quasi-steady behavior and inertance effect on phase angle

relationships ...... 225

5.7.4. Summary ...... 232

5.8. Conclusions from numerical study of in-line pulse tube refrigerator ...... 233

Chapter 6 : Numerical Studies of Co-Axial type Pulse Tube Refrigerators ...... 236

6.1. Introduction ...... 236

6.2. Problem geometry ...... 236

6.3. Boundary, initial and porous media conditions ...... 238

6.3.1. Boundary conditions ...... 239

6.3.2. Initial conditions ...... 239 xiv

6.3.3. Porous media conditions ...... 240

6.3.4. Numerical scheme ...... 241

6.4. Results and discussion ...... 241

6.4.1. Temporal processes in the system ...... 241

6.4.2. Porous media cold head region ...... 247

6.4.3. Spatio-temporal temperature and velocity variations ...... 250

6.4.4. Steady-periodic multi-dimensional effects ...... 253

6.5. Chapter summary and conclusions ...... 254

Chapter 7 : Summary and Conclusions ...... 257

7.1. Overall summary ...... 257

7.2. Specific conclusions and impact of findings ...... 258

7.2.1. Numerical studies of acoustic resonators ...... 258

7.2.2. Experimental characterization of cryogenic orifice and inertance type PTRs

...... 261

7.2.3. Numerical studies of cryogenic orifice and inertance PTRs ...... 264

7.3. Future work and recommendations for continuing research ...... 268

7.3.1 Recommendations for future work: Acoustic resonator studies ...... 269

7.3.2 Recommendations for future work: PTR studies ...... 270

Bibliography ...... 274

Appendix A ...... 284 xv

A.1. DeltaEC code used to design the Mk-I PTR ...... 284

Vita ...... 289

xvi

List of Tables

Table 3.1: Dimensions of acoustic resonator geometries simulated ...... 58

Table 3.2: List of cases simulated for model validation with published analytical and numerical results (cylinder geometry) ...... 63

Table 3.3: Reference non-dimensional numbers used in non-dimensional form of the governing equations (cylinder resonator) ...... 63

Table 3.4: Resultant acoustic non-dimensional numbers for the cylinder resonator geometry ...... 71

Table 3.5: List of cases simulated for model validation with published analytical and numerical results (cone geometry) ...... 75

Table 3.6: Reference non-dimensional numbers used in non-dimensional form of the governing equations (cone resonator) ...... 75

Table 3.7: Resultant acoustic non-dimensional numbers for the cone resonator geometry

...... 80

Table 4.1: Dimensions and tube materials used in experimental OPTR system ...... 103

Table 4.2: Heat-exchanger and regenerator mesh properties ...... 104

Table 4.3: List of cases studied in the experimental studies of the OPTR ...... 110

Table 4.4: Dimensions and tube materials used in experimental IPTR system ...... 122

Table 4.5: Heat-Exchanger and Regenerator mesh properties ...... 123

Table 4.6: List of experimental cases investigated with Mk-II cryocooler ...... 126

Table 4.7: List of experimental cases investigating the effect of driver input power (Mk-

II)...... 127 xvii

Table 5.1: Dimensions and boundary conditions of the simulated system ...... 145

Table 5.2: List of cases simulated for validation of numerical model with Mk-I experimental cryocooler ...... 147

Table 5.3: Porous Media parameters used in the simulation ...... 147

Table 5.4: Dimensions and boundary conditions of the simulated system with thermal mass of the components incorporated in the model (schematic in Figure 5.2b) ...... 156

Table 5.5: Dimensions and boundary conditions of the simulated system ...... 166

Table 5.6: List of the cases simulated (operating frequency) ...... 167

Table 5.7: Cases of the various conditions (Taper angle and Hot heat-exchanger diameter) simulated ...... 190

Table 5.8: Dimensions of the IPTR system (Mk-I) simulated and time invariant thermal boundary conditions ...... 213

Table 5.9: Porous media parameters used in the simulation ...... 215

Table 5.10: Cases of the various conditions simulated (Mk-I) ...... 216

Table 6.1: Dimensions and boundary conditions of the simulated system (Space

Cryocooler) ...... 238

Table 6.2: List of the cases simulated (Length of Regenerator) ...... 240

xviii

List of Figures

Figure 1.1: Schematic of a cylindrical resonator (i.e. constant cross-section) ...... 2

Figure 1.2: Standing wave profiles a cylindrical resonator at the (a) 1st harmonic

(fundamental natural frequency), (b) 2nd harmonic and (c) 3rd harmonic [4] ...... 5

Figure 1.3: Applications of cryocoolers and cryogenics plotted as a function of their operating temperatures and the refrigeration/cooling power required [9] ...... 8

Figure 1.4: Schematic of a PTR showing the main components: A- Pressure wave generator (yellow arrow), B- Aftercooler HX, C- Regenerator, D- Cold HX, E- Pulse

Tube, F- Warm HX and G- phase control mechanism (orifice and/or inertance and compliance volume) ...... 12

Figure 1.5: Schematics of acoustic resonators (a) cylinder, (b) horn-cone, (c) cone and

(d) bulb [17] ...... 15

Figure 2.1: Temporal pressure profiles at the closed end of a cylindrical resonator with the opposite end vibrating at the fundamental resonant frequency of the resonator [17] . 23

Figure 2.2: Schematic of a sorption-pumped 3He – 4He dilution refrigerator [26] ...... 28

Figure 2.3: Schematic of a NASA ADR system developed for operation in the 2 K – 10

K region [26] ...... 30

Figure 2.4: Schematic of cryogenic recuperative refrigerators (a) Joule-Thomson and (b)

Brayton cryocoolers [9] ...... 32

Figure 2.5: Schematic of cryogenic regenerative refrigerators (a) Stirling, (b) Pulse Tube and (c) Gifford-McMahon cryocoolers [9] ...... 32 xix

Figure 2.6: The reverse Brayton (refrigeration) cycle (a) schematic of the process and (b)

T-s diagram showing the ideal and real thermodynamic states of the cycle [28] ...... 36

Figure 2.7: P-V diagram of an ideal reverse Stirling (refrigeration) cycle [29] ...... 38

Figure 2.8: Schematic of different types of pulse tube cryocoolers (a) U-tube, (b) Co- axial and (c) In-line [12] ...... 46

Figure 2.9: Acoustic streaming patterns (a) external: outside a cylinder oscillating in a quiescent medium [60] and (b) internal: inside a cylinder that has oscillatory flow within it generated by a loudspeaker [61] ...... 50

Figure 3.1: Schematic of acoustic resonator geometries simulated (a) cylinder and (b) cone ...... 57

Figure 3.2: Transient pressure variation at the rigid end of the cylindrical resonator (a) evolution from the start of the simulation to quasi-steady state (case 1), (b) comparison of current model (case 1, solid line) with past studies (Chester’s semi-analytical model

‘delta symbol’ and Aganin et al.’s numerical model ‘diamond symbol’) and (c) comparison of current model (cases 2- solid line and case 3- dotted line) with past studies at higher piston amplitudes (Goldshtein et al.’s numerical model ‘diamond’ and ‘delta’ symbols) ...... 66

Figure 3.3: Transient pressure variation at the rigid end of the cylindrical resonator (at x

= 0, r = R1) from the start of the simulation (a) case 4, (b) case 5 and (c) case 6 ...... 67

Figure 3.4: Comparison of temporal pressure variation at the rigid end of the cylindrical resonator near quasi-steady state (solid lines) with the results of Aganin et al. (symbol)

(a) case 4, (b) case 5 and (c) case 6 ...... 68 xx

Figure 3.5: Cycle averaged temperature contours and velocity streamlines (acoustic streaming) in the cylindrical resonator near quasi-steady state for different operating frequencies (a) 100.809 Hz (case 1, Reacoustic = 459.03), (b) 97.389 Hz (case 4, Reacoustic =

186.15) and (c) 107.641 Hz (case 6, Reacoustic = 153.63) ...... 73

Figure 3.6: Temporal evolution of gas pressure at the rigid end of the cone resonator

(Fig. 1b) filled with CO2 and excited at (a) 1000 Hz (below resonance) and (b) 1187 Hz

(the actual resonant frequency, case 7) ...... 78

Figure 3.7: Transient pressure variation at the rigid end of the conical resonator near quasi-steady state for (a) case 7, (b) case 8 and (c) case 9 ...... 81

Figure 3.8: Pressure variation at the rigid end of the conical resonator near quasi-steady state for (a) case 10, (b) case 11 and (c) case 12 ...... 82

Figure 3.9: Comparison of the ratio of transient pressure variation to mean operating pressure at the rigid end of the conical resonator in CO2 (solid line) and Ar (dashed line) for different piston amplitudes (a) 10 µm (case 7 and case 10), (b) 50 µm (case 8 and case

11) and (a) 100 µm (case 9 and case 12) ...... 84

Figure 3.10: Comparison of the ratio of transient density variation near the rigid end of the conical resonator to the initial density in CO2 (solid line) and Ar (dashed line) for different piston amplitudes (a) 10 µm (case 7 and case 10), (b) 50 µm (case 8 and case

11) and (a) 100 µm (case 9 and case 12) ...... 85

Figure 3.11: Comparison of the ratio of transient temperature variation near the rigid end of the conical resonator to the initial temperature in CO2 (solid line) and Ar (dashed line) for different piston amplitudes (a) 10 µm (case 7 and case 10), (b) 50 µm (case 8 and case

11) and (a) 100 µm (case 9 and case 12) ...... 86 xxi

Figure 3.12: Radial variation of the cycle-averaged gas temperature in the conical resonator at quasi-steady state for a piston amplitude of 10 µm operating with CO2 (case

7) near (a) the fixed end of the resonator (x = 0.001 m) and (b) the piston (x = 0.13 m) . 88

Figure 3.13: Cycle averaged temperature contours in the conical resonator near quasi- steady state for a piston amplitude of 50 µm operating with two different gases (a) CO2

(case 6) and (b) Ar (case 9) ...... 89

Figure 3.14: Cycle averaged temperature contours and velocity vectors (acoustic streaming) in the conical resonator near quasi-steady state for CO2 operating at different values of piston amplitude (a) 10 µm (case 7), (b) 50 µm (case 8) and (c) 100 µm (case 9)

...... 90

Figure 3.15: Frequency response curves of the pressure ratio at the rigid end of the dissonant conical resonator operating with CO2 and a piston amplitude of 50 µm for increasing frequency (square symbol) and decreasing frequency (diamond symbol) ...... 93

Figure 3.16: Comparison of the shift in resonance frequency with increasing piston amplitude observed in a cone shaped resonator operating with CO2 (solid line and square symbol, cases 7 – 9, Table 3.5) and with Ar (dashed line and delta symbol, cases 10 – 12,

Table 3.5) ...... 96

Figure 3.17: Comparison of (a) pressure ratio (pmax/pmin) and (b) temperature ratio

(Tmax/Tmin) as a function of the acoustic Reynolds number observed in a cone shaped resonator operating with CO2 (solid line and square symbol, cases 7 – 9) and Ar (dashed line and delta symbol, cases 10 – 12) ...... 97

Figure 4.1: A photograph of the experimental OPTR fabricated (see Table 4.1 for a listing of the components) ...... 102 xxii

Figure 4.2: Images of (a) the interior of the aftercooler heat-exchanger (left) and the regenerator (right) and (b) close-up image of the regenerator woven wire-mesh screen 105

Figure 4.3: Simplified schematic of the inside of the linear motor pressure wave generator [7] ...... 106

Figure 4.4: Image of the vacuum chamber built for the Mk-I cryocooler enclosing parts of the regenerator and pulse tube and the entire cold heat-exchanger ...... 107

Figure 4.5: Images of the cold heat-exchanger when the cryocooler is operating (a) in ambient conditions (notice the frost formation, Tgas in the cold heat-exchanger ~ 200 K) and (b) inside the vacuum chamber (notice no large scale frost formation, Tgas ~ 185 K)

...... 109

Figure 4.6: Effects of operating frequency on system performance: (a) Temporal variation of gas temperature in the cold heat-exchanger, (b) Pressure amplitude in the transfer tube and the quasi-steady gas temperature in the cold heat-exchanger as a function of the operating frequency (cases 1 – 5 in Table 4.3) ...... 112

Figure 4.7: Effects of mean operating pressure: (a) Temporal variation of gas temperature in the cold heat-exchanger, (b) Pressure amplitude in the transfer tube and the quasi-steady gas temperature in the cold heat-exchanger as a function of the mean pressure (cases 6 – 10 in Table 4.3) ...... 114

Figure 4.8: Effects of pressure amplitude: (a) Temporal variation of gas temperature in the cold heat-exchanger, (b) Pressure amplitude in the transfer tube and the quasi-steady gas temperature in the cold heat-exchanger of the OPTR as a function of the applied input power to the linear motor ...... 115 xxiii

Figure 4.9: The flow coefficient Cv of the needle valve orifice (NOSHOK 101-MMB) plotted as a function of the number of open turns of the needle [89] ...... 117

Figure 4.10: The quasi-steady gas temperature in the cold heat-exchanger and the pressure amplitude in the transfer tube at different values of the orifice flow coefficient

Cv ...... 118

Figure 4.11: Image of experimental Mk-II IPTR housed in vacuum chamber ...... 121

Figure 4.12: Temporal evolution of the gas temperature in the middle of the cold heat- exchanger (component E) for the two inertance tubes radii studied (a) rIT = 0.1524 cm and (b) rIT = 0.1930 cm ...... 128

Figure 4.13: Comparison of the transient variation of the gas temperature in the cold heat-exchanger at comparable values of inertance ‘M’ for the two inertance tube radii studied ...... 129

Figure 4.14: Comparison of the performance of the Mk-II PTR as a function of the inertance ‘M’ for the two inertance tube radii studied (a) Gas temperature in the cold heat-exchanger (dash lines indicate trends) and (b) Pressure amplitude in the transfer tube

...... 131

Figure 4.15: Transient variation of the gas temperature in the cold heat-exchanger for different values of input electrical power (rIT = 0.1930 cm) ...... 133

Figure 4.16: Variation of gas temperature in the cold heat-exchanger and pressure amplitude in the transfer tube as a function of the input electrical power to the linear motor (dash line indicates trend) ...... 134

Figure 5.1: Comparison of current model with past computational studies ...... 143 xxiv

Figure 5.2: A schematic for the OPTR geometry used in the numerical simulations (a) no wall thickness and (b) wall thickness and/or thermal mass of component’s flanges considered ...... 145

Figure 5.3: (a) Temporal variation of the gas and solid temperature at the exit of the cold heat-exchanger for the entire simulation, (b) Gas and Solid Temperature and Pressure profiles at the exit of the cold heat-exchanger at the start of the simulations (case 1, Table

5.2) ...... 150

Figure 5.4: Gas Temperature v/s Solid Temperature: (a) Temperature profiles at the exit of the cold heat-exchanger at a later point in the simulations, (b) Temporal variation of the cycle-averaged gas and solid temperature at the exit of the cold heat-exchanger .... 151

Figure 5.5: Axial distribution of the gas temperature (along the axis of symmetry) at four different instants in cycle 3301 for case 1 (Table 5.2), (a) components B, C, D, E, F, G and H and (b) components I and J ...... 152

Figure 5.6: Axial distribution of the axial component of velocity ‘ux’ (along the axis of symmetry) at four different instants in cycle 3301 for case 1 (Table 5.2), (a) Components

B, C, D, E, F, G and H and (b) components I and J ...... 153

Figure 5.7: Comparison of near-quasi-steady cycle-averaged experimentally (cases 8, 9 and 10, Table 4.3) and computationally (cases 1, 2 and 3, Table 5.2) obtained gas temperature ...... 154

Figure 5.8: Comparison of simulation results (gas and solid temperature) at the exit of the cold heat-exchanger (x = 19.1 cm, r = 0.4699 cm) for case 1 (Table 5.2) with and without component wall thickness ...... 157 xxv

Figure 5.9: Comparison of the cycle-averaged (3301st cycle) gas temperature in the pulse tube section of the OPTR for the simulations (a) without wall thickness considered (case

1, Table 5.2) and (b) with wall thickness and flange mass considered ...... 158

Figure 5.10: Cycle-averaged gas temperature in the pulse tube section of the OPTR for

(a) prescribed initial condition near quasi-equilibrium and (b) at quasi-equilibrium (~

3600 sec) ...... 160

Figure 5.11: Comparison of temporal simulation results and experimental results (case

10, Table 4.3) in the center of the cold heat-exchanger (x = 16.6 cm, r = 0.0 cm) at (a) early time in the simulation and (b) near quasi-steady state ...... 161

Figure 5.12: Comparison of experimental results (cases 11 – 14, Table 4.3) and quasi- steady state simulation results at the center of the cold heat-exchanger (x = 16.6 cm, r =

0.0 cm) ...... 163

Figure 5.13: Grid structure in (a) the cold heat-exchanger to the Orifice valve, (b) the cold heat-exchanger and (c) the hot heat-exchanger and the orifice valve ...... 169

Figure 5.14: Temporal evolution of the temperature at the entrance to the pulse tube for all cases studied for (a) 0 – 18 seconds and (b) 17.5 – 18 seconds ...... 170

Figure 5.15: Temperature and pressure profiles at the exit of the cold heat-exchanger during the initial cycles (cycles 1 – 5) of the OPTR operation for case 1 ...... 171

Figure 5.16: Temperature and pressure profiles at the exit of the cold heat-exchanger during the initial cycles (cycles 1 – 5) of the OPTR operation for case 3 ...... 172

Figure 5.17: Temporal evolution of the gas and solid temperatures at the exit of the cold heat-exchanger for cases 1, 3 and 5 for (a) 0 – 18 seconds and (b) 17.5 – 18 seconds .. 173 xxvi

Figure 5.18: Comparison of the phase-angle shift of axial velocity with respect to the pressure along the axis of symmetry in the pulse tube of the OPTR ...... 174

Figure 5.19: Performance map of the OPTR simulated over the frequency range on 15 –

100 Hz ...... 175

Figure 5.20: Distribution of the temperature (along the axis of symmetry) at four instants in cycle 330 for case 1 (Table 5.6). (a) Components B, C, D, E, F, G and H (see Figure

5.2 and table 5.5) and (b) components I and J ...... 176

Figure 5.21: Distribution of the temperature (along the axis of symmetry) at four instants in cycle 630 for case 3 (Table 5.6). (a) Components B, C, D, E, F, G and H and (b) components I and J ...... 177

Figure 5.22: Distribution of the axial component of velocity ‘ux’ (along the axis of symmetry) at four instants in cycle 630 for case 3 (Table 5.6). (a) Components B, C, D,

E, F, G and H and (b) components I and J ...... 178

Figure 5.23: Cycle-averaged temperature and velocity-streamlines in the pulse tube section (case 1, Table 5.6) ...... 179

Figure 5.24: Cycle-averaged temperature and velocity-streamlines in the pulse tube section (case 3, Table 5.6) ...... 180

Figure 5.25: Cycle-averaged temperature and velocity streamlines in the pulse tube section (case 4, Table 5.6) ...... 181

Figure 5.26: Cycle-averaged temperature and velocity streamlines in the pulse tube section (case 5, Table 5.6) ...... 181

Figure 5.27: Cycle and radially averaged enthalpy flow in the pulse tube section of the

OPTR along the axis of symmetry for cases 1 and 3 (Table 5.6) ...... 183 xxvii

Figure 5.28: Cycle-averaged gas temperature in the pulse tube section of the OPTR along the centerline axis of the system for cases 1 and 3 (Table 5.6) ...... 184

Figure 5.29: General schematic of the OPTR geometries considered (a) Geometry A and

(b) Geometry B ...... 189

Figure 5.30: Schematic of the sign convention used to determine the taper angle of the pulse tube and the diameter of the hot heat-exchanger for (a) Geometry A and (b)

Geometry B ...... 191

Figure 5.31: Temporal evolution of the temperature at the entrance to the pulse tube for geometry- A (cases 1, 2, 3 and 4, Table 5.7) for (a) 0 – 19 seconds and (b) 18.75 – 19 seconds ...... 195

Figure 5.32: Temporal evolution of the temperature at the entrance to the pulse tube for geometry- B (cases 1, 5, 6 and 7, Table 5.7) for 0 – 19 seconds ...... 197

Figure 5.33: Evolution of the gas and solid temperatures as a function of time at the exit of the cold heat-exchanger for (a) Cases 1, 2 and 3 (geometry- A) from 0 – 18 seconds and (b) Case 1 (geometry- A) from 0 – 0.15 seconds ...... 199

Figure 5.34: Performance map of the two OPTRs simulated ...... 200

Figure 5.35: Distribution of the temperature (along the axis of symmetry) at four instants in cycle 650 (components E, F and G) for (a) Case 1 (geometry- A) and (b) Case 3

(geometry- A) ...... 202

Figure 5.36: Distribution of the temperature (along the axis of symmetry) at four instants in cycle 650 (components E, F and G) for (a) Case 2 (geometry- A) and (b) Case 6

(geometry- B) ...... 203 xxviii

Figure 5.37: Cycle-averaged temperature and velocity streamlines in the pulse tube section (case 1 – geometry- A) ...... 205

Figure 5.38: Cycle-averaged temperature and velocity streamlines in the pulse tube section (case 2 – geometry- A) ...... 206

Figure 5.39: Cycle-averaged temperature and velocity streamlines in the pulse tube section (case 4 – geometry- A) ...... 207

Figure 5.40: Cycle-averaged temperature and velocity streamlines in the pulse tube section (case 5 – geometry- B) ...... 208

Figure 5.41: Cycle-averaged temperature and velocity streamlines in the pulse tube section (case 6 – geometry- B) ...... 209

Figure 5.42: Cycle-averaged temperature and velocity streamlines in the pulse tube section (case 8 – geometry- B) ...... 210

Figure 5.43: Schematic of IPTR (Mk-I) simulated ...... 213

Figure 5.44: Temporal variation of the gas temperature in the cold heat-exchanger for the first three cases simulated at a constant value of inertance ‘M’ (a) from near-equilibrium initial condition and (b) near quasi-steady state ...... 219

Figure 5.45: Comparison of gas and solid temperature at the exit of the cold heat- exchanger for cases 1 and 2 simulated ...... 220

Figure 5.46: Temporal variation of the cycle-averaged gas temperature in the cold heat- exchanger for the IPTR operating with a constant inertance tube radius from near- equilibrium initial condition (Note: case numbers refer to Table 5.10) ...... 221

Figure 5.47: Comparison of the transient variation of the axial component of the gas velocity at the center of the regenerator as a function of the cycle time for (a) constant xxix inertance value (different tube radii) and (b) different inertance values (constant tube radius) ...... 222

Figure 5.48: Comparison of the transient variation of the pressure and axial component of the gas velocity at the center of the regenerator for (a) case 1, (b) case 6 and (c) case 9

...... 223

Figure 5.49: Comparison of the temporal variation of the acoustic power ‘PV’ in one cycle at (a) entrance to the regenerator, (b) exit of the regenerator and (c) entrance to the inertance tube ...... 225

Figure 5.50: Variation of the gas temperature in the cold heat-exchanger at quasi-steady state and the pressure amplitude in the transfer tube as a function of the inertance ...... 227

Figure 5.51: Comparison of quasi-steady transport properties as a function of the inertance (a) pressure amplitude at various locations in the regenerator and (b) pressure amplitude and axial direction velocity amplitude at the center of the regenerator ...... 228

Figure 5.52: Comparison of phase angles as a function of the inertance (a) mass flow rate-pressure phase angle in the regenerator and (b) velocity-pressure phase angle at center of regenerator and entrance to inertance tube ...... 230

Figure 5.53: Comparison of (a) cycle-average mass flow rate, (b) cycle-average enthalpy flow and (c) cycle-average acoustic power as a function of inertance at three locations in the regenerator ...... 231

Figure 6.1: Schematic of the co-axial type OPTR geometry simulated ...... 237

Figure 6.2: Comparison of the cool-down behavior of the OPTR at the entrance to the pulse tube for cases 1 and 2 ...... 242 xxx

Figure 6.3: Pressure, gas temperature and solid temperature profiles at the start of the simulation for case 1 ...... 243

Figure 6.4: Temporal evolution of the gas temperature at the exit of the regenerator in case 1 and case 2 for (a) 0 - 22 sec and (b) early time (0 - 0.5 sec) ...... 244

Figure 6.5: Comparison of the pressure variation at the exit of the regenerator for case 1 and case 2 ...... 245

Figure 6.6: Comparison of temporal evolution of the cycle-averaged gas temperature at three locations in the OPTR (coordinates are in 'mm') for case 1 ...... 246

Figure 6.7: Cyclic variation of gas temperature at three locations in the cold end of the

OPTR (coordinates in 'mm') for case 1 ...... 247

Figure 6.8: Comparison of the cool-down performance for the three cases studied (a) up to 2 seconds of simulated time and (b) predicted values ...... 248

Figure 6.9: Comparison of gas temperature at three locations in the OPTR (coordinates in 'mm') for case 3 ...... 249

Figure 6.10: Gas Temperature in the various components of the OPTR at four instants in the 2201st cycle for case 1 (a) transfer tube to the regenerator (r = 0.0095 m) and (b) cold head to the inertance tube (r = 0.00 m) ...... 251

Figure 6.11: Axial component of velocity ‘ux’ in the various components of the OPTR at four instants in the 2201st cycle for case 1 (a) transfer tube to the regenerator (r = 0.0095 m) and (b) cold head to the compliance volume (r = 0.00 m) ...... 252

Figure 6.12: Cycle-averaged gas temperature and secondary streaming velocity vectors in the pulse tube region for case 1 ...... 253

xxxi

xxxii

Abstract

Effects of Acoustic and Fluid Dynamic Interactions in Resonators: Applications in Thermoacoustic Refrigeration Dion Savio Antao Prof. Bakhtier Farouk

Thermoacoustic refrigeration systems have gained increased importance in cryogenic cooling technologies and improvements are needed to increase the efficiency and effectiveness of the current cryogenic refrigeration devices. These improvements in performance require a re-examination of the fundamental acoustic and fluid dynamic interactions in the acoustic resonators that comprise a thermoacoustic refrigerator. A comprehensive research program of the pulse tube thermoacoustic refrigerator (PTR) and arbitrarily shaped, circular cross-section acoustic resonators was undertaken to develop robust computational models to design and predict the transport processes in these systems. This effort was divided into three main focus areas: (a) studying the acoustic and fluid dynamic interactions in consonant and dissonant acoustic resonators, (b) experimentally investigating thermoacoustic refrigeration systems attaining cryogenic levels and (c) computationally studying the transport processes and energy conversion through fluid-solid interactions in thermoacoustic pulse tube refrigeration devices.

To investigate acoustic-fluid dynamic interactions in resonators, a high fidelity computational fluid dynamic model was developed and used to simulate the flow, pressure and temperature fields generated in consonant cylindrical and dissonant conical resonators. Excitation of the acoustic resonators produced high-amplitude standing waves in the conical resonator. The generated peak acoustic overpressures exceeded the initial undisturbed pressure by two to three times. The harmonic response in the conical xxxiii resonator system was observed to be dependent on the piston amplitude. The resultant strong acoustic streaming structures in the cone resonator highlighted its potential over a cylindrical resonator as an efficient mixer.

Two pulse tube cryogenic refrigeration (PTR) devices driven by a linear motor (a pressure wave generator) were designed, fabricated and tested. The characterization of the systems over a wide range of operating conditions helped to better understand the factors that govern and affect the performance of the PTR. The operating frequency of the linear motor driving the PTR affected the systems’ performance the most. Other parameters that resulted in performance variations were the mean operating pressure, the pressure amplitude output from the linear motor, and the geometry of the inertance tube.

The effect of the inertance tube’s geometry was controlled by a single parameter labeled the “inertance”. External/ambient conditions affected the performance of the cryocoolers too. To prevent the influence of the ambient conditions on the performance, a vacuum chamber was fabricated to isolate the low temperature regions of the PTR from the variable ambient atmosphere. The experiments provided important information and guidelines for the simulation studies of the PTR that were carried out concurrently.

A time-dependent high fidelity computational fluid dynamic model of the entire

PTR system was developed to gain a better understanding of internal interactions between the refrigerant fluid and the porous heat-exchangers in its various components and to facilitate better design of PTR systems based on the knowledge gained. The compressible forms of the conservation of mass, momentum and energy equations are solved in the gas and porous media (appropriate estimation of fluid dynamics in heat- exchangers) regions. The heat transfer in the porous regions is governed by a thermal xxxiv non-equilibrium heat transfer model that calculates a separate gas and solid temperature and accounts for heat transfer between the two. The numerical model was validated using both temporal and quasi-steady state results obtained from the experimental studies.

The validated model was applied to study the effects of different operating parameters

(frequency, pressure and geometry of the components) on the PTR’s performance. The simulations revealed interesting steady-periodic flow patterns that develop in the pulse tube due to the fluctuations caused by the piston and the presence of the inertance tube.

Similar to the experiments, the simulations provided important information that help guide the design of efficient PTR systems.

Chapter 1 : Introduction

1.1. Background

This thesis reports the interactions between acoustic waves and the properties of the fluid (viscosity and conductivity) in which these waves propagate. The focus of this treatise is placed on how these interactions are harnessed to the purpose of energy conversion, specifically refrigeration. The information contained within this chapter will put into context the motivations and objectives of the current research and will provide a better understanding of the results presented in the following chapters.

1.1.1. Acoustics

The generation, transmission and reception of energy in the form of vibrational

(mechanical) waves in a medium is defined as acoustics [1]. As Kinsler et al. [1] explain, the elastic restoring force of atoms/molecules in a media that have been displaced from their equilibrium condition in combination with the inertia of the system enables the components of the media to participate in oscillatory motions that generate and transmit acoustic waves. In a fluid these acoustic/sound waves are observed as pressure waves and the acoustic/pressure waves travel at the speed of sound in that fluid. An acoustic wave is a mechanical longitudinal wave whose direction of propagation is the same as

(parallel to) the direction of oscillation/vibration. In the following paragraphs of this sub- 2 section of chapter 1, common acoustic terms/definitions that will be used in the following chapters are introduced and explained to better orient the reader with the discussion.

Speed of sound or acoustic speed is the speed at which an acoustic wave travels in the medium. In general, the speed of sound in a media is governed by the following equation:

p a   (1.1) where, a is the speed of sound in the media (m/s), p is the pressure (Pa) and ρ is the density (kg/m3). For ideal gases, this reduces to:

a  RTi (1.2)

where, γ is the ratio of specific heats of the gas (γ = cp/cv), Rgas is the gas constant and T is the temperature of the gas in Kelvin.

Figure 1.1: Schematic of a cylindrical resonator (i.e. constant cross-section)

An acoustic resonator is a hollow device (tube or pipe of arbitrary shape) in which oscillations are induced in the fluid contained within it by exciting/vibrating one of the ends. Figure 1.1 above shows a schematic of a cylindrical resonator (circular or rectangular cross-section). The length of the cylinder is normalized by its length and 3 hence the piston is at a normalized distance of x* = 0. The opposite end of the acoustic resonator may be open or closed (blocked). In Figure 1.1, the opposite end is closed at x*

= 1. If one end of an acoustic resonator is excited and the opposite end is open, the resultant wave produced is a travelling wave (no reflection). If the opposite end is blocked/closed, the incident wave upon reaching the end of the resonator gets reflected back, thus creating a reflected wave. The reflected wave and the incident wave interfere.

When the reflected wave and the incident wave have the same frequency and amplitude

(assuming no attenuation of the wave), the interference of the two waves travelling in opposite directions creates a stationary wave called a standing wave.

The reflected wave always has a frequency which is dependent on the length of the resonator and the speed of sound of the fluid (eqs. 1.1 and 1.2 above) in the resonator.

Hence when the frequency of excitation matches the frequency of this reflected wave, a standing wave results and the acoustic resonator is said to have achieved acoustic resonance. Acoustic resonance is defined as the condition when the system (the fluid in the resonator) absorbs the most energy when it is excited at one of its natural frequencies.

The consequence of acoustic resonance is that the pressure and velocity variations inside the resonator are at their maximum when the system has achieved resonance. This can be better understood by considering how standing waves are formed. When the incident and reflected waves in the resonator have the same frequency the resultant wave (due to interference between the two waves) has a higher amplitude than its component waves.

If the frequencies were dissimilar, interference of the two waves would result in a wave with diminished amplitudes. 4

The natural frequencies of an acoustic resonator are the frequencies at which the fluid in the resonator vibrates once it has been excited. It is also the frequency at which the acoustic resonator responds to the input with the maximum output and the frequency at which standing waves are generated in a resonator closed at the end opposite the end which is being excited/vibrated. This natural frequency (also seen as the frequency of the reflected wave above) is dependent on the speed of sound in the fluid and the length of the acoustic resonator. The lowest natural frequency of vibration in an acoustic resonator is called the fundamental natural frequency [1]. For a cylindrical resonator, the natural frequency can be calculated as follows:

a fn (1.3) res 2L

where, fres is the natural frequency, a is the speed of sound, L is the length of the resonator and n is an integer (n = 1, 2, 3, ....). For n = 1, the frequency is called the fundamental natural frequency.

All integer multiples of the fundamental natural frequency are called harmonics

[1] and by that extension, the fundamental natural frequency (n = 1 in eq. 1.3) is called the first harmonic. As can be seen below in Figure 1.2, the standing wave generated has nodes (locations of minimum particle displacement/pressure variation) and antinodes

(locations where the particle displacement/pressure variation are maximum). This pattern of standing waves (combinations of nodes and antinodes) is called the normal mode or mode. The first mode corresponds to the standing wave pattern observed at the first harmonic or the fundamental natural frequency. This is true for all resonators and this fundamental mode has two pressure antinodes (near the end wall and the vibrating wall) 5 and one pressure node (half wavelength resonator). In a cylindrical resonator with one wall vibrating, the pressure antinodes have the same pressure amplitude and the pressure node exists at the center of the resonator [2-4]. Figure 1.2 shows the pressure profiles in the cylindrical resonator as a function of the normalized cylinder length and similar profiles exist of the velocity profiles. However, there exists a minor difference between the two profiles. At the pressure node, there exists a velocity antinode and at the pressure antinodes, there are velocity nodes (velocity goes to zero at the walls).

Figure 1.2: Standing wave profiles a cylindrical resonator at the (a) 1st harmonic

(fundamental natural frequency), (b) 2nd harmonic and (c) 3rd harmonic [4] 6

1.1.2. Thermoacoustics

Pressure waves in fluids can be generated by either mechanical or thermal effects.

As stated above, pressure waves travelling in a medium at the speed of sound of that medium are known as acoustic waves. In fluids, pressure waves that are generated due to a rapid expansion or compression of the fluid due to thermal effects are characterized as thermally generated acoustic waves or thermoacoustic waves. The study of this phenomenon (involving both thermodynamics and acoustics) is broadly classified as thermoacoustics. The term “thermoacoustics” includes all effects in acoustics in which heat conduction and entropy variations of the gaseous medium play a role [5]. All acoustics in gases in which diffusive effects are considered belong within the field of thermoacoustics.

An example of thermoacoustics that is common to most people is the naturally occurring phenomenon of thunder. Without elaborating on the specifics of its generation and propagation, lightning can be labeled as a thermal plasma discharge [6]. The operative word in the previous sentence (as it concerns thermoacoustics) is ‘thermal’.

This high temperature gas channel causes a rapid change in the density and pressure of the gas surrounding it. This rapid increase in the pressure creates a pressure or acoustic wave that travels at the speed of sound and we hear this pressure/acoustic wave as thunder. So based on this example of lightning and thunder, thermoacoustics can be explained as the conversion of changes in temperature to changes in pressure (or acoustics). The opposite is true too, with changes in pressure leading to changes in 7 temperature. Talking creates pressure oscillations in the audible range (~ 20 Hz – 20 kHz) and these pressure oscillations result in temperature oscillations too. However, these temperature oscillations are on the order of 10-2 – 10-4 oC which are not noticeable variations [5]. Unlike lightning and thunder, thermoacoustic devices harness rapidly oscillating (high frequency) temperatures or pressures and convert them to pump work

(engine) or cooling work (refrigerators).

1.1.3. Cryogenics

The term “cryogenics” may be defined as the branch of physics that deals with the production of extremely low temperatures. Typically, the cryogenic limit is defined as

120 K (– 153 oC) [7]. The boiling points for most gases lie around/below this temperature of 120 K and hence, this value of 120 K is used as the limit for cryogenics.

Helium has the lowest boiling point at 4.23 K [8], hydrogen has its boiling point at 20.39

K [8], nitrogen at 77.36 K [8], oxygen at 90.19 K [8] and methane at 111.67 K [8].

Cryogenic refrigerators or cryocoolers are refrigeration devices capable of achieving temperatures below the cryogenic limit. The various applications of cryogenics and cryocoolers are summarized in Figure 1.3 below. 8

Figure 1.3: Applications of cryocoolers and cryogenics plotted as a function of their

operating temperatures and the refrigeration/cooling power required [9]

Some of the applications of cryogenics and cryocoolers seen in Figure 1.3 above are discussed below:

a. Liquefaction of gases: Gas liquefaction and separation were the first uses of cryocooler technology [7]. The Linde O2 liquefier (c. 1895) and the Collins He liquefier

(c. 1946, < 4K temperatures) are the first examples of cryocooler applications.

Liquefaction of gases is important even today. Applications which require liquefied gas include storage of liquefied natural gas (LNG) and other fuel (commercial gas utility), 9

liquid oxygen (LOX) and liquid hydrogen (LH2) used as oxidizer and fuel respectively on spacecrafts and to harness gases (specifically oxygen) from the atmosphere on other planets [10].

b. Superconductivity: One of the major uses of cryogenic technology and cryocoolers today is in the field of superconductivity. Typical cryocoolers used are dilution refrigeration systems that are capable of sub-Kelvin level cooling (mK range).

Applications of superconductivity include magnetic resonance imaging (MRI) and dipole magnets used in particle accelerators like the large Hadron collider (LHC). The discovery of materials that behave like superconductors at higher temperatures (range of

60 – 80 K) has led to research in the field of high temperature superconductivity (HTS).

These HTS materials are proposed as components on fully superconducting HTS machines on next generation aircraft [11].

c. Cooling optics (Infrared detectors, etc.): Military tactical applications, specifically “night vision” equipment are one of the major application areas of cryocoolers. Typical night vision sensors or cameras are equipped with low cooling power (0.5 – 1.0 W @ 60 – 80 K) cryocoolers. Space cryocoolers are another application area that require cooling of optics/detectors [12, 13]. These include cooling of detectors for atmospheric infrared imaging and on infrared and X-ray telescopes.

d. Cryopumping and cryosurgery: Cryopumps are used to trap/condense gases and vapors so as to achieve higher levels of vacuum. However, cryopumps are/must be used as the last stage of vacuum (high and ultra-high levels) pumping since cryocooled surface that condenses/traps the gases and vapors saturates. This saturation occurs faster 10 at low levels of vacuum and can decrease the pumping power. These cryopumps are an important part of the semiconductor processing industry [9]. Cryosurgery is the application of cryocoolers for tissue/tumor ablation and as catheters (cryo-catheters).

Common cryogens used in some of the cryogenic applications listed above are liquid helium (below 4.23 K) and liquid nitrogen (below 77.36 K). However, cryogens are not capable of being used as the source of indefinite cryogenic temperature levels for all the applications above. For this reason and due to the limited availability of liquid cryogens like liquid helium and the need for equipment capable of liquefying gases, cryogenic capable refrigerators (cryocoolers) are intrinsic to sustainability of the cryogenic engineering.

1.2. Dissertation statement

This dissertation details a project that includes the experimental investigations of the thermoacoustic refrigeration phenomena with the development of computational models and numerical simulations to predict the behavior and performance of these thermoacoustic refrigerators and wave-shaped acoustic resonators. The overall message of this dissertation can be stated as follows: ‘thermoacoustic pulse tube refrigerators are complex systems that require multi-dimensional numerical models to accurately predict their performance and wave-shaped resonators can enhance the performance of these thermoacoustic refrigeration systems’. The approach of this engineering thesis includes the development of models to predict the nature or transport processes in these acoustic and thermoacoustic systems and an analysis of the models’ predictions and 11 accompanying experimental results to provide guidelines for the design and operation of efficient thermoacoustic devices.

1.3. Motivation for the dissertation research

The present study was motivated by the need for high efficiency pulse tube cryocoolers and has two main aspects with regards to this general motivation and impact.

The first aspect deals with the study of thermoacoustic Pulse Tube

Refrigerator/Cryocooler (PTR) systems. The second aspect deals with the investigation of fundamental acoustic phenomena in ‘wave shaped’ acoustic resonators to for applications in efficient thermoacoustic refrigeration systems.

1.3.1. Development of accurate multi-dimensional PTR models

A PTR is a type of thermoacoustic refrigerator and is comprised of various heat- exchangers (HXs) and pressure-mass flow phase controlling mechanisms/components1.

Figure 1.4 below shows the main components of the PTR. The PTR is driven by a pressure wave generator (A, yellow arrow) and adjacent to the pressure wave generator is the aftercooler HX (B). This heat-exchanger is required to maintain the compressed gas at room temperature. After the aftercooler HX is the regenerator (C) where the main energy separation occurs and across the length of the regenerator, a temperature gradient

1 The description of the PTR given above is brief and intended to provide a better understanding of the motivations discussed below. A comprehensive discussion of the PTR and its operation is provided in the following chapter. 12 develops. The next component is the pulse tube (E) which is a hollow tube and acts as a thermal buffer between the cold HX (D) on one end and the warm HX (F) on the other.

Beyond the warm HX are various phase controlling mechanisms (G). These phase controlling mechanisms include an orifice and/or an inertance and a compliance volume which convert the PTR system into a pseudo-travelling wave system.

Figure 1.4: Schematic of a PTR showing the main components: A- Pressure wave

generator (yellow arrow), B- Aftercooler HX, C- Regenerator, D- Cold HX, E- Pulse

Tube, F- Warm HX and G- phase control mechanism (orifice and/or inertance and

compliance volume)

The current method of designing PTR systems relies on a linear form of the one dimensional wave equation that is solved for the various components in the PTR. The most prominently used models are DeltaEC developed at Los Alamos National

Laboratories by Greg Swift and his team [14] and SAGE developed by David Gideon

[15, 16]. While these models may have the advantage of being fast and fairly accurate, they do not take into consideration the multi-dimensional and non-linear aspects of the transport processes in PTR systems. All cryocooler systems built have transport 13 processes occurring in multiple dimensions in space and hence 1-D models cannot predict the fluid dynamic behavior in these systems. Multi-dimensional computational fluid dynamic (CFD) models provide the added advantage of being more accurate, robust and economical in time and capital (as compared to experiments) and can provide insight into the transport processes of a system that even careful experiments cannot.

The first motivation of this study is to develop a multi-dimensional computational fluid dynamic (CFD) model that can be used to accurately predict the behavior and performance of a thermoacoustic PTR. This involves addressing the presence of heat- exchanger material in the system and how it affects the flow fields, accounting for heat transfer between the operating fluid (refrigerant gas) and the solid surfaces including the heat-exchanger material and the walls of the components and accurately predicting the acoustic nature of the system. In order to have confidence in the results predicted by the developed model, they must be validated with experimentally obtained values. This adds another component to the research tasks, i.e., to design, fabricate, instrument and test a physical thermoacoustic PTR. The results of the experimental characterization of the

PTR’s performance will be use to validate the developed CFD model. Since most of the applications of contemporary cryocoolers (discussed earlier in this chapter) deal with an operating temperature range of 60 – 80 K or the liquid nitrogen region, building an experimental cryocooler capable of liquid nitrogen temperatures was an added motivation.

14

1.3.2. Investigation of wave-shaping for cryocooler performance enhancement

A simple cylindrical resonator (a component used in most PTR systems) has limitations to the output response observed at one end when the opposite end is acoustically excited. The limitations arise in the formation of shock-like structures which lead to large amounts of energy dissipation in the form of heat [4, 17]. Figure 1.5 below shows schematics of acoustic resonators of four different shapes. Figure 1.5a shows the simple consonant cylindrical resonator (similar to Figure 1.1) and Figures 1.5b, 1.5c and

1.5d show schematics of “wave-shaped” dissonant resonators2. Hence wave-shaped resonators may be described as acoustic devices with non-cylindrical shapes (i.e., variable cross-section resonators).

2 Consonant and dissonant resonators will be discussed in detail in the following chapter. 15

Figure 1.5: Schematics of acoustic resonators (a) cylinder, (b) horn-cone, (c) cone and

(d) bulb [17]

Wave-shaping is a novel technique of using dissonant acoustic resonators to convert an acoustic input to high energy un-shocked non-linear acoustic output. The essence of wave-shaping lies in the fact that in a dissonant resonator, high pressure ratio non-linear acoustic waves with shock-less structures can be generated for the same input power as a resonant cylindrical system (being operated at the same experimental conditions). This phenomenon allows for higher efficiencies and more available acoustic power. The pressure output at the small end of a cone shaped resonator when the large end is excited at the fundamental resonant frequency is not sinusoidal (the crest and trough are not symmetric about the mean pressure). The reason the output is considered 16 non-linear is because the input wave is a sinusoidal function and the resultant output at the small end is similar to an inverted rectified sinusoid.

Wave-shaped acoustic resonators have potential applications as pumping systems, mixers in the chemical process industry and thermoacoustic energy systems (engines and refrigerators). When applied to thermoacoustic energy systems, wave-shaping has the capability to generate higher outputs for a fraction of the input cost/power that traditional thermoacoustic systems designed and built with consonant cylindrical components require. The cylindrical components of the PTR (Figure 1.4) can be replaced by the wave-shaped resonators seen in Figures 1.5b, 1.5c and 1.5d. This can change the way thermoacoustic energy systems are designed. However, despite all the attractive applications of wave-shaped acoustic resonators, there is a fundamental obstacle to the use of such systems; there are no design guidelines or formulae that can be used to build these devices. The second motivational thrust for this research work is the development of a multi-dimensional CFD model that can be used to predict and study the behavior of wave-shaped acoustic resonators. An obvious extension to the development of the CFD model is to provide design guidelines for future investigators which were lacking in the past and to use the model to investigate possible application areas for this novel and exciting phenomenon.

1.4. Objectives of the dissertation research

The objectives of the current research were to develop comprehensive computational models to accurately predict and study the behavior of thermoacoustic 17 pulse tube refrigeration systems and wave-shaped acoustic resonator systems and to validate the numerical modes with detailed experiments. The objective was to perform the research based on a set of synergistic experiments and numerical simulations that can be used to understand the behavior and evaluate the performance of these complex systems. The developed models were applied to study acoustically driven viscous flows in PTR systems and wave-shaped acoustic resonators and to investigate the effects of a variety of operating conditions on the performance of these systems.

The objectives for the research were to:

a. Design, fabricate, instrument and test a pulse tube refrigeration system capable of liquid nitrogen temperatures (< 77.5 K)

b. Characterize the experimental pulse tube refrigeration system under a variety of operating conditions to understand the parameters that affect its behavior and evaluate its performance under the different operating conditions

c. Develop a numerical model to accurately predict the performance of a pulse tube refrigeration system by accounting for the presence of heat-exchangers which affect the fluid dynamics and heat transfer characteristics of the system and validate the model with experimental results

d. Incorporate the non-equilibrium nature of heat transfer in the heat-exchangers by accounting for energy exchange between the gas (refrigerant fluid) and the solid

(metal heat-exchanger material) components in the model. 18

e. Apply the numerical model to understand the behavior of the PTR system by studying various phenomena in the system and optimizing its performance by incorporating geometric design improvements

f. Develop a high fidelity multi-dimensional computational fluid dynamic model of to investigate non-linear wave-shaping phenomena in acoustic resonators

g. Apply the developed numerical model to investigate the wave-shaping phenomenon and provide guidelines for the design of such acoustic resonators as well as investigate possible application areas of wave-shaped acoustic resonators

To a significant extent, the objectives set at the onset were met by the present research study. The completed research and how it meets the objectives listed above is discussed in the following chapters.

1.5. Overview of the dissertation

The organization of this dissertation is as follows. Chapter 1 introduces the thesis topic and discusses the motivations and objectives of the research. Chapter 2 provides introductory information and background to familiarize the reader with acoustics resonator systems and cryogenic refrigerators in context of the current research. This includes a discussion of consonant and dissonant resonators briefly introduced in chapter

1, an introduction to the different types of cryocoolers and a more in-depth explanation of the PTR which is the focus of this study. In this thesis, the term cryocooler and cryogenic refrigerator are used interchangeably. Chapter 2 also includes a detailed 19 literature review of the past research performed on consonant and dissonant acoustic resonators and PTR systems.

The presentation of results begins in chapter 3. For the most part, the chapters are self-contained with introductions, experimental setup/numerical model and simulation conditions, results and discussion and a brief set of conclusions corresponding to each study. Chapter 3 introduces the numerical model developed to investigate acoustic and thermal-fluid interactions in consonant cylindrical resonators and dissonant conical resonators. Insight is provided into possible applications of the dissonant resonators and design guidelines are provided on how the resonant frequency can be estimated and choice of operating fluids. In chapter 4, the experimental studies of an orifice type pulse tube refrigerator (OPTR, Mk-I) and an inertance type pulse tube refrigerator (IPTR, Mk-

II) are presented. The experimental study details the procedure followed to design, fabricate, instrument and characterize the two PTRs’ performances over a variety of operating conditions. The numerical studies of the in-line pulse tube refrigeration systems are presented in chapter 5. This includes a detailed description of the numerical model developed and the validation of the numerical model with some of the experimental results presented in chapter 4. Chapter 5 also presents the application of the numerical model of the PTR to investigate and study the effects of various operating parameters like operating frequency, pressure amplitude and geometries of the pulse tube and inertance tube components on the behavior and performance of the PTR system. In chapter 6, the numerical model of the PTR is applied to study a co-axial type geometry of the PTR which is typical in space cryocoolers used for applications in gas liquefaction and optics and electronics cooling. Finally Chapter 7 summarizes and concludes the 20 research presented in this dissertation with a proposal and discussion of research tasks that may be undertaken in the future to improve and better understand the thermoacoustic phenomena and its applications in energy systems.

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Chapter 2 : Background and Literature Review

2.1. Acoustic resonators: consonant and dissonant systems

As explained in section Chapter 1 (1.1.1. Acoustics), resonance in acoustic resonators is characterized by terms like natural frequencies, harmonics and normal modes. Based on these definitions, acoustic resonators are characterized or categorized into two types: consonant resonators and dissonant resonators (Note: All acoustic resonators discussed beyond this point have circular cross-sections). As stated earlier, the fundamental natural frequency (1st harmonic frequency) is always the same as the frequency of the fundamental mode. This is true for all resonators, consonant or dissonant. However, when the higher order harmonic frequencies (2nd, 3rd, 4th, etc., basically integer multiples of the fundamental natural frequency) overlap (are the same as) with the frequencies of the higher order normal modes (2nd, 3rd, 4th, etc., recall Figure

1.2), the resonator is defined as a consonant resonator. If these higher order harmonics are not the same as the frequencies of the corresponding normal modes (i.e., the frequencies of the normal mode are not integer multiples of the fundamental mode), the resonator is defined as a dissonant resonator.

2.1.1. Consonant cylindrical resonators

In general, no real resonators can be termed as consonant [17]. This is due to the inherent non-ideal (real) nature of physical resonators (geometric imperfections and 22 energy dissipation) [17]. However, a cylindrical resonator (hollow circular cross-section tube/pipe) is generally called a consonant resonator due to the fact that the harmonics

(integer multiples of the fundamental resonant frequency) for the resonator coincide with the (natural) frequencies of the normal modes. The harmonic frequency response of a cylindrical resonator typically follows the following equation (repeated from Chapter 1 for convenience):

a fn (2.1) res 2L

where, the fres is the resonant frequency, n is an integer (n = 1, 2, 3, 4, ...), a is the speed of sound in the fluid and L is the length of the resonator. By simple observation of the equation above and the figures of the normal modes (Figure 1.2), one can see that the number of pressure nodes in the resonator is equal to n and the number of pressure antinodes is equal to n+1.

2.1.1.1. Limitations of consonant cylindrical resonators

A cylindrical resonator (a component used in most thermoacoustic refrigeration systems) has limitations to the output response observed at one end when the opposite end is acoustically excited. The limitations arise in the formation of shock-like structures and micro-shock-like structures (Figure 2.1) which lead to large amounts of energy dissipation in the form of heat. Furthermore, past studies have shown that the maximum pressure ratios (pmax/pmin) that can be attained in cylindrical resonators are between 1.3 to 23

1.5 and at these pressure ratios the cylindrical resonators have reached acoustic saturation

[4, 17].

Figure 2.1: Temporal pressure profiles at the closed end of a cylindrical resonator with

the opposite end vibrating at the fundamental resonant frequency of the resonator [17]

2.1.2. Dissonant wave-shaped resonators

All acoustic resonators (except the cylindrical acoustic resonator) are dissonant resonators. Hence, all arbitrary shaped resonators, which include cone shapes, bulb shapes, exponential shapes, etc. are dissonant resonators. This implies that the higher order harmonics do not coincide with the frequencies of the higher order normal modes.

The implication of this is that higher order natural frequencies are not integer multiples of the fundamental natural frequency and hence the resonance spectrum of dissonant resonators cannot be mapped by the simple formula stated in eq. 2.1 above for consonant cylindrical systems. What is more important is that the fundamental natural frequency 24 too cannot be calculated using the above formula. Hence pure experimentation and instrumentation of a dissonant acoustic resonator is the only method of obtaining its frequency spectrum.

In the following sub-section, some past research (experimental and numerical) on dissonant resonators (various shapes) are reviewed in context of the research reported in this thesis.

2.1.2.1. Experimental and numerical studies of dissonant wave-shaped resonators

As opposed to cylindrical consonant resonators, wave-shaped resonators with non-uniform cross-sections can have a variety of shapes [4, 17]. The phenomenon of wave-shaping was first reported by Lawrenson et al. (experimental) [17] and Ilinskii et al. (numerical) [4] in 1998. The different wave-shaped resonators studied in the past include the cone [4, 17-20], the horn-cone [17-19], the bulb [17-19], the cosine (1/2 and

3/4) [21] and the exponentially expanding resonators [22].

The numerical study by Ilinskii et al. [4] was based on one-dimensional nonlinear gas dynamic equations for an ideal gas to address the wave-shaping phenomenon. In the model developed, the resonator is excited by an ‘entire resonator drive’ similar to the experiments of Lawrenson et al. [17]. The losses in the simulation domain are introduced via bulk viscosity, however losses in the boundary layer are not considered. The model predicted the nonlinear behavior of wave-shaped resonators (cone and bulb) and a qualitative comparison of the model’s results were made with the experimental results of

Lawrenson et al. [17]. This model was subsequently modified (Ilinskii et al. [23]) to 25 include the effects of boundary layer losses on the system’s performance. The losses were incorporated by adding a source term in the continuity equation to account for the mass flow from the boundary layer into the resonator volume. The losses predicted by the model were then compared to experimental data and good agreement was shown between the two.

Since the first studies of the wave-shaping phenomenon by Lawrenson et al. [17] and

Ilinskii et al. [4], there have been several analytical/asymptotic studies [18, 19, 24, 25] that have predicted the wave-shaping phenomenon in a variety of resonator shapes. The most common shapes are the cone and bulb shaped resonators. Hamilton et al.[25] developed an analytical (asymptotic) model for the natural frequencies of a resonator with ‘slowly’ varying cross-section (i.e., resonators close to cylindrical in shape) by solving the Webster horn equation. They additionally developed an analytical result for the amplitude-frequency response curve and the fundamental mode nonlinear resonance frequency shift using Lagrangian mechanics. Hamilton et al. [24] recently extended their previous model to include resonators of any shape (the previous model had the restriction of ‘slowly’ varying cross-section resonators). The new analytical model was validated against the results of Ilinskii et al. [4] and had the added benefit of being far more computationally cheaper than the direct numerical simulations of the one-dimensional gas dynamic equations solved by Ilinskii et al. [4]. A similar asymptotic theory was developed in Eulerian coordinates by Mortell and Seymour [19].

Luo et al. [22] studied wave-shaping in an exponentially expanding resonator both numerically and experimentally. The numerical model was based on the solution of the one-dimensional nonlinear wave equation (using the Galerkin method). The model 26 predicted higher pressure ratios for higher flare constants of the exponentially expanding resonator and a shift in the resonance frequency with an increase in the flare constant.

Using the model developed, they compared the effects of ‘entire resonator drive’ and the

‘piston drive’, but found insignificant differences between the two.

Chun and Kim [21] developed a one-dimensional fluid dynamic model (solving the conservation of mass, momentum and energy equations) and applied the model to study wave-shaping in conical and cosine shaped resonators (similar to the bulb shaped resonator). They showed that the 1/2 cosine shaped resonator generated the highest pressure ratios. Even though their model solves the energy equation, the effects of heat generation within the system are not discussed.

Some of the proposed applications of wave-shaped dissonant resonators are process reactors for chemical and pharmaceutical industries, separation or mixing chambers, oil-less compressors and pumps and for conversion of combustible fuels into electric power (RMS pulse combustion) [17]. Another set of applications lie in the field of thermoacoustic cryocooling. The need for compact, space worthy cryocoolers has resulted in the need to research high frequency thermoacoustic refrigerators. High frequency offers the advantages of a compact system and fast cool-down times from room temperature [9]. Wave-shaping can be used either as the drive mechanism (instead of a linear motor which is frequency limited or a thermoacoustic engine which has large components) of a purely thermoacoustic high frequency cryocooler or it can be used to modify current thermoacoustic cryocooler components (which are cylindrical) so as to enhance the cryocooler performance by taking advantage of the high pressure ratios attained in wave-shaped dissonant resonators. 27

The past studies have made assumptions in order to obtain a solution and predict the wave-shaping phenomenon. These include one-dimensional governing equations, not accounting for the effects of the boundary layer, linear approximations and assuming an isothermal fluid inside the resonator. Additionally, the models do not provide any insight on how to design wave-shaped resonators and calculate the resonant frequencies. Of the various geometries studied in the past, the conical geometry was shown to generate relatively high pressure ratios and is the least complicated “wave-shaped” geometry to manufacture (compared to bulb, horn-cone or exponentially expanding geometries which would require computer numerical control (CNC) manufacturing techniques).

2.2. Cryogenic refrigeration systems

As stated in chapter 1, refrigeration systems capable of cryogenic level cooling are called cryogenic refrigerators or cryocoolers. There are many types of cryocoolers used for the various applications listed in chapter 1. These cryocoolers have different operating principles and based on the application, different temperature regimes of operation. In this sub-section of chapter 2, some of the cryocooler types are briefly discussed and the cryocooler of importance to the research reported here (i.e., the PTR) is discussed in detail.

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2.2.1. Dilution refrigeration

This refrigerator works on the principle that cooling is produced when the two isotopes of Helium (i.e., 3He and 4He) cross the phase boundary separating them [26].

3He is pumped through 4He in a mixing chamber and flows through 4He to a higher temperature chamber (called a ‘Still’) where 3He is fractionally distilled and separated from 4He. For space applications, these dilution refrigerators work on a continuous loop and 3He is recycled. Figure 2.2 below shows the conceptual operation of a sorption- pumped single-cycle dilution cryocooler.

Figure 2.2: Schematic of a sorption-pumped 3He – 4He dilution refrigerator [26]

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2.2.2. Adiabatic demagnetization refrigeration (ADR)

An ADR system works on the principle of the magnetocaloric effect. The magnetocaloric effect is a phenomenon in which a reversible change in temperature of a suitable material is caused by exposing the material to a changing magnetic field. The advantage of ADR systems is that they have high efficiency and an infinite lifetime (due to the lack of moving parts and no vibration) [27]. The figure below (Figure 2.3) shows the schematic of an ADR system built at NASA for operation between 2 K and 10 K. 30

Figure 2.3: Schematic of a NASA ADR system developed for operation in the 2 K – 10

K region [26]

2.2.3. Sorption refrigeration

3He is condensed in a bath of 4He (due to its lower boiling point) and then pumped to 0.3 K by adsorption onto charcoal. A typical cycle lasts till 3He runs out/is depleted [26]. The 3He is recycled by re-heating the charcoal. Typical sorption coolers 31 can provide μW cooling at 0.3 K for 5 – 8 days and the recycling period lasts for ~ 10 –

20 hours.

2.2.4. Gas cycle refrigeration

The above three methods listed have the advantages of being stable and vibration free coolers at temperatures near absolute zero (~ μK – mK range). However, a common shortcoming is that they require a base temperature in the 2 – 10 K range. Unless liquid helium is available, the above three cryogenic refrigeration systems cannot operate. Even liquid helium is obtained by liquefying gaseous state helium and refrigeration methods/systems are required that are capable of reducing the temperature of the gas below the 2 – 4 K range required to liquefy it. The best method to get to the 2 – 10 K range from room temperature (~ 295 – 300 K) is to use a gas cycle refrigeration system.

There are five types of gas-cycle refrigerators used today [9, 12]. These are broadly classified into two different cycles of operation: recuperative cycle systems

(Figure 2.4) and regenerative cycle systems (Figure 2.5). In the following sub-sections of this chapter, a brief introduction and operation of the various gas cycle refrigeration systems are presented, however the focus is on the evolution of the research in pulse tube refrigeration. 32

Figure 2.4: Schematic of cryogenic recuperative refrigerators (a) Joule-Thomson and (b)

Brayton cryocoolers [9]

Figure 2.5: Schematic of cryogenic regenerative refrigerators (a) Stirling, (b) Pulse Tube

and (c) Gifford-McMahon cryocoolers [9] 33

2.2.4.1. Recuperative cryocoolers

The first category of gas cycle refrigeration systems is the recuperative refrigeration system. These recuperative refrigeration systems are steady flow systems and may be used in open cycle form or closed cycle form (where the refrigerant is re- cycled by incorporating a compressor into the system). The two types of cryogenic recuperative refrigerators are the Joule-Thomson (J-T) and the Brayton cycle cryocoolers.

The steady pressure and the steady flow of gas in these cryocoolers allow them to use large gas volumes anywhere in the system with little adverse effects except for larger radiation heat leaks if the additional volume is at the cold end [12]. This makes it is possible to transport the refrigeration effect (cold gas/liquid) to any number of distant locations after the gas has expanded and cooled. Thus, the cold end can be separated from the compressor by any distance and this greatly reduces the electromagnetic interference

(EMI) and vibration associated with the compressor. The only drawback of these systems is that oil removal equipment is required in these cryocoolers at the warm end of the heat exchanger to remove any traces of oil from the working gas before it is cooled in the heat exchanger. The removal of oil is necessitated by the fact that oil will freeze at cryogenic temperatures and clog the system (unlike conventional refrigerators operating near ambient temperature). Below, the J-T and Brayton cryocoolers are discussed in more detail. 34

2.2.4.1.1. Joule-Thomson cryocoolers

The Joule-Thomson cryocoolers work on the Joule-Thomson effect. The Joule-

Thomson effect describes a temperature change (cooling) when a high-pressure gas expands through a flow impedance (orifice, valve, capillary, porous plug), often referred to as the JT valve. The expansion occurs with no heat input or production of work, thus, the process occurs at a constant enthalpy. The heat input occurs after the expansion and is used to warm up the cold gas or to evaporate any liquid formed in the expansion process. Cooling in a Joule-Thomson expansion occurs only with real gases and at temperatures below the inversion curve (above the inversion temperature, an increase in temperature is observed). In fact, for a given pressure change the amount of cooling increases as the temperature is lowered and reaches a maximum around the critical point.

Typically, nitrogen or argon is used in JT coolers, but pressures of 20 MPa (200 bar) or more on the high pressure side are needed to achieve reasonable cooling. Such high pressures are difficult to achieve and require special compressors with limited lifetimes.

The main advantage of JT cryocoolers is the fact that there are no moving parts at the cold end. The cold end can be miniaturized and provide a very rapid cool-down. This rapid cool-down (a few seconds to reach 77 K) has made them the cooler of choice for cooling infrared sensors used in missile guidance systems [12].

A disadvantage of the JT cryocooler is the susceptibility to plugging by moisture of the very small orifice. Another disadvantage is the low efficiency when used in a closed cycle mode (mainly due to low compressor efficiencies). 35

2.2.4.1.2. Brayton cryocoolers

In Brayton cryocoolers cooling occurs as the expanding gas does work (generally on an expansion engine or an expansion turbine supported on gas bearings). The heat absorbed with an ideal gas in the reverse Brayton cycle (see Figure 2.6 below) is equal to the work produced (conservation of energy). This provides the Brayton cycle with higher efficiencies. This along with the fact that it does not require very high pressure makes it better than the Joule-Thomson cycle. The Brayton cycle is commonly used in large liquefaction plants. For small Brayton cryocoolers the challenge is fabricating miniature turbo-expanders (expansion engines) that maintain high expansion efficiency. The working fluid used in the turbo-Brayton cryocoolers (i.e. the Brayton cryocooler using an expansion turbine) is usually neon when operating above 35 K, but helium is required for lower temperatures. 36

Figure 2.6: The reverse Brayton (refrigeration) cycle (a) schematic of the process and

(b) T-s diagram showing the ideal and real thermodynamic states of the cycle [28]

An advantage of the Brayton cryocooler is the very low vibration associated with rotating parts in the system (turbo-expanders and centrifugal compressors). This low vibration is often required with sensitive telescopes in satellite applications. The main disadvantage of the low-pressure operation of the miniature Brayton systems is that it requires relatively large and expensive heat exchangers to be fabricated.

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2.2.4.2. Regenerative refrigerators

The other category of gas cycle refrigeration systems is regenerative refrigeration.

These refrigerators operate on the reverse Stirling cycle (see Figure 2.7 below), with oscillating pressures and mass flows in the cold head (as compared with the steady flow recuperative cycle systems). The oscillating pressure can be generated with a pressure oscillator (a valve-less compressor used in Stirling and Pulse Tube cryocoolers), with valves that switch the cold head between a low and high pressure source (for the Gifford-

McMahon (GM) cryocooler) or with a thermoacoustic engine [9]. For the GM cryocooler a conventional compressor with inlet and outlet valves is used to generate the high and low pressure sources. This compressor is usually an oil lubricated compressor and oil removal equipment can be placed in the high-pressure line where there is no pressure oscillation. The use of valves greatly reduces the efficiency of this system.

Pulse tube cryocoolers can use either source of pressure oscillations (i.e. a pressure oscillator like a speaker and linear motor or a compressor). The main heat exchanger in the regenerative cycles is called a regenerator. The regenerator works as a heat capacitor.

The incoming hot gas transfers heat to the matrix of the regenerator, where the heat is stored for a half cycle in the heat capacity of the matrix. In the second half of the cycle the returning cold gas, flowing in the opposite direction through the same channel, absorbs heat from the matrix and returns the matrix to its original temperature before the cycle is repeated. To achieve the very high surface areas required for enhanced heat transfer in a regenerator, stacked fine-mesh screen or packed spheres are used as regenerator material. 38

Figure 2.7: P-V diagram of an ideal reverse Stirling (refrigeration) cycle [29]

2.2.4.2.1. Gifford McMahon and Stirling type cryocoolers

The Stirling cycle was first used as a refrigerator to produce ice in 1834. Initially air was used as the working fluid in these early regenerative systems. The application of

Stirling refrigerators to cryogenics happened by accident when in 1946 the Stirling engine was run in reverse with a motor at a Dutch company and was found to liquefy air at the cold tip [30]. The engine used helium as the working fluid, since earlier work at the company showed helium to give much improved performance to the engines.

In a regenerative cycle cryocooler system, the pressure oscillation (produced by the prime mover) would cause the temperature to oscillate and produce no refrigeration.

However, in the Stirling and GM cryocoolers a second moving component is used. This moving component is called the displacer and is required to separate the heating and 39 cooling effects by causing motion of the gas in the proper phase relationship with the pressure oscillation. When the displacer moves towards the cold end, the helium gas is displaced to the warm end of the system through the regenerator. The piston in the compressor then compresses the gas, and the heat of compression is removed by heat exchange with the ambient. Next the displacer is moves away from the cold end and thus the gas is pulled/displaced through the regenerator to the cold end of the system. The piston then expands the gas, and the gas in the cold end too is expanded and cooled. This cold gas absorbs heat from the system it is cooling before the displacer again moves towards the cold end, thus forcing the gas back to the warm end through the regenerator and the cycle repeats. There is little pressure difference across the displacer but there is a large temperature difference. This low pressure drop across the displacer (which is comparable to the pressure drop in the regenerator) exists because there is a clearance between the displacer and the tube it oscillates in. Most Stirling cryocoolers have the regenerator inside the displacer thus forming a co-axial system with the advantage of producing a single cylinder with a convenient cold finger at the end. The main difference between the GM type cryocooler and the Stirling cryocooler is the presence of a GM-type compressor with rotary valves in the GM type system which switches the cold heat exchanger between high and low pressure sources. The presence of these rotary valves greatly reduces its efficiency.

The motion of the piston and the displacer are nearly sinusoidal. The correct phasing occurs when the volume variation in the cold expansion space (end of the regenerator and beginning of the cold heat exchanger) leads the volume variation in the warm compression space by about 90º. With this condition the mass flow or volume 40 flow through the regenerator is approximately in phase with the pressure. Without the displacer in the Stirling cycle the mass flow leads the pressure by 90º and no refrigeration occurs. Though the moving piston (displacer) causes both compression and expansion of the gas, net power input is required to drive the system. The moving displacer reversibly extracts net work from the gas at the cold end and transmits it to the warm end where it contributes some to the compression work.

Stirling cryocoolers have been used for a long time in cooling infrared sensors for tactical military applications in such equipment as tanks and airplanes. The long history of the Stirling cryocooler in cooling infrared equipment has resulted in many specifications being tailored to the geometry characteristics of the Stirling cryocooler.

The only disadvantage when compared to Joule-Thomson systems is that they cannot provide very fast cool down times. Gifford–McMahon (GM) cryocoolers were first developed in 1960 [31]. They began to be used in the 1980s for the cooling of charcoal adsorbers to about 15 K in cryopumps. Recently, the GM cryocooler started being used for the cooling of radiation shields in MRI equipment and for reducing the boil-off rate of the liquid helium that was used to maintain the superconducting magnet at 4.2 K.

The main disadvantage of both the Stirling and GM systems is the presence of the displacer which is a source of vibration, has a limited lifetime, and contributes to axial heat conduction as well as to a shuttle heat loss and thus reduces overall efficiency.

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2.2.4.2.2. Pulse tube refrigerators/cryocoolers

In this subsection on Pulse Tube cryocoolers, we first present a chronological summary and timeline to briefly present the evolution of pulse tube refrigeration to its present state today and then the theory and operation of the pulse tube refrigerator is discussed.

The pulse tube was first discovered and reported by Gifford and Longsworth in

1964 [32, 33]. They named the pulse tube refrigerator so because the displacer (Stirling and GM cryocoolers) was replaced by a hollow tube. The word pulse is derived from the fact that pressure pulses are used to compress and expand the gas (converted to heating and cooling respectively). This initial form of the pulse tube refrigerator is today called a

Basic Pulse Tube refrigerator (BPT) and consisted of an aftercooler, a regenerator, cold heat exchanger, pulse tube and a hot heat exchanger. Initial experiments using Helium as the working gas resulted in cold temperatures of around 169 K for a single stage and 123

K for a double stage cryocooler.

Traditional understanding of acoustic devices suggested that BPTs in general required heat transfer between a solid boundary and the working gas. This necessitates the need to have spacing between the solid boundaries on the order of the thermal and viscous diffusion lengths. This limits enthalpy flow because the amplitudes of the various thermodynamic and flow properties and the phase angle are restrained due to the fact that transverse diffusion governs the dynamics. In 1984, Mikulin [34] demonstrated that the phase and amplitude relation between velocity and temperature can be managed by controlling the boundary conditions at the end of the pulse tube. Mikulin placed an orifice and a reservoir (compliance/surge) volume at the end of the BPT thus allowing a 42 finite gas flow. The presence of the orifice changes the phase angle between the velocity and temperature at the cold end and increases enthalpy flow at the hot end. His initial experiments attained 105 K with air as the working gas. This was the first Orifice Pulse

Tube refrigerator (OPTR). The OPTR results in lower temperatures, increased cooling and higher efficiencies than the BPT.

Finally, Ray Radebaugh at NIST (Boulder, CO) began working on the OPT as it is known today. His initial work recorded temperatures around 60 K [35]. Recent variations of the OPTR include the addition of an extra orifice creating a double-inlet pulse tube and the addition of an inertance tube. The double-inlet allows the gas to be compressed from the hot end of the tube too, thus increasing the phase angle and reducing the flow through the regenerator (which reduces enthalpy flow losses). The term inertance is a combination of the words inertia and inductance (analogous to the inductance in an electrical circuit) [36]. The idea of using an inertance tube is to use the inertia of the gas to provide an added phase shift in a long tube. The inertance tube can be used instead of an orifice or in addition to one. Marquardt and Radebaugh reported the highest efficiency when using a combination of the orifice and the inertance tube

[10].

In the Stirling and GM regenerative cryocoolers, the appropriate phasing between the pressure wave and the velocity or mass flow rate is maintained by the displacer piston located in the cold end. This displacer piston can be controlled by an external motor

(linear motor) or it can be a spring-mass system whose mass (of the displacer piston) and spring constant are carefully chosen to provide adequate phasing. However, this moving displacer placed in the cold end leads to vibration and early failure of the displacer 43 components. The advantage of the PTR is that there are no moving parts in the cold end with the moving displacer piston replaced by a pulse tube and a phasing mechanism. The added advantage is that there is no need for an external motor to control the displacer piston’s motion either.

The phasing mechanism performs the function of maintaining the appropriate phase relationships between the pressure wave and the flow rate (or velocity). Proper phasing must exist in the regenerator for the PTR to operate optimally. Since the regenerator is made up of a porous substance (stacked wire-mesh screen), the flow in the regenerator is primarily governed by Darcy’s Law for flow in porous media [37]. Flow governed by Darcy’s law follows that the pressure drop over a porous structure is directly proportional to the flow rate of the fluid. As the flow rate through the regenerator (the porous region) increases, the pressure drop proportionally increases. Hence the goal in a

PTR is to minimize the flow rate through the regenerator while maintaining high pressure amplitude values and consequently high values of acoustic power PV (V is the volumetric flow rate). However being an ‘AC’ system, the acoustic power is also dependent on the phase angle between the axial component of the velocity and the pressure (Pac =

P×V×Cosθ, where θ is the phase angle). This implies for a constant value of acoustic power, if the phase angle at the center of the regenerator is minimized, the velocity (and flow rate) and the pressure drop are minimized.

Various phasing mechanisms have been used in the past [12, 38]. The proper gas motion in phase with the pressure is achieved by the use of an orifice (in an OPTR) and/or an inertance tube, along with a reservoir volume to store the gas during a half cycle. The reservoir volume is large enough that negligible pressure oscillation occurs in 44 it during the oscillating flow. The first method used was the orifice valve and compliance volume which was used to convert the standing wave BPTR into a pseudo travelling wave system [34, 35]. In an orifice, due to the purely resistive nature of the flow impedance, the flow rate and the pressure are in phase [12]. This causes the phase angle at the center of the regenerator to be extremely large (~ 40 – 60°) and hence the flow rate through the regenerator is very high (for a constant acoustic power). To reduce this flow rate, a different phase shifting mechanism was developed. An extra orifice and a by-pass line that connects the warm end of the pulse tube and the warm end of the regenerator were added to the system, creating a double-inlet pulse tube [39]. The flow rate through the regenerator is reduced using the by-pass, but DC flow is generated through the pulse tube from its warm end to its cold end. This DC flow leads to a decrease in the performance of the PTR. The last phasing mechanism used in PTR cryocoolers is the inertance tube. The inertance tube is a tube placed between the warm end of the pulse tube and the compliance volume. Using the inertance tube, the losses in the regenerator are minimized when the flow lags the pressure at the entrance to the inertance tube by about 50 – 60° [13]. Consequently, the flow in the cold end of the regenerator lags the pressure and the flow in the warm end leads the pressure (~ 30° in both cases) which results in the flow and pressure being in phase at the center of the regenerator [13]. The inertance tube can be used to replace the orifice phasing shifting mechanism or in addition to it. Marquardt and Radebaugh reported the highest efficiency for the pulse tube refrigerator when using a combination of the orifice and the inertance tube [10]. In the current study, the inertance tube is the sole phase shifting mechanism used. 45

The entire cycle in a PTR can be simplified to the following four steps [12]: (1)

The piston in the compressor compresses the gas in the pulse tube. (2) Due to compression, the gas is heated and is at a higher pressure than the average in the reservoir. This high pressure gas then flows through the orifice into the reservoir and exchanges heat with the ambient through the heat exchanger at the warm end of the pulse tube. The flow stops when the pressure in the pulse tube is reduced to the average pressure. (3) The piston in the compressor expands the gas adiabatically in the pulse tube. (4) The cold low-pressure gas (due to expansion) in the pulse tube is forced toward the cold end by the gas flow from the reservoir into the pulse tube through the orifice. As the cold gas flows through the heat exchanger at the cold end of the pulse tube it picks up heat from the object being cooled. The flow stops when the pressure in the pulse tube increases to the average pressure.

The main function of the pulse tube is to insulate the processes at its two ends.

The pulse tube must be large enough that gas flowing from the warm end traverses only part way through the pulse tube before flow is reversed. Similarly, the flow entering from the cold end never reaches the warm end. Gas in the middle portion of the pulse tube never leaves the pulse tube and forms a temperature gradient that insulates the two ends. The overall function of the pulse tube is to transmit hydrodynamic or acoustic power in an oscillating gas system from one end to the other across a temperature gradient with a minimum of power dissipation and entropy generation. 46

Figure 2.8: Schematic of different types of pulse tube cryocoolers (a) U-tube, (b) Co-

axial and (c) In-line [12]

There are three different geometries that have been used with pulse tube cryocoolers as shown in Figure 2.8. The inline arrangement is the most efficient because it requires no void space at the cold end to reverse the flow direction nor does it introduce turbulence into the pulse tube from the flow reversal. Turbulence can be minimized in the pulse tube by adding a flow straightner each at the beginning and end of the pulse tube. The disadvantage is the possible awkwardness associated with having the cold plate located between the two warm ends. The most compact arrangement and the one most like the geometry of the Stirling cryocooler is the coaxial arrangement. That geometry has the potential problem of a mismatch of temperature profiles in the regenerator and in the pulse tube that would lead to steady heat flow between the two components and a reduced efficiency. Early pulse tube cryocoolers were not nearly as 47 efficient as Stirling cryocoolers, but advances in recent years have brought pulse tube refrigerators to the point of being the most efficient of all cryocoolers.

2.3. Modeling of PTR systems

1-D computational models have been widely used for modeling thermoacoustic devices. Swift et al. [5, 14, 40] developed a 1-D code for the entire PTR system (and other thermoacoustic engines and refrigerators) based on Rott’s [41] linear acoustic equations. While, the 1-D codes provide relatively good estimations of various operating parameters of the PTR (dimensions, operating frequencies, etc.); they use idealistic assumptions and do not reflect the multi-dimensional nature of the flow and transport inside the PTR systems. Other past computational studies of the PTR system include the work by Chao et al. [42] and Ju, et al. [43] where a 1-D system is assumed and the various gas dynamic equations are solved with losses being incorporated into the system using friction factor and heat transfer coefficients. What is notable about these studies is that they assume thermal equilibrium in the various heat-exchangers and do not account for axial heat conduction in the various heat-exchangers, most notably the regenerator where heat transfer from the warm to the cold end within the regenerator matrix itself is a loss mechanism. Other 1-D models include phasor type models [44-47], gas dynamic models [48, 49] and linear network type [50].

Lee [36, 51] developed a set of 2-D differential equations for use in describing the steady secondary flows generated by the periodic compression and expansion of an ideal gas in pulse tubes. The equations were used to obtain an insight into the physics of the 48 pulse tube in a basic pulse tube (BPTR) and an orifice pulse tube refrigerator (OPTR) for what is known as the thermally strong case.

More recently, Flake and Razani [52] carried out an axisymmetric analysis of a

BPTR and a PTR and showed cycle-averaged flow fields in the pulse tube. Cha et al.

[53] studied two IPTR systems based on the geometry of the pulse tube (for two values of

L/D ratio). They showed the formation of instantaneous vortical structures in the pulse tube for the small L/D case which had a negative effect on the cooling performance of the

IPTR due to the mixing of flow in the pulse tube.

Ashwin et al. [54] used a thermal non-equilibrium model in the porous media

(heat exchangers and regenerator) and considered a finite wall thickness for the various components of the IPTR. The effect of a finite wall thickness was found to increase the steady state temperature at the cold end of the pulse tube due to the heat conduction along the walls of the pulse tube from the hot end to the cold end.

More recently, there has been a push to build PTRs that work efficiently at higher frequencies [54-57]. The use of high frequency oscillations allows the system to be comparatively small in size. These smaller sized systems have niche applications in the space industry where localized low power (< 1 W) cooling systems with extremely fast cool-down times are required.

49

2.4. Acoustic streaming and streaming in PTRs

Acoustic streaming can be defined as steady convection which is driven by oscillatory phenomenon [58] in bounded channels. This “acoustic streaming” is a second order flow that results due to non-zero cycle-averaged velocities that exist in real acoustic resonator systems. The secondary streaming flow is superimposed on the oscillatory flow field. Typical acoustic streaming velocities are more than an order of magnitude smaller than the instantaneous oscillatory velocity flow field in the resonator [59]. The direction of acoustic streaming flow is mostly dc (uni-directional) in nature as compared to the ac nature of the oscillatory flow. Acoustic streaming has applications in mixing channels and enhanced convective heat transfer, etc. Figure 2.9a below shows the streaming patterns that appear near the external surface of a cylinder oscillating in a quiescent medium (here glycerin-water mixture). Figure 2.9b shows similar streaming patterns on the inside of an acoustic resonator tube that is excited by a loudspeaker. 50

Figure 2.9: Acoustic streaming patterns (a) external: outside a cylinder oscillating in a quiescent medium [60] and (b) internal: inside a cylinder that has oscillatory flow within

it generated by a loudspeaker [61]

Even though acoustic streaming has its advantages, in a PTR, the occurrence of streaming in the pulse tube leads to a deterioration of its performance. This is due to the basic function of the PTR, which is as stated earlier “to isolate the cold and warm ends from each other”. This is accomplished by the formation of a buffer zone in the center of the tube. This buffer zone (visualized as a plug of gas) oscillates within the pulse tube, but never leaves either end. The presence of a streaming or secondary

The presence of streaming in the pulse tube has been hypothesized [36, 51, 52,

62-64] to cause re-circulation in the pulse tube. Earlier studies of streaming [36, 51] showed unicellular cells in the pulse tube that circulated gas from the warm end of the pulse tube to the cold end. This leads to degradation in performance of the PTR. 51

Lee et al. [65] first proposed that acoustic streaming can be suppressed in a PTR by using a pulse tube that has a slight taper. Later Olson and Swift [66, 67] developed a relation to predict the optimum taper angle required suppress streaming in the pulse tube.

These equations were developed based on Rott’s [68] general method of calculating boundary-layer streaming based on standing wave phasing between the pressure and velocity. Rott’s equations had to be modified because the PTR is a travelling wave device and Rott’s equations were developed for a tube of constant cross-sectional area.

The Olson-Swift equation predicts the location at which streaming in the pulse tube of a

PTR is suppressed. In the studies by Olson and Swift [66, 67] and Swift et al. [69], the

Olson-Swift equation was used to design an experimental system whose performance was tested. However, the Olson-Swift relation is a 1-D simplification of a multi-dimensional problem and is applicable only in the limit of low amplitude oscillations. The concept of acoustic streaming suppression in pulse tubes too is not well known or visualized.

Further investigation of the acoustic streaming phenomenon in the pulse tube section of

PTR is required to study both, the effects of the acoustic streaming on the performance of the PTR and how this acoustic streaming can be suppressed.

2.5. Identification of research areas

As stated earlier, thermoacoustic refrigeration systems have gained increased importance in cryogenic cooling technologies in the recent past. However, improvements are needed to increase the efficiency and effectiveness of the current state of the art cryogenic refrigeration devices. These improvements in performance can be achieved by 52 superior design and performance prediction models. The current available models for design and performance prediction can be categorized as follows:

a. 1-D models for design: The most common and widely used design models for

PTR systems are DeltaEC [14] and SAGE [15]. These are system level 1-D

models that predict the performance of system by solving for the

thermoacoustic behavior in each of the PTR’s components. Another common

design model specific to the regenerator component is called the REGEN [70]

software developed at the National Institute of Standards and Technology

(NIST) in Boulder, CO. The advantage of these 1-D models is that they

provide quick design solutions (each simulation run takes about 5 – 10

minutes). However, they do not account for the multi-dimensional nature of

the flow and heat transfer in the PTR’s components.

b. Multi-dimensional models for performance prediction: There have been a few

published multi-dimensional models that have been used to predict the

performance of PTR systems [52-54, 71]. The advantage of these models

over the more common 1-D models is their multi-dimensional nature which

enables the study of flow and heat transfer processes in the PTR. However

the models have never been validated and most do not account for the non-

equilibrium nature of heat transfer in the various heat-exchanger and

regenerator components (i.e., the gas and solid temperatures are not equal).

An important application of multi-dimensional models is in PTR design. The

reported multi-dimensional models have not been used to investigate the 53

thermoacoustic phenomena in the PTR or use such knowledge for the design

of effective and efficient PTR systems.

With the ability to generate un-shocked high amplitude pressure waves, wave- shaped acoustic resonators demonstrate the possibility of improving the performance of

PTR systems. The research performed in the past on the wave-shaping phenomena has resulted in numerous models that have been able to predict the performance of such resonators and demonstrate their non-linear nature [4, 19-22, 24]. The available models are one-dimensional and do not consider the inherent multi-dimensional nature of real systems. The high amplitude pressure and velocity waves in these standing wave resonators coupled with the oscillatory nature of the flow in them requires a multi- dimensional model to predict the transport processes within it. Additionally, these models do not account for the temperature changes in these wave-shaped resonators which affect the flow and acoustic phenomena within. The most significant drawback of the available models and theory is the lack of design guidelines for design of wave- shaped acoustic resonators. Accurate design of wave-shaped resonators is vital to its application in PTRs.

A comprehensive research program was undertaken to overcome the shortcomings of the past research in the field of the pulse tube thermoacoustic refrigeration and wave-shaped acoustic resonators. The goal was to develop robust computational models to design and predict the transport processes in these systems and provide useful design guidelines based on the predictions of the models. To accomplish this, three main focus areas were chosen: (a) studying the acoustic and fluid dynamic interactions in consonant and dissonant acoustic resonators, (b) experimentally 54 investigating thermoacoustic refrigeration systems attaining cryogenic levels and (c) computationally studying the transport processes and energy conversion through fluid- solid interactions in thermoacoustic pulse tube refrigeration devices.

A numerical model of the wave-shaped acoustic resonator was developed and validated. The validated model was used to investigate the effect of operating fluid and driver amplitude on its performance. The results predicted the non-linear nature of such systems as observed in the past. Additionally, the results of the model were used to provide guidelines on how to calculate the resonant frequencies of these wave-shaped resonators. A numerical model of the PTR was also developed and validated with results from the experimental PTR (that was designed, fabricated and tested). The validated model was used to understand the thermoacoustic nature of PTR systems (effects of operating frequency and inertance tube geometry). The simulations also revealed interesting steady-periodic (acoustic streaming) flow patterns that develop in the pulse tube due to the fluctuations caused by the piston and the presence of the inertance tube.

These steady-periodic flow patterns are undesirable and suppression of this acoustic streaming was achieved by modifying the geometry of the pulse tube. The developed models can be combined to design pulse tube refrigeration systems enhanced by wave- shaping technology.

55

Chapter 3 : Numerical Studies of Transport Phenomena in Acoustic Resonators3

3.1. Introduction

As stated earlier in chapter 1, resonators can be classified into two groups with

respect to harmonic and natural frequencies, i.e., dissonant and consonant types. The

fundamental natural frequency of a system is the lowest frequency at which the resonator

responds to the input driving force with a maximum amplitude output (also called the

fundamental resonant frequency). The harmonic frequencies are integer multiples of the

fundamental natural frequency. In dissonant resonators the harmonics (integer multiples

of the fundamental natural frequency) do not coincide with the higher order natural

frequencies of the resonator (frequencies of the normal mode or modal frequencies) [17].

On the other hand, consonant resonators are systems for which the harmonic frequencies

coincide with the corresponding natural frequencies of that resonator [17, 72]. The

wave-shaping technique of using dissonant acoustic resonators (with non-equidistant

frequency spectrum) [17, 59] to convert linear acoustic input to high energy un-shocked

nonlinear acoustic output can be effectively used to increase the pressure ratio in the

resonator. The essence of wave-shaping lies in the fact that in a dissonant resonator, high

pressure ratio (compression ratio) nonlinear acoustic waves with shock-less structures can

be generated. When operated under similar experimental conditions and for the same

input power supplied to the driver mechanism (piston), a resonant cylindrical system

3 The results presented in this chapter can be found in, “Antao, D. S. and Farouk, B., ‘High amplitude nonlinear acoustic wave driven flow fields in cylindrical and conical resonators’, The Journal of the Acoustical Society of America, 2013, (in press)” 56 produces shock-like structures [17]. Most dissonant wave-shaped resonators studied to date have circular cross-sections with axially varying diameter and are axisymmetric.

In this chapter, high amplitude standing waves in a cylindrical consonant and a conical dissonant resonator are investigated numerically using a high fidelity compressible axisymmetric computational fluid dynamic model. The conservation equations of fluid flow (conservation of mass, momentum and energy) are solved for the fluid domain to accurately capture the transport processes within the resonators. The model is validated using past numerical results of standing waves in cylindrical consonant resonators. The nonlinear nature of the harmonic response of the conical resonator system is investigated for two different working fluids (carbon dioxide and argon) operating at various values of piston amplitude. The results presented in this chapter closely follow a paper under review [73].

3.2. Geometry of the consonant and dissonant resonators

In this chapter, we consider acoustic resonators with two different geometries: (a) cylindrical resonator (the ‘cylinder’ geometry) and (b) a dissonant conical resonator (the

‘cone’ geometry). Schematics of the geometries studied are shown in Figure 3.1a

(cylinder) and Figure 3.1b (cone). 57

Figure 3.1: Schematic of acoustic resonator geometries simulated (a) cylinder and (b)

cone

The dimensions of the two geometries are listed in Table 3.1. The dimensions of the cylinder geometry considered were based on past experimental and numerical studies of cylindrical acoustic resonators. The dimensions of the dissonant cone geometry were chosen such that the experimental “wave-shaped” acoustic resonators (based on dimensions used in the numerical study) can be easily manufactured. The cone angle of this resonator was chosen to be 15o which was similar to the cone angle of the ‘cone’ resonators used in the studies by Lawrenson et al. [17] and Ilinskii et al. [59]. The aspect ratio (L/D) for the cylinder geometry studied (Figure 3.1a) is 68.07 and that of the cone geometry studied (Figure 3.1b) is 5.03 (based on the mean diameter of the two ends of the cone geometry). 58

Table 3.1: Dimensions of acoustic resonator geometries simulated

Length ‘L’ Radius (cm) Angle ‘θ’ Resonator Geometry (cm) (Degrees) R1 R2

Cylindrical 170.180 2.500 2.500 0 (‘Cylinder’)

Conical (‘Cone’) 13.356 0.866 4.445 15

3.3. Numerical model of the acoustic resonator

The numerical model of the acoustic resonator is described in this following section. Specific details about the governing equations solved, the initial and boundary conditions and the numerical scheme used are provided.

3.3.1. Governing equations

The conservation equations for the gas (air, carbon dioxide (CO2) and argon (Ar)) within the system undergoing periodic compression and expansion are given as follows:

 ()u  0 t (3.1)

()u ()uu   p  t ij (3.2)

h  p 0 ()uuh  ()) k T  (  tt0 ij (3.3) 59

p  RT (3.4) where,  is the density, u is the (r – x) velocity vector, p is the pressure, T is the gas/fluid temperature, R is the ideal gas constant, k is the thermal conductivity of the fluid and the total enthalpy is h0 given by

p 1 2 h i   u 0  2   i cT (3.5) 1 p 1 2 T [] h u c 0  2   where, i is the internal energy and c is the specific heat capacity of the fluid.

The gases studied (air, CO2 and Ar) are assumed to behave like an ideal gas (eq.

3.4) and their viscosity (µ), specific heat capacity (c) and thermal conductivity (k) are considered to be temperature dependent. The temperature dependent properties are obtained from the NIST database [8].

The conservation equations of mass, momentum and energy for the gas are non- dimensionalized to investigate the effects of Reynolds number (governed by piston amplitude, piston frequency and fluid properties) on the transport processes in the resonators. The superscript * denotes non-dimensional variables and the subscript i indicates the properties evaluated at the initial temperature and pressure. The variables in the governing equations were non-dimensionalized in the following manner: 60

u x u u*** , x , tpiston t u piston 

***pT h0 p,   , T  , h0  pi i T i h i (3.6)  k * , k iik

where, upiston is the maximum velocity amplitude of the piston, µ is the dynamic viscosity of the fluid and δν (the viscous penetration depth) and a (the speed of sound) are given by:

2  ,  2  f   (3.7)  pi a RTi i where, ν is the kinematic viscosity of the fluid and ω is the angular frequency and γ is the ratio of specific heats.

The non-dimensionalized governing equations are as follows (variables with * indicate non-dimensional variables):

* ()**u  0 t* (3.8)

** () u *****11 *2() u u   p   ij (3.9) t  Mref Re ref 61

 **h *  0  * * *1 * *R p 2 pi * * **()() uuh0  ()M k T   ref   ij ttReref Prc 2

(3.10) where, the following reference non-dimensional numbers result

uupiston piston iicc   Reref , M ref  , Pr   (3.11)  a kii k

where, δν (the viscous penetration depth) is known (eq. 3.7 above) and c is the specific heat capacity of the gas.

3.3.2. Numerical scheme

The numerical scheme for solving the governing equations is based on the finite volume approach. The continuity, momentum and energy equations are solved for the fluid using the central difference scheme. The motion of the piston is captured by a moving grid scheme near the piston wall. The re-meshing scheme used in the simulations is the Transfinite Interpolation scheme [74].

A 2nd order Crank-Nicholson scheme (with a blending factor of 0.7) is used for the time derivatives in the continuity, momentum and energy equations. The time-step

(Δt) for the simulations is dependent on the geometry simulated. For the cylinder geometry, ‘Δt’ is chosen to be 1.1476 × 10-5 s. This value of ‘Δt’ (~ 850 Δt/cycle) is sufficient to accurately predict the motion of the piston and the pressure waves generated.

This time-step obeys the CFL condition with a Courant number (C = (a×Δt)/Δx, where ‘a’ is the acoustic speed for the given conditions) between 0.3 and 0.4. For the cone 62 geometry, ‘Δt’ is chosen to be 1.0 × 10-6 s. This time-step allows for sufficient temporal resolution of each cycle (~ 500 – 1500 Δt/cycle). An overall convergence criterion is set for all the variables at 10-4 in the iterative implicit numerical solver.

Due to the symmetry of the problem geometry, only one-half of the resonators’ geometry (Figure 3.1a and Figure 3.1b) is considered for the simulations. Both the cylindrical and conical geometries are studied with non-uniform structured grid. A total of 32000 grid points are used in the simulations of the cylinder resonator geometry

(similar to the study by Aganin et al. [75]) and a total of 11033 grid points are used in the simulations of the conical resonator geometry. The number of grid points chosen is dependent on the grid spacing required to maintain the above mentioned Courant number and the non-uniformity of the grid distribution ensured finer mesh near the walls to capture the effects of the boundary layer. Typical grid size near the wall boundaries is about 200 µm for the cylindrical geometry and between 20 µm – 30 µm for the conical geometry. The governing equations along with the boundary conditions are solved using

CFD-ACE+ [74].

3.4. Numerical model validation with past computational studies

The numerical model of the acoustic resonator is validated using results from previously published analytical and numerical studies of cylindrical. For comparison of the resonator model’s predictions to past published results, six cases were considered

(cases 1 – 6, Table 3.2). The first two cases correspond to the resonator excited at its resonant frequency (at the given pressure and temperature) and different values of piston 63 amplitude. The following four cases (cases 3 – 6, Table 3.2); correspond to excitation frequencies below and above resonance and are simulated to show the behavior of the cylindrical resonator when excited at frequencies other than its resonant frequency.

Table 3.2 below lists the cases studied with the cylindrical geometry.

Table 3.2: List of cases simulated for model validation with published analytical and

numerical results (cylinder geometry)

Case Frequency of Piston Piston Amplitude Wall Temperature No. (Hz) (μm) (K)

1 100.809 3165 293

2 100.809 5417 293

3 104.900 5417 293

4 97.389 2740 303

5 103.540 2740 303

6 107.641 2740 303

Table 3.3 lists the values of the reference non-dimensional numbers/parameters used in the non-dimensional form of the governing equations for the various cases studied.

Table 3.3: Reference non-dimensional numbers used in non-dimensional form of the

governing equations (cylinder resonator)

Case Reynolds Number Mach Number Prandtl Number No. (Reref) (Mref) (Pr)

1 28.951 0.0058 0.72 64

2 49.551 0.0100 0.72

3 50.546 0.0104 0.72

4 23.908 0.0048 0.72

5 24.652 0.0051 0.72

6 25.135 0.0053 0.72

3.4.1. Initial conditions for the numerical model

For the cylindrical resonators, the temperature and pressure in the system are assumed to be constant everywhere at the start of the simulation. The values of the temperature (equal to the wall temperature in Table 3.2) and mean pressure used for each case is 101.325 kPa.

3.4.2. Boundary conditions for the numerical model

For the validation of the model with past published analytical and numerical results, the piston for the cylinder geometry is located at x = L and r = R2. The piston is modeled as a reciprocating wall having an oscillatory velocity. The velocity of the piston is defined by the function, u = A0ωcos(ωt) = upistoncos(ωt), where A0 is the maximum displacement of the piston (piston amplitude), ω is the angular frequency (ω = 2πf), f is the frequency of operation and upiston = A0ω. Table 3.2 (given earlier) specifies the frequency (f) and maximum piston displacement (A0) boundary condition used for each of the cases simulated. 65

The wall boundaries at the rigid end and along the length of the resonator are maintained at a constant temperature (isothermal). The value of the wall temperature

(Tw) used for each case simulated is listed in Table 3.2. The isothermal boundary condition is the closest approximation to an experiment and has been shown in past studies to have the best agreement with experimental results [76, 77]. The piston wall is maintained adiabatic.

3.5. Validation results: Temporal pressure variation

In case 1 simulated, the resonant frequency fres of the resonator studied (L =

170.18 cm) is 100.809 Hz (for a temperature of 293 K). Figure 3.2a shows the ratio of the temporal evolution of the pressure measured at the rigid wall of the resonator (at x =

0, r = R1) to the initial mean operating pressure pi. The system appears to have achieved quasi-steady state around 0.15 – 0.2 sec. The distinct saw-tooth like pressure wave profile (characteristic of the 1st harmonic in a cylindrical acoustic resonator) is apparent from the plot in Figure 3.2a. A comparison between the results of the current model

(solid line), the results of Chester’s theoretical model (delta symbol) [2] and the results from Aganin et al.’s numerical model (diamond symbol) [75] is shown in Figure 3.2b.

In Figure 3.2c, the results of the current model (cases 2 and 3) are compared with the results of the numerical model of Goldshtein et al. [78] at higher values of piston amplitudes. It is important to note here that the resonant frequency of the cylinder resonator does not vary even though the piston amplitude is varied between case 1 and case 2 (only the pressure amplitude changes). 66

The temporal structure of the pressure waves (saw-tooth profile) predicted by the current model matches well with the results of all three studies, however the agreement with the results of Aganin et al. is better. The minor differences observed with the predictions of Chester, Aganin et al. and Goldshtein et al. are due to the assumptions in their studies (i.e., inviscid fluid and no boundary/wall effects which lead to larger pressure amplitudes).

Figure 3.2: Transient pressure variation at the rigid end of the cylindrical resonator (a) evolution from the start of the simulation to quasi-steady state (case 1), (b) comparison of

current model (case 1, solid line) with past studies (Chester’s semi-analytical model

‘delta symbol’ and Aganin et al.’s numerical model ‘diamond symbol’) and (c) comparison of current model (cases 2- solid line and case 3- dotted line) with past studies

at higher piston amplitudes (Goldshtein et al.’s numerical model ‘diamond’ and ‘delta’

symbols) 67

The behavior of an acoustic resonator off resonance is interesting. When the piston vibrates at a frequency different from the resonant frequency of the resonator;

‘beats’ are observed in the temporal pressure variation at the rigid wall. Beats occur due to the interaction of two waves moving at different frequencies [1]. The difference between these two dissimilar frequencies is called the ‘beat frequency’ (number of beats per second) and in the case of a resonator, the beat frequency is the difference between the frequency of the piston (vibrating at a frequency different from the resonant frequency) and the resonant frequency of the resonator. These beats die out over time, however the data obtained from the beating pressure waves can be used to accurately predict the resonant frequency of the acoustic resonator. This technique will be applied later in this chapter to accurately predict the resonant frequency of non-cylindrical resonators. In Figure 3.3, the temporal evolution of pressure waves at the rigid end of the resonator are plotted for cases 4 (Figure 3.3a), 5 (Figure 3.3b) and 6 (Figure 3.3c).

Figure 3.3: Transient pressure variation at the rigid end of the cylindrical resonator (at x

= 0, r = R1) from the start of the simulation (a) case 4, (b) case 5 and (c) case 6 68

In Figure 3.4a, Figure 3.4b and Figure 3.4c, the quasi-steady state results for the three cases (4, 5 and 6 respectively) are compared with the results from Aganin et al. [75] and the comparisons (temporal structures of the waves) are very good. For these three cases the resonator is excited by a piston operating at frequencies different from the resonant frequency. In case 4, the frequency used is 5% lower than the resonant frequency (at 303 K) and the corresponding values for cases 5 and 6 are 1% higher and

5% higher than the resonant frequency (at 303 K) respectively. The beating visible in

Figure 3.3a and Figure 3.3c demonstrates that the resonator is being excited at frequencies different from its resonant frequency. The only difference between the results of Aganin et al. and the current model is that the amplitude of the waves predicted by the current model is lower. This reduced amplitude is due to the effects of the wall and the viscosity of the gas which were neglected in that previous study.

Figure 3.4: Comparison of temporal pressure variation at the rigid end of the cylindrical

resonator near quasi-steady state (solid lines) with the results of Aganin et al. (symbol)

(a) case 4, (b) case 5 and (c) case 6 69

In Figure 3.3b and Figure 3.4b, the structures of the waveforms appear similar to those of waveforms observed at resonance, i.e., the saw-tooth like pressure wave profiles are visible. However, a minor difference between a pressure wave at resonance (Figure

3.2b) and one off resonance (Figure 4.4b) is that the saw-tooth profile is not as sharp.

The plots in Figure 3.4a and Figure 3.4c compare well (qualitatively) with the data presented by Cruikshank [3]. Cruikshank’s results show that an inverted ‘rectified sine- wave’ like structure (with the cusp of the wave on top) exists for frequencies above resonance (Figure 3.4c); and below resonance (Figure 3.4a), the waves exhibit a partially skewed ‘rectified sine-wave’ structure (with the cusp of the wave at the bottom).

Cruikshank experimentally observed that the pressure waveforms above and below the resonant frequency are asymmetrical and the higher frequency produces a higher value of peak-to-peak pressure. These observations match well with the predictions from the current numerical model.

3.6. Acoustic streaming in a cylinder resonator

Acoustic streaming in resonators is important for a variety of application (as discussed earlier in chapter 2) and at time it can hinder the performance of acoustic systems. Hence it is important to study the cycle-steady streaming phenomena in acoustic resonators. Past studies investigated acoustic streaming in cylindrical resonators operating at their resonant frequency [59, 79-81]. In this section, cycle averaged temperature contours and velocity vectors (acoustic streaming) in the cylindrical resonator excited at frequencies above, below and on resonance are presented. Favre 70 averaging is used to obtain the cycle averaged velocity values in the computational domain.

The results for the various cases are also characterized as a function of the maximum acoustic Reynolds number [82]. The acoustic Reynolds and Mach numbers are defined as follows:

uumax max Re , M  (3.12) acoustic acoustic a

where, umax is the maximum axial velocity amplitude of the gas in the resonator (obtained from the simulations at quasi-steady state). The acoustic Reynolds and Mach numbers for the cases studying the cylindrical and cone resonator geometries are listed in Table

3.4 below.

Ohmi and Iguchi [83] calculated the critical or transition Reynolds number for oscillatory flows in a pipe. Ohmi and Iguchi’s correlation shows that the critical

Reynolds number for oscillatory flow depends on the frequency of oscillation [83]. The correlation and experimental results of Ohmi and Iguchi match well with the published experimental results of Merkli and Thomann [82] and Hino et al. [84]. The correlation for the critical Reynolds number for oscillatory flow is shown below in eq. 4.13. This correlation is derived from Ohmi and Iguchi’s correlation using equations and relations provided in Hino et al. [84]. The critical Reynolds number for oscillatory flow is given by: 71

1 D 7 Reacoustic, crit  305 (3.13) 

where, D is the diameter of the resonator and δν is the viscous penetration depth. Based on this correlation above, the calculated critical Reynolds numbers for the six cases studied are listed in Table 3.4 below. When comparing the acoustic Reynolds number with its corresponding critical value (see Table 3.4), it can be seen that the flow is always laminar for all the cases simulated. The normalized frequency for the cylindrical resonator Ω is given by:

 f  (3.14) 0 fres

where, f is the frequency of the oscillating piston, fres is the resonant frequency of the cylinder of length L (fres = a/2L) and ω0 is the resonant angular frequency of the resonator

(ω0 = 2πfres). The maximum value of the acoustic Reynolds number is observed at the resonant frequency where the amplitudes of the thermodynamic properties in the system

(pressure, velocity, temperature, etc.) are at their highest values. Table 3.4 below lists the acoustic non-dimensional numbers that result from the simulations.

Table 3.4: Resultant acoustic non-dimensional numbers for the cylinder resonator

geometry

Normalized Acoustic Reynolds Critical Acoustic Acoustic Mach Case Frequency Number Reynolds Number Number No. (Reacoustic,crit) (Ω) (Reacoustic) (Macoustic)

1 1.00 459.029 662.724 0.0926 72

2 1.00 640.531 662.724 0.1292

3 1.04 501.146 664.610 0.1031

4 0.95 186.154 658.272 0.0374

5 1.01 388.192 661.158 0.0804

6 1.05 153.625 662.995 0.0324

The velocity streamlines in Figure 3.5a, 3.5b and 3.5c depict the cycle averaged flow patterns in the cylindrical resonators for cases 1 (116th cycle), 4 (112th cycle) and 6

(124th cycle) respectively (Note: The vertical axis ‘r’ has been exaggerated to show the two-dimensional nature of the flow in the resonator). In case 1 (Figure 3.5a), the excitation frequency is the resonant frequency of the resonator and the acoustic Reynolds number is the highest compared to those of the resonators operating at off-resonant frequencies (cases 4 (Figure 3.5b) and 6 (Figure 3.5c)). For the system excited at its resonant frequency, distinct counter rotating cells are visible. The two non-interacting loops have smaller sub-loops that show similar structures to previous studies where inner and outer streaming cells were observed [59]. 73

Figure 3.5: Cycle averaged temperature contours and velocity streamlines (acoustic streaming) in the cylindrical resonator near quasi-steady state for different operating 74

frequencies (a) 100.809 Hz (case 1, Reacoustic = 459.03), (b) 97.389 Hz (case 4, Reacoustic =

186.15) and (c) 107.641 Hz (case 6, Reacoustic = 153.63)

In Figure 3.5a, the maximum and minimum values of streaming velocity in the axial direction are 0.585 m/s and -0.515 m/s (the negative sign indicates flow in the negative x-direction) respectively for case 1, however the corresponding values for cases

4 (Figure 3.5b) and 6 (Figure 3.5c) are ~ 0.15 and -0.10 m/s. The maximum and minimum velocities in the radial direction are 0.07 m/s and -0.08 m/s for case 1 and 0.02 m/s and -0.09 m/s for cases 4 and 6. This difference in streaming velocities shows the enhancement of acoustic streaming in a resonator when it is excited at its resonant frequency.

3.7. Numerical studies of cone shaped resonators: Results and discussion

For the dissonant conical resonator, six cases are studied (cases 7 – 12). As seen in past studies of “wave-shaped” cone resonators, the resonant frequency of the resonator is dependent on the gas, the cone angle and the amplitude of the driver. The six cases studied here correspond to two different gases (carbon dioxide and argon) excited at their resonant frequencies and at various amplitudes of the piston driver (10 µm, 50 µm and

100 µm). Since the cone angle is fixed in this study, the resonant frequency is only dependent on the gas and driver amplitude. Table 3.5 below lists the cases studied with the cylindrical geometry. 75

Table 3.5: List of cases simulated for model validation with published analytical and

numerical results (cone geometry)

Case Operating Frequency of Piston Piston Amplitude Wall Temperature No. Fluid (Hz) (μm) (K)

7 CO2 1187 10 300

8 CO2 1217 50 300

9 CO2 1239 100 300 10 Ar 1844 10 300

11 Ar 1887 50 300

12 Ar 1935 100 300

Table 3.6 lists the values of the reference non-dimensional numbers/parameters used in the non-dimensional form of the governing equations for the various cases studied.

Table 3.6: Reference non-dimensional numbers used in non-dimensional form of the

governing equations (cone resonator)

Case Operating Reynolds Number Mach Number Prandtl Number No. Fluid (Reref) (Mref) (Pr)

7 CO2 0.948 0.0003 0.78

8 CO2 4.799 0.0014 0.78

9 CO2 9.685 0.0029 0.78 10 Ar 0.902 0.0004 0.67

11 Ar 4.563 0.0018 0.67

12 Ar 9.242 0.0038 0.67 76

3.7.1. Initial and boundary conditions for the numerical model

The initial conditions and boundary conditions used for the simulation of the cone resonator (Figure 3b) are described below.

3.7.1.1. Initial conditions

Past experimental and numerical studies of a cone-shaped resonator have shown that as the amplitude of the piston is increased, a frequency sweep of the piston oscillating frequency exhibits hysteresis (i.e., increasing and decreasing the frequency results in different values of the fundamental resonant frequency). Hysteresis indicates that the initial condition affects the resonant frequency of a cone resonator at high piston amplitudes. To account for this phenomenon, we use two different types of initial conditions in this sub-section. For cases 7 and 10 where the piston amplitude is small, the initial condition used was similar to the initial condition applied for the simulations of the cylindrical resonator, i.e., uniform pressure and uniform temperature in the entire domain (T = 300 K and P0 = 501.325 kPa). For the other cases (8, 9, 11 and 12) frequency sweeps (multiple simulation runs where the frequency of the piston is changed/increased with each run until the resonant frequency is identified) were used to identify the resonant frequency. The initial condition for each case was the quasi-steady state results (pressure, density, temperature and velocity) of the previous simulation run. 77

3.7.1.2. Boundary conditions

Similar to the cylindrical resonator, the piston for the cone geometry is located at x = L and r = R2. The piston is modeled as a reciprocating wall having an oscillatory velocity. The velocity of the piston is defined by the function, u = A0ωcos(ωt) = upistoncos(ωt), where A0 is the maximum displacement of the piston (piston amplitude), ω is the angular frequency (ω = 2πf), f is the frequency of operation and upiston = A0ω. Table

3.5 (given earlier) specifies the frequency (f) and maximum piston displacement (A0) boundary condition used for each of the cases simulated.

The wall boundaries at the rigid end and along the length of the resonator are maintained at a constant temperature (isothermal). The value of the wall temperature

(Tw) used for each case simulated is listed in Table 3.5. The piston wall is maintained adiabatic.

4.7.2. Results: Un-steady processes in the dissonant cone resonator

The temporal behavior of the pressure, temperature and velocity waves in a resonator can be used to evaluate the applicability of the resonator for various applications. This temporal behavior also can be used to provide design guideline for the resonator system. The biggest hurdle that must be overcome when working with dissonant resonators (as discussed earlier) is that there are no set guidelines on how to design such systems. In this sub-section of the chapter, a method on predicting the resonant frequency of a cone resonator is suggested and discussed. Furthermore, the 78 temporal behavior of the transport properties in the system is characterized as a function of the operating fluid and the amplitude of the piston driver.

In Figures 3.6a and 3.6b, the pressure at the rigid end of the conical resonator

(Figure 3.1b) is plotted as a function of time. The frequency of the vibrating piston

(larger end, r = R2) in Figure 3.6a is 1000 Hz. The value of 1000 Hz was chosen because it was calculated to be the fundamental resonant frequency (fres) of a cylindrical resonator of the same length (L = 13.356 cm) operating at a temperature of 300 K and a pressure of

501.325 kPa (CO2).

Figure 3.6: Temporal evolution of gas pressure at the rigid end of the cone resonator

(Fig. 1b) filled with CO2 and excited at (a) 1000 Hz (below resonance) and (b) 1187 Hz

(the actual resonant frequency, case 7)

The 'beats' visible in Figure 3.6a clearly show that the system is operating off resonance and based on the definition of beats and beat frequency fbeat (number of beats 79 per second) introduced earlier, the resonant frequency of the cone resonator can be estimated (i.e., adding the beat frequency to the frequency of the piston’s motion). The actual resonant frequency for the cone resonator fact,res (in a cylinder fact,res = fres) obtained from this analysis is found to be 1000 + 187 = 1187 Hz. In Figure 3.6b, the plot of temporal pressure evolution at the rigid end of the conical resonator excited at its actual resonant frequency (f = 1187 Hz, case 7) is shown. At this frequency, the peak-to-peak pressure is at the maximum for the given piston amplitude (A0 = 10 µm). This method is applied to find the resonant frequency of the conical resonator filled with Ar and excited by a piston with amplitude A0 = 10 µm. A simple formula to calculate the actual resonant frequency (fact,res) of a conical resonator is:

fact, res f res f beat (3.15)

where, fres is the resonant frequency for a cylindrical resonator of the same length and fbeat is the beat frequency. The normalized frequency for the cone resonator used in this study

‘Ωco’ (listed in Table 3.7 below) is given by:

fact, res co  (3.16) fres

Table 3.7 below also lists the acoustic Reynolds and acoustic Mach numbers (eq.

3.12) and the critical acoustic Reynolds number for oscillatory flow. A point of note; for the cone resonator cases considered in this sub-section, the diameter used in eq. 3.13 is the mean of the diameters at the two ends of the resonator. As with the cases of the cylindrical resonator, all the acoustic Reynolds number values are below the critical values calculated from the correlation of Ohmi and Iguchi [83]. 80

Table 3.7: Resultant acoustic non-dimensional numbers for the cone resonator geometry

Cone Normalized Acoustic Reynolds Critical Acoustic Acoustic Mach Case Frequency Number Reynolds Number Number No. (Reacoustic,crit) (Ωco) (Reacoustic) (Macoustic)

7 1.187 162.512 933.566 0.0483

8 1.235 663.756 936.214 0.1985

9 1.257 834.295 937.395 0.2528

10 1.523 79.735 926.884 0.0335

11 1.546 432.553 927.884 0.1923

12 1.600 529.865 930.152 0.2476

The plots of transient pressure evolution (near quasi-steady state) at the rigid end of the conical resonator filled with CO2 are shown in Figure 3.7 (a – c). Figure 3.7a corresponds to a piston amplitude (A0) of 10 µm (case 7), Figure 3.7b to an amplitude of

50 µm (case 8) and Figure 3.7c to an amplitude of 100 µm (case 9). 81

Figure 3.7: Transient pressure variation at the rigid end of the conical resonator near

quasi-steady state for (a) case 7, (b) case 8 and (c) case 9

Even for the system operating at the first harmonic frequency, the pressure profiles in a conical resonator do not have the saw-tooth like structure observed in cylindrical resonators. At lower amplitudes (A0 = 10 µm) the structure of the wave is close to a sinusoid. However, at higher amplitudes (A0 = 50 µm and 100 µm), the pressure waves are no longer symmetrical about the ambient/mean pressure value (the difference between the peak pressure at the crest of the wave and ambient pressure is much greater than the corresponding difference between the trough of the wave and the ambient pressure).

The structure of the waves resembles the inverted ‘rectified sine wave’ structure

(with the cusp at the top) observed by Cruikshank [3] in cylindrical resonators excited at frequencies above their resonant frequency. Another important observation from a comparison of the results of these three cases is that as the piston amplitude is increased, the actual resonant frequency ‘fact,res’ of the resonator shifts to higher values. This is 82 termed as resonance hardening [17, 59] (i.e., the fundamental resonant frequency of the resonator increases with an increase in the amplitude) and is not observed in cylindrical resonators where resonant shifts due to temperature or changes in piston amplitude are fairly small (less than 1 Hz).

In Figure 3.8 (a – c), the pressure wave profiles are plotted for the cone resonator operating with Ar as the working fluid near quasi-steady state. These pressure waves have similar structures to the pressure wave profiles in CO2 (Figure 3.7 (a – c)). At lower piston amplitudes, the structure of the wave is close to a sinusoid (Figure 3.8a) and at higher amplitudes as seen in Figures 3.8b and 3.8c, the temporal variation of the pressure waves has an inverted ‘rectified sine wave’ structure.

Figure 3.8: Pressure variation at the rigid end of the conical resonator near quasi-steady

state for (a) case 10, (b) case 11 and (c) case 12

Due to its higher speed of sound, the resonant frequency for cylinder filled with

Ar will be higher than that of a cylinder filled with CO2. For a cylindrical resonator filled with Ar, length L = 13.356 cm, operating at a temperature of 300 K and a pressure of 83

501.325 kPa the value of the calculated resonant frequency is 1208.9 Hz. The normalized frequency for the cone resonator Ωco for cases 10, 11 and 12 are 1.523, 1.546 and 1.600 respectively (based on eq. 3.16). These values of non-dimensional frequency are large compared to the corresponding values in CO2. Additionally, for the conical resonator operating with CO2 (cases 7, 8 and 9) the values of peak-to-peak pressure are approximately 116 kPa, 549 kPa and 781 kPa, whereas the corresponding values in Ar

(cases 10, 11 and 12) are approximately 131 kPa, 855 kPa, and 1282 kPa respectively.

In Figures 3.9a, 3.9b and 3.9c, the ratio of transient pressure variation to mean operating pressure (p/pi) at the rigid end of the cone resonator near quasi-steady state is compared for the two gases studied (CO2 and Ar). The pressure profiles are plotted as a function of the cycle period due to the difference in the actual frequencies for the different cases. The ratio of transient pressure to mean operating pressure is higher for the cone resonator operating with Ar (2.6 – 3.5) compared to CO2 (1.8 – 2.2). In the case of Ar the entire pressure wave shifts up with the troughs of the waves having ratios close to 1.0. This implies that the values of pressure in the troughs of the wave are close to the initial mean operating pressure of the system. This is only possible if there is considerable heating of the gas within the resonator which results in an increase in the mean operating pressure. This increase in temperature implies that the heat generated due to the oscillations is not being removed by the cooling at the wall boundaries

(isothermal boundary condition with Tw = 300 K).

In addition to the pressure variations, due to the ideal gas assumption, the density and the temperature oscillate too. In Figure 3.10, the ratio of the transient density variations to the initial density for CO2 and Ar are compared. 84

Figure 3.9: Comparison of the ratio of transient pressure variation to mean operating pressure at the rigid end of the conical resonator in CO2 (solid line) and Ar (dashed line) for different piston amplitudes (a) 10 µm (case 7 and case 10), (b) 50 µm (case 8 and case

11) and (a) 100 µm (case 9 and case 12) 85

Figure 3.10: Comparison of the ratio of transient density variation near the rigid end of

the conical resonator to the initial density in CO2 (solid line) and Ar (dashed line) for

different piston amplitudes (a) 10 µm (case 7 and case 10), (b) 50 µm (case 8 and case

11) and (a) 100 µm (case 9 and case 12)

The structures of the temporal variations in the density (Figure 3.10) are similar to the variations in the pressure (i.e., the inverted rectified sinusoid, due to the ideal gas assumption). The temporal variation of the temperature (Figure 3.11) will have a similar profile due to the assumption of an ideal gas in the resonator. This implies that the peak temperature attained in every cycle (compression stroke of the piston) is approximately

1.5 – 2 times the value of the lowest temperature in that cycle (expansion stroke of the 86 piston). This leads to an increase in the average temperature of the gas inside the resonator from cycle to cycle. Over time, this leads to an accumulation of heat at the small end of the resonator and the increase in temperature cannot be balanced by the heat being removed at the walls (given the high frequency of operation).

Figure 3.11: Comparison of the ratio of transient temperature variation near the rigid end of the conical resonator to the initial temperature in CO2 (solid line) and Ar (dashed

line) for different piston amplitudes (a) 10 µm (case 7 and case 10), (b) 50 µm (case 8

and case 11) and (a) 100 µm (case 9 and case 12) 87

The reason for the higher temperature values in Ar can be explained by the comparing the ratio of specific heats ‘γ’ for the two gases (1.66 for Ar and 1.28 for CO2).

The higher value of γ for Ar implies that the resonator operating with Ar will have higher pressure fluctuations compared to the system operating with CO2. This will lead to a larger increase in temperature and mean operating pressure for the system with Ar over time.

3.7.3. Results: Cycle-steady behavior in the dissonant cone resonator

The 2D numerical model of the cone resonator developed in this study enables for the first time the study of multi-dimensional effects in such wave-shaped systems. It has been suggested that wave-shaped resonators can be used for various applications including mixing chambers and chemical reactors. However there has been a lack of conclusive evidence, both experimental (due to the high pressure nature of the system) and numerical (due to the absence of multi-dimensional models) that proves these possibilities. In this subsection of the thesis, the cycle-steady thermal and fluid behavior of the cone resonator is studied under a variety of operating conditions (operating fluid, operating frequency and piston amplitude).

The plots in Figure 3.12 show the cycle-averaged radial variation of the gas temperature in the cone resonator near the fixed end (Figure 3.12a) and near the piston

(Figure 3.12b) for case 7. At lower piston amplitudes, the temperature rise in the resonator is small with the cycle averaged temperature near quasi-steady state. The temperature rise is mainly near the small end of the cone resonator (where the pressure 88 fluctuations are greater) and that the temperature near the piston (i.e., near the large end) is close to the initial temperature. The maximum change in the temperature of the gas in the resonator occurs near the wall (thermal boundary layer).

Figure 3.12: Radial variation of the cycle-averaged gas temperature in the conical resonator at quasi-steady state for a piston amplitude of 10 µm operating with CO2 (case

7) near (a) the fixed end of the resonator (x = 0.001 m) and (b) the piston (x = 0.13 m)

Figure 3.13a shows the cycle averaged temperature contours (619th cycle) in the conical resonator with CO2 and a piston amplitude of A0 = 50 µm (case 8) near quasi- steady state (t = 0.5 s) and Figure 3.13b shows corresponding cycle averaged temperature contours for Ar (562nd cycle, case 11). In Ar, the cycle averaged temperature is ~ 540 –

580 K in the small end of the cone resonator and the corresponding value for CO2 is ~

340 – 380 K. The temporal temperature profiles (Figure 3.11) show that the non-linear temperature oscillations occur only in the small end of the resonator and this explains why the high temperature zones are near the small end of the cone resonators (Figure

3.13 below). 89

Figure 3.13: Cycle averaged temperature contours in the conical resonator near quasi-

steady state for a piston amplitude of 50 µm operating with two different gases (a) CO2

(case 6) and (b) Ar (case 9)

The cycle averaged temperature contours and the cycle averaged velocity vectors

(acoustic streaming) in the cone resonator with CO2 near quasi-steady state are shown in

Figure 3.14. Figure 3.14a shows the results for case 7 (1188th cycle), Figure 3.14b for case 8 (619th cycle) and Figure 3.14c for case 9 (630th cycle). The temperature at the small end of the cone resonator increases as the piston amplitude is increased.

At the lower piston amplitude, the structure of the pressure and temperature waves are almost sinusoidal with no “nonlinearity” observed and hence the quasi-steady state temperature in the entire resonator has increased very little from the initial condition of Ti = 300 K. However at the higher piston amplitude values a nonlinear temperature profile is observed (i.e., the inverted ‘rectified sine wave’ structure) and this leads to heating in the small end of the cone resonator as explained earlier. 90

Figure 3.14: Cycle averaged temperature contours and velocity vectors (acoustic streaming) in the conical resonator near quasi-steady state for CO2 operating at different values of piston amplitude (a) 10 µm (case 7), (b) 50 µm (case 8) and (c) 100 µm (case 9)

The acoustic streaming flow seen in Figures 3.14a, 3.14b and 3.14c shows interesting distribution patterns that vary as the piston amplitude is increased. For case 7

(Figure 3.14a), the maximum and minimum values of streaming velocity in the axial direction are 0.432 m/s and -0.410 m/s (the negative sign indicates flow in the negative x- direction) respectively, the corresponding values for cases 8 (Figure 3.14b) are 3.329 m/s and -4.855 m/s and those for case 9 (Figure 3.14c) are 4.454 m/s and -6.736 m/s respectively. In the above figures (3.14a, 3.14b and 3.14c), a vector length of 1.0 cm corresponds to a velocity magnitude of 2.0 m/s. The maximum instantaneous velocities vary between 67.55 m/s (case 9) and 12.79 m/s (case 7) and the corresponding minimum 91 values vary between -65.35 m/s (case 9) and -12.96 m/s (case 7). These values of instantaneous and cycle averaged velocity listed above result in peak normalized axial streaming velocities (normalized by peak instantaneous velocity in the axial direction) being ~ 0.07 – 0.10, however, the bulk of the normalized streaming velocities are between 0.01 – 0.02. The maximum and minimum streaming velocities in the radial direction are 0.158 m/s and -0.168 m/s for case 7, 0.865 m/s and -1.314 m/s for case 8 and 1.155 m/s and -1.818 m/s for case 9. The streaming velocity values listed above are the maximum streaming velocities possible for a given piston amplitude since the system is being excited at its resonant frequency (as observed in the cylindrical resonators). As the piston amplitude is increased, the strength of the streaming velocity also increases.

We also observe a change in the structure of the acoustic streaming flow- the streaming cells increase in size as the piston amplitude is increased.

In conical resonators, much higher acoustic streaming velocities are possible at lower piston amplitudes and better mixing is observed in conical resonators (larger number of streaming cells) as compared to cylindrical resonators. For the same input amplitude, a conical resonator can provide better mixing of the operating fluid and introduces the possibility of using conical resonators as efficient mixers.

3.8. Non-linear phenomena in dissonant cone resonators

The nonlinear behavior of cone shaped resonators includes the shift in resonant frequency as the piston amplitude changes and the non-sinusoid pressure waves observed at the rigid end of a resonator excited by a sinusoidal input. Additionally as mentioned 92 earlier, the hysteresis observed in frequency response curves is characteristic of dissonant resonator systems and has been documented in past experimental studies [17]. The frequency response curves for a conical resonator operating in CO2 with a piston amplitude of 50 µm (case 8) are plotted in Figure 3.15. The curves are plotted for increasing piston oscillating frequency (square symbol) and decreasing piston oscillating frequency (diamond symbol). In an experimental system, to tune an acoustic resonator to its resonant frequency, the frequency is varied until resonance is achieved (i.e., the peak- to-peak pressure or pressure ratio is at its maximum). To incorporate this in the numerical simulations, the initial condition was varied for each run. For the ‘increasing frequency’ curve, the simulations were started at a frequency far from resonance (~ 30 –

40 Hz below resonance) and for the ‘decreasing frequency’ curve, the simulations were started at a frequency above resonance. 93

Figure 3.15: Frequency response curves of the pressure ratio at the rigid end of the

dissonant conical resonator operating with CO2 and a piston amplitude of 50 µm for

increasing frequency (square symbol) and decreasing frequency (diamond symbol)

Once the simulation reaches a quasi-steady state, the quasi-steady state simulation data is used as an initial condition for the following simulation run at the next frequency

(closer to the resonant frequency). This method is followed until resonance is achieved, i.e., maximum peak-to-peak pressures or pressure ratios are observed. The ‘increasing frequency’ and ‘decreasing frequency’ curves do not coincide. The maximum value of pressure ratio observed is different for both the curves. For the ‘increasing frequency’ curve, the maximum pressure ratio predicted is 2.45 and is observed at a frequency of 94

1235 Hz (Ωco = 1.235) and for the ‘decreasing frequency’ curve, the maximum pressure ratio predicted by the simulations is 1.72 as is observed at 1202 Hz (Ωco = 1.202). Past experimental studies of conical resonators have shown this trend of the ‘increasing frequency’ curve having a higher peak pressure ratio than the ‘decreasing frequency’ curve [17]. This hysteresis does not occur in cylindrical resonators or in conical resonators excited at lower piston amplitudes.

The plot in Figure 3.16 shows the variation in the normalized resonant frequency of the cone resonator (eq. 3.16) with increasing piston amplitude for the two gases studied (i.e., CO2, cases 7 – 9 and Ar, cases 10 – 12). As can be seen, as the piston amplitude increases the resonance frequency shifts to higher values. This behavior is termed as ‘hardening’ and has been observed experimentally in dissonant conical resonators [17]. The term ‘hardening’ comes from the nonlinear behavior of a mechanical oscillator (mass-spring) system where the spring stiffness is dependent on the amplitude (hardening behavior implies increasing spring stiffness with increase in displacement amplitude) [85]. The shift in resonance frequency for both gases is fairly linear over the piston amplitudes investigated. For CO2 from 10 µm to 100 µm the shift is about 70 Hz and the corresponding shift for Ar is 93 Hz. A stronger hardening behavior exists in Ar when compared to CO2.

The coefficient of nonlinearity β suggested by Lawrenson et al. [17] is as follows:

 1  1 (3.17) 2 95

which is dependent on the ratio of specific heats of the gas γ. For CO2 (γ = 1.28) the value of β is 1.14 and that of Ar (γ = 1.66) is 1.33. According to Lawrenson [17], higher values of β result in higher nonlinear behavior. The results of the simulations in Ar predict larger shifts in resonance frequency as well as larger pressure ratio values.

In Figure 3.16, the shift in resonance frequency for Ar is much higher than for

CO2 and hence there are larger increases in the frequency for Ar, which leads to larger changes in the acoustic Reynolds number over the same piston amplitude variation. As the piston amplitude is increased, the acoustic Reynolds number increases, however, the piston amplitude has a greater influence on the acoustic Reynolds number in Ar compared to CO2 (Table 3.7 seen earlier). The larger change in the values of Reacoustic as a function of piston amplitude for Ar is due to the fact that it is dependent on the value of

δν (eq. 3.6) which is affected by both the frequency and viscosity of the gas. 96

Figure 3.16: Comparison of the shift in resonance frequency with increasing piston amplitude observed in a cone shaped resonator operating with CO2 (solid line and square symbol, cases 7 – 9, Table 3.5) and with Ar (dashed line and delta symbol, cases 10 – 12,

Table 3.5)

Finally, the plots in Figure 3.17a compare the pressure ratios (pmax/pmin) at the rigid end of the cone resonator as a function of the acoustic Reynolds number (Reacoustic) for the two gases studied (CO2 and Ar). For comparable Reacoustic values, the plots for Ar show much larger values of pressure ratio, but as shown in the previous sub-section, with

Ar there is a considerable increase in the mean temperature of the gas in the system. This can also be seen in Figure 3.17b where the temperature ratio (Tmax/Tmin) is plotted as a 97

function of the acoustic Reynolds number. For comparable values of Reacoustic, the temperature ratio in Ar is much higher than the corresponding value in CO2 and this can be a limiting factor for use of Ar in various applications even though the pressure ratios are much higher.

Figure 3.17: Comparison of (a) pressure ratio (pmax/pmin) and (b) temperature ratio

(Tmax/Tmin) as a function of the acoustic Reynolds number observed in a cone shaped resonator operating with CO2 (solid line and square symbol, cases 7 – 9) and Ar (dashed

line and delta symbol, cases 10 – 12)

3.9. Summary of numerical studies on acoustic resonators and brief conclusions

The nonlinear effects observed in conical acoustic resonators make designing such systems challenging. Past numerical and experimental studies do not provide guidelines on how to design such systems and find their optimum operating conditions.

In the present study, nonlinear, high amplitude standing waves in cylindrical and non- cylindrical (cone) circular resonators are investigated numerically using a high fidelity 98 compressible axisymmetric computational fluid dynamic model. The conservation equations of fluid flow are solved in the fluid domain to accurately capture the transport

(flow and heat transfer) processes within these resonators. The numerical model in the present study solves the energy equation for the fluids in the resonators and this introduces the effects of temperature on the transport in the resonators. The model is validated using past numerical results of standing waves in cylindrical resonators. The nonlinear nature of the harmonic response of the conical resonator system is investigated for two different working fluids (carbon dioxide and argon) operating at various values of piston amplitude. The results in the cone resonator demonstrate how the temperature distribution within it affects the pressure and flow fields depending on the gas and the driver amplitude studied.

Using beat theory [1], it is possible to accurately predict the resonant frequency of conical resonators operating at low piston/driver amplitudes. At higher amplitudes, the resonant frequency shifts to higher frequencies and this ‘resonance hardening’ behavior has been observed in past numerical and experimental studies of cone resonators. In addition to the resonance hardening, the structure of the wave is observed to change when the piston amplitude is increased. At low piston amplitudes, the structure of the pressure waves is close to being sinusoidal. However at higher piston amplitudes the shape and structure of the wave change to the inverted ‘rectified sinusoid’ with the cusps of the waveform on the top. The use of argon (or any noble gas since they have high ratios of specific heat) may not be suitable for many applications due to the heat generated in the system- we observed that the heat generated in the resonator is not sufficiently removed at the walls of the resonator which are being cooled (isothermal boundary condition). 99

The overall heating of the system with argon as the operating fluid also raises the mean operating pressure in the system. Based on the results from this study it is possible to suggest that nitrogen may be a more suitable option to either carbon dioxide or argon.

The evolution of the temperature of the gas in the resonator and the driver operating amplitude should be considered too. It is also observed that the quasi-steady streaming observed in the conical resonators is dependent on the piston amplitude. As the piston amplitude is increased, the large numbers of small acoustic streaming cells that are observed at lower piston amplitudes coalesce into larger streaming structures with higher streaming velocities. The acoustic streaming observed in conical resonators is different to that observed in the cylindrical resonators. One of the differences in the two acoustic streaming patterns is that the structures of the streaming profiles are different, with large organized streaming cells seen in the cylindrical resonator (in the current study) compared to the smaller streaming cells observed in the conical resonator. This difference in patterns possibly arises due to the differences in excitation frequencies with higher resonance frequencies resulting in larger number of smaller streaming cells. The high streaming velocities observed in conical dissonant resonators and the large number of small streaming cells indicates that efficient mixing will possible in conical resonators.

100

Chapter 4 : Experimental Studies of a Pulse Tube Refrigerator4

4.1. Introduction

As stated in the motivation section of chapter 1, there are no experimentally validated computational models of the pulse tube refrigerator. Based on this motivation, two objectives were set for the experimental studies of the pulse tube cryocooler. The first objective was to design, fabricate and test a cryogenic PTR capable cooling in the liquid nitrogen temperature range (i.e., ~ 77 K). The second objective was to characterize the PTR over a wide variety of operating parameters. The goal of the second objective was to gain a better understanding of how the PTR operates and the extent to which the operating parameters affect its performance. A byproduct of the successful completion of the above two stated objectives help achieve another objective, which is model validation against experimental results.

In keeping with the objectives, two cryogenic capable pulse tube refrigerators were designed, fabricated and tested. This chapter reports the entire process of design, fabrication and testing of each of the systems. The lessons learned from building the first

PTR were applied to design and build a second system which was capable of liquid nitrogen level temperatures. Each of the two sections (characterization of the two

4 The results presented in this chapter can be found in the papers: a. Antao, D. S. and Farouk, B., “Experimental and numerical investigations of an orifice type cryogenic pulse tube refrigerator”, Applied Thermal Engineering, v. 50, n. 1, pp. 112 – 123, 2013 b. Antao, D. S. and Farouk, B., “Experimental and numerical characterization of the inertance effect on pulse tube refrigerator performance”, Journal of Applied Physics, (under review), 2013 101 different cryocoolers) reported in this chapter are self-contained with separate experimental methods, results and discussion sections and the two cryocoolers were used to study effects of different operating parameters on the PTR’s performance.

4.2. Characterization of an orifice type PTR (Mk-I)

This section reports the experimental studies carried out on the first pulse tube refrigerator built [86]. The PTR was labeled the Mk-I system and its main purpose was to provide a better understanding of the PTR’s performance when operated under different operating conditions. The following subsections detail the design, fabrication and testing steps of the experimental study. The choices of components, component material and the operating conditions tested are justified. A vacuum chamber was built and assembled after repeated efforts to achieve the designed performance (temperature in the cold heat-exchanger) of the Mk-I PTR fell short of the mark. The design, fabrication and motivation for using the vacuum chamber are also discussed below.

4.2.1. Mk-I Orifice type PTR geometry and fabrication

The OPTR was designed using a computer code DeltaEC [14]. DeltaEC is a 1-D model that allows the user to design the dimensions of a thermoacoustic cryocooler based on the constraints of the experimental setup (pressure, frequency, heat-exchanger and regenerator matrix material parameters and boundary conditions). The pressure wave generator used in the experiments was a QDrive twin-STAR linear motor type pressure 102 wave generator. The linear motor has a swept volume of 15.55 cubic centimeters and a maximum operating pressure of 2.5 MPa. DeltaEC derives its solution by solving a 1-D wave equation based on a shooting method. Hence, the initial dimensions and input parameters provided to the code have to be carefully chosen. Empirical correlations were used from a review paper by Radebaugh [87] to generate these initial values. Based on the swept volume of the pressure wave generator, the mean operating pressure, the frequency of operation and the regenerator and heat exchanger characteristics, DeltaEC provided the lengths of the various components5. Figure 4.1 shows an image of the experimental OPTR system.

Figure 4.1: A photograph of the experimental OPTR fabricated (see Table 4.1 for a

listing of the components)

Due to the large temperature gradients over small lengths of the system and use of high pressure gases (helium), it is important to keep in mind the thermal properties of the materials used to fabricate the various components. The heat exchangers were made from copper tubing (due to its high thermal conductivity) and the other components of the

OPTR were made of 316-type stainless steel tubing (stainless steel exhibits relatively low

5 See Appendix A (section A.1) for a sample of the DeltaEC code used to design the Mk-I OPTR. 103 thermal conductivity and high strength in the temperature range of 350 – 100 K).

Stainless steel exhibits decreasing thermal conductivity with decreasing temperature and copper exhibits the opposite trend [88]. The various components were connected by flange couplings. The flanges were welded on to the component tubing (stainless steel regenerator, cold heat-exchanger and pulse tube sections) or soldered on to the component tubing (copper aftercooler and hot heat-exchanger sections). Each flange had a diameter of 4.45 cm (~ 1.75”) and a thickness of 0.38 cm (~ 0.15”). Indium wire O- rings were found to provide the best seal for the system operating with high pressure

Helium and large temperature gradients. The orifice used in the system was a NOSHOK brass needle valve (101-MMB) capable of a maximum flow coefficient of 0.42.

The dimensions of the OPTR obtained from DeltaEC predictions and the materials used in the fabrications are summarized in Table 4.1 below.

Table 4.1: Dimensions and tube materials used in experimental OPTR system

Radius Length Wall thickness Component* Tube Material (cm) (cm) (cm)

Transfer Tube (B) 0.85 4.0 0.8 Stainless Steel

Aftercooler (C) 0.85 3.0 0.107 Copper

Regenerator (D) 0.85 6.0 0.107 Stainless Steel

Cold Heat-Exchanger (E) 0.47 5.0 0.165 Stainless Steel

Pulse Tube (F) 0.47 23.0 0.165 Stainless Steel

Hot Heat-Exchanger (G) 0.47 3.0 0.165 Copper

Orifice (H) - - - Brass

Inertance Tube (I) 0.193 150.0 0.125 Copper 104

Compliance Volume (J) 2.6 14.9 0.5 Cast Iron

* The letters in parentheses correspond to the location of the components in Figure 4.1

The aftercooler, cold and hot heat-exchanger tubing were filled with stacked copper woven square-mesh screen. The mesh screen in the heat-exchangers acts as the primary heat-exchanger between the gas refrigerant and the tube walls. The regenerator tubing houses the regenerator material which is a high thermal capacity and low thermal conductivity permeable material. In the current system, the regenerator is composed of stacked stainless steel woven square-mesh screen. The properties of the woven square- mesh screen used in the heat-exchanger are listed below in Table 4.2. Figure 4.2a below shows an image of the interior of aftercooler heat-exchanger (copper wire mesh inside a copper tube) on the left and the regenerator on the right. The pores in the copper mesh are clearly visible due to the coarse copper wire mesh used. Figure 4.2b shows a close-up image of the stainless steel screen used in the regenerator.

Table 4.2: Heat-exchanger and regenerator mesh properties

Wire Diameter Component Mesh Material Mesh Size/Count* (cm)

Heat-Exchangers Copper 20 × 20 0.04064

Regenerator Stainless Steel 325 × 325 0.003556

* Number of wires per inch 105

Figure 4.2: Images of (a) the interior of the aftercooler heat-exchanger (left) and the

regenerator (right) and (b) close-up image of the regenerator woven wire-mesh screen

4.2.2 Experimental operating conditions

The pressure wave generator was connected to a signal generator (BK Precision

4011A) and an amplifier (Crown CE1000). The signal generator is capable of providing

± 5.0 V sine waves up to a frequency of 5.0 MHz. For the OPTR system in the study, the pressure wave generator has an upper limit for the operating frequency at 100 Hz.

Between the amplifier and the pressure wave generator, there were two multi-meters used to measure the voltage (in parallel) and the current (in series) going into the pressure wave generator. These measurements were used to calculate the Apparent Input Power to the pressure wave generator. Figure 4.3 below shows a simplified schematic of the internal components of the linear motor pressure wave generator. 106

Figure 4.3: Simplified schematic of the inside of the linear motor pressure wave

generator [7]

An Omega pressure transducer (PX-309-500G5V) was placed at the exit of the pressure wave generator. An Omega K-type thermocouple (1/16” un-grounded probe type) thermocouple was used to measure the gas temperature in the cold heat-exchanger using a compression fitting to prevent leaks from the high pressure system. The thermocouple has an accuracy of around ± 1 K below 273 K.

The two heat-exchangers (i.e., the aftercooler and the hot heat-exchanger) were enclosed in water-jackets. Tap water at ~ 22 – 25 oC was run through the water-jacket heat-exchangers at ~ 1 liter/min (total). The temperature of the tap water varies between

18 – 19 oC in the winter to 22 – 25 oC in the summer. This creates a bit of variation in the cryocooler performance; however this variation is within the uncertainty of the thermocouples (i.e., < 1 oC) and can be further decreased by controlling the flow rate of the coolant through the heat-exchanger water-jackets. However, the ambient 107 surroundings of the cryocooler affect its performance more than the coolant water supplied to the two warm heat-exchangers. The environment of the room where the cryocooler was tested was never stable (changes in temperature and humidity). To create a stable environment (irrespective of the conditions prevalent in the room) around the sensitive sections of the cryocooler (i.e., the sections that get the coldest, the regenerator, the cold heat-exchanger and the pulse tube), a vacuum chamber was built. The vacuum chamber was used to reduce convective heat transfer losses observed for the OPTR operating in unstable ambient conditions. Only the regenerator, cold heat-exchanger and part of the pulse tube were enclosed in a vacuum chamber (Figure 4.4).

Figure 4.4: Image of the vacuum chamber built for the Mk-I cryocooler enclosing parts

of the regenerator and pulse tube and the entire cold heat-exchanger

The vacuum chamber was capable of reaching low levels of vacuum ~ 20 – 30

Torr. High and ultra-high vacuum (< 10-6 Torr) can further decrease the heat transfer losses; however those levels of vacuum were beyond the capabilities of the available 108 vacuum pump and vacuum chamber. The presence of humidity/water vapor results in the formation of frost/ice on the cold regions of the OPTR (see Figure 4.5a below). To purge the vacuum chamber of any water vapor, N2 gas was passed through the vacuum chamber prior to running an experiment. To reduce the radiation heat transfer, the components of the OPTR in the vacuum chamber were wrapped in a reflective foil. The reflective foil used in this current study was Aluminized Mylar. The advantages of using Aluminized

Mylar are its high reflectivity and low thermal conductivity. By employing the vacuum chamber for insulation, two major differences were observed: (a) the frost forming on the cold heat-exchanger and the cold regions of the regenerator and pulse tube (Figure 4.5a) was eliminated (Figure 4.5b) and (b) the temperature in the cold heat-exchanger decreased from ~ 170 – 180 K when operating in ambient conditions to ~ 150 K when operating with the vacuum chamber. 109

Figure 4.5: Images of the cold heat-exchanger when the cryocooler is operating (a) in ambient conditions (notice the frost formation, Tgas in the cold heat-exchanger ~ 200 K)

and (b) inside the vacuum chamber (notice no large scale frost formation, Tgas ~ 185 K)

Before an experiment was run, the OPTR was purged of any residual gases (using the vacuum pump). Then the system was charged with Helium to the required operating pressure. N2 gas was passed through the vacuum chamber at a pressure of 5 – 10 psig.

N2 was run through the system for ~ 15 minutes. Next the vacuum pump was turned on and N2 was allowed to flow through the vacuum chamber for an additional 15 minutes with the vacuum pump running. For the final step of purging the vacuum chamber, the vacuum pump was run for a further 15 minutes after closing the N2 supply before the experiment was started. After the purging process, the vacuum pump was operated for the entire experiment (~ 60 – 70 minutes) to maintain vacuum pressures in the vacuum chamber during the experiment. This procedure was followed before each experiment and was performed to ensure the highest vacuum and the lowest amount of water vapor 110 with the given equipment. The results for the various experiments performed are discussed in the next sub-section.

4.2.3. Results: Characterization of the orifice type PTR

Table 4.3 summarizes the parameters used to experimentally characterize the performance of the OPTR. The experiments conducted provide a thorough study of the

OPTR and the factors that affect its performance. The steady-state temperature results reported here are at 60 minutes of operation. After 60 minutes, the temporal variation of the gas temperature was fairly small.

Table 4.3: List of cases studied in the experimental studies of the OPTR

Mean Flow Case Frequency Voltage Current Apparent Power Pressure Coefficient of No. (Hz) (V) (A) (Volt-Amperes) (MPa) Orifice 'Cv'

1 55 28.9 5.18 149.702 1.81 0.42

2 60 35.1 4.26 149.526 1.81 0.42

3 62 37.7 4.00 150.80 1.81 0.42

4 65 34.9 4.30 150.07 1.81 0.42

5 70 25.7 5.86 150.602 1.81 0.42

6 65 19.2 6.31 121.152 0.74 0.42

7 65 23.7 5.13 121.581 1.05 0.42

8 65 28.2 4.27 120.414 1.41 0.42

9 65 32.0 3.74 119.68 1.74 0.42

10 65 32.2 3.77 121.394 2.19 0.42 111

11 65 16.0 2.02 32.32 2.20 0.42

12 65 24.8 3.00 74.40 2.20 0.42

13 65 34.2 4.00 136.8 2.20 0.42

14 65 41.2 5.04 207.648 2.20 0.42

15 65 41.0 5.01 205.41 2.23 0.42

16 65 41.1 5.01 205.911 2.23 0.35

17 65 41.0 5.02 205.82 2.23 0.20

4.2.3.1. Optimum frequency of operation

An important input characteristic of an OPTR is the operating frequency. Each

OPTR has an optimum/resonant frequency at which it operates the best. At the optimum frequency, all the phase relationships between the pressure and mass flow rate are at the optimum. This frequency varies from system to system and is dependent on the pressure wave generator, the regenerator and heat-exchanger matrices and the length of the various components in the system.

To study the frequency dependence of the system, the OPTR was operated at a constant mean pressure of 1.81 MPa, a constant value of the apparent input power (150

Volt-Amperes) and a constant orifice valve flow coefficient of Cv = 0.42. The OPTR was run at five different values of operating frequency, 55, 60, 62, 65 and 70 (cases 1 – 5 respectively in Table 4.3). For the given system, values of operating frequency below 55

Hz and above 70 Hz resulted in current values above 8 A which is beyond the capacity of the present linear motor pressure wave generator. 112

Figure 4.6a shows the temporal variation of gas temperature in the cold heat- exchanger (between the regenerator and the pulse tube) region for the above cases. The lowest gas temperature attained after 60 minutes of operating the OPTR was 125.74 K.

This temperature was obtained for an operating frequency of 65 Hz. Figure 4.6b shows the quasi-steady gas temperature in the cold heat-exchanger for the OPTR at the various frequencies studied (error bars are based on the American Society of Mechanical

Engineers (ASME) suggested uncertainty analysis methods [89, 90]). Also seen at the optimum frequency (65 Hz), the OPTR not only reaches its lowest temperature, but the cool-down time is the fastest. It is also interesting to note that the operating pressure amplitude at the optimum frequency is the highest (input power being constant). This trend in the pressure amplitude indicates the system is running at its optimum for the given input power.

Figure 4.6: Effects of operating frequency on system performance: (a) Temporal

variation of gas temperature in the cold heat-exchanger, (b) Pressure amplitude in the

transfer tube and the quasi-steady gas temperature in the cold heat-exchanger as a

function of the operating frequency (cases 1 – 5 in Table 4.3) 113

4.2.3.2. Effect of mean operating pressure

The next set of experiments was run to study the effect of mean operating pressure on the performance of the system. The OPTR was operated at a constant frequency of 65 Hz (the optimum frequency of the system), a constant apparent input power (121 Volt-Amperes) and a constant orifice valve flow coefficient (Cv = 0.42). The experiments were conducted at 0.74 (~ 100 psig), 1.05, 1.41, 1.74 and 2.19 (~ 320 psig)

MPa (cases 6 – 10 respectively in Table 4.3).

An increase in the mean operating pressure increases the density and the thermal conductivity of the gas. These enhance the heat transfer in the OPTR system and lead to improved cooling. The experimentally obtained gas temperatures at the cold heat- exchanger are shown in Figures 4.7a and 4.7b. When the operating pressure is increased and input power to the system is maintained constant, the pressure amplitude in the system is increased. Figure 4.7b shows this increasing trend of pressure amplitude as the mean operating pressure is increased. Due to this increase in the pressure amplitude, the acoustic power density is increased (for a constant frequency of operation). It is apparent that the cold temperature in the OPTR decreases as the mean pressure in the system increases.

It is interesting to note that the temperature appears to asymptotically reach a plateau (between 1.7 MPa and 2.2 MPa the difference in the steady-state temperature is ~

2 K). Helium has the highest thermal conductivity for an inert gas and it has a high ratio of specific heats (γ = 1.6). A high value of γ implies that adiabatic expansion of the gas 114 provides maximum cooling and adiabatic compression of the gas provides maximum heating. Very high mean operating pressures lead to high pressure amplitudes (for a system operating at constant input power). Due to this high ratio of specific heats for helium, an increase in pressure and pressure amplitude will lead to very high temperatures at the inlet of the regenerator. These high temperatures were also observed during the present experiments - the temperature of the wall of the transfer tube was above ambient (it was warm) even though the adjacent aftercooler was being cooled by flowing water. This leads to a degradation of performance of the system at very high mean operating pressures and pressure amplitudes. Better aftercooler heat-exchangers will be required to further improve the performance of the system.

Figure 4.7: Effects of mean operating pressure: (a) Temporal variation of gas

temperature in the cold heat-exchanger, (b) Pressure amplitude in the transfer tube and

the quasi-steady gas temperature in the cold heat-exchanger as a function of the mean

pressure (cases 6 – 10 in Table 4.3) 115

4.2.3.3. Effect of operating pressure amplitude

To study the effect of operating pressure amplitude on the performance of the

OPTR, the OPTR was operated at a constant frequency of 65 Hz, constant value of mean operating pressure 2.2 MPa and a constant flow coefficient of the orifice valve Cv = 0.42.

To control the operating pressure amplitude, the input power to the linear motor pressure wave generator was varied. The experiments were conducted at pressure amplitudes of

0.118, 0.178, 0.200 and 0.232 MPa (cases 11 – 14 respectively in Table 4.3). Table 4 has the corresponding values of the apparent input power to the linear motor.

Figure 4.8: Effects of pressure amplitude: (a) Temporal variation of gas temperature in the cold heat-exchanger, (b) Pressure amplitude in the transfer tube and the quasi-steady gas temperature in the cold heat-exchanger of the OPTR as a function of the applied input

power to the linear motor

Figure 4.8a shows the temporal variation of the gas temperature in the cold heat- exchanger at the various values of input power studied. As expected, the increase in the 116 applied input power results in an improvement of the performance of the OPTR (Figure

4.8b). The temporal plots for gas temperature also show that the cool-down time decreases with an increase in the applied input power.

4.2.3.4. Effect of flow coefficient of the orifice valve

The opening size of the orifice valve (NOSHOK 101-MMB) can affect the performance of the OPTR. Hence, to test this effect on its performance, the OPTR was operated at a constant frequency of 65 Hz, constant value of mean operating pressure

2.23 MPa and a constant input power to the linear motor of 205 Volt-Ampere. For a needle-type orifice valve, the opening is quantified by a flow coefficient Cv (larger opening sizes correspond to larger values of Cv). The values of flow coefficient studied were 0.20, 0.35 and 0.42 (cases 15 – 17 respectively in Table 4.3). The figure below

(Figure 4.9) shows the flow coefficient Cv plotted as a function of the number of turns of the needle valve. The values of Cv listed above were achieved in the experiment by opening the valve by the corresponding number of turns. 117

Figure 4.9: The flow coefficient Cv of the needle valve orifice (NOSHOK 101-MMB)

plotted as a function of the number of open turns of the needle [91]

After its introduction, the inertance tube has become the more preferred component to maintain the proper phase relationships in the system. However, it has been shown that using both the orifice and inertance tube can improve the performance for the OPTR. In the current experiment, a needle type orifice valve is used. The use of a needle valve enables the variation of the orifice flow coefficient so as to find the optimum value “sweet spot”. Figure 4.10 shows the performance of the OPTR as a function of the orifice flow coefficient. 118

Figure 4.10: The quasi-steady gas temperature in the cold heat-exchanger and the

pressure amplitude in the transfer tube at different values of the orifice flow coefficient

Cv

The best performance was observed when the flow coefficient is at its maximum

(Cv = 0.42) and the orifice is fully open. As the orifice was closed, the performance of the system deteriorated. Below a flow coefficient of 0.15 (for this particular needle valve), the decrease in performance was very large. This can be explained by two possible reasons. First, a decrease in the flow coefficient (closing the valve) leads to a decrease in the mass flow through the orifice and hence decreases the cooling. Second, the constriction of the orifice opening leads to an increase in velocity and hence possible jetting can occur in the pulse tube which causes mixing of flow in the pulse tube that leads to deterioration in the performance. The high value of velocity in the orifice region 119 is observed in the computational model results and is discussed in the following section.

Closing the orifice valve also leads to an increase in the pressure amplitude in the system.

This is observed in the cool-down times for the OPTR to reach cryogenic temperature

(120 K). They are 36, 26 and 28 minutes for cases 15, 16 and 17 respectively (Table

4.3). Even though the value of Cv = 0.42 has the best performance, due to the increase in pressure amplitude for Cv = 0.35, the cool-down time to 120 K was smaller for Cv = 0.35 as compared to Cv = 0.42.

4.3. Investigation of the inertance effect on a PTR (Mk-II)

The characterization of the Mk-I cryocooler provided a lot of insight into the operation of a PTR and enabled the investigation of the effects of various operating conditions on the PTR performance. However the Mk-I PTR was not optimized with the major areas of performance loss being the thermal mass of the components and the inadequate performance of the vacuum chamber built. One major component of the PTR that was not investigated in the Mk-I system was the inertance tube and its effect on the performance of the system. Hence to improve a system’s performance, study the effect of the inertance and achieve the goal of near liquid nitrogen temperatures, a new PTR was designed, fabricated and tested [92]. This new PTR (an inertance type PTR, i.e., no orifice valve) was labeled the Mk-II system.

In this section details about the geometry of the cryocooler (Mk-II), its fabrication, instrumentation and the experimental operating conditions are provided. The 120 characterization of the inertance effect on the performance of the system is investigated too.

4.3.1. Mk-II inertance type PTR geometry and fabrication

Similar to the Mk-I system, the PTR was designed using a computer code

DeltaEC [14]. DeltaEC is a 1-D model that allows the user to design the dimensions of a thermoacoustic cryocooler based on the constraints of the experimental setup (pressure, frequency, heat-exchanger and regenerator matrix material parameters and boundary conditions). The pressure wave generator used in the experiments was a QDrive twin-

STAR linear motor type pressure wave generator. The linear motor has a swept volume of 15.55 cm3 and a maximum operating pressure of 2.5 MPa. The input parameters provided to the DeltaEC code are estimated using the empirical correlations from a review paper by Radebaugh [87]. Based on the swept volume of the pressure wave generator, the mean operating pressure, the frequency of operation and the regenerator and heat exchanger characteristics, DeltaEC provided the lengths of the various components. Figure 4.11 shows an image of the experimental OPTR system. 121

Figure 4.11: Image of experimental Mk-II IPTR housed in vacuum chamber

Due to the large temperature gradients over small lengths of the system and use of high pressure gases (helium), it is important to keep in mind the thermal properties of the materials used to fabricate the various components. The heat exchangers were made from copper (due to its high thermal conductivity), the regenerator was made from 316- type stainless steel tubing with flanges welded on to it and the pulse tube was made from titanium tubing and titanium flanges. Most of the components of the PTR were connected by flange couplings. The regenerator and pulse tube experience large temperature gradients across their two ends and heat transfer along the component’s surface must be minimized. Hence they are made from stainless steel and titanium respectively due to the low thermal conductivity and high strength of these materials.

Each flange has a diameter of 5.08 cm (~ 2.0”) and a thickness of 0.38 cm (~ 0.15”). 122

Indium wire O-rings were found to provide the best seal for the system operating with

high pressure helium and large temperature gradients. The inertance tubes are made from

commercially available copper tubing and the compliance volume (not shown in Figure

4.11) is a commercially available spherical 4L stainless steel float.

The dimensions of the IPTR obtained from DeltaEC predictions and the materials

used in the fabrications are summarized in Table 4.4 below. The compliance volume

(component J) used is a spherical volume and hence only the radius is listed below in

Table 4.4.

Table 4.4: Dimensions and tube materials used in experimental IPTR system

Radius Length Wall thickness Component* Material (cm) (cm) (cm)

Transfer Tube (B) 0.8500 5.70 0.58 Aluminum

Aftercooler Heat-Exchanger (C) 1.0224 6.37 0.32 Copper

Regenerator (D) 1.0224 5.90 0.26 Stainless Steel

Cold Heat-Exchanger (E) 0.5829 1.00 2.40 Copper

Pulse Tube (F) 0.5829 15.35 0.17 Titanium

Warm Heat-Exchanger (G) 0.5829 3.30 0.49 Copper

Connector (H) 0.2300 4.80 - Stainless Steel

Inertance Tube (I) -a - a - a Copper

Compliance Volume (J) 9.8475 - - Stainless Steel

* The letters in parentheses correspond to the location of the components in Figure 4.11 a see Table 4.6

The aftercooler, cold and hot heat-exchanger were filled with stacked copper

woven square-mesh screen. The mesh screen in the heat-exchangers acts as the primary 123

heat-exchanger between the gas refrigerant and the tube walls. The finer mesh copper

screens used in the heat-exchangers at the two ends of the pulse tube (components E and

G) act as flow straightners. The regenerator tubing houses the regenerator material which

is a high thermal capacity and low thermal conductivity permeable material. In the

current system, the regenerator is composed of stacked stainless steel woven square-mesh

screen. The properties of the woven square-mesh screen used in the heat-exchanger are

listed below in Table 4.5.

Table 4.5: Heat-Exchanger and Regenerator mesh properties

Component Mesh Material Mesh Size/Count Wire Diameter (cm)

Aftercooler Heat-Exchanger Copper 20 × 20 0.04064

Cold and Warm Heat-Exchangers Copper 100 × 100 0.01143

Regenerator Stainless Steel 325 × 325 0.003556

4.3.2 Experimental operating conditions

The pressure wave generator was connected to a signal generator (BK Precision

4011A) and an amplifier (Crown CE1000). The signal generator is capable of providing

± 5.0 V sine waves up to a frequency of 5.0 MHz. For the IPTR system in the study, the

Q-drive linear motor pressure wave generator has an upper limit for the operating

frequency at 100 Hz. A wattmeter (Powertek, ISW 8000) was connected between the

amplifier and the pressure wave generator to measure the RMS values of the delivered

input power, the applied voltage, the applied current and the phase angle between the

voltage and the current. 124

An Omega pressure transducer (PX-309-500G5V) was placed at the exit of the pressure wave generator to measure the instantaneous pressure. An Omega K-type thermocouple (1/16” un-grounded probe type) thermocouple was used to measure the gas temperature at the exit of the pressure wave generator and an E-type probe thermocouple was used to measure the gas temperature in the cold heat-exchanger. Compression fittings were used to insert the probes into the cryocooler and prevent leaks from the high pressure system. The thermocouples have an accuracy of around ± 1 K below 273 K.

The two heat-exchangers (i.e., the aftercooler and the hot heat-exchanger) were enclosed in aluminum water-jackets. Tap water at ~ 18 – 22 oC was run through the water-jacket heat-exchangers at ~ 1 – 1.5 liter/min (total). The regenerator, cold heat- exchanger and part of the pulse tube were enclosed in a vacuum chamber. The vacuum chamber was used to reduce convective heat transfer losses observed for the IPTR operating in ambient conditions. The vacuum chamber was capable of reaching low levels of vacuum ~ 2 – 4 Torr. High and ultra-high levels of vacuum (< 10-6 Torr) are beyond the capabilities of the available vacuum chamber, however past studies have shown that those high levels of vacuum can improve the performance of the cryocooler.

No radiation shield was used in the experiments.

Before an experiment was run, the PTR was purged of any residual gases using the vacuum pump and the PTR was filled with high pressure helium gas. The vacuum pump was then connected to the vacuum chamber and the chamber was purged for about

30 minutes before an experiment was started. At this point, the vacuum levels in the chamber were close to the steady state value of ~ 2 – 4 Torr. After this purging process, the vacuum pump was operated for the entire experiment (60 minutes) to maintain 125 vacuum pressures in the vacuum chamber during the experiment. This procedure was followed before each experiment and was performed to ensure the highest vacuum possible. The results for the various experiments performed are discussed in the following sub-section.

4.3.3. Results: Characterization of the orifice type PTR

The performance of the Mk-II experimental system described above is characterized over the different inertance values (inertance tube length and diameter) and the electrical input power supplied to the linear motor pressure wave generator. The results from the experimental characterization are discussed below in this sub-section.

The mean pressure of helium in the experimental IPTR was always 2.24 MPa and the operating frequency was 62 Hz.

Table 4.6 lists the various inertance tube geometries studied (combinations of length and diameter). The value of inertance ‘M’ is given by the formula [1, 93]:

m 0l M 2 2 (4.1) 2 r rIT  IT

where, rIT is the radius of the inertance tube, m is the mass of the gas in the inertance tube, ρ0 is the mean initial density obtained from the NIST 12 database [8] (T = 295 K, P0

= 2.24 MPa) and l is the length of the inertance tube. 126

Table 4.6: List of experimental cases investigated with Mk-II cryocooler

Case No. IT Length (cm) IT Radius (cm) Input Power (W) Inertance (Pa-s2/m3)

1 30.00 0.1524 100.11 148741.0204

2 60.00 0.1524 100.55 297482.0409

3 119.00 0.1524 101.21 590006.0477

4 146 0.1524 100.36 723872.9661

5 179 0.1524 100.38 887488.0886

6 96.00 0.1930 100.63 296657.9909

7 144.00 0.1930 100.82 444986.9863

8 192.00 0.1930 100.98 593315.9818

9 230.00 0.1930 100.05 710743.1032

10 288.00 0.1930 100.32 889973.9727

As can be seen in Table 4.6 above, the input electrical power was maintained fairly constant during the experiments. In addition to experimentally studying the effects of the ‘inertance’ by varying the inertance tube length and diameter, the effects of input electrical power to the linear motor driver were also studied. Four different values of input power to the linear motor pressure wave generator were studied. These cases studied are listed below in Table 4.7. Above 130 W, for the given combination of inertance tube length and diameter (case 9) and mean operating pressure, the electrical current supplied to the linear motor is too high and the dual opposed pistons “knock” each other. Hence the maximum operating input electrical power used (for case 9) was ~

125 W. For each of the cases listed in Tables 4.6 and 4.7, the experiment was run two times to ensure that the results obtained were accurate. The difference of the final steady 127 state temperature between the two runs for each case was always within the uncertainty of the thermocouple probe (~ 1 K).

Table 4.7: List of experimental cases investigating the effect of driver input power (Mk-

II)

Case No. IT Length (cm) IT Radius (cm) Input Power (W)

9a 50.80

9b 75.41 230.00 0.1930 9c 100.05

9d 125.59

4.3.3.1. Effect of inertance tube geometry

The temporal change in the gas temperature in the cold heat-exchanger (i.e., the cool-down characteristics) is an important performance characterization criterion for a

PTR system. Figure 4.12 below shows the cool-down characteristics of the Mk-II PTR for the different cases studied. By 60 minutes of operation, the PTR has reached quasi- steady state for all the cases. At the optimum inertance value, the PTR system has the fastest cool-down time and the lowest quasi-steady state temperature. This optimum inertance value (independent of the inertance tube radius) is about 710 – 720 kPa-s2/m3 for both the inertance tubes studied. 128

Figure 4.12: Temporal evolution of the gas temperature in the middle of the cold heat-

exchanger (component E) for the two inertance tubes radii studied (a) rIT = 0.1524 cm

and (b) rIT = 0.1930 cm

It is important to note that if the geometry of the inertance tube is changed (i.e., different inertance tube radius), the PTR system will have the same performance if the value of inertance ‘M’ is maintained constant (by varying the length of the tube). This is evidenced by the plots in Figure 4.13 below where the cool-down characteristics are compared for cases where the inertance tube radii are different, but the value of inertance

‘M’ is comparable. The minor differences are due to the fact that the value of inertance is not identical for the two cases. 129

Figure 4.13: Comparison of the transient variation of the gas temperature in the cold

heat-exchanger at comparable values of inertance ‘M’ for the two inertance tube radii

studied

The performance map of the Mk-II PTR is plotted in Figure 4.14a as a function of the inertance ‘M’ for the two inertance tube radii studied. The two different inertance tubes show a similar trend in the quasi-steady state temperature achieved, with the optimum value of inertance ‘M’ being ~ 700 kPa-s2/m3. However minor differences are observed in the results of the two inertance tubes. These differences possibly occur due to the difference in the pressure amplitude values. As can be seen in Figure 4.14b, the pressure amplitude values increase with an increase in the inertance values for both inertance tubes. This occurs even though the input electrical power to the linear motor

(which indirectly controls the pressure amplitude) is maintained constant for all the 130 experiments (Table 4.6). More importantly, the pressure amplitude in the transfer tube for the larger inertance tube (rIT = 0.1930 cm) is larger than the corresponding value in the smaller inertance tube (rIT = 0.1524 cm) for comparable values of inertance and input power. This larger pressure amplitude is a possible reason for the difference observed between the results seen in Figure 4.14a. Figure 4.14b also compares the pressure amplitudes in the transfer tube for the various cases studied, at the start of the experiment

(~ 30 sec) and the corresponding value at quasi-steady state/equilibrium (~ 60 minutes).

The figure clearly shows the pressure amplitude near quasi-equilibrium has decreased from the initial value. In addition to the decrease in pressure amplitude, the mean pressure in the transfer tube increases (due to an increase in the gas temperature) by ~ 20

– 30 kPa. This decrease in pressure amplitude is observed in CFD simulations of the cryocooler too and can be attributed to the drop in pressure experienced by the gas flow in porous media (heat-exchangers and regenerator). 131

Figure 4.14: Comparison of the performance of the Mk-II PTR as a function of the

inertance ‘M’ for the two inertance tube radii studied (a) Gas temperature in the cold heat-exchanger (dash lines indicate trends) and (b) Pressure amplitude in the transfer tube

The large difference in quasi-steady state temperature observed above (Figure

4.14a) at inertance values above the optimum can be explained with an observation by

Gardner and Swift [93]. They state that the length of inertance tubes for low power systems (like the Mk-II cryocooler) cannot be usefully increased beyond a limit of λ/2π, where λ = a/f is the wavelength, a is the speed of sound in helium and f is the operating frequency. For the current system with operating conditions: P0 = 2.24 MPa, f = 63 Hz,

Tamb = 295 K and a = 1020.336 m/s, the maximum useful length (λ/2π) is about 2.58 m.

As can be seen in Table 4.6 above, the length of the inertance tube for case 10 is 2.88 m and the corresponding length for case 5 (similar inertance M = 890 kPa-s2/m3) is 1.79 m.

This value of 2.88 m (case 10) is greater than the maximum useful length 2.58 m suggested by Gardner and Swift [93] and hence the performance of the cryocooler decreases much more for case 10 compared to case 5. They also suggest a lower limit for 132 the radius of the inertance tube being a value larger than when the dissipative effects dominate the inertial effects (i.e., rIT ~ δν, where δν is the viscous penetration depth).

4.3.3.2. Effect of input electrical power

The input electrical power to the linear motor pressure wave generator controls the displacement of the dual-opposed pistons in the motor. As the input power is increased, the piston displacement is increased and consequently, the pressure amplitude is increased. The cool-down characteristics of the Mk-II PTR system operating with an inertance value of 710 kPa-s2/m3 (case 9, rIT = 0.1930 cm) and at different values of input electrical power are shown in Figure 4.15. As the input electrical power is increased, the pressure amplitude in the transfer tube increases (Figure 4.16) and hence the cool-down time of the gas in the cold heat-exchanger decreases and the cryocooler reaches quasi-steady state faster. 133

Figure 4.15: Transient variation of the gas temperature in the cold heat-exchanger for

different values of input electrical power (rIT = 0.1930 cm)

Another important observation from Figure 4.16 below is that when the input power (and consequently the pressure amplitude) increases, the decrease in pressure amplitude between the initial value and that at quasi-equilibrium increases. This observation can be explained by considering the flow in the PTR system. In the PTR system, the flow (and pressure amplitude) is controlled by the displacement of the pistons in the linear motor. As the input power to the linear motor is increased, the piston displacement increases and hence the flow rate of gas in the system is increased. This increase in flow rate will lead to a proportional increase in the pressure drop observed

(Darcy’s Law explained earlier) across the porous zones and over time will lead to a larger decrease in the pressure amplitude observed experimentally (Figure 4.16 below). 134

Figure 4.16: Variation of gas temperature in the cold heat-exchanger and pressure

amplitude in the transfer tube as a function of the input electrical power to the linear

motor (dash line indicates trend)

An important accomplishment for the Mk-II cryocooler is evident in Figure 4.16 too. The cryocooler attained liquid nitrogen temperatures (~ 77 K) when operated at the maximum driving power of 125 W. This was an important goal that was set at the start of the dissertation study.

4.4. Chapter summary and brief conclusions

This chapter reports the experimental studies carried out to characterize the PTR operating under a variety of conditions and understand its response to key operating 135 parameters. Two cryogenic capable PTRs were designed, built and tested. The first PTR built (Mk-I) was an orifice type PTR and was capable of achieving cryogenic temperatures in ~ 25 – 30 minutes and the lowest temperature achieved was 114.5 K.

The performance of this Mk-I system was characterized under various operating parameters including the operating frequency, mean pressure, pressure amplitude and the orifice flow coefficient. The vacuum chamber built for the system was capable of ~ 30

Torr. Even though the vacuum levels were low, the vacuum chamber served the purpose of isolating the cold sections of the PTR from the ambient, thus allowing it to achieve cryogenic levels. The second PTR was designed and built with the goal of achieving near liquid nitrogen temperatures and this was achieved when the temperature in the cold heat- exchanger reached 77 K. This second PTR (Mk-II) was an inertance type PTR and it was used to investigate the effects of the inertance on the performance of the cryocooler. The design of the vacuum chamber for the Mk-II PTR was considered when the entire setup was being designed and built. The result was the vacuum chamber achieving vacuum levels between 2 and 4 Torr. Another important improvement over the Mk-I PTR was that the cryocooler was capable of achieving cryogenic levels in 10 – 15 minutes of operation (reducing the time to 120 K in about half).

At the culmination of the experimental studies, the outcome of the study can be categorized into two areas. The first outcome was expected and part of the motivation to perform the experimental study, i.e., a better understanding of the PTR’s performance under a variety of operating conditions was realized and the results of the PTR experimental study were vital in the validation of the numerical model developed. The second outcome of the experimental studies was the experience gained from running the 136 experimental PTR system. This experience was invaluable in the development of the numerical model as will be evidenced in the following chapter where an understanding of the effects of the thermal mass of the components and the coolant water flow-rate, a realistic cool-down time for the system and the pressure drop measurements in the transfer tube helped improve the numerical model substantially. This second outcome was not the primary motivation for the experimental study; however it proved to be just as important.

137

Chapter 5 : Numerical Investigation of Flow and Heat Transfer Processes in an In-line

Pulse Tube Refrigerator6

5.1. Introduction

1-D computational models have been widely used for modeling thermoacoustic

devices. However, they use idealistic assumptions and do not reflect the multi-

dimensional nature of the flow and transport inside the PTR systems. What is notable

about these studies is that they assume thermal equilibrium in the various heat-

exchangers and do not account for axial heat conduction in the various heat-exchangers,

most notably the regenerator where heat transfer from the warm to the cold end within

the regenerator matrix itself is a loss mechanism. Some multi-dimensional models of the

PTR do exist and they include semi-analytical models and computational fluid dynamic

models. The semi-analytical models of require far too many assumptions including

assuming a negligible velocity component in the radial direction. The use of CFD

models reduces the number of assumptions and constraints imposed by the 1-D and semi-

6 The results presented in this chapter can be found in the papers: a. Antao, D. S. and Farouk, B., “Experimental and numerical investigations of an orifice type cryogenic pulse tube refrigerator”, Applied Thermal Engineering, v. 50, n. 1, pp. 112 – 123, 2013 b. Antao, D. S. and Farouk, B., “Computational Fluid Dynamics Simulations of an Orifice type Pulse Tube Refrigerator: Effects of Operating Frequency”, Cryogenics, v. 51, n. 4, pp. 192 – 201, 2011 c. Antao, D. S. and Farouk, B., “Numerical simulations of transport processes in a pulse tube cryocooler: Effects of taper angle”, International Journal of Heat and Mass Transfer, v. 54, n. 21 – 22, pp. 4611 – 4620, 2011 d. Antao, D. S. and Farouk, B., “Experimental and numerical characterization of the inertance effect on pulse tube refrigerator performance”, Journal of Applied Physics, (under review), 2013 138 analytical 2-D models developed in the past. Flake and Razani’s [52] axisymmetric analysis of a basic PTR (i.e., a PTR with no flow in the warm end of the pulse tube) showed cycle-averaged axial velocity contours in the pulse tube, however no significant insight was gained from that study. The multi-dimensional models of Cha et al. [53] and

Ashwin et al. [54] the latter of which used a thermal non-equilibrium model in the porous media zones, offer an important improvement of the previous 1-D and 2-D semi- analytical work.

In this chapter, the numerical model of the PTR that was developed is introduced and the model is applied to study the flow and transport processes in the cryocooler. The numerical model incorporates the compressible form of the conservation equations for the gas (mass, momentum and energy) in the fluid and porous media domains and solves a separate energy equation for the solid material in the porous media zones. The numerical model is validated against temporal and quasi-steady results and this is the first such validation of a CFD model (to the best knowledge of the author). The validated model is used to simulate the effects of operating frequency, pressure amplitude and the inertance tube geometry. The presence of acoustic streaming in the pulse tube which is detrimental to the performance of the PTR is also observed and this acoustic streaming is shown to be suppressed when the geometry of the pulse tube is modified. This chapter is organized as follows; the numerical model is first introduced, then the model validation studies are reported and finally, the model is used to study the effects of various operating parameters on the PTR’s operation and its performance.

139

5.2. Governing equations of PTR numerical model

The flow and heat transfer simulation model incorporates the fluid dynamic equations of conservation of mass (continuity equation), momentum (Navier-Stokes equation) and energy in the fluid domain. The various heat exchangers and the regenerator are modeled as porous media with the relevant solid properties of the regenerator and the heat exchanger materials (stainless steel and copper respectively).

The mass, momentum and energy conservation equations [74, 94, 95] for the porous media (solid) regions are solved simultaneously with the gas phase equations for the

PTR.

The conservation equations for the gas phase (helium) within the system undergoing periodic compression and expansion are given as follows:

 ()u  0 t (5.1)

()u ()uu   p  t ij (5.2)

h  p 0 ()uuh  ()) k T  (     S tt0 f f ij f (5.3)

where  is the density, u is the (r – z) velocity vector and the total energy is h0 given by

p 1 2 h i   u 0  2   (5.4) and the gas-phase temperature is 140

1 p 1 2 T [] h u f c 0  2   (5.5)

The last term in the gas-phase energy equation is only applicable for the porous media zones (where the porosity ε is < 1.0):

hfp A() T p T f S  (5.6) f 

The value of the convective heat transfer coefficient hfp for the gas-solid interfaces is calculated from a heat transfer correlation for oscillating flow in regenerators developed by Tanaka et al. [96]. The equation is given by:

0.67 k8 r V h  0.33 f f h p r fp  (5.7) 4rAh s, p

where, rh is the hydraulic diameter of the porous media material (in the case of wire mesh, it can be calculated from the diameter of the wire), Vp is the volume of the solid material in the porous media, As,p is the total surface area of the porous media, µ is the viscosity of the fluid and ρr is the ratio of the density of the fluid to the average density of the fluid in the porous media.

The refrigerant helium is assumed to be an ideal gas and its viscosity and thermal conductivity are considered to be temperature-dependent. The temperature dependent properties were obtained from the NIST database [8]. The specific heat was kept constant since it does not vary considerably in the temperature range anticipated (i.e. 370 K – 80

K). 141

The mass and momentum equations for the porous media zones (Darcy-

Forchheimer model) [54, 74] are given as follows:

  ()u  0 t (5.8)

 u 2 3     CF  ()()uu    p   ij  u  u u (5.9) t  

where, κ is the permeability of the porous zone and CF is the quadratic drag factor.

For the solid matrix, the energy equation is given by:

Tp   ()TS  t pp (5.10)

hfp A() T f T p S  (5.11) p 1 where Γ is the diffusivity of the solid porous media and A is the heat transfer area per unit volume. The value of Γ for copper (in the heat exchangers) is 1.17 x 10-4 m2/s and that for stainless steel (in the regenerator) is 1.18 x 10-5 m2/s.

In the heat exchangers the solid phase material considered was copper (recall

Figure 4.2a in Chapter 4). The density and specific heat of copper were assumed to be

8950 kg/m3 and 380 J/kg-K respectively and temperature dependent thermal conductivity values were considered. Similarly for the regenerator (considered as porous media), stainless steel (recall Figure 4.2a and Figure 4.2b in Chapter 4) was considered as the solid phase material and the density and specific heat were assumed to be 7810 kg/m3 and 142

460 J/kg-K respectively and the temperature dependent thermal conductivity values were considered.

5.3. Code validation with past computational fluid dynamic studies

To test the validity of the current model developed, we compared the model predictions to a previously published work where an IPTR was considered. (case 1 of

Cha et al. [53]). A total of 4710 grid points were used in the simulation (please refer to the paper by Cha et al. [28] for further details on the geometry, dimensions and boundary conditions used). A time step of 3.67 × 10-4 s was used for the simulations (~ 80 time- steps/cycle). The thermal equilibrium assumption was employed here (i.e. the gas temperature and solid temperature are assumed equal) in keeping with the work of Cha et al. The simulations were run for 15 seconds. The gas temperature evolution at the exit of the cold heat exchanger in the current model is compared with that published as Case

1 by Cha et al. (Figure 5.1) [53]. As can be seen in Figure 5.1, the present model compares favorably with the published work. The banded temperature profile indicates the periodic fluctuations of the gas temperature due to oscillatory flow within the flow domain. 143

Figure 5.1: Comparison of current model with past computational studies

5.4. Code validation with experimental studies

In addition to the comparison with previously published computational predictions, the model is validated against experimental results from the Mk-I cryocooler that was fabricated and characterized. In this sub-section, a comparison of the experimental results and the computational predictions shows the importance of the components’ thermal mass on the performance of the cryocooler [86].

144

5.4.1. Problem geometry

Figure 5.2 depicts the experimental geometry simulated (i.e. an inline OPTR Mk-I system). Only half the geometry shown in Figures 5.2a and 5.2b was simulated (the axisymmetric assumption) due to the cylindrical nature of an actual system and to save on computation time. The OPTR simulated has the same components as the Mk-I experimental system (Figure 4.1). These include a compression chamber with a moving piston (which models the motion of the piston in the linear motor pressure wave generator), a transfer tube, an aftercooler (the first red hatched region), a regenerator

(blue cross-hatched region), a pulse tube with a heat exchanger at each end (the other two red hatched regions), an orifice (a simple obstruction to the flow), an inertance tube and a compliance volume. The schematic in Figure 5.2b shows the geometry of the OPTR considered in the simulations when the wall thickness was included in the computational domain. The thermal mass of the flanges (Figure 4.1) were approximated by considering additional thickness of the walls. The material properties used for the walls of the various components was based on the respective components’ materials in the experimental system (Table 5.1). 145

Figure 5.2: A schematic for the OPTR geometry used in the numerical simulations (a)

no wall thickness and (b) wall thickness and/or thermal mass of component’s flanges

considered

5.4.2. Numerical model conditions

The flow and heat transfer governing equations solved were described above in section 5.2. of this chapter. The specific initial, boundary and porous media conditions used in the simulations reported in this section (5.4. Code validation with experimental studies) are discussed below.

Table 5.1: Dimensions and boundary conditions of the simulated system 146

Boundary condition No. Component Radius (cm) Length (cm) along the outer wall

A Compression Chamber 3.0 1.1 Adiabatic

2 B Transfer Tube 0.85 4.0 hc = 20 W/m -K

C Aftercooler 0.85 3.0 Tw = 293 K

D Regenerator 0.85 6.0 Adiabatic

E Cold Heat-Exchanger 0.47 5.0 Adiabatic

F Pulse Tube 0.47 23.0 Adiabatic

G Hot Heat-Exchanger 0.47 3.0 Tw = 293 K

H Orifice Valve 0.0965 0.4 Adiabatic

I Inertance Tube 0.193 150.0 Adiabatic

J Compliance Volume 2.6 14.9 Adiabatic

5.4.2.1. Boundary conditions

Table 5.1 above also specifies the heat transfer boundary conditions used at the surface boundaries of the various components.

The piston is modeled as a reciprocating wall having an oscillatory velocity. The velocity of the piston is defined by the function, u = A0ωCos (ωt), where A0 is the maximum displacement of the piston and ω is the angular frequency (ω = 2πf) and f is the frequency of operation (set at 65 Hz based on the experimental study performed).

Table 5.2 has the values of the maximum displacement of the piston A0 for the cases studied. These values were chosen so as to generate pressure amplitudes (in the transfer tube) as close to experimental values as possible. 147

Table 5.2: List of cases simulated for validation of numerical model with Mk-I

experimental cryocooler

Case Mean Pressure Piston displacement Pressure Amplitude No. (MPa) A0 (cm) (MPa)

1 2.2 0.280 0.40

2 1.74 0.320 0.38

3 1.41 0.345 0.34

5.4.2.2. Initial conditions

At the start of the simulation, the temperature in the system is assumed to be 300

K everywhere. Table 5.2 lists the initial mean operating pressure condition (specified as an initial condition) for each of the cases studied.

5.4.2.3. Porous media conditions

The following table (Table 5.3) lists the porous media parameters used in the governing equations of the porous media. The values for porosity, permeability and drag factor were calculated based on correlations for woven square-mesh screen where the weaving causes no inclination of the wires [97, 98].

Table 5.3: Porous Media parameters used in the simulation

2 Component Porosity ‘ε’ Permeability ‘κ’ (m ) Drag Factor ‘CF’

Heat-Exchangers 0.774 4.08 × 10-8 0.2 148

Regenerator 0.72 9.70 × 10-11 0.3

5.4.2.4. Numerical scheme

The numerical scheme for solving the governing equations is based on the finite volume approach. The continuity, momentum and energy equations are solved for the fluid as well as the porous media using a 2nd order upwind scheme. The motion of the piston is captured by a moving grid scheme near the piston wall in the compression space

(component ‘A’ in Figure 5.2). The re-meshing scheme used in the simulations is the

Transfinite Interpolation scheme [74].

A 2nd order Crank-Nicholson scheme (with a blending factor of 0.7) is used for the time derivatives in the continuity, momentum and energy equations. The Crank-

Nicholson scheme provided a better prediction of the performance of the OPTR compared with the Euler 1st order time-marching scheme used in past studies [53, 54].

The time-step size was determined by allowing 80 time-steps/cycle, which was sufficient to accurately simulate the problem (Δt = 1.923077 × 10-4). An overall convergence criterion was set for all the variables at 10-4 in the iterative implicit numerical solver.

Mass and energy conservation were verified in the simulations.

Due to the symmetry of the problem geometry, only one-half of the domain

(Figure 5.2) was considered for the simulations. A hybrid (structured-unstructured) grid system is used in the simulations. Structured grid (non-uniform orthogonal mesh) is used in all components except the cold heat exchanger at the beginning of the pulse tube and 149 the orifice valve. A total of 3719 grid points were used in the simulations. The governing equations and the boundary conditions were solved using CFD-ACE+ [74].

5.4.3. Model validation: Simulation results and discussion

In this sub-section, the results from the model validation are presented and discussed. The validation of the model is done with results obtained from section 4.2 in chapter 4 where the effects of the mean operating pressure on the PTR’s performance is investigated. The results presented below include temporal variations of the gas temperature and pressure and the solid temperature, quasi-steady state temperature in the cold heat-exchanger and cycle-averaged flow and temperature fields in the pulse tube.

5.4.3.1. Transient processes in the system

The cool-down behavior of the PTR is very important to its operation. Figure

5.3a shows this temporal evolution of the gas and solid temperature at the exit of the cold heat exchanger and inlet to the pulse tube section (i.e. x = 19.1 cm, r = 0.4699 cm) for the first case studied. The banded profiles are cyclic variation of the gas temperature in the system. This cyclic nature of both the gas and solid temperature is highlighted in Figure

5.3b. The gas temperature decreases rapidly at the start; however the decrease in temperature is exponential over time. The solid temperature oscillates too, however the oscillations have much lower amplitude as compared to the gas temperature. This lower amplitude is due to the high density and thermal mass of the solid media. 150

Figure 5.3: (a) Temporal variation of the gas and solid temperature at the exit of the cold

heat-exchanger for the entire simulation, (b) Gas and Solid Temperature and Pressure profiles at the exit of the cold heat-exchanger at the start of the simulations (case 1, Table

5.2)

Figure 5.4a compares the gas and solid temperatures at a later point in the simulations (~ 50 sec of simulation time). The difference in the solid and gas temperatures are clearly visible. Another important point of note is that the solid temperature lags the gas temperature by a phase angle. In the current study, the value of the phase angle difference between the gas and solid temperatures is about 81o. In a later section of this chapter, the phase angle between the solid and gas temperature will be shown to depend on the operating frequency. 151

Figure 5.4: Gas Temperature v/s Solid Temperature: (a) Temperature profiles at the exit

of the cold heat-exchanger at a later point in the simulations, (b) Temporal variation of

the cycle-averaged gas and solid temperature at the exit of the cold heat-exchanger

It can be clearly seen that the values of gas temperature in cases 2 and 3 (Table

5.2) are comparatively higher than that in case 1 (Figures 5.4a and 5.4b). This emphasizes the importance of high values of mean operating pressure in an OPTR system. The higher values of mean operating pressures also lead to faster cool-down times. This is similar to the trends observed in the experimental studies. In the following section, a one-to-one comparison between the computational and experimental results is provided.

5.4.3.2. Spatial temperature and velocity profiles

In Figures 5.5a and 5.5b, we show the axial distribution of the gas temperature in the system along the axis of symmetry (i.e. at r = 0 in the simulations) for case 1 (Table

5.2). The four plots in the figure indicate the temperature profiles at four points in the cycle (3301st ~ 50 sec) i.e. π/2, π, 3π/2 and 2π. The temperature is shown from the 152 transfer tube region up to the compliance volume. As can be seen in Figure 5.5b, beyond the hot heat exchanger, the temperature barely oscillates. However, the mean temperature in the inertance tube is fairly high (320 – 330 K). Cooling of the inertance tube can possibly lead to improvement in the performance of the OPTR system. The temperature profiles along the regenerator and the heat exchangers (porous media) barely oscillate. This indicates that the system is in a steady cyclic mode.

Figure 5.5: Axial distribution of the gas temperature (along the axis of symmetry) at four different instants in cycle 3301 for case 1 (Table 5.2), (a) components B, C, D, E, F,

G and H and (b) components I and J

In Figures 5.6a and 5.6b, the axial component of velocity in the system along the axis of symmetry (i.e. r = 0 in the simulations) for case 1 (Table 5.2) is shown. The four plots in the figure indicate the velocity profiles at four points in the cycle (3301st ~ 50 sec) i.e. π/2, π, 3π/2 and 2π. The velocity barely oscillates in the middle of the cryocooler i.e. from the aftercooler (0.051 m) to the hot heat exchanger (0.44 m). This is due to the presence of porous media in most of that region. However, in the orifice and inertance 153 tube regions (~ 0.44 – 1.95 m in Figure 5.6b), the flow has a very high velocity due to their fairly small cross sectional area. The high velocities in the vicinity of the pulse tube can lead to jetting and mixing of flow in the pulse tube. This jetting or mixing of flow in the pulse tube can adversely affect the cooling performance of the OPTR.

Figure 5.6: Axial distribution of the axial component of velocity ‘ux’ (along the axis of symmetry) at four different instants in cycle 3301 for case 1 (Table 5.2), (a) Components

B, C, D, E, F, G and H and (b) components I and J

5.4.3.3. Comparison of experimental and computational results

The computational results with idealized (no loss) boundary conditions given in

Table 5.1 showed rapid cooling. The experimental studies however showed a slower cooling response mainly due to the heat losses/gains through exposed boundaries and finite wall thicknesses of each component. 154

In Figure 5.7 the experimentally obtained gas temperatures (for cases 8, 9 and 10,

Table 4.3) at ~ 3600 sec are compared with computationally obtained cycle-averaged results (cases 1, 2 and 3, Table 5.2) with the idealized boundary conditions (Table 5.1), albeit at ~ 50 sec. Both the experiments and the computations show fairly similar trends, i.e. lower cycle-averaged gas temperatures at higher values of mean operating pressure.

Even though the trends are similar, there is a difference between the experimental and the computational results and more importantly, the cool-down times (3600 sec vs. 50 sec) are different.

Figure 5.7: Comparison of near-quasi-steady cycle-averaged experimentally (cases 8, 9

and 10, Table 4.3) and computationally (cases 1, 2 and 3, Table 5.2) obtained gas

temperature 155

The simulations predicted the temperatures at ~ 50 sec but the experimental time to attain the temperatures was orders of magnitude higher (~ 3600 sec). This large difference was due to the assumptions made in the computations. These assumptions include, zero wall thickness of the components and absence of flanges (i.e. neglecting thermal mass of the system) [9] and idealized thermal boundary conditions (adiabatic and isothermal surfaces).

5.4.3.4. Effects of component wall thickness

In order to more accurately model the experimental system, an additional set of two simulations were performed by considering finite wall thickness of the components with flanges and realistic thermal boundary conditions. The component dimensions

(Table 5.1) and initial conditions used were the same as case 1 (Table 5.2). The wall thickness and the boundary conditions used are listed below in Table 5.4. In the first simulation run, a finite wall thickness was added for all the components from the aftercooler heat-exchanger (component C) to the inertance tube (component I) (Figure

5.2b). The values of wall thickness and the solid properties for the components’ walls were based on the experimental system (listed in Table 4.1 and Table 4.2). The thermal boundary conditions at all the adiabatic surfaces were updated to convective heat transfer boundary conditions and the aftercooler and hot heat-exchanger wall temperature values were changed from Tw = 293 K to Tw = 296 K (the measured coolant water temperature).

The convective heat transfer coefficients for the regenerator, pulse tube and the cold heat- exchanger (all stainless steel) were based on natural convection heat transfer in low 156

2 vacuum (hc ~ 1.0 W/m -K). The heat transfer coefficient was estimated based on a correlation by Churchill and Chu [99] using properties from the NIST 12 database [8]. In the second simulation run, the thermal mass of the various components’ flanges were considered in addition to the wall thickness (Figure 5.2b). This was accomplished by increasing the wall thickness of the various components (from the previous run) to include the volume (and thermal mass) of the flanges. In both simulations, radiation heat transfer was neglected because radiation was minimized in the experiment by wrapping the components in aluminized-mylar.

Table 5.4: Dimensions and boundary conditions of the simulated system with thermal

mass of the components incorporated in the model (schematic in Figure 5.2b)

Wall Wall Thickness Boundary Component Thickness including Flange condition along (cm) Thermal Mass(cm) the outer wall

Compression - - Adiabatic Chamber

2 Transfer Tube - - hc = 20 W/m -K

Aftercooler 0.107 0.214 Tw = 296 K

2 Regenerator 0.107 0.214 hc = 1.0 W/m -K

Cold Heat- 0.165 0.330 h = 1.0 W/m2-K Exchanger c

2 Pulse Tube 0.165 0.330 hc = 1.0 W/m -K

Hot Heat- 0.165 0.330 T = 296 K Exchanger w

Orifice Valve - - Adiabatic

2 Inertance Tube 0.125 0.125 hc = 20 W/m -K

Compliance - - Adiabatic 157

Volume

Figure 5.8 compares the gas temperatures for case 1 (Table 5.2) simulated earlier

(P0 = 2.2 MPa) and the simulations with wall thickness incorporated. Due to the added thermal mass of the system with component wall thickness, this new simulation took longer time to reach steady state (similar to an experimental system) and its quasi-steady temperature will be higher than the one predicted by the earlier simulations with idealized conditions (Table 5.1). Additionally, incorporating the mass of the flanges in the simulations, further differences are observed in Figure 5.8.

Figure 5.8: Comparison of simulation results (gas and solid temperature) at the exit of

the cold heat-exchanger (x = 19.1 cm, r = 0.4699 cm) for case 1 (Table 5.2) with and

without component wall thickness 158

In Figure 5.9, the cycle-averaged gas temperature in the pulse tube section of the

OPTR is compared for the simulations during the 3301st cycle of operation (~ 50 sec) without wall thickness (Figure 5.9a) and with wall thickness and flange mass (Figure

5.9b) considered.

Figure 5.9: Comparison of the cycle-averaged (3301st cycle) gas temperature in the pulse tube section of the OPTR for the simulations (a) without wall thickness considered

(case 1, Table 5.2) and (b) with wall thickness and flange mass considered

In Figure 5.9a (without wall thickness) the gas temperature fields show one- dimensional structure (a linear temperature variation from cold end to warm end) with three distinct zones, i.e. a cold zone near the cold heat-exchanger, a warm zone near the warm heat-exchanger and a buffer zone in the center isolating the two ends. The temperature fields in Figure 5.9b show three similar zones, however the temperature fields have a two-dimensional structure. This two-dimensional structure is due to the 159 heat transfer between the gas and the solid walls that is imposed by the inclusion of the wall thickness in the simulations. The two-dimensional structure also shows a visible thermal boundary layer which will affect the flow fields in the pulse tube and cause steady-secondary flow in the pulse tube region (streaming phenomena) which may lead to further deterioration of the performance of the system.

In order to predict the system’s performance near quasi-equilibrium, the computational model was run with a prescribed initial condition that represents an expected quasi-steady temperature distribution. Figure 5.10a shows the prescribed gas temperature distribution in the pulse tube section of the cryocooler. With this modified initial condition, the simulations were run for until a quasi-stead state was achieved.

Figure 5.10b shows the cycle averaged gas temperature in the cryocooler at quasi- equilibrium (~ 3600 seconds) 160

Figure 5.10: Cycle-averaged gas temperature in the pulse tube section of the OPTR for

(a) prescribed initial condition near quasi-equilibrium and (b) at quasi-equilibrium (~

3600 sec)

Figure 5.11a below compares the early temporal variation of the temperature in the center of the cold heat-exchanger for the simulations with the temperature measured experimentally (measured at the center of the cold heat-exchanger). The simulations were done with no wall thickness (case 1, Table 5.2), with wall thickness and with wall thickness and flange mass. 161

Figure 5.11: Comparison of temporal simulation results and experimental results (case

10, Table 4.3) in the center of the cold heat-exchanger (x = 16.6 cm, r = 0.0 cm) at (a)

early time in the simulation and (b) near quasi-steady state

The difference between the computational results and the experimental results are quite small when the effects of wall thickness and flange mass are included in the computational model. It is important to note that considering only the component wall thickness is not sufficient to accurately predict the temporal temperature variation observed in the experiments. The various components are coupled/joined using flange couplings which add to the thermal mass of the system and incorporating the mass of the flanges in the simulations enables the one-to-one temporal comparison of the gas temperatures shown in Figure 5.11a. The temporal comparison in Figure 5.11b is for the experiments and simulation results near quasi-steady state (or quasi-equilibrium). The results match-up fairly well even near quasi-steady state and this highlights the importance of considering the thermal mass of the components when simulating the performance of the OPTR. 162

After validating the computational model for the transient response of the OPTR at an early time and near quasi-steady state (Figures 5.11a and 5.11b), the effect of pressure amplitude on the performance of the OPTR near quasi-steady state was studied with the computational model (considering realistic boundary conditions and thermal mass for the components). In the experimental studies the effects of pressure amplitude on the performance of the OPTR were demonstrated. The OPTR system operating at high pressure amplitudes was shown to have better performance (lower quasi-steady state temperatures) than the system operating at lower pressure amplitudes. In the experiment, the pressure amplitude is controlled by controlling the input power to the linear motor, however in the simulations; the pressure amplitude is controlled by adjusting the piston amplitude. The values of pressure amplitude studied were, 0.118, 0.178, 0.200 and 0.232

MPa (same as the experimental studies, cases 11 – 14, Table 4.3), the corresponding piston amplitudes are, 0.084, 0.1268, 0.1425 and 0.1653 cm and the mean operating pressure was 2.2 MPa. Quasi-steady state (quasi-equilibrium) is assumed because the change in gas temperature is less than 0.005 K from cycle to cycle. Figure 5.12 shows a comparison of the performance of the experimental measurements (diamond symbols) with the cycle-averaged quasi-steady state simulation results (square symbols). The experimental and simulation results compare well and show the effects of increasing pressure amplitude on the performance of the OPTR. 163

Figure 5.12: Comparison of experimental results (cases 11 – 14, Table 4.3) and quasi-

steady state simulation results at the center of the cold heat-exchanger (x = 16.6 cm, r =

0.0 cm)

The trends of the temporal gas temperature evolution in Figures 5.8 and 5.11a, demonstrate the detrimental effect that a component’s large thermal mass has on the performance (cool-down time) of an OPTR system. This thermal mass includes not just the wall thickness of each component that previous experimental studies have suggested, but the contribution made by the flanges that are used to couple each of these components. The computational model additionally identifies the effects of component wall thickness on the temperature and flow fields within the pulse tube section of the

OPTR. The added component wall thickness clearly leads to the formation of thermal boundary layers and de-stratification of the ideal 1-D pulse tube. The computational 164 model predictions clearly advocate the reduction in thermal mass of the cryocooler components in order to enhance its performance. This is possible by reducing wall thickness and/or flange thickness and using alternate methods of component coupling.

From the simulation results it may be inferred that adiabatic boundary conditions are desirable in the regenerator, cold heat-exchanger and pulse tube sections. This can be realized in an actual/experimental system by improving the vacuum chamber used to insulate the cryocooler (compared to the current vacuum chamber capable of only ‘low vacuum’).

5.4.4. Summary

The developed axisymmetric time-dependent computational fluid dynamics model was used to simulate the flow and temperature fields in an OPTR at various values of mean operating pressure. The computational results show trends similar to the experimentally observed performance of the OTPR where higher pressure amplitudes result in lower temperatures and faster cool-down times. To accurately predict the performance of the OPTR system it is important to account for the thermal mass of the system and the losses that occur due to the high thermal mass of the various components.

This effect is included in the model by adding a wall thickness and the flange coupling’s mass for the various components. The simulation prediction with this addition of the wall thickness is demonstrated to closely resemble the temporal performance of the experimental system. This modified model provides a one-to-one temporal comparison between the simulations (CFD) and the experiments of an OPTR for the first time. 165

5.5. Effects of frequency on pulse tube refrigerator performance

Due to the thermoacoustic nature of the system, the system has a specific operating frequency at which it performs the best. In an experimental system, this depends on a variety of factors like the pressure wave generator (linear motor), the mesh screen used and their orientation and the dimensions of the various components (as seen in chapter 4). Most of these effects can be captured in a CFD simulation of the OPTR system. In this section of chapter 5, the effects of operating frequency on the flow and transport processes in the OPTR system are discussed [100].

5.5.1. Problem geometry

Figure 5.2a depicts the geometry studied (i.e. an inline OPTR system) and only one half the geometry shown in Figure 5.2a is simulated (the axisymmetric assumption).

The OPTR consists of a compression chamber (which includes the moving piston), the transfer tube, aftercooler (the first red hatched region), regenerator (blue cross-hatched region), pulse tube with two heat exchangers at its ends (the other two red hatched regions), an orifice (a simple obstruction to the flow), an inertance tube and the compliance volume. This geometry has the same components as the experimental Mk-I system, however the component dimensions are different.

Table 5.5 summarizes the various dimensions of the problem geometry and the time-invariant thermal boundary conditions in the simulations. 166

Table 5.5: Dimensions and boundary conditions of the simulated system

Boundary Radius Length No. Component condition along (m) (m) the outer wall

A Compression Chamber 0.01904 0.0075 Adiabatic

B Transfer Tube 0.0031 0.101 hc = 20 W/m-K

C Aftercooler 0.008 0.02 Tw = 300 K D Regenerator 0.008 0.058 Adiabatic

E Cold Heat-Exchanger 0.006 0.0057 Adiabatic

F Pulse Tube 0.005 0.06 Adiabatic

G Hot Heat-Exchanger 0.008 0.01 Tw = 300 K H Orifice Valve 0.000425 0.003 Adiabatic

I Inertance Tube 0.00085 0.681 Adiabatic

J Compliance Volume 0.026 0.13 Adiabatic

6.5.2. Numerical model conditions

This sub-section details the boundary, initial and porous media conditions used in the model and discusses the numerical model operating parameters (time-step, convergence criteria, number of grid points, etc.).

6.5.2.1. Boundary conditions

Table 5.5 (given earlier) specifies the heat transfer boundary conditions used at the surface boundaries of the various components. At all the walls, the no-slip velocity 167 boundary condition is applied. The piston is modeled as a reciprocating wall having an oscillatory velocity. The velocity of the piston is defined by the function, u = A0ωCos

(ωt), where A0 is the invariant maximum displacement of the piston (4.511 mm) and ω is the angular frequency (ω = 2πf) and f is the frequency of operation. Since the objective of this part of the thesis focuses on investigating the effects of the operating frequency, the value of frequency is varied keeping all other conditions constant. Five different frequency values are studied (listed in Table 5.6 below)

Table 5.6: List of the cases simulated (operating frequency)

Driver Operating Frequency Case No. ‘f’ (Hz)

1. 15 2. 25 3. 34 4. 60 5. 100

5.5.2.2. Initial conditions

At the start of the simulation, the temperature in the system is assumed to be 300

K everywhere. The initial pressure in the system is set to 3.1 MPa.

5.5.2.3. Porous media conditions

The heat-exchangers have thermal properties of copper and the regenerator has the corresponding values of stainless steel (see section “5.2. Governing equations and 168

PTR numerical model” for the corresponding values of thermal conductivity and specific heat used). For the simulations in this sub-section of Chapter 5, the porosity ε is considered to be equal to 0.69 [53, 101] in all the porous zones. Similarly, the permeability κ and the quadratic drag factor CF are the same for all the porous zones and have values of 1.06 ×10−10 m2 [101] and 0.55 respectively.

5.5.2.4. Numerical scheme

The time-step size was determined by allowing 80 time-steps/cycle, which is sufficient to accurately simulate the problem. The time-step for each case simulated is dependent on the frequency of operation. An overall convergence criterion is set for all the variables at 10-4 in the iterative implicit numerical solver. A hybrid (structured- unstructured) grid system is used in the simulations. Figure 5.13a shows the grid structure in the region between the cold heat-exchanger and the orifice valve. Figures

5.13b and 5.13c show the grid system at the cold heat-exchanger – pulse tube and the hot heat-exchanger – orifice valve junctions. A total of 4701 grid points were used in the simulations. 169

Figure 5.13: Grid structure in (a) the cold heat-exchanger to the Orifice valve, (b) the

cold heat-exchanger and (c) the hot heat-exchanger and the orifice valve

5.5.3. Results and discussion

In the subsequent sub-sections, results for the spatio-temporal evolution of the flow and temperature fields in the OPTR (Figure 5.2a) are presented. The effects of operating frequency on the performance is studied, in particular, the phenomena of acoustic streaming in the pulse tube section of the OPTR.

5.5.3.1. Temporal processes in the system

The predicted cool-down time of the cold end of the OPTR considered (Figure

5.2a) from room temperature (300 K) is presented next. Figure 5.14a shows the cool- down time i.e. the temperature-history of the gas at the exit of the cold heat-exchanger into the pulse tube (r = 0.005 m) for the five cases (f = 15, 25, 34, 60 and 100 Hz) studied

(Table 5.6). The band-like trends visible in the plots are the cyclic structures of the temperature variation over the time period displayed (i.e. 0 – 18 sec). The best 170 performance is obtained at a frequency of 34 Hz. The simulations indicated that the system operating at 34 Hz not only indicates a much faster cool-down time, but also a lower quasi steady-state temperature and the worst performance are observed at a frequency of 15 Hz. The effect of the various operating frequencies on the transport processes in the OPTR is discussed further in section.

Figure 5.14: Temporal evolution of the temperature at the entrance to the pulse tube for

all cases studied for (a) 0 – 18 seconds and (b) 17.5 – 18 seconds

Figure 5.14b shows the transient gas temperature in an expanded time scale near the end of the simulations (i.e. 17.7 – 18 sec) for all of the five cases. The periodic variations in the temperature (that make up the band-like structures) are evident. The gas temperature is found to vary with time even at long times (i.e. a steady periodic mode). It is also interesting to note that the amplitude of gas temperature oscillation decreases with an increase in the frequency of the operating driver piston. 171

Figure 5.15: Temperature and pressure profiles at the exit of the cold heat-exchanger

during the initial cycles (cycles 1 – 5) of the OPTR operation for case 1 172

Figure 5.16: Temperature and pressure profiles at the exit of the cold heat-exchanger

during the initial cycles (cycles 1 – 5) of the OPTR operation for case 3

A similar trend can be observed in the solid (porous media) temperatures. Figures

5.15 and 5.16 show the pressure, gas and solid temperature profiles at the entrance of the pulse tube (r = 0.005 m) at the beginning of the simulation process (the first five cycles are shown) for cases 1 and 3 (i.e. 15 and 34 Hz driver frequency) respectively. The pressure exhibits a fairly sinusoidal pattern; however the gas temperature profile is distorted. The solid (porous media) temperature also appears to be near-sinusoidal. The solid material having a much higher thermal mass does not respond as rapidly to changes in the gas temperature profile. This figure also shows a steep decrease/drop in the temperature at the beginning of the process. When comparing the two figures (5.15 and

5.16) for cases 1 and 3 respectively, it is evident that the system operating in case 3 (i.e. 173 at 34 Hz) has a much faster cool-down time and indicates good performance during the initial stages of the OPTR’s operation.

Figure 5.17: Temporal evolution of the gas and solid temperatures at the exit of the cold

heat-exchanger for cases 1, 3 and 5 for (a) 0 – 18 seconds and (b) 17.5 – 18 seconds

Figure 5.17a shows the temporal variation of the gas and solid temperatures at the exit of the cold heat-exchanger/entrance to the pulse tube (r = 0.005 m) for cases 1, 3 and

5. As stated earlier, as the frequency of the driver increases, the amplitude of oscillation of the solid temperature decreases. At higher frequencies, the gas temperature oscillates rapidly such that the time for heat transfer between gas and solid is decreased considerably. Figure 5.17b shows the oscillatory nature of the gas and sold temperatures at the exit of the cold heat exchanger. The band-like structures that appear in Figure

5.17a are due to the cyclic temperature variations. The phase angle difference between the gas and the solid temperatures is also observed in Figure 5.17b. The solid temperature lags the gas temperature. This phase angle difference is dependent on the 174 frequency of operation of the driver. In our simulations, the solid temperature lags the gas temperature by 31.49°, 41.99°, 49.5°, 62.99° and 85.5° for the cases 1, 2, 3, 4 and 5 respectively. These phase angle differences were predicted at the exit of the cold heat exchanger after 18 seconds of operation. At higher frequencies, the gas temperature oscillates much faster, hence the phase angle lag between the solid and gas temperatures increases with an increase in the operating frequency of the driver.

Figure 5.18: Comparison of the phase-angle shift of axial velocity with respect to the

pressure along the axis of symmetry in the pulse tube of the OPTR

The phase angle (θ) of the axial velocity (mass-flow rate) with respect to the pressure along the axis of symmetry in the pulse tube is shown in Figure 5.18. Figure

5.18 provides a comparison between the phase shift in the pulse tube for case 1 (15 Hz) 175 and case 3 (34 Hz). The positive values of the phase shift indicate that the pressure leads the mass flow rate by an angle θ. It is interesting to note that the value of the phase shift is not constant along the length of the pulse tube. The smaller values of phase angle for case 3 (34 Hz) indicate that the mass-flow rate and pressure are more in phase for a frequency of 34 Hz compared with a frequency of 15 Hz. Hence, a frequency of 34 Hz will provide a better performance for the given system.

Figure 5.19: Performance map of the OPTR simulated over the frequency range on 15 –

100 Hz

The predicted cycle-averaged gas temperature at the end of the cold heat exchanger (r = 0.0) for the five cases studied is shown as a function of the operating frequency in Figure 5.19. It is evident from the figure that an optimum frequency for this 176 particular system lies around 34 Hz. In the following sections, we provide detailed results of the flow and pressure fields in the pulse tube to explain the frequency dependency.

5.5.3.2. Spatio-temporal temperature and velocity variations

In this sub-section, we present the results of the spatial and temporal variations in the gas temperature for the case 1 (Table 5.6, 15 Hz) and that with case 3 (Table 5.6, 34

Hz).

Figure 5.20: Distribution of the temperature (along the axis of symmetry) at four instants in cycle 330 for case 1 (Table 5.6). (a) Components B, C, D, E, F, G and H (see

Figure 5.2 and table 5.5) and (b) components I and J

In Figures 5.20a and 5.20b, the axial distribution of the temperature in the system along the line of symmetry (i.e. r = 0 in the simulations) for case 1 (Table 5.6, 15 Hz) studied is shown. In the figures the point x = 0.0 m corresponds to the entrance to the transfer tube and x = 1.0687 m corresponds to the end-wall of the compliance volume.

The four plots in the figure show the temperature profiles at four equally-spaced intervals within a cycle (330th cycle) i.e. at 0, (π/2), π and (3π/2) radians. As can be seen in Figure 177

5.20b, beyond the hot heat exchanger, the temperature oscillates negligibly and its value lies between 290 and 300 K. The temperature profile along the regenerator and the heat exchangers (porous media) show negligible oscillation, i.e. ~ 1 – 2 K. In these regions the gas interacts with the porous media (via the non-equilibrium coupling of the energy equations) which has a high thermal mass and thus the amplitude of the temperature oscillations are damped.

Figure 5.21: Distribution of the temperature (along the axis of symmetry) at four instants in cycle 630 for case 3 (Table 5.6). (a) Components B, C, D, E, F, G and H and

(b) components I and J

The gas temperature distribution in the system operating at 34 Hz (i.e. case 3,

Table 5.6) along the axis of symmetry is shown in Figures 5.21a and 5.21b. Here, the temperature profiles are plotted at four instants (0, (π/2), π and (3π/2) radians) in the 630th cycle. The temperature in the porous media has negligible oscillation similar to case 1

(Figures 5.20a and 5.20b). The main difference between Figures 5.20a and 5.21a is the temperature oscillation along the pulse tube. Figure 5.21a clearly shows the thermal 178 separation (cold and hot regions separated by a buffer zone) in the pulse tube which is not evident in Figure 5.20a.

Figure 5.22: Distribution of the axial component of velocity ‘ux’ (along the axis of symmetry) at four instants in cycle 630 for case 3 (Table 5.6). (a) Components B, C, D,

E, F, G and H and (b) components I and J

In Figures 5.22a and 5.22b, the distributions of the axial velocity in the system along the line of symmetry (i.e. r = 0 in the simulations) are shown for case 3 (Table 5.6).

The four plots in the figure indicate the velocity magnitude profiles at four instants in the cycle (630th). The velocity oscillation in the middle of the cryocooler is negligible. This is due to the presence of the porous media. However, in the transfer tube (0 – 0.1 m), in the orifice (0.252 – 0.255 m) and in the inertance tube (Figure 5.22b), the flow velocities are high due to their fairly small cross sectional area. As one approaches the compliance volume, the flow dampens out once again. 179

6.5.3.3. Steady-periodic multi-dimensional effects

In this section, we address the steady-periodic flow properties of the OPTR in the pulse tube region. In Figures 5.23, 5.24, 5.25 and 5.26, the time-averaged temperature and velocity profiles in the pulse tube are shown for case 1 (Table 5.6, 330th cycle), case

th th th 3 (Table 5.6, 630 cycle), case 4 (Table 5.6, 920 cycle) and case 5 (Table 5.6, 1900 cycle) respectively. At this point in the simulation (i.e. ~18 sec) the system is assumed to have reached a quasi-steady periodic state and the variation of the fluid temperature from cycle to cycle is small.

Figure 5.23: Cycle-averaged temperature and velocity-streamlines in the pulse tube

section (case 1, Table 5.6) 180

Figure 5.24: Cycle-averaged temperature and velocity-streamlines in the pulse tube

section (case 3, Table 5.6)

Figure 5.23 shows the cycle-averaged temperature field and streamlines in the pulse tube of the OPTR (x = 0.1922 to x = 0.2522) operating at 15 Hz (case 1, Table 5.6).

We observe a uni-cellular structure which most likely leads to significant mixing through the cycle-averaged streaming process. Figure 5.24 shows the cycle-averaged temperature field and streamlines in the pulse tube for case 3 (Table 5.6, 34 Hz). While the flow field in the pulse tube is oscillatory the cycle-averaged flow field shows remarkable organization. The cycle-averaged flow field is characterized by two sets of counter- rotating vortex pairs. The maximum streaming velocity in the axial direction is around

0.99 m/s and the minimum is also around - 0.99 m/s (the negative sign indicates flow in the negative x-direction), however the bulk of the streaming velocity is around 0.1 and -

0.1 m/s. The maximum and minimum velocities in the radial direction are 0.5032 m/s and -0.1373 m/s. As seen in Figures 5.19 and 5.21 earlier, a frequency of 34 Hz (case 3,

Table 5.6) has the best performance among the cases studied. This is evident from the bi- cellular streaming structures visible in Figure 5.24. These bi-cellular structures help 181 create a buffer zone at the center of the pulse tube and help maintain a temperature gradient across the two ends and isolating them from each other. A discussion emphasizing the importance of this form of streaming is provided later in this section.

Figure 5.25: Cycle-averaged temperature and velocity streamlines in the pulse tube

section (case 4, Table 5.6)

Figure 5.26: Cycle-averaged temperature and velocity streamlines in the pulse tube

section (case 5, Table 5.6)

In Figures 5.25 and 5.26 the cycle averaged gas temperature field and streamlines are shown for the 60 Hz (case 4, Table 5.6) and 100 Hz (case 5, Table 5.6). In Figures 182

5.25 and 5.26, the streaming rolls/cells are much smaller than those observed in case 3

(operating at 34 Hz, Table 5.6). The acoustic power density of a system is proportional to the frequency of the oscillations in the system [55]. As the frequency of oscillations in the system increase, the size of the system needs to decrease in order to maintain the same high acoustic power density. In the case of the present system, the components were kept at the same dimensions for all the cases. This leads to larger stagnant volumes

(excess void spaces) in the pulse tube as compared to the system operating at the lower frequency (case 3, Table 5.6), thus decreasing the efficiency and effectiveness of the refrigerator. The other factors that may be preventing the system from providing the optimum performance at 60 and 100 Hz are the lengths and diameters of the inertance tube and the orifice valve respectively. 183

Figure 5.27: Cycle and radially averaged enthalpy flow in the pulse tube section of the

OPTR along the axis of symmetry for cases 1 and 3 (Table 5.6)

The cycle averaged enthalpy flow in the pulse tube along the axis of symmetry is shown in Figure 5.27. The averaging is done over the cross-sectional area of the pulse tube and over a single cycle. The cycle averaged enthalpy flows are compared for case 1 and case 3 (Table 5.6). As can be seen in Figures 5.23 and 5.24, very high cycle-steady streaming velocities are present along the axis of symmetry. The direction of these high streaming velocities along the axis of symmetry dictates the direction of the average enthalpy flow. In the cold section of the pulse tube (x = 0.1922 – 0.226), the streaming velocity is in the negative x-direction, hence the average enthalpy flow in this direction is negative. Conversely, the positive streaming velocities in the warm section of the pulse tube (x = 0.226 – 0.2522) correspond to positive values of average enthalpy flow in this 184 region. The higher values of enthalpy flow for case 3 (Table 5.6) are indicative of the superior performance of the OPTR operating at a frequency of 34 Hz compared with 15

Hz.

Figure 5.28: Cycle-averaged gas temperature in the pulse tube section of the OPTR

along the centerline axis of the system for cases 1 and 3 (Table 5.6)

Figure 5.28 shows the time-averaged gas temperature in a single cycle along the centerline axis of the pulse tube for cases 1 and 3 (Table 5.6). It compares the gas temperatures in the pulse tube for the cases with the best and worst performances (i.e. 15

Hz and 34 Hz) observed. The bi-cellular streaming pattern observed 34 Hz compared to the fairly uni-cellular structure 15 Hz enables the separation of the cold and hot zones in the pulse tube. This is evident in the temperature profiles of the two cases. With the bi- 185 cellular structure observed for an operating frequency of 34 Hz, the pulse tube is able to maintain a rather steep temperature gradient across its two ends as compared to a lower frequency (i.e., 15 Hz).

A buffer zone is visible (between 0.222 m and 0.236 m in Figure 5.24) within the pulse tube. The presence of these sets of symmetric re-circulating rolls near the cold and the hot ends generate a stagnant zone within the pulse tube. Past work in this area has shown only streaming structures that occupy the entire pulse tube [51, 52], i.e. single cell structures where the entire flow of gas in the pulse tube is circulated by the single streaming cell structure. In that case (where single cell structures are predicted in the pulse tube), streaming is detrimental to the performance of the PTR. A similar effect is seen in case 1 (Table 5.6) of the present study where the system is operating at 15 Hz (see

Figure 5.23). Case 1 (Table 5.6) happens to also have the worst performance of the all the cases simulated. In that case, there is no definite bi-cellular structure, but more of a random single cellular structure which leads to the poor performance of the system.

6.5.4. Summary

The PTR numerical model is used to simulate the evolution fluid dynamic and heat transfer characteristics in an OPTR at different operating frequencies. The flow and temperature fields in the gas-phase and the porous media zones were modeled by the

Navier-Stokes and the Darcy-Forchheimer equations respectively. The thermal non- equilibrium heat transfer model was used to simulate the temperature fields in the heat exchanger matrices. The simulations indicate that the temperature of the solid heat 186 exchanger material (porous media) lags the temperature of the gas in the porous media by a phase angle that increases with an increase in the operating frequency. For the given system’s dimensions and orifice opening, a frequency of around 34 Hz provides the optimum cooling. The temperature profile along the length of the pulse tube is non- linear. At higher frequencies (34 Hz and above), the pulse tube can be divided into three zones/regions. However, these three zones are not perfectly stratified and non-uniform radial temperature profiles are observed across the cross-section of the pulse tube. Cycle- averaged temperature and velocity fields in the pulse tube section of the OPTR system indicate the presence of steady counter-rotating streaming recirculation patterns. The non-uniform profiles of temperature in the pulse tube are attributed to the presence of these steady streaming rolls. At the optimum frequency of 34 Hz, a bi-cellular structure is observed (two counter-rotating streaming patterns are formed at each of the two ends of the pulse tube). We propose that the presence of the bi-cellular structure (observed at higher frequencies) enhances the performance of the OPTR by creating a dead zone between the cold and warm ends of the pulse tube. This dead zone (buffer region) effectively isolates the cold and warm ends of the pulse tube. However, at the lower frequencies these streaming rolls degenerate to a single re-circulating cell – thus leading to reduced performance in the OPTR system.

5.6. Tapering the pulse tube for acoustic streaming suppression

Acoustic streaming can be defined as steady convection which is driven by oscillatory phenomenon [58] in bounded channels. Acoustic streaming has applications 187 in mixing channels, etc. However, in a PTR, the occurrence of streaming in the pulse tube leads to a deterioration of its performance. The function of the pulse tube as stated earlier is to isolate the cold and warm ends from each other. This is accomplished by the formation of a buffer zone in the center of the tube. This buffer zone (visualized as a plug of gas) oscillates within the pulse tube, but never leaves either end. Hence, it is vital to study and investigate methods of suppressing (or eliminating) acoustic streaming flows in the pulse tube section of PTRs. This sub-section of Chapter 6, a method of streaming suppression is discussed and evidence of streaming suppression is presented in the cycle- averaged flow and temperature fields of the pulse tube section [102].

5.6.1. Suppression of streaming

The presence of streaming in the pulse tube has been hypothesized [36, 51, 52,

62-64] to cause re-circulation in the pulse tube. Earlier studies of streaming [36, 51] showed unicellular cells in the pulse tube that circulated gas from the warm end of the pulse tube to the cold end. This leads to degradation in performance of the PTR.

However, as seen in the previous sub-section of this chapter (section “5.5. Effects of frequency on pulse tube refrigerator performance”), it is possible to use acoustic streaming to isolate the two ends of the pulse tube. The presence of counter rotating bi- cellular streaming cells in the pulse tube (one each in the warm and cold ends) creates a secondary buffer zone at the center of the pulse tube. This cycle-independent secondary buffer zone helps to isolate the cold and warm ends of the pulse tube. However, in most pulse tube refrigerators, acoustic streaming is detrimental to its performance. 188

Lee et al. [65] first proposed that acoustic streaming can be suppressed in a PTR by using a pulse tube that has a slight taper. Later Olson and Swift [66, 67] developed a relation to predict the optimum taper angle required suppress streaming in the pulse tube.

These equations were developed based on Rott’s [68] general method of calculating boundary-layer streaming based on standing wave phasing between the pressure and velocity. Rott’s equations had to be modified because the PTR is a travelling wave device and Rott’s equations were developed for a tube of constant cross-sectional area.

The Olson-Swift equation predicts the location at which streaming in the pulse tube of a

PTR is suppressed. In the studies by Olson and Swift [66, 67] and Swift et al. [69], the

Olson-Swift equation was used to design an experimental system whose performance was tested.

However, the Olson-Swift relation is a 1-D simplification of a multi-dimensional problem and is applicable only in the limit of low amplitude oscillations. The concept of acoustic streaming suppression in pulse tubes too is not well known or visualized. In this following sub-section of chapter 6, the performance of a PTR with a tapered pulse tube is studied and acoustic streaming suppression in tapered pulse tubes is examined.

5.6.2. Problem geometry

Figure 5.29 depicts the generalized geometry (i.e. an inline OPTR system) for the two different geometries studied; Geometry A and Geometry B. Only half the geometry shown in Figure 5.29 is simulated (the axisymmetric assumption) to save on computation time. The OPTR consists of a compression chamber (which includes the moving piston, 189 component A), the transfer tube (B), aftercooler (the first red hatched region, C), regenerator (blue cross-hatched region, D), pulse tube (F) with two heat-exchangers at its ends (the other two red hatched regions, E and G respectively), an orifice (a simple obstruction to the flow, H), an inertance tube (I) and the compliance volume (J).

Figure 5.29: General schematic of the OPTR geometries considered (a) Geometry A and

(b) Geometry B

Table 5.5 given in the previous section, summarizes the various dimensions of the problem geometry (except components F and G) and the time-invariant boundary 190 conditions (all components) in the simulations. The mathematical boundary conditions at the various components’ surfaces will be explained in detail later in this sub-section.

Table 5.7: Cases of the various conditions (Taper angle and Hot heat-exchanger

diameter) simulated

Simulation Case Taper Angle Radius of Hot Heat Geometry No. θ (o) Exchanger (mm)

1 0 8.0 2 1 8.0 A 3 2 8.0 4 -1 8.0

5 0 5.0 6 1 3.9527 B 7 2 2.90474 8 -1 6.0473

The simulations in this study are divided into two distinct sets of geometry which are labeled as A and B. The different types of geometry are based on the taper angle (θ) of the pulse tube and the diameter of the hot heat-exchanger. Table 5.7 summarizes the various cases studied and the diameter of the hot heat-exchanger. Geometry- A comprises the first four cases (i.e. Case 1 – Case 4, Table 5.7 above) that are simulated based on the geometry shown in the Figure 5.29a and the dimensions in Tables 5.5 and

5.7. The sign conventions used for the taper angles in this study is shown by a schematic in Figure 5.30a. A positive taper angle indicates that the cold end of the pulse tube has a larger diameter as compared to the warm end. Conversely, a negative taper angle 191 indicates that the cold end of the pulse tube has a smaller diameter as compared to the warm end of the pulse tube.

Figure 5.30: Schematic of the sign convention used to determine the taper angle of the

pulse tube and the diameter of the hot heat-exchanger for (a) Geometry A and (b)

Geometry B

For Geometry- B, the same four taper angles were studied (i.e. Case 5 – Case 8) and the geometry used for system B is shown in Figure 5.29b. The difference between

Geometry- A (Figure 5.29a) and Geometry- B (Figure 5.29b) is that for B the effect of a change in the diameter of the Hot Heat Exchanger (component G in Figure 5.29a) is studied when the pulse tube is tapered. For Geometry- B, when the pulse tube is tapered, the diameter of the hot heat-exchanger is maintained the same as the warm end of the 192 pulse tube attached to it (so as to maintain continuity along the pulse tube without any sudden expansions into the hot heat-exchanger- see Figure 5.30b). This is different from cases 1 – 4 (i.e. Geometry- A) where the diameter of the hot heat-exchanger is maintained constant at 0.016 m (Figure 5.30a).

The main results and findings of all the cases studied are shown and discussed in the results sections below. In the subsequent sections of this sub-section, the cases will be referred to by the case number, the geometry and the angle (e.g., Case 1 (A [0])).

5.6.3. Boundary, initial and porous media conditions

Details of the time-invariant boundary conditions, initial conditions and porous media conditions used in the model are provided below. This sub-section also provides information on the operating parameters of the numerical model.

5.6.3.1. Boundary conditions

Table 5.5 (given earlier) specifies the heat transfer boundary conditions used at the surface boundaries of the various components. The cold heat-exchanger is always maintained adiabatic in order to study the best (no-load) performance of the OPTR. The piston is modeled as a reciprocating wall having an oscillatory velocity. The velocity of the piston is defined by the function, u = A0ωCos (ωt), where A0 is the maximum displacement of the piston (4.511 mm, same as the previous sub-section where the effects of the operating frequency was investigated) and ω is the angular frequency (ω = 2πf) and f is the frequency of operation. For this study (all cases), the frequency is set at 34 193

Hz. This value of frequency was chosen based on the study of the effect of the frequency on the performance of the cryocooler in the previous sub-section of this chapter (section

“5.5. Effects of frequency on pulse tube refrigerator performance”).

5.6.3.2. Initial conditions

At the start of the simulation, the temperature in the system is assumed to be 300

K everywhere. The initial pressure in the system is set to 3.1 MPa.

5.6.3.3. Porous media conditions

In the porous media regions, the three heat exchangers (components C, E and G in

Figure 5.29) are copper with density 8950 kg/m3, constant specific heat 380 J/kg-K (since it does not exhibit large variation in the temperature range studied, i.e. 100 – 350 K) and temperature dependent thermal conductivity. The regenerator has properties of 316 stainless steel, with density 7810 kg/m3, constant specific heat 460 J/kg-K and temperature dependent thermal conductivity. For the present simulations, ε is considered to be equal to 0.69 [53, 101] and in the rest of the domain (gas phase) ε has a value of 1.0.

The permeability κ is considered to be 1.06 ×10−10 m2 [101] and the quadratic drag factor

CF has a value of 0.55.

5.6.3.4. Numerical scheme

The time-step size was determined by allowing 80 time-steps/cycle, which is sufficient to accurately simulate the problem. Based on the aforementioned criterion, the time-step used was found to be 3.67647×10-4 s. An overall convergence criterion is set for all the variables at 10-4 in the iterative implicit numerical solver. A hybrid 194

(structured-unstructured) grid system is used in the simulations. A total of 4600 – 4900 grid points were used in the simulations. The number of grid points chosen was dependent on the taper angle of the pulse tube and the diameter of the hot heat-exchanger.

5.6.4. Results and discussion

In the following sub-sections, results for spatio-temporal evolution of the flow and temperature fields in the OPTR (Figure 5.29) are presented.

5.6.4.1. Transient processes in the pulse tube refrigerator

The cool-down time from room temperature (300 K) as predicted by the simulation at the cold end of the different geometries of the OPTRs considered is presented next. 195

Figure 5.31: Temporal evolution of the temperature at the entrance to the pulse tube for

geometry- A (cases 1, 2, 3 and 4, Table 5.7) for (a) 0 – 19 seconds and (b) 18.75 – 19

seconds

6.6.4.1.1. Standard OPTR (geometry- A)

Figure 5.31a shows the cool-down time i.e. the temperature-history of the gas at the exit of the cold heat-exchanger into the pulse tube (x = 192.2 mm and r = 5.0 mm) for the first four cases studied (θ = 0°, 1°, 2° and -1°). These cases are studied for the first type of geometry i.e., geometry- A. The plots that can be seen in the figure are cyclic structures of the temperature variation over the time period displayed (i.e. 0 – 18 sec) and hence appear like banded structures. The best performance is obtained for a taper angle of 0° (a straight pulse tube) and the worst performance was predicted for a taper angle of

2°. In section 7.3.3.3., we take a look at the transport phenomena in the pulse tube that affect the performance of the OPTR.

In Figure 5.31b, the temperature at the exit of the cold heat-exchanger (x = 192.2 mm and r = 5.0 mm) is shown at the end of the simulation (18.75 – 19 s). Here, the 196 cyclic variations of the gas temperature are clearly visible. These cyclic structures are generated due to the oscillatory nature of the system. The solid temperature of the cold heat-exchanger exhibits a similar structure (with lower amplitude of oscillation) and the results are discussed later in this section.

5.6.4.1.2. Variable diameter hot heat exchanger OPTR (geometry- B)

In general, schematics of OPTRs in previous studies [66, 67, 69] that investigated the effect of the taper angle of the pulse tube indicate that the diameter of the hot heat- exchanger has been maintained the same as the diameter of the warm end of the pulse tube (i.e. the tapered end). In these studies, the tapering of the pulse tube was shown to have a positive effect on the performance of the OPTR. The results and the performance trends observed for geometry- A in the present study are different from those observed by

Swift et al. [66, 67, 69]. In the study by Swift et al. streaming suppression and enhanced performance are observed. The performance trends observed for geometry- A are similar to the performance trends that were observed by Yang [103], however, there are no schematics of the OPTR used in the experimental study by Yang [103]. To investigate this difference in performance observed between the results of Geometry- A and the results of Swift et al., four cases were simulated (i.e. geometry- B, case 5 – case 8). For these additional four cases, the diameter of the hot heat-exchanger is maintained the same as the diameter of the warm end of the pulse tube for the same four taper angles studied in geometry- A (θ = 0°, 1°, 2° and -1°). 197

Figure 5.32: Temporal evolution of the temperature at the entrance to the pulse tube for

geometry- B (cases 1, 5, 6 and 7, Table 5.7) for 0 – 19 seconds

In Figure 5.32 the cycle averaged gas temperature (at x = 192.2 mm and r = 5.0 mm) is plotted as a function of time. The difference between the temperatures for the different cases is fairly small for geometry- B, hence the cycle-averaged temperature has been plotted instead of the oscillatory temperature profiles seen in Figure 5.31a and 5.31b

(overlap of the various plots make it difficult to distinguish between them). By changing the size of the hot heat-exchanger to coincide with the diameter of the warm end of the pulse tube, the performance is shown to improve. This improved performance in geometry- B can be attributed to the reduction in fluid dynamic losses that occur in geometry- A due to the sudden expansion from the pulse tube to the hot heat-exchanger

(the traditional back-step problem in fluid mechanics). The simulations predict that the 198 taper angle of 1° has the best performance and that the taper angle of -1° has the worst performance. This trend is different from what is observed for geometry- A. We take a closer look at this new trend in a following sub-section, where the cycle steady periodic flow and temperature fields in geometry- A and geometry- B are compared.

5.6.4.1.3. Gas temperature v/s solid temperature

The uniqueness of the computational model used in the present study is the application of a thermal non-equilibrium assumption in the porous media regions. As seen in section “5.2. Governing equations and PTR numerical model” of this chapter, the gas and solid equations are coupled via a convective heat transfer correlation (equation

5.7) [96]. Due to this coupling and the large difference in the densities and specific heats for the two media, their respective temperatures are different. In the previous section of this chapter (section “5.5. Effects of Frequency on Pulse Tube Refrigerator

Performance”), the solid temperature was found to lag the gas temperature by a phase angle difference (for a frequency of 34 Hz) of 49.5° at quasi-steady state. 199

Figure 5.33: Evolution of the gas and solid temperatures as a function of time at the exit

of the cold heat-exchanger for (a) Cases 1, 2 and 3 (geometry- A) from 0 – 18 seconds

and (b) Case 1 (geometry- A) from 0 – 0.15 seconds

In Figure 5.33a, the gas and solid temperatures are plotted for the first three cases

(i.e. geometry- A). As expected the solid temperature follows a similar trend as the gas temperature. However, due to the high density of the solid porous media, the gas temperature has higher amplitudes of oscillation. In Figure 5.33b, the gas and solid temperatures are shown as a function of time at the start of the simulations (i.e. for the first five cycles) for case 1. Here the difference between the gas and solid temperatures as well as the phase angle difference between the two are clearly visible.

5.6.4.1.4. System performance

The performance of the system is characterized by the quasi-steady state temperature in the cold heat-exchanger section of the cryocooler. This performance is affected by the geometry of the pulse tube section of the system. The temperature values 200 reported here are ‘no-load’ values due to the fact that the wall boundary condition of the cold heat-exchanger in this study is adiabatic.

Figure 5.34: Performance map of the two OPTRs simulated

The predicted cycle-averaged gas temperature at the end of the cold heat exchanger (r = 0.0) for the eight cases studied is shown as a function of the taper angle of the pulse tube in Figure 5.34. For geometry- A it is evident that a straight pulse tube is required for the OPTR to operate at its optimum. However, geometry- B exhibits a different trend where a taper angle of 1° is shown to provide the best performance.

The trend of decreased performance as the positive taper angle is increased (i.e., cases 2, 3 for geometry- A and case 7 for geometry- B) as observed in Figure 5.34 can be explained by the decrease in the volume of the pulse tube and the larger expansion into 201 the hot heat-exchanger when the pulse tube is tapered. The decrease in the volume of the pulse tube also leads to a deterioration of the ideal stratified nature of the gas in the pulse tube. Due to the decrease in volume, the buffer zone at the center of the pulse tube is forced out of the pulse tube during the cycle, thereby not performing the function of a pulse tube (to insulate the two ends). As the volume is decreased there is not enough gas in the pulse tube to create the three required zones i.e., the cold zone, the warm zone and the buffer zone in between.

However, we observe a deviation from this in case 6 (a 1° taper angle with a smaller diameter hot heat-exchanger). Despite the decrease in volume of the pulse tube, we observe that as the pulse tube is tapered, there is an improvement in the performance of the OPTR. The only difference between geometry- A and geometry- B is the diameter of the hot heat-exchanger; however we observe different performance characteristics

(based on the pulse tube taper angle) for the two. A more in-depth study into the factors that lead to this difference is required. In the following sections, the effect of the OPTR geometry on cycle-average streaming phenomena in the pulse tube and how this streaming phenomenon affects the performance of the OPTR is discussed.

5.6.4.2. Spatial temperature variation in the pulse tube

In Figure 5.35a and 5.35b, the axial distribution of the gas temperature is shown in the system along the axis of symmetry (i.e. r = 0 in the simulations) for case 1 (A [0]) and case 3 (A [2]) respectively. The four plots (plotted across components E, F and G in

Figure 5.30) show the temperature profiles at four equally-spaced intervals within a cycle

(650th cycle) at the end of the simulations i.e. at π/2, π, 3π/2 and 2π radians. 202

Figure 5.35: Distribution of the temperature (along the axis of symmetry) at four instants in cycle 650 (components E, F and G) for (a) Case 1 (geometry- A) and (b) Case

3 (geometry- A)

In Figure 5.35a and 5.35b, it is apparent that the gas temperature in the porous regions does not oscillate much. In Figure 5.35a (case 1, Table 5.7), three clear zones are visible in the pulse tube (i.e. from 0.1922 m to 0.2522 m). The first is the cold zone

(0.1922 m – 0.215 m) where the temperature does not oscillate much during the cycle i.e., the temperature oscillations are ~ 10 – 20K. The next zone is the buffer zone at the center of the pulse tube (0.215 m – 0.245 m) where high oscillations in the gas temperature are observed i.e., the temperature oscillations are ~ 80 – 110K. The last zone is the warm zone near the warm end of the pulse tube (0.245 m – 0.2622 m). In the warm zone too, the temperature oscillations are fairly large. Comparatively, Figure 5.35b (case

3, Table 5.7) does not exhibit a similar structure. Here, in the cold end of the pulse tube the gas temperature undergoes very high oscillations (~ 70 – 100K) compared to the other regions where the temperature oscillations are fairly small (~ 20 – 50K). These high 203 oscillations in the cold end of the pulse tube do not allow the temperature of the cold end to decrease and leads to high oscillations in the solid temperature over a cycle (~ 20K) compared to case 1 when the solid temperature oscillations are ~ 5K over the cycle.

Figure 5.36: Distribution of the temperature (along the axis of symmetry) at four instants in cycle 650 (components E, F and G) for (a) Case 2 (geometry- A) and (b) Case

6 (geometry- B)

Figure 5.36a and 5.36b are similar to Figure 5.35a and 5.35b. Here, we compare the axial temperature distribution inside the pulse tube when the diameter of the heat exchanger is changed keeping all other parameters constant i.e., for case 2 (A [1]) and case 6 (B [1]). The plots in Figure 5.36a have shapes similar to those in Figure 5.35a.

However, the buffer zone and warm zones are larger in size compared to Figure 5.35a. In

Figure 5.36b the temperature profile along the axis is fairly linear from end of the cold zone (~ 0.21 m) to the warm end of the pulse tube. 204

The axial variation of the gas temperature provides an important insight into the structure of the gas inside the pulse tube. However, due to the two dimensional (often three dimensional) nature of the flow in the pulse tube, it is also important to study the 2-

D variation of the temperature inside the pulse tube. In the following section we look at the cycle-averaged temperature and velocity profiles (steady secondary streaming) in the pulse tube.

5.6.4.3. Cycle-averaged multi-dimensional transport processes in the pulse tube

In this section, we address the steady-periodic transport processes of the OPTR in the pulse tube region. In Figures 5.37, 5.38, 5.39, 5.40, 5.41 and 5.42 the cycle-averaged

(650th cycle) temperature and velocity profiles in the pulse tube are shown for the case 1

(A [0]), case 2 (A [1]), case 4 (A [-1]), case 5 (B [0]), case 6 (B [1]) and case 8 (B [-1]) respectively (all cases correspond to Table 5.7). At this point in the simulation (i.e. ~19 sec) the system is assumed to have reached a quasi-steady periodic state, the variation of the fluid temperature from cycle to cycle is small and the change in cycle averaged flow velocity from cycle to cycle is negligible. 205

Figure 5.37: Cycle-averaged temperature and velocity streamlines in the pulse tube

section (case 1 – geometry- A)

In Figure 5.37 we see the cycle-averaged temperature field and the cycle-averaged velocity vectors in the pulse tube for case 1 (x = 0.1922 to x = 0.2522). Even though the flow field in the pulse tube is oscillatory the cycle-averaged flow fields show remarkable organization. The cycle-averaged flow field is characterized by two sets of counter- rotating streaming cells (one set on each side of the buffer zone). These bi-cellular structures help create a buffer zone at the center of the pulse tube and help maintain a temperature gradient across the warm and the cold ends and isolate them from each other.

The buffer zone is visible approximately at the center (between 0.222 m and 0.236 m in

Figure 5.37) of the pulse tube. The presence of these sets of symmetric re-circulating rolls near the cold and the hot ends generate a stagnant zone at the center of the pulse tube. This is different from past work in this area where a single streaming structure occupies the entire pulse tube [51, 52], i.e. the entire flow of gas in the pulse tube is circulated by the single streaming cell structure. Compared to the unicellular streaming 206 structures [36, 51, 52] that are detrimental to the performance of an OPTR, the bi-cellular structures observed in this study help isolate the warm and cold ends of the pulse tube.

The maximum streaming velocity in the axial direction is around 0.99 m/s and the minimum is around -0.99 m/s (the negative sign indicates flow in the negative x- direction), however the bulk of the streaming velocity is around 0.1 and -0.1 m/s. The maximum and minimum velocities in the radial direction are 0.5032 m/s and -0.1373 m/s.

In the figure, a velocity of 1.333 m/s corresponds to a vector of length 1.0 cm. This convention is followed for all the vector plots that follow, i.e. Figures 5.38, 5.39, 5.40,

5.41 and 5.42.

Figure 5.38: Cycle-averaged temperature and velocity streamlines in the pulse tube

section (case 2 – geometry- A) 207

Figure 5.39: Cycle-averaged temperature and velocity streamlines in the pulse tube

section (case 4 – geometry- A)

In Figures 5.38 and 5.39 the cycle averaged gas temperature field and streamlines are shown for geometry- A with taper angles of 1° (case 2) and -1° (case 4) respectively.

Figures 5.38 and 5.39 show bi-cellular streaming structures similar to those observed in

Figure 5.37 (case 1). Even though the bi-cellular structures appear similar, the sizes of these structures vary for the different cases studied. For θ = 1°, the streaming cells on the cold and warm ends occupy equal lengths of the pulse tube, however for θ = -1°, the streaming cells on the cold end of the pulse tube are much larger than those on the warm end of the pulse tube. The magnitude of the streaming velocity is higher for θ = 1° (case

2) and for θ = -1° (case 4) than for θ = 0° (case 1). This higher streaming velocity for cases 2 and 4, especially along the axis of symmetry causes a larger amount of circulation within the pulse tube. The cycle averaged temperature fields in Figure 5.38 and Figure

5.39 for θ = 1° (case 2) and θ = -1° (case 4), show poor temperature stratification along the axis of symmetry. This poor stratification is due to the presence of large streaming rolls with comparatively higher streaming velocity on cold and warm ends of the pulse 208 tube. This is the reason why for geometry A, θ = 0° has a better performance compared to θ = 1° and θ = -1°.

Figure 5.40: Cycle-averaged temperature and velocity streamlines in the pulse tube

section (case 5 – geometry- B)

As observed in Figure 5.34, geometry- B performs better than geometry- A for the same operating conditions. The only difference between the two systems is the diameter of the hot heat-exchanger. Figures 5.40, 5.41 and 5.42 show the cycle-averaged temperature field and streamlines in the pulse tube of geometry- B (x = 0.1922 to x =

0.2522) for case 5 (θ = 0°), case 6 (θ = 1°) and case 8 (θ = -1°) respectively. 209

Figure 5.41: Cycle-averaged temperature and velocity streamlines in the pulse tube

section (case 6 – geometry- B)

An important difference between the cycle-averaged temperature fields for geometry- A (Figures 5.37, 5.38 and 5.39) and those for geometry- B (Figures 5.40, 5.41 and 5.42) is the absence of leakage in the warm region of the pulse tube along the axis of symmetry. The absence of leakage along the axis of symmetry can be attributed to a reduction/suppression of steady secondary streaming in the warm end of the pulse tube.

In Figures 5.40, 5.41 and 5.42, there are no streaming cells in the warm region of the pulse tube. The vectors that can be seen in the buffer region have streaming velocity magnitudes that are at least one order of magnitude lower than streaming velocity magnitude in the cold end of the pulse tube. In the warm end, the magnitude of the velocity of the streaming vectors shown (in Figures 5.40, 5.41 and 5.42) is at least two orders of magnitude lower than the streaming velocity magnitude near the cold end. 210

Figure 5.42: Cycle-averaged temperature and velocity streamlines in the pulse tube

section (case 8 – geometry- B)

Thus it is clearly apparent that maintaining the diameter of the warm end of the pulse tube and the hot heat-exchanger the same, helps prevent/suppress the acoustic streaming phenomenon in the warm end of the pulse tube. It is also shown that a positive taper angle on the pulse tube is only useful when the diameter of the warm end of the pulse tube and the hot heat-exchanger are maintained equal. While tapering the pulse tube does not eliminate secondary streaming in the pulse tube, it helps decrease the magnitude of the streaming velocity (i.e., it helps suppress streaming). However, if construction of an OPTR requires different diameters for the pulse tube and the hot heat- exchanger (or cold heat-exchanger), a straight pulse tube provides the best performance.

5.6.5. Summary

A numerical model was developed to simulate the transport and heat transfer phenomena in an OPTR. The spatio-temporal flow and temperature fields in the gas- 211 phase and the porous media zones were modeled by the Navier-Stokes and the Darcy-

Forchheimer equations respectively. The thermal non-equilibrium heat transfer model was used to simulate the temperature fields in the heat exchanger matrices. Of the two geometries studied, geometry- B is shown to perform better. A straight pulse tube (θ =

0°) is shown to provide the best performance when heat-exchangers at the ends of the pulse tube have diameters larger than the diameter of the pulse tube (i.e., for geometry-

A). Steady secondary velocity streaming in the pulse tube is shown to affect the performance of the OPTR. For geometry- A, tapering the pulse tube leads to higher streaming velocities and hence a reduction in the performance of the OPTR. However, considerable suppression of acoustic streaming is achieved within the pulse tube when diameter of the heat-exchanger at the warm end of the pulse tube has the same diameter as the pulse tube (i.e., for geometry- B). By reducing the diameter of the heat-exchanger, the magnitude of the streaming velocity is decreased by one or two orders of magnitude.

In this new system (geometry- B), a positive taper angle of θ = 1° is shown to achieve the best streaming suppression (lowest streaming velocity) and thus achieve the best performance. These results are of direct application in the design of OPTRs for cryogenic applications.

5.7. Effects of the inertance tube on pulse tube refrigerator performance

The application of the inertance tube and its role in the PTR was discussed in chapter 2. In this section, a study of the effects of the inertance tube on the performance of an IPTR is presented [92]. The validated CFD model of the Mk-I system (section 212

“5.4. Code validation with experimental studies”) is used to predict the effects of the inertance on the phase relationships (between the pressure and velocity) and the system performance. The wall thickness of the various components is not considered in this analysis. This is based on the fact that the thermal mass of the walls was not observed to affect the pressure and velocity profiles in the inertance tube and regenerator sections of the cryocooler which are studied in this sub-section.

5.7.1. Problem geometry

Figure 5.43 below shows the geometry studied (i.e. an inline IPTR system). Only half the geometry shown in Figure 5.43 was simulated (the axisymmetric assumption) due to the cylindrical nature of an actual system and to save on computation time. The

IPTR simulated has the same components and dimensions (see Table 5.8 below) as the

Mk-I experimental system studied (the orifice valve is replaced by a diffuser cone in this study). These include a compression chamber with a moving piston, a transfer tube, an aftercooler (the first red hatched region), a regenerator (blue cross-hatched region), a pulse tube with a heat exchanger at each end (the other two red hatched regions), a diffuser cone to connect hot heat-exchanger to the inertance tube, an inertance tube and a compliance volume. 213

Figure 5.43: Schematic of IPTR (Mk-I) simulated

Table 5.8: Dimensions of the IPTR system (Mk-I) simulated and time invariant thermal

boundary conditions

Radius Length No. Component Thermal Boundary Condition (cm) (cm)

A Compression Chamber 3.00 1.1 Adiabatic

2 B Transfer Tube 0.85 4.0 hc = 20 W/m -K

C Aftercooler Heat-Exchanger 0.85 3.0 Tw = 293 K D Regenerator 0.85 6.0 Adiabatic

E Cold Heat-Exchanger 0.47 5.0 Adiabatic

F Pulse Tube 0.47 23.0 Adiabatic

G Hot Heat-Exchanger 0.47 3.0 Tw = 293 K

r1 = 0.193 H Diffuser Cone 0.4 Adiabatic a r2 = rIT

a a I Inertance Tube rIT - Adiabatic J Compliance Volume 2.60 14.9 Adiabatic

a See Table 5.10 214

5.7.2. Boundary, initial and porous media conditions

The initial and boundary conditions adopted in the numerical model for the simulations reported in this section of the thesis are elaborated below. The porous media conditions used in the regenerator and heat-exchanger sections are also discussed.

5.7.2.1. Boundary conditions

Table 5.8 (given earlier) specifies the thermal boundary conditions used at the surface boundaries of the various components. The piston is modeled as a reciprocating wall having an oscillatory velocity. The velocity of the piston is defined by the function, u = A0ωCos (ωt), where A0 is the maximum displacement of the piston (piston amplitude), ω is the angular frequency (ω = 2πf) and f is the frequency of operation. For the current simulations (all cases of cryocooler Mk-I), the frequency f is set at 65 Hz and the piston amplitude A0 is set as 1.75 mm. These values of frequency and piston amplitude are maintained the same as the experimental and numerical studies of the Mk-I cryocooler system.

5.7.2.2. Initial conditions

The initial pressure in the system is set to 2.2 MPa for all the cases. In order to predict the system’s performance near quasi-equilibrium, the computational model was run with a prescribed initial condition that represents an expected quasi-steady temperature distribution (cold heat-exchanger at 100 K, the aftercooler and warm heat- 215 exchangers at 300 K and a linear gradient across the regenerator and pulse tube). With this initial condition, the simulations were run for until a quasi-stead state was achieved.

5.7.2.3. Porous media conditions

In the porous media regions, the three heat exchangers (components C, E and G in

Figure 5.43) have properties of copper with density 8950 kg/m3, constant specific heat

380 J/kg-K and a thermal conductivity of 470 W/m-K. The regenerator (component D in

Figure 5.43) has properties of 316 stainless steel, with density 7810 kg/m3, constant specific heat 460 J/kg-K and a thermal conductivity of 10 W/m-K.

Table 5.9 lists the porous media parameters used in the governing equations of the porous media. The values for porosity, permeability and drag factor are the same as those used in a previous study (section “5.4. Code validation with experimental studies”) where the cryocooler model was validated against temporal and quasi-steady experimental results [86].

Table 5.9: Porous media parameters used in the simulation

Component Material Porosity Permeability (m2) Drag Factor

Heat-Exchangers Copper 0.774 4.08 × 10-08 0.20

Regenerator Stainless Steel 0.720 9.70 × 10-11 0.30

216

5.7.2.4. Numerical scheme

The time-step (Δt) for the simulations was chosen to be 1.1476 × 10-5 s. This value of ‘Δt’ (~ 80 Δt/cycle) is sufficient to accurately predict the motion of the piston and the pressure waves generated. An overall convergence criterion is set for all the variables at 10-4 in the iterative implicit numerical solver. Due to the symmetry of the problem geometry, only one-half of the resonators’ geometry (Figure 5.43) was considered for the simulations. The total number of grid points in the system was between 3500 and 5000 depending on the inertance tube geometries studied. A hybrid meshing scheme was used with unstructured mesh used in the cold heat-exchanger due to the high temperature gradient observed there.

5.7.3. Results and discussion

The results from the numerical study of the inertance effect on the performance of the Mk-I cryocooler are presented in this sub-section. The various cases studied

(inertance tube length and radius) are listed below in Table 5.10.

Table 5.10: Cases of the various conditions simulated (Mk-I)

Inertance Tube Length Inertance Tube Radius Inertance ‘M’ Case No. 2 3 (cm) ‘rIT’ (cm) (Pa-s /m )

1 93.50 0.1524 458454.60

2 150.00 0.1930 458597.09

3 217.60 0.2324 458734.56

4 46.75 0.1524 229227.30 217

5 160.70 0.1524 787953.52

6 170.70 0.1524 836986.09

7 187.00 0.1524 884180.42

8 210.0 0.1524 1029684.12

9 233.8 0.1524 1146381.66

The purpose of this numerical study is to identify the best operating conditions of the

IPTR and to investigate the role of the inertance tube in achieving the optimum operating conditions. The second case (case 2, Table 5.10) represents the dimensions of the Mk-I experimental PTR system studied in an earlier sub-section of this chapter (section “5.4.

Code validation with experimental studies”) and in the experimental study [86]. Cases 1,

2 and 3 (Table 5.10) are for similar values of inertance, but different combinations of length and diameter (see eq. 4.1 in chapter 4). The values of inertance tube radius are chosen based on commercially available copper tubing. The rest of the cases (i.e., cases

4 – 9, Table 5.10) are for the same radius of the inertance tube as case 1; however the lengths have been changed from case 1 in order to study the effects of different values of inertance on the performance of the system.

In the following sub-sections, results for temporal evolution of the flow and temperature fields in the IPTR (Figure 5.43) and the quasi-steady state cycle-averaged system properties are presented and the effects of the inertance tube’s geometric parameters on the performance of the system are discussed. 218

5.7.3.1. Temporal transport processes in the cryocooler

The transient behavior of an IPTR is very important to its operation, as stated earlier. Figure 5.44a shows this temporal variation of the gas temperature at the exit of the cold heat exchanger and inlet to the pulse tube section (i.e. x = 19.1 cm, r = 0.4699 cm) for the first three cases studied (i.e., constant value of inertance with tubes of different radii). The banded profiles are cyclic variation of the gas temperature in the system. The cyclic temperature profiles are not purely sinusoidal in nature (Figure 5.44b) and the amplitude of temperature oscillation varies between 2 K and 4 K. The amplitudes of oscillation are dependent on the pressure amplitudes at that location (ideal gas assumption) and non-sinusoidal oscillations are due to secondary streaming flow (since the pressure exhibits purely sinusoidal behavior) [100, 102]. For a constant value of inertance (changing both the length and radius of the inertance tube), the change in performance of the IPTR is fairly small (~ 2 – 4 K difference between cases 1, 2 and 3,

Table 5.10). A similar trend is observed in the experimental study reported here, where negligible performance differences are observed for a constant (comparable) value of inertance ‘M’ (eq. 4.1). 219

Figure 5.44: Temporal variation of the gas temperature in the cold heat-exchanger for

the first three cases simulated at a constant value of inertance ‘M’ (a) from near-

equilibrium initial condition and (b) near quasi-steady state

The numerical model of the system is a thermal non-equilibrium model. Due to this, the gas and solid temperature values in the porous zones are different. Figure 5.45 plots the gas and solid temperature at the exit of the cold heat-exchanger (x = 19.1 cm, r

= 0.4699 cm) for case 1 and case 2. The difference between the solid and gas temperature values are clearly visible in this plot. The differences stem from the difference in the thermal heat capacities of the two phases. The larger thermal mass of the solid leads to smaller cyclical variations in its temperature compared to the gas temperature. The temporal profile of the solid temperature is fairly sinusoidal compared to the gas temperature profile which is affected by flow (instantaneous and steady) variations in the pulse tube which is adjacent to the cold heat-exchanger. 220

Figure 5.45: Comparison of gas and solid temperature at the exit of the cold heat-

exchanger for cases 1 and 2 simulated

Since the smallest inertance tube radius (rIT = 0.1524 cm) resulted in the best performance (lowest temperature achieved), this inertance tube radius was used to study the effects of the inertance (i.e., changing the length and keeping the radius constant).

When the value of the inertance is varied, a larger variation in the performance of the

IPTR is observed (Figure 5.46, similar to the variation observed in the experiments reported in chapter 4). As can be seen in Figure 5.46 where the transient cycle-average temperature is plotted, the Mk-I PTR with rIT = 0.1524 cm exhibits the best performance for an inertance value of M = 836986.09 Pa-s2/m3 (case 6, Table 5.10) with a quasi-steady state gas temperature ~ 40 K. 221

Figure 5.46: Temporal variation of the cycle-averaged gas temperature in the cold heat-

exchanger for the IPTR operating with a constant inertance tube radius from near-

equilibrium initial condition (Note: case numbers refer to Table 5.10)

In addition to affecting the cool-down time, a change in the inertance changes the flow and pressure profiles in the system. Figure 5.47 plots the axial component of the gas velocity at the center of the regenerator (x = 11.1 cm, r = 0.0 cm) as a function of the cycle time for the various cases simulated. For a constant value of inertance (cases 1 – 3,

Table 5.10), the temporal axial velocity profiles remain un-changed (Figure 5.47a).

However, when the inertance is varied, both the amplitude of the axial velocity and the structure of its temporal variation change. As discussed earlier, when the flow rate through the regenerator (a porous region) increases, the losses including the pressure drop increase. Hence the goal in a PTR is to minimize the flow rate while maintaining a high 222 pressure amplitude and consequently a high acoustic power (PV, where V is the volumetric flow rate). This can be achieved by minimizing the phase angle between the pressure and the flow rate at the center of the regenerator which in turn minimizes the flow rate at the center of the regenerator.

Figure 5.47: Comparison of the transient variation of the axial component of the gas

velocity at the center of the regenerator as a function of the cycle time for (a) constant

inertance value (different tube radii) and (b) different inertance values (constant tube

radius)

Figure 5.48 below shows plots of the temporal variations of the axial component of the gas velocity and the pressure as a function of the cycle time near quasi-steady state for case 1 (Figure 5.48a), case 6 (Figure 5.48b) and case 9 (Figure 5.48c). In Figure

5.48a it can be seen that the velocity amplitude is fairly large and the velocity leads the pressure by a small phase angle. At the optimum value of inertance (case 6, Figure 5.

48b), the phase angle between the velocity and the pressure is close to zero and the velocity amplitude is further decreased. Finally, beyond the optimum value of inertance 223 the velocity amplitude has further decreased (leading to a further decrease in the flow associated losses). However, the pressure now leads the velocity by a large phase angle and hence the acoustic power will be further decreased (due to its dependence on Cosθ).

Figure 5.48: Comparison of the transient variation of the pressure and axial component of the gas velocity at the center of the regenerator for (a) case 1, (b) case 6 and (c) case 9

The instantaneous acoustic power for a single cycle at quasi-steady state is plotted in Figure 5.49 for the various cases simulated. The plots in Figure 5.49a and Figure

5.49b are for the acoustic power predictions at the entrance and exit of the regenerator respectively (x = 8.1 cm and x = 14.1 cm, r = 0.0 cm). At the entrance to the regenerator 224

(Figure 5.49a), the instantaneous acoustic power values are highest at the smaller values of inertance. At these smaller values of inertance, the gas velocity is higher than at larger values of inertance (as seen in Figure 5.47 above) and this contributes to the large values of instantaneous acoustic power. Due to these higher values of gas velocity (and flow rate), the losses in the regenerator are higher and hence the reduction in acoustic power is much higher for the cases where the value of inertance is smaller than the optimum value.

In Figure 5.49c, the instantaneous acoustic power is plotted at the entrance to the inertance tube for the same four cases as Figure 5.49a and Figure 5.49b. Here the values of instantaneous acoustic power are much higher than the values in the regenerator. This is due to the high gas velocity values at the entrance to the inertance tube (due to the smaller cross-sectional area compared to the heat-exchanger before it). It is interesting to note that when Figure 5.49a and Figure 5.49c are compared, the temporal acoustic power profiles are in phase at the entrance to the regenerator (for different values of inertance), however as the value of inertance is increased the profiles move further out of phase at the entrance to the inertance tube. 225

Figure 5.49: Comparison of the temporal variation of the acoustic power ‘PV’ in one cycle at (a) entrance to the regenerator, (b) exit of the regenerator and (c) entrance to the

inertance tube

5.7.3.2. Quasi-steady behavior and inertance effect on phase angle relationships

The temporal results presented in the previous sub-section offer an insight into the transient behavior of the PTR which is important during the cool-down and warm-up phases of the cryocooler operation. However, to understand the performance of the cryocooler and how the inertance tube can affect its operation, the quasi-steady behavior in the regenerator and inertance tube must be inspected. In this sub-section, we present 226 quasi-steady state (near equilibrium) results and discuss the effect of the inertance on the system’s performance.

In Figure 5.50 below the variation of the quasi-steady state temperature in the cold heat-exchanger and the pressure amplitude in the transfer tube are plotted as a function of the inertance. This figure summarizes the performance of the PTR simulated with respect to the inertance. As can be seen, the lowest temperature (and best performance) is achieved at ~ 800 kPa-s2/m3, however the reason for this optimum performance is not intuitive from the plot below. In general, the highest pressure amplitude in the transfer tube corresponds with the lowest temperature achieved (and the fastest cool-down time as seen in the experimental results reported). However in Figure

5.50, at the optimum inertance the pressure amplitude in the transfer tube is not the highest. 227

Figure 5.50: Variation of the gas temperature in the cold heat-exchanger at quasi-steady

state and the pressure amplitude in the transfer tube as a function of the inertance

This observation can be explained by comparing the pressure amplitude at different locations in the regenerator and the pressure drop over the length of the regenerator. Figure 5.51a below shows the pressure amplitude at the entrance, center and exit of the regenerator plotted against the value of inertance. The pressure drop is highest for the lower inertance values and as the inertance approaches the optimum (~ 800 kPa- s2/m3), the pressure drop decreases. This high pressure drop at lower inertance values is due to the high values of velocity (as seen in Figure 5.51b). In the discussion of the experimental results, it was mentioned that due to the presence of porous media, the flow follows Darcy’s Law. The result is that higher pressure drop values are observed over the length of the porous media region (in this case the regenerator) when the flow rate or 228 velocity values are higher. In Figure 5.51b it can be seen that the pressure amplitude at the center of the regenerator is highest near the optimum value of inertance. Hence, the lowest temperature and the fastest cool-down time are observed at this value of inertance due to the combination of a lower velocity (and flow rate) and the highest pressure amplitude at the center of the regenerator.

Figure 5.51: Comparison of quasi-steady transport properties as a function of the

inertance (a) pressure amplitude at various locations in the regenerator and (b) pressure

amplitude and axial direction velocity amplitude at the center of the regenerator

The mass flow rate is calculated from the product of the density, velocity and cross-sectional area of the regenerator. The resultant phase angle between the instantaneous mass flow rate and the instantaneous pressure is plotted as a function of the inertance in Figure 5.52a for three different locations along the axis of symmetry of the regenerator. A positive phase angle indicates that the mass flow rate (and velocity) leads the pressure and conversely, a negative phase angle indicates that the pressure leads the mass flow rate. As can be seen in Figure 5.52a, at the entrance of the regenerator the flow rate lags the pressure for all the inertance values studied and at the exit of the 229 regenerator the flow rate leads the pressure for most of the inertance values (the exception being the largest inertance value). It is interesting to note that the maximum phase change occurs over the second half of the regenerator’s length. The velocity- pressure phase angles are plotted in Figure 5.52b as a function of the inertance at the center of the regenerator and the entrance to the inertance tube. At the optimum value of inertance, the velocity-pressure phase angle at the center of the regenerator is close to zero (which is the optimum) and the corresponding value at the entrance to the inertance tube is about 76° (velocity leading pressure). Though the magnitude of the phase angle values reported here are consistent with past published results [104], the opposite trend/direction is evident here. For example, the optimum phase angle values at the entrance, center and exit of the regenerator (as reported by Radebaugh et al. [13, 104]) are 30°, 0° and -30° and the value at the inertance tube entrance is ~ -60°. However, the optimum phase angle relationships in the current study are -13°, 2° and 27° in the regenerator and 76° at the entrance to the inertance tube. 230

Figure 5.52: Comparison of phase angles as a function of the inertance (a) mass flow

rate-pressure phase angle in the regenerator and (b) velocity-pressure phase angle at

center of regenerator and entrance to inertance tube

The cycle-average mass flow rate, enthalpy flow and acoustic power are plotted as a function of the inertance in Figure 5.53 for three different locations in the regenerator. The magnitude of the cycle-average mass-flow rate is about an order of magnitude lower than the instantaneous values (~ 10 – 15 g/s) and this is consistent with acoustic streaming phenomena reported in the past [59, 105]. The cycle-average values at the entrance to the regenerator are positive indicating streaming flow from the warm end to the cold end and the corresponding values at the exit of the regenerator indicate flow in the opposite direction. Even though there is streaming flow in the regenerator, the magnitudes of the flow are very small and at the center of the regenerator, there is almost no streaming flow (~ 10-3 – 10-4 g/s). Additionally, the streaming flow rates approach a minimum near the optimum inertance value. A similar trend is observed in the regenerator for the cycle-average enthalpy flow; however the positive enthalpy flow 231

(from warm to cold end) at the entrance to the regenerator is much higher that the enthalpy flow in the opposite direction at the cold end of the regenerator.

Figure 5.53: Comparison of (a) cycle-average mass flow rate, (b) cycle-average

enthalpy flow and (c) cycle-average acoustic power as a function of inertance at three

locations in the regenerator

In Figure 5.53c, the cycle-average acoustic power is plotted as a function of the inertance tube at the entrance, center and exit of the regenerator. The trends of the cycle- average acoustic power as a function of the inertance are similar to those of the cycle- 232 average mass flow rate and enthalpy flow. At the optimum inertance, the acoustic power drop over the length of the regenerator is close to the minimum and the cycle-average acoustic power at the center of the regenerator is the maximum.

5.7.4. Summary

The numerical study reported here investigates the effects of the inertance

(combinations of length and diameter of the inertance tube) on the performance of an

IPTR. In the CFD model, the continuity, momentum and energy equations are solved for both the refrigerant gas (helium) and the porous media regions (the regenerator and the three heat-exchangers). The effects of different diameter of the inertance tubes and different values of inertance on the performance of the system are investigated. The value of inertance is maintained constant by changing both the diameter and the length of the inertance tube and the inertance is varied by maintaining the diameter of the tube constant and changing the length. Maintaining the inertance constant does not significantly affect the performance of the system (2 – 5 K difference in the three cases studied, similar to the experiments). When the value of inertance is changed (keeping the diameter constant and varying the length of the inertance tube), larger variations in the performance are observed (~ 50 – 60 K). Increasing the inertance increased the pressure amplitude at the center of the regenerator and hence an improvement in the performance was observed. In addition to decreasing the pressure amplitude, the velocity amplitude at the center of the regenerator is decreased. At the optimum inertance, the phase angle 233 difference between the pressure and the velocity at the center of the regenerator is the minimum and the pressure amplitude is the maximum.

5.8. Conclusions from numerical study of in-line pulse tube refrigerator

In this chapter, the numerical model of the PTR that was developed is introduced, validated and then applied to investigate the effects of various operating parameters on the performance of the cryocooler. The numerical model solves the compressible form of the conservation of mass (continuity) equation, the conservation of momentum (Navier-

Stokes) equation and the conservation of energy for the gas. The heat-exchangers are assumed to be porous media and in these porous zones, the conservation equations account for the porous media parameters (porosity, permeability and quadratic drag factor) and the losses associated with porous media, i.e., the viscous dominated losses

(Darcy term) and the inertia dominated losses (Forchheimer term). In addition to solving for the fluid dynamics, the temperature of the solid porous media material is also calculated from the solid energy equation. The solid and gas temperatures interact via convective heat transfer relations. The numerical model is validated against temporal and quasi-steady experimental results.

When the numerical model was applied to investigate the effects of operating pressure amplitude on the performance of the PTR, the simulations predicted trends that were similar to those observed in the experimental studies. The validated model was also used to study the effects of operating frequency on the transport processes in the cryocooler. While an optimum operating frequency was predicted similar to the 234 experiments, the simulations predicted interesting cycle-averaged flow patterns in the pulse tube. These acoustic streaming patterns had structures that were dependent on the frequency of operation. The counter-rotating bi-cellular streaming structures observed at the higher frequencies helped prevent direct mixing of the gas in the cold and warm ends of the pulse tube. The counter-rotating streaming structures created a secondary buffer zone in the center of the pulse tube.

Using the model, suppression of the streaming structures was demonstrated when the pulse tube was tapered by a small angle (< 2o). The suppressed streaming velocities had magnitudes that were approximately two-four orders of magnitude smaller than the instantaneous oscillatory flow and more than an order of magnitude smaller than the corresponding magnitudes when a straight pulse tube was used. Finally, the numerical model is used to investigate the effects of the “inertance” on the transport within the system and the overall performance of the PTR. The “inertance” is dependent on the refrigerant fluid and the dimensions of the inertance tube (i.e., its diameter and length).

The inertance is shown to affect the pressure and velocity profiles in the regenerator section of the PTR. At the optimum inertance, the pressure amplitudes are the highest and the velocity/mass flow-rate amplitudes are the lowest. Reductions in the amplitudes of velocity lead to a decrease in the flow related losses for the regenerator and this helps improve the performance. In the lower limit, when the inertance is maintained constant

(the tube diameter and length are varied to enable a constant inertance), the performance does not change (2 – 4 K differences are observed). However as the length of the inertance tube increases (caused by an increase in the tube diameter), there are large deviations in the performance even at constant inertance values. These large deviations 235 can be explained by the fact that inertance tubes have an optimum length beyond which the performance of the PTR suffers. This length is dependent on the gas (speed of sound in the gas) and the frequency of operation.

236

Chapter 6 : Numerical Studies of Co-Axial type Pulse Tube Refrigerators7

6.1. Introduction

Space cryocoolers are compact cryocoolers that have fast cool-down times, high durability and low vibration. Due to these limitations, pulse tube cryocoolers have proven to be very good space cryocoolers. Recently, there has been a push to build PTRs that work efficiently at higher frequencies [54-57]. The use of high frequency oscillations allows the system to be comparatively small in size. These smaller sized systems have niche applications in the space industry where localized low power (< 1 W) cooling systems with extremely fast cool-down times are required. Hence, a high frequency co-axial cryocooler is the best choice for a pulse tube cryocooler with space applications. In this chapter, a high frequency co-axial type OPTR is studied for application in space cryocooling [106].

6.2. Problem geometry

Figure 6.1 depicts the geometry studied (i.e. a co-axial design OPTR system). The geometry was chosen for its compactness – for possible space applications. Only half the geometry shown in Figure 6.1 is simulated (axisymmetric geometry). The OPTR consists of a compression chamber (which includes the moving piston), the transfer tube,

7 The results presented in this section can be found in the paper “Farouk, B. and Antao, D. S., ‘Numerical analysis of an OPTR: Optimization for space application’, Cryogenics, v. 52, n. 4 – 6, pp. 196 – 204, 2012” 237 aftercooler (the first red hatched region), regenerator (blue cross-hatched region), pulse tube with two heat exchangers at its ends (the other two red hatched regions), an orifice

(a simple obstruction to the flow), an inertance tube and the compliance volume.

Figure 6.1: Schematic of the co-axial type OPTR geometry simulated

Table 6.1 summarizes the various dimensions of the problem geometry and the time-invariant boundary conditions in the simulations. The letters in the first column of

Table 6.1 correspond to the components in Figure 6.1. The mathematical boundary conditions at the various components’ surfaces are explained in detail in a following section. 238

Table 6.1: Dimensions and boundary conditions of the simulated system (Space

Cryocooler)

Boundary No. Component Radius (cm) Length (cm) condition along the outer wall

A Compression Chamber 2.00 0.85 Adiabatic

B Transfer Tube 0.31 10.1 hc = 20 W/m-K

C Aftercooler 0.80 2.00 Tw = 293 K D Regenerator 0.80 * Adiabatic E Cold Head - 0.50 Adiabatic

F Cold Heat-Exchanger 0.50 0.50 Adiabatic G Pulse Tube 0.50 6.00 Adiabatic

H Hot Heat-Exchanger 0.50 1.00 Tw = 293 K I Orifice Valve 0.0425 0.30 Adiabatic

J Inertance Tube 0.085 25.00 Adiabatic

K Compliance Volume 2.60 13.00 Adiabatic

* See Table 6.2 for values of regenerator length

6.3. Boundary, initial and porous media conditions

The details of the boundary, initial and porous media conditions are provided in this section. The numerical scheme used in this particular study too is discussed.

239

6.3.1. Boundary conditions

Table 6.1 (given earlier) specifies the heat transfer boundary conditions used at the surface boundaries of the various components. The piston is modeled as a reciprocating wall having an oscillatory velocity. The velocity of the piston is defined by the function, u = A0ωCos (ωt), where A0 is the maximum displacement of the piston

(4.511 mm) and ω is the angular frequency (ω = 2πf) and f is the frequency of operation.

For the current simulations (both cases), the frequency is set at 100 Hz. This value of frequency was chosen based on the need to study high frequency pulse tube cryocoolers for space applications. At this frequency, the fluid (Stokes) boundary layer and the thermal boundary layer are ~ 120 μm and ~ 140 μm respectively. These boundary layer thicknesses are used to calculate the acoustic Reynolds numbers in the system.

6.3.2. Initial conditions

At the start of the simulation, the temperature in the system is assumed to be 300

K everywhere. The initial pressure in the system is set to 3.1 MPa.

Table 6.2 specifies the variable parameters in the three cases studied. In case 1, the length of the regenerator is maintained at 6.0 cm and the cold head (component E in

Figure 6.2) is assumed to be a gas filled region. In case 2, the length of the regenerator is increased to 7.5 cm. In case 3, the regenerator length is maintained at 6.0 cm, however the cold head is assumed to be a porous media region. In this porous media region, the porosity is assumed to be 0.8, the permeability 1 × 10-7 m2 and the quadratic drag factor 240

0.15. The porous media material is assumed to be copper and non-equilibrium heat transfer is assumed between the gas and solid in this region too.

Table 6.2: List of the cases simulated (Length of Regenerator)

Case Length of Regenerator Cold Head No. (cm) Region

1. 6.00 Gas 2. 7.50 Gas 3. 6.00 Porous Media

6.3.3. Porous media conditions

In the porous media regions, the three heat exchangers (components C, E and G in

Figure 6.1) are copper with density 8950 kg/m3, constant specific heat 380 J/kg-K (since it does not exhibit large variation in the temperature range studied, i.e. 100 – 350 K) and temperature dependent thermal conductivity. The regenerator has properties of 316 stainless steel, with density 7810 kg/m3, constant specific heat 460 J/kg-K and temperature dependent thermal conductivity. For the present simulations, ε is considered to be equal to 0.69 [53, 101] and in the rest of the domain (gas phase) ε has a value of 1.0.

The permeability κ is considered to be 1.06 ×10−10 m2 [101] and the quadratic drag factor

CF has a value of 0.3917.

241

6.3.4. Numerical scheme

The time-step size was determined by allowing 80 time-steps/cycle, which is sufficient to accurately simulate the problem. Based on the aforementioned criterion, the time-step used was found to be 1.25×10-4 s for the chosen frequency of 100 Hz. An overall convergence criterion is set for all the variables at 10-4 in the iterative implicit numerical solver. A total of 6591 grid points were used for case 1 and case 3 in the simulations and a total of 7396 grid points were used for case 2 (to account for the larger regenerator). A typical grid size for the number of grid points used is ~ 100 – 300 μm.

6.4. Results and discussion

In the following sub-sections, results for spatio-temporal evolution of the flow and temperature fields in the co-axial type OPTR (Figure 6.1) are presented.

6.4.1. Temporal processes in the system

Figure 6.2 shows the temporal variation of the gas temperature at the exit of the cold heat-exchanger into the pulse tube (x = 1.0001 cm and r = 0.499 cm) for the first two cases (Table 6.2). The band-like trends visible in the plots are the cyclic structures of the temperature variation over the time period displayed (i.e. 0 – 22 sec). The simulations predict a better performance for case 1 (shorter regenerator section). This is indicated by the faster cool-down time and lower quasi steady-state temperature. 242

Figure 6.2: Comparison of the cool-down behavior of the OPTR at the entrance to the

pulse tube for cases 1 and 2 243

Figure 6.3: Pressure, gas temperature and solid temperature profiles at the start of the

simulation for case 1

To illustrate and better visualize the band-like structures, the gas temperature, solid temperature and pressure are plotted in Figure 6.3 in an expanded time scale at the start of the simulations (i.e. the first ten cycles) for case 1 (x = 10.001 mm and r = 4.99 mm). The cyclic nature of the temperature is clearly evident in Figure 7.46. The solid

(porous media) temperature follows a similar oscillatory trend as the gas temperature.

The pressure exhibits a fairly sinusoidal pattern; however the gas temperature profile is distorted. The solid (porous media) temperature also appears to be near-sinusoidal. Due to the high density and thermal mass of the solid region, the solid temperature does not respond as rapidly to changes in the gas temperature profile. This results in the solid temperature that lags the gas temperature by a fixed phase angle. As seen in the previous chapter, the phase angle difference between the gas and solid temperatures is dependent 244 on the operating frequency of the system. At higher frequencies, the phase angle difference is much higher. For case 1, the phase angle difference is 85.5o and for case 2 the phase angle difference is 90o.

Figure 6.4: Temporal evolution of the gas temperature at the exit of the regenerator in

case 1 and case 2 for (a) 0 - 22 sec and (b) early time (0 - 0.5 sec)

Figure 6.4a shows the temporal variation of the gas temperature at the exit of the regenerator (x = 0.499 cm, r = 1.200 cm) from 0 to 22 seconds of simulation time. The gas temperature at the exit of the regenerator follows a similar trend to the gas temperature evolution at the entrance to the pulse tube. However the gas temperature behavior at the start of the simulation is interesting. Figure 6.4b shows the temporal evolution of the gas temperature at the start of the simulations for the first two cases at the exit of the regenerator. The gas temperature appears to rise at first before decreasing.

This is an effect of the void volume/gas region in the cold head. During each cycle of the

OPTR, the gas in the cold head is compressed and expanded. Thus there is a large 245 variation in the gas temperature and this creates a temperature variation between the exit of the regenerator (cold end of the regenerator) and the cold end of the pulse tube. This large variation in gas temperature affects both the exit of the regenerator (Figure 6.4b) and the entrance to the pulse tube. We find that this effect can be reduced by adding a porous substance in this region. Figure 6.5 shows the pressure variation at the exit of the regenerator (x = 0.499 cm, r = 1.200 cm) for the first two cases studied. As expected, the longer regenerator in case 2 results in a larger pressure drop as compared to case 1 (when the piston amplitude is kept constant).

Figure 6.5: Comparison of the pressure variation at the exit of the regenerator for case 1

and case 2 246

Figure 6.6: Comparison of temporal evolution of the cycle-averaged gas temperature at

three locations in the OPTR (coordinates are in 'mm') for case 1

Figure 6.6 compares the cycle-averaged temperature at three locations in the

OPTR. The three locations are the entrance to the pulse tube (x = 1.001 cm, r = 0.499 cm), the entrance to the cold heat-exchanger (x = 0.499 cm, r = 0.499 cm) and the exit of the regenerator (x = 0.499 cm, r = 1.200 cm). It is evident from the figure that the major cooling occurs at the entrance to the pulse tube and the exit of the regenerator. The void volume between the regenerator and the cold heat-exchanger (i.e. in the cold head) allows the temperature in the cold head to oscillate. The amplitude of these oscillations is on the order of ~ 15 – 20 K (see Figure 6.7). Figure 6.7 compares oscillatory nature of the gas temperature in the three locations of the OPTR. The gas temperature is non-sinusoidal; however the temperature profiles at the entrance of the cold heat-exchanger (dashed red 247 line) and the exit of the regenerator (solid green line) are distorted. This distortion can be explained by the nature of the flow in the cold head region. Due to the flow reversal (~

180o) when going from the regenerator to the pulse tube and vice versa and the fact that the cold head is a void gas filled region, the flow is highly rotational and there is mixing of cold and warm gas. This leads to the distorted cyclic temperature profiles.

Figure 6.7: Cyclic variation of gas temperature at three locations in the cold end of the

OPTR (coordinates in 'mm') for case 1

6.4.2. Porous media cold head region

To reduce the void volume in the cold head region of the OPTR, the cold head is filled with porous filler that is highly porous and highly permeable. The porous media conditions are described in an earlier section of this chapter (see section 6.4.2. Initial 248 conditions). In addition to reducing the void volume, the porous media prevents excessive mixing of the flow in the cold head region of the OPTR. In this section we compare the transient performance of this system with a porous media cold head (case 3) with the transient performance of the OPTR system for case 1 and case 2.

Figure 6.8: Comparison of the cool-down performance for the three cases studied (a) up

to 2 seconds of simulated time and (b) predicted values

Figure 6.8a shows the cool-down behavior (up to 2 seconds of simulation time) for the three cases studied. The performance of the OPTR system in the Case 3 configuration shows a remarkable improvement over the first two cases. The gas temperature in case 3 is lower than the gas temperature in case 1 by about 25 – 30 K.

This clearly shows that the replacement of the void cold head region by a porous media region results in an improvement of the performance of the system. Figure 6.8b shows the predicted temperature evolution for the three cases simulated. The predictions

(shown by dashed curves) are based on extrapolation of the simulated data based on an exponential decay curve. 249

Figure 6.9: Comparison of gas temperature at three locations in the OPTR (coordinates

in 'mm') for case 3

In Figure 6.9 we compare the gas temperature profiles in three locations in the cold region of the OPTR system. These results are interesting as they show the effect that the porous media has on the gas oscillations. Firstly, the gas temperature oscillations are fairly small (~ 1 – 3 K) at the entrance to the cold heat-exchanger (dashed red line) and the exit of the regenerator (solid green line). The importance of the addition of the porous media is emphasized when comparing Figures 6.7 and 6.9. The high temperature oscillations in the cold head and the exit of the regenerator (Figure 6.7 – case 1) degrade the performance of the OPTR system. The second effect of the porous media is reduction in the rotational nature of the flow in the cold head region. This is evident in the gas temperature profiles at the entrance to the cold heat-exchanger (dashed red line) and the 250 exit of the regenerator (solid green line). In Figure 6.9, these lines have a fairly sinusoidal shape in addition to reduced amplitudes of oscillation. In comparison, in

Figure 6.7 the same temperature profiles are distorted and have high amplitudes of oscillation. These results (for case 3) are shown for a fairly short time (0 – 2 seconds of simulation time), but the nature and shape of the temperature profiles will not change as the simulations approach steady-periodic state.

6.4.3. Spatio-temporal temperature and velocity variations

In this sub-section, the spatial and temporal variations of the gas temperature and the axial velocity in the OPTR. Since the geometry is co-axial in nature, the results are shown along two different radii. In Figure 6.10a the gas temperature is shown for the transfer tube, aftercooler and the regenerator (at r = 0.0095 m). Figure 6.10b shows the gas temperature variation from the cold head to the middle of the transfer tube (r = 0.00 m, i.e. the axis of symmetry). Only part of the transfer tube is shown because the temperature variation in the entire transfer tube is small. The results are plotted at four instants in the 2201st cycle for case 1. The four plots in the figure show the temperature profiles at 0, (π/2), π and (3π/2) radians. The temperature in the transfer tube (x = 0 – 0.1 m in Figure 6.10a) oscillates with a large amplitude due to the compression and expansion of the gas in the compression chamber. The temperature profile along the aftercooler and the regenerator (x = 0.1 – 0.181 m in Figure 6.10a) and the two heat- exchangers in Figure 6.10b (x = 0.005 – 0.01 m and x = 0.07 – 0.08 m) show negligible oscillation, i.e. ~ 1 – 2 K. In these regions the gas interacts with the porous media (via 251 the non-equilibrium coupling of the energy equations) which has a high thermal mass and hence we observe a damped temperature oscillation. In the pulse tube, the temperature oscillations at the four instants (0, (π/2), π and (3π/2)) are similar and the temperature profile over the length of the pulse tube is linear. This trend in the pulse tube is encouraging as it suggests thermally stratified pulse tube and hence a lack of mixing in that region. As stated earlier, the gas temperature in the orifice, the inertance tube (x =

0.08 m onwards in Figure 6.10b) and the compliance volume oscillates negligibly and its value lies between 285 and 295 K.

Figure 6.10: Gas Temperature in the various components of the OPTR at four instants in the 2201st cycle for case 1 (a) transfer tube to the regenerator (r = 0.0095 m) and (b) cold

head to the inertance tube (r = 0.00 m) 252

Figure 6.11: Axial component of velocity ‘ux’ in the various components of the OPTR

at four instants in the 2201st cycle for case 1 (a) transfer tube to the regenerator (r =

0.0095 m) and (b) cold head to the compliance volume (r = 0.00 m)

Figure 6.11a shows the variation in the axial velocity vector in the transfer tube, aftercooler and the regenerator (at r = 0.0095 m) and Figure 6.11b shows the axial velocity variation from the cold head to the compliance volume (r = 0.00 m, i.e. the axis of symmetry). Similar to the plots for gas temperature in Figures 6.10a and 6.10b, the results are plotted at four instants in the 2201st cycle for case 1. The four plots in the figure show the axial component of the velocity profiles at 0, (π/2), π and (3π/2) radians.

The velocity oscillation in the middle of the cryocooler is negligible due to the presence of the porous media which restricts the flow. However, in the transfer tube (0 – 0.1 m), in the orifice (0.252 – 0.255 m) and in the inertance tube (Figure 6.11b), the flow velocities are high due to their fairly small cross sectional area. The high velocities in the orifice region and the transfer tube can lead to jetting phenomena and non-uniformities in the flow in the pulse tube [107]. 253

6.4.4. Steady-periodic multi-dimensional effects

In this section, the steady-periodic flow properties in the pulse tube section of the

PTR are addressed. In the earlier chapters, the effects of operating frequency [100] and taper angle of the pulse tube [102] on the steady secondary streaming flow patterns in the pulse tube section of an OPTR have been discussed. The generation of counter rotating bi-cellular streaming structures was proposed to aid in maintaining the buffer zone in the pulse tube region of the cryocooler [100]. Suppression of secondary streaming was demonstrated [102] by tapering the walls of the pulse tube. In this section, the results are shown for the 2201st cycle of operation for case 1. At this point in the simulation (i.e. ~

22 sec) the system is assumed to have reached a quasi-steady periodic state and the variation of the fluid temperature from cycle to cycle is small.

Figure 6.12: Cycle-averaged gas temperature and secondary streaming velocity vectors

in the pulse tube region for case 1

Figure 6.12 shows the cycle-averaged temperature field and cycle-averaged velocity vectors in the pulse tube for case 1. The instantaneous flow field in the pulse tube is oscillatory; however the cycle-averaged flow field that is superimposed on the 254 instantaneous flow shows remarkable organization. The maximum streaming velocity in the axial direction is around 0.129 m/s and the minimum is also around - 0.131 m/s (the negative sign indicates flow in the negative x-direction), however the bulk of the streaming velocity is around 0.001 and -0.001 m/s. The maximum and minimum velocities in the radial direction are 0.0138 m/s and -0.0161 m/s. In Figure 6.12, a velocity of 0.2 m/s corresponds to a vector of length 1.0 cm. It is clearly apparent from the figure that the maximum streaming velocities are at the ends of the pulse tube and along the boundaries of the boundary walls of the pulse tube. Figures 6.10b and 6.12 clearly demonstrate that the temperature gradient across the length of the pulse tube is fairly linear, thereby insulating the two ends. The radial temperature distribution in the pulse tube too is fairly uniform, except for a small region near the warm end (x = 0.058 –

0.064 m) where minor variations in the radial temperature distribution are visible. It can thus be concluded that for this case, the cycle-averaged secondary streaming velocity has no detrimental effect on the performance of the pulse tube region and the OPTR as a whole.

6.5. Chapter summary and conclusions

A computational fluid dynamic model was developed to predict the transport characteristics of a co-axial type OPTR for the first time. The co-axial OPTR simulated is the preferred configuration for space applications due to its compact nature and the high frequency of operation enables small component dimensions and fast cool-down times. A thermal non-equilibrium heat transfer model was used to simulate the 255 temperature fields in the heat exchanger matrices. The solid temperature as predicted by the non-equilibrium heat transfer model shows a lag between the gas and solid temperatures which can be attributed to the high thermal mass of the solid media. The simulations predict better cool-down times for the shorter regenerator due to the larger pressure drop observed in the longer regenerator. The void volume in the cold head region adversely affects the performance of the OPTR due to the large temperature oscillations in this region and the mixing of cold gas from the entrance of the pulse tube and the warm gas in the cold head. Another reason for the decreased performance is the low thermal conductivity of the gas (compared to a solid) which acts as a heat sink and further increases the cool-down time. A simple solution is the addition of a porous solid that causes low pressure drop (slots or mesh). In the simulations, this is modeled by a porous media region with high porosity and permeability and a low drag factor. At an early stage of the simulations (up to ~ 2 sec.), the addition of the porous media region in the cold head leads to a drastic improvement in the performance of the system and a faster cool-down time. The results also show the temperature and velocity distribution across the length of the OPTR. The effect of the high velocities and velocity fluctuations observed in the orifice and the inertance tube on the flow in the pulse tube region has not been investigated in this chapter but may provide important insights into the nature of flow in the pulse tube. Additional cycle-average gas temperature and velocity vectors in the pulse tube region indicate the presence of acoustic streaming in the pulse tube.

However, the magnitude of the streaming velocity in most of the pulse tube is approximately two to three orders of magnitude smaller than the magnitude of the 256 instantaneous velocity and hence streaming is predicted to have a small effect on the performance of the current OPTR system.

257

Chapter 7 : Summary and Conclusions

7.1. Overall summary

Recalling the dissertation statement introduced in chapter 1, the research reported in the subsequent chapters show that “thermoacoustic pulse tube refrigerators are complex systems that require multi-dimensional numerical models to accurately predict their performance and wave-shaped resonators can enhance the performance of these thermoacoustic refrigeration systems”. Based on the motivations of the research work, the objectives set were divided into two main areas: (a) to gain a better understanding of

PTR systems through a synergistic set of experimental and numerical studies which required the development of a multi-dimensional numerical model of the PTR and (b) to investigate the wave-shaping phenomenon for possible thermoacoustic applications by developing a multi-dimensional model that will aid in the design of wave-shaped resonator systems and improve the understanding of their operation.

In keeping with these objectives, this dissertation reports research that was completed in order to meet the set goals. Chapter 3 reports the development and validation of the numerical model of consonant and dissonant acoustic resonators as well as design criteria (resonant frequency, operating fluid and excitation amplitudes). Studies of the two Mk-I and Mk-II experimental cryocoolers are reported in chapter 4. The reported studies discuss the experiments conducted and the corresponding results from the characterization of the two cryocoolers under a variety of operating conditions

(operating pressure, operating frequency, inertance tube geometry, etc.). Chapters 5 and 258

6 discuss the numerical model development of the PTR system and report the numerical model validation studies performed. These two chapters also discuss the results from the studies where the numerical model of the PTR was used to investigate the performance of the PTR and to better understand how its performance can be optimized (suppression of acoustic streaming in the pulse tube, controlling the pressure-mass flow rate phase angle using the inertance, etc.). The specific conclusions of each of the above studies are presented in the next section of this chapter and the impact of the research is discussed.

Based on that discussion, ideas for continuing and future work are presented in the section following the conclusions.

7.2. Specific conclusions and impact of findings

In this section, the specific conclusions and impact of the research reported in the previous chapter are discussed in detail. The conclusions in this chapter are divided into three sub-sections as follows: (a) numerical studies of acoustic resonators, (b) experimental characterization of cryogenic orifice and inertance type PTRs and (c) numerical studies of cryogenic orifice and inertance PTRs.

7.2.1. Numerical studies of acoustic resonators

As discussed in the earlier chapters, the nonlinear effects observed in conical acoustic resonators make designing such systems challenging. Past numerical and experimental studies do not provide guidelines on how to design such systems and find 259 their optimum operating conditions. In this study, nonlinear, high amplitude standing waves in cylindrical and non-cylindrical (cone) circular resonators were investigated numerically using a high fidelity compressible axisymmetric computational fluid dynamic model.

In the conical dissonant resonator the model predicts the nonlinear behavior of the system. Using beat theory [1], it is possible to accurately predict the resonant frequency of conical resonators operating at low piston/driver amplitudes. At higher amplitudes, the resonant frequency shifts to higher frequencies and this ‘resonance hardening’ behavior has been observed in past numerical and experimental studies of cone resonators. In addition to the resonance hardening, the structure of the wave is observed to change when the piston amplitude is increased. At low piston amplitudes, the structure of the pressure waves is close to being sinusoidal. However at higher piston amplitudes the shape and structure of the wave change to the inverted ‘rectified sinusoid’ with the cusps of the waveform on the top. When the effects of different gases are investigated, gases with higher ratios of specific heat (in the current study, argon) show more nonlinear behavior which may be quantified by the coefficient of nonlinearity as suggested by past research studies on the subject. The use of argon (or any noble gas since they have high ratios of specific heat) which was studied, may not be suitable for many applications due to the heat generated in the system. The overall heating of the system with argon as the operating fluid also raises the mean operating pressure in the system. Based on the results from this study it is possible to suggest that nitrogen may be a more suitable option to either carbon dioxide or argon. The value of the ratio of specific heats for nitrogen is 1.4 (between the values for CO2 and Ar) and hence it will offer better peak-to- 260

peak pressures and pressure ratios than CO2, but will not generate as much heat as Ar. It was also observed that the quasi-steady streaming observed in the conical resonators is dependent on the piston amplitude. The acoustic streaming observed in conical resonators is different to that observed in the cylindrical resonators. The two main differences noticed are that (a) the acoustic streaming patterns/structures observed have different profiles (organized streaming cells in cylindrical resonators v/s the smaller disorganized streaming cells in conical resonators) and (b) high streaming velocities are observed in conical dissonant resonators compared to cylindrical resonators operating at similar driver amplitudes. The presence of strong streaming structures in conical dissonant resonators imply that better mixing is possible in conical resonators and introduces the possibility of using conical resonators as efficient mixers.

The specific impact of this study on the thermoacoustics community can be summarized in the following points:

 This study reports the first multi-dimensional numerical model of dissonant

acoustic resonators where effects of the fluid temperature on the operating

performance of the resonators are considered.

 A method for calculating the fundamental resonant frequency of the dissonant

conical resonator based on beat theory is presented. Given that the main

disadvantage of dissonant resonators was the lack of a method to “design” them,

this is a major impact.

 The study also reports for the first time the presence of acoustic streaming

structures in dissonant acoustic resonators and powerful nature of these streaming

structures (high streaming velocities). These structures show that acoustic 261

streaming in conical resonators can be harnessed for highly efficient convective

mixing.

7.2.2. Experimental characterization of cryogenic orifice and inertance type PTRs

Past research studies do not offer insight into how operating conditions (operating frequency, mean operating pressure, amplitude of pressure wave, etc.) govern the performance of the PTRs. Hence, to better understand the operation of the PTR and how various operating parameters affect its performance, an experimental PTR was designed, fabricated and tested. The experimental investigation of the cryogenic PTRs provides a thorough experimental performance characterization of both the orifice type and inertance type PTR systems.

The first system built, the Mk-I orifice PTR was capable of a minimum cold heat- exchanger temperature of 114.5K. The use of a vacuum chamber was vital to the system reaching performance values near its potential, however better vacuum levels would have resulted in further improvements to the PTRs performance. The operation of this Mk-I cryocooler was characterized under a variety of operating conditions like different operating frequencies, mean operating pressure values, pressure amplitudes and the opening of the orifice valve (given that it was an orifice type PTR). The optimum operating frequency was experimentally observed at 65 Hz and as the operating pressure in system was increased, the performance of the system is improved. This effect of an improvement of the performance at higher values of mean operating pressure is due to the increased density and the increased thermal conductivity of the gas. Additionally, the 262 increasing the mean operating pressure had the effect of increasing the pressure amplitude when the input power to the motor was maintained constant. A similar improvement in the performance was observed when the pressure amplitude was increased. The orifice valve was also shown to have effects on cooling attained; however the effect is not large in a PTR system with an inertance tube since the inertance tube controls the phasing.

Though a detailed characterization of the Mk-I PTR performance was carried out, the experimental system was deemed to be far from the optimum possible. The lessons learned from designing, fabricating and testing the Mk-I system were invaluable to both the second system designed and built and the numerical studies of the PTR. One of the major problems with the Mk-I cryocooler was that the various components had large wall thickness which is detrimental to the system. This was emphasized when the numerical model was applied to predict the performance of the experimental cryocooler. Another major drawback of the Mk-I system was the vacuum levels achieved in the vacuum chamber were poor (~ 20 – 30 Torr). The Mk-II inertance type PTR was designed and fabricated. Specific improvements include reducing the wall thickness of both the regenerator and pulse tube components, switching the stainless steel pulse tube for a titanium tube (lower thermal conductivity than stainless steel), decreasing the length of the cold heat-exchanger by redesigning it and improving the vacuum chamber capable of

2 – 4 Torr vacuum levels.

The Mk-II cryocooler was capable of attaining cold heat-exchanger temperatures in the liquid nitrogen range (~ 77 K). The Mk-II PTR was used to investigate the effects of the inertance (combinations of length and diameter of the inertance tube) on the 263 performance of the IPTR. The cryocooler was operated at a mean pressure of 2.24 MPa and at a frequency of 63 Hz (experimentally determined optimum). The inertance was varied from ~ 150 kPa-s2/m3 to 900 kPa-s2/m3 (by varying the length and diameter of the inertance tubes) and the optimum value of inertance was about 710 kPa-s2/m3. This optimum value of inertance was independent of the diameter of the inertance tube used.

Below the optimum inertance value, the performance was fairly consistent with differences in the quasi-steady state temperatures being about 2 – 5 K for similar inertance values (different tube diameter). However above the optimum inertance for similar inertance values (different tube diameter), the difference between the gas temperature values at quasi-steady state was about 10 – 15 K. This larger discrepancy at higher inertance (and length) values in the results of the two different diameter inertance tubes used is due to the fact that the length of inertance tubes for low power systems (like the Mk-II cryocooler) cannot be usefully increased beyond a limit. This limit is estimated at approximately λ/2π, where λ = a/f is the wavelength, a is the speed of sound in the operating fluid (helium) and f is the operating frequency.

The specific impact of this study on the thermoacoustic cryocooler community can be summarized in the following points:

 This study reports the first thorough characterization of the PTR system. The

characterization includes the effects of operating frequency, mean operating

pressure, operating pressure amplitude and the orifice flow coefficient on the

performance of the PTR.

 The importance of vacuum insulation to achieve cryogenic temperatures was

highlighted in this study. 264

 The study also reports for the first time effects of the inertance on the PTR

performance by considering the inertance as a single parameter that affects the

PTR’s performance. The value of this parameter inertance is derived from the

dimensions of the inertance tube and hence different combination of inertance

tube diameter and length values can be used to achieve the same performance.

7.2.3. Numerical studies of cryogenic orifice and inertance PTRs

The lack of accurate and experimentally validated multi-dimensional numerical models of PTRs motivated this part of the research. An axisymmetric time-dependent computational fluid dynamics model was developed to predict the transport phenomena in PTRs. The flow and temperature fields in the gas-phase and the porous media zones were modeled by the Navier-Stokes and the Darcy-Forchheimer equations respectively.

A thermal non-equilibrium model was used to predict the differences in the gas and solid temperatures in the porous media zones. The solid temperature (in the porous zones) has much lower amplitudes of oscillation due to its large thermal mass and lags the gas temperature oscillations by a phase angle, dependent on the operating frequency. To accurately predict the performance of the OPTR system it is important to account for the thermal mass of the system and the losses that occur due to the high thermal mass of the various components. The numerical model was used to simulate the flow and temperature fields in an OPTR at various values of operating pressure amplitude (similar to the experiments). These results were compared to the results of the experimental 265 study, the trends were similar and the differences in the results were within the limits of experimental error.

When the numerical model was used to study the effects of operating frequency on the PTR’s performance, the temperature profile along the length of the pulse tube was observed to be non-linear. At higher frequencies, the pulse tube was divided into three zones/regions that were not perfectly stratified. Cycle-averaged temperature and velocity fields in the pulse tube section of the OPTR system indicated the presence of steady counter-rotating acoustic streaming recirculation patterns. These steady streaming rolls lead to non-uniform radial temperature profiles in the pulse tube. The bi-cellular streaming structures observed at higher frequencies enhance the performance of the PTR by creating a dead zone between the cold and warm ends of the pulse tube. However, at the lower frequencies the bi-cellular acoustic streaming rolls degenerate to a single re- circulating cell – thus leading to reduced performance in the PTR system. This acoustic streaming was suppressed by tapering the pulse tube section by small taper angles (< 2o).

However, only tapering the pulse tube does not lead to suppression of the acoustic streaming. In order to suppress streaming, the heat-exchanger at the tapered end must have a diameter equal to the pulse tube diameter at that end. By maintain the diameter of the heat-exchanger equal to the pulse tube diameter; the magnitude of the streaming velocity is decreased by one or two orders of magnitude. What is more important is that by suppressing the streaming in the pulse tube, the performance of the PTR improved.

The multi-dimensional CFD model was used to investigate the effects of different diameter inertance tubes and different values of the parameter “inertance” on the performance of the PTR. The value of inertance is maintained constant by changing both 266 the diameter and the length of the inertance tube and the inertance is varied by maintaining the diameter of the tube constant and changing the length. Similar to the experiments with the Mk-II cryocooler, maintaining the inertance constant does not significantly affect the performance of the system. However, by varying the inertance

(constant diameter and varying length), larger variations in the performance are observed

(~ 50 – 60 K). With increasing inertance, the pressure amplitude at the center of the regenerator increased and the velocity amplitude at that location decreased. This decrease in velocity amplitude (and flow rate) leads to a decrease in the flow related losses in the regenerator component (Darcy’s Law). Additionally, at the optimum inertance, the phase angle difference between the pressure and the velocity at the center of the regenerator is the minimum. This decrease in the phase angle at the optimum inertance leads to larger acoustic power values at the center of the regenerator and lower acoustic power losses compared to other inertance values. At the optimum inertance, the flow leads the pressure at the cold end of the regenerator and warm end of the pulse tube.

This is different from previous studies where the flow was shown to lag the pressure at these two locations by comparable values.

Finally the CFD model of the PTR was applied to simulate the transport processes and the performance of a co-axial type PTR. The co-axial orifice type PTR simulated is the preferred configuration for space applications due to its compact nature and the high frequency of operation simulated enables small component dimensions and fast cool- down times. The void volume in the cold head region was shown to adversely affect the performance of the PTR. This is due to the large temperature oscillations in this region and the mixing of cold gas from the entrance of the pulse tube and the warm gas in the 267 cold head. A simple solution to damp these temperature oscillations is to add some porous material in the cold head region while maintaining low pressure drop conditions across the porous material. In the simulations, this improvement is modeled by a porous media region with high porosity and permeability and a low drag factor. The addition of the porous media region in the cold head leads to a drastic improvement in the performance of the system and a faster cool-down time. Since this is an oscillatory flow system, acoustic streaming (which is unavoidable) was observed in the pulse tube region.

However, the magnitude of the streaming velocity in most of the pulse tube was approximately two to three orders of magnitude smaller than the magnitude of the instantaneous velocity and hence streaming was predicted to have a small effect on the performance of the current OPTR system.

The specific impacts of this aspect of the dissertation (i.e., numerical simulations of the PTR) on the thermoacoustic cryocooler community can be summarized in the following points:

 This study reports the first experimentally validated (temporal and quasi-steady

state) computational fluid dynamic model of the PTR.

 Acoustic streaming in the pulse tube was shown using a numerical model for

the first time. This was enabled by the multi-dimensional nature of the model.

Acoustic streaming was shown to be dependent on the frequency and suppression

of this streaming by tapering the pulse tube was shown for the first time.

 The multi-dimensional numerical model of the PTR was used to study a co-

axial type PTR for the first time. This is important because most PTR used in 268

space applications (liquefaction of gases and HTS applications) have co-axial type

geometries.

 The numerical model was used to show how the inertance tube’s geometry

affects the performance of the cryocooler by affecting the phase relationships in

the regenerator section.

7.3. Future work and recommendations for continuing research

Significant progress was made in the area of thermoacoustics and thermoacoustic cryocooling with the research reported in this dissertation. These significant contributions include the development of experimentally validated numerical models of the pulse tube refrigerator and consonant and dissonant acoustic resonators. That being said, there is always is room for improvement. Numerical models and experimental systems can be improved, the parameters being investigated can be changed to better understand the problem and the methods of solution can be improved to make the solution process more efficient. To further advance the knowledge and understanding of thermoacoustic cryocoolers the following research tasks are recommended. Research on some of the recommended tasks suggested below is already underway and preliminary results have been obtained; however these results have not been included in this dissertation. The recommended tasks are organized into two categories.

269

7.3.1 Recommendations for future work: Acoustic resonator studies

The numerical model of the consonant and dissonant acoustic resonators has provided important insight into the thermal-fluid interactions inside the resonators.

However, the model can be used to further investigate the transport phenomena in wave- shaped dissonant acoustic resonators.

One of the unknowns is the effect that different gases will have on the performance of the resonator. Carbon dioxide and argon were studied in chapter 3; however gases like nitrogen or oxygen may have more desirable properties depending on the application. In addition to studying pure fluids (gas), the effects of gas mixtures may prove to be better, where the desirable properties of each component gas are harnessed to maximize the output.

Another interesting and important study would be to investigate the effect of the cone angle on the harmonic response and pressure output of the resonator. Intuitively, it can be assumed that larger cone angles will generate higher pressure amplitude waves in the small end; however how this variation in the cone angle affects the harmonic response of the resonator is unknown. In addition to varying the cone angle, other researchers have proposed different wave-shaped resonators like the horn cone, the bulb and the exponentially expanding resonator. The developed numerical model can be modified to study these various shapes and identify the most efficient of the designs.

Visualization of the flow structures inside a consonant cylindrical resonator have been performed by numerous researchers in the past. The same cannot be said about dissonant resonators (cone or other shapes). Flow visualization studies in dissonant 270 acoustic resonators will improve the understanding of acoustic streaming driven convective transport inside the resonator. This can further validate the proposed application of using dissonant acoustic resonators as efficient mixers. The challenge of such a study lies in the nature of wave-shaping. Non-linear output in the pressure waves has been observed only at high pressures. Given that flow inside acoustic resonators is sensitive to the wall roughness, the visualization of flow structures inside acoustic resonators with curved surfaces and operating with high pressure gases is indeed challenging. In addition to flow visualization, velocity measurements inside dissonant resonators have never been attempted or reported. The best method of attempting velocity measurements would be with constant temperature anemometers given the visualization challenges posed in these high pressure systems (with regards to PIV methods).

7.3.2 Recommendations for future work: PTR studies

The recommended future work for PTR systems can be divided into two areas: experimental studies and numerical/computational studies. Each of the areas is discussed separately below.

The experimental investigation techniques reported in cryocooler literature are fairly advanced. Gaining insight from the experimental and numerical studies reported in this dissertation, three areas of need for experimental studies arise. The first is improved instrumentation of in the regenerator, pulse tube and inertance tube entrance and exit sections are required. The numerical and experimental studies of the inertance effect on 271 the performance of the PTR showed that the phase angle between the velocity (or flow- rate) and the pressure controls the performance of the cryocooler. This phase angle is in turn regulated by the “inertance”. This phase angle difference can be calculated based on simultaneous measurement of the axial velocity and pressure at different locations along the length of the regenerator and pulse tube sections and the entrance and exit of the inertance tube. These measurements in high pressure systems like the PTR operating with difficult to seal gases like Helium is a challenge. The task is made more difficult by the fact that the regenerator and pulse tube components are sensitive to the any excess thermal mass which then affects the performance. The pulse tube is also sensitive to disturbances in its flow with the possibility of undesirable mixing occurring within it.

The second proposed future work task is visualization of the flow fields in the pulse tube. The numerical studies of the PTR showed acoustic streaming structures in the

PTR component and those structures were shown to affect the performance of the cryocooler. There are only two reported studies in literature that have attempted such a task, however the results weren’t very successful given that the temperature differences between the cold and warm ends of the pulse tube were about 50 K. Such a flow visualization study of the pulse tube section operating under cryogenic conditions will be an important addition to the cryocooler literature. Finally, the proposed wave-shaping of acoustic resonators to enhance cryogenic refrigeration in PTRs has never been attempted.

Components that can be wave-shaped include the regenerator and the transfer tube.

Wave-shaping the regenerator will need precision machining of the screen mesh inside it; however, a wave-shaped regenerator can possibly improve the performance of the cryocooler with larger pressure amplitudes generated with reduced flow rates. Wave- 272 shaping can be used to create a driver mechanism that drives the cryocooler, replacing the linear motor or other pressure wave generator.

The numerical model of the PTR has been applied to investigate the effects of a variety of operating parameters on the transport phenomena in the PTR. As with any numerical model, the method of solution can be improved. A proposed improvement to the numerical model is to incorporate automated calculation of temperature dependent thermal conductivity for the solid material that comprises the porous media zone. The current model (as it is) can be used to study how wave-shaping of the components affects the performance of the PTR (similar to the proposed experimental work). The current model can be applied to investigate novel PTR configurations like a split regenerator (i.e., regenerators operating in parallel). The idea derives from an electrical circuit. When two resistors are connected in parallel, the voltage across them is the same; however, the current is distributed between the two. If the resistors have the same resistance, the current is evenly distributed. Importing these electrical analogies to a fluid/acoustic system, the pressure is equivalent to the voltage and the mass flow-rate or velocity to the current. By connecting regenerators in parallel, the mass flow is split between the two, but the pressure drop across both is the same. This leads to a decrease in the flow associated losses in the regenerator (the flow losses are proportional to the velocity/mass flow-rate). This reduction in the flow losses of the regenerator will lead to an improvement in the performance of the PTR.

The model can be possibly extended to include the modeling of the actual linear motor (electro-magnetic motion driven flow) which is connected to an external circuit that would calculate the transient current and voltage supplied to the linear motor. 273

During the experiments it was noticed that the current and voltage drawn by the linear motor varied as the cryocooler cooled down. This can be explained by the change in the impedance of the gas as it heats up and cools down. This change in impedance causes a change in the current drawn by the linear motor (self-correcting/impedance matching within the linear motor). Developing a coupled external circuit, linear motor and PTR model will help predict these variations in the power drawn by the linear motor and the added advantage will be that the COP and efficiency calculations of the PTR can be obtained directly from the model (electric power to acoustic power conversion and subsequent acoustic power to refrigeration power conversion). In addition to modeling the linear motor, another avenue of research is to develop a model of a thermoacoustic engine. The thermoacoustic engine can be used in lieu of the linear motor and can act as a pressure wave generator itself. Developing a model of the thermoacoustic engine has two advantages. First, the thermoacoustic engine has no moving parts and combining it with a PTR results in an engine-refrigeration system that has potentially no maintenance costs because of the lack of any moving parts that can wear down over time. The second advantage is that the travelling wave thermoacoustic engine has components similar to the PTR (i.e., three ambient heat-exchangers, a regenerator and a pulse tube) and the rest of the components are acoustic resonators (like the compliance volume and inertance tube).

274

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284

Appendix A

In this appendix, a sample of the DeltaEC codes used to design the two PTRs is provided. For the actual design of the system, the DeltaEC code for the cryocooler section was coupled to a section that models the linear motor pressure wave generator8 and so the outputs obtained are appropriate only for the linear motor used in experiments reported in this dissertation. For the sake of brevity, only the cryocooler parts of the

DeltaEC code are presented below.

A.1. DeltaEC code used to design the Mk-I PTR

The code below contains information for the main Mk-I PTR components. The code is in the form that appears in the DeltaEC program interface.

---0--- BEGIN

1.6895E+ a Mean P Pa 06 63.00 b Freq Hz

300.25 c Tbeg K G

1.6895E+ d p' Pa 05 -116.00 e Ph(p) deg

-1.2160E- m3/ f U' G 03 s -105.83 g Ph(U) deg G

8 The linear motor section of the code was provided by Phil Spoor of Chart Industries 285

gas helium type ---1--- Transfer DUCT Tube 2.2700E- 1.6895E+ a Area m2 A p' Pa 04 05 5.3409E- Mst b Perim m -116 B Ph(p) deg 02 r 1a 1.0000E- 1.2150E- m3/ c Length m C U' 03 03 s Ph(U -106.08 D deg ) 101.11 E Htot W

stainle Solid 101.10 F Edot W ss type ---2--- Aftercoole SX r HX 1.6634E+ sameas 1a a Area m2 A p' Pa 05 0.7361 b VolPor -116.18 B Ph(p) deg

3.0000E- 1.1502E- m3/ c Length m C U' 02 03 s 2.8340E- Ph(U d rh m -114.64 D deg 04 ) -87.684 e HeatIn W G 13.426 E Htot W

= 300.00 f SolidT K 95.629 F Edot W 2H 300.25 G GasT K

Solid Solid copper 300.00 H K type T ---3--- Regenerat STKSCREEN or 1.2505E+ sameas 1a a Area m2 A p' Pa 05 0.69 b VolPor -114.27 B Ph(p) deg

6.0000E- 5.6959E- m3/ c Length m C U' 02 04 s 4.3180E- Ph(U d rh m -134.06 D deg 05 ) 286

0.200 e ksFrac 13.426 E Htot W

33.509 F Edot W

300.25 G TBeg K

stainle Solid 139.97 H TEnd K ss type ---4--- SX Cold HX

6.9330E- 1.0424E+ a Area m2 A p' Pa 05 05 0.7361 b VolPor -109.99 B Ph(p) deg

5.0000E- 5.8211E- m3/ c Length m C U' 02 04 s 2.8340E- Ph(U d rh m -140.86 D deg 04 ) 7.494 e HeatIn W G 20.921 E Htot W

= 140.00 f SolidT K 26.041 F Edot W 25H 139.97 G GasT K

Solid Solid copper 140.00 H K type T ---5--- Pulse STKDUCT Tube 1.0216E+ sameas 4a a Area m2 A p' Pa 05 2.9525E- Mst b Perim m -111.32 B Ph(p) deg 02 r 5a 7.2747E- m3/ 0.23 c Length m C U' 04 s 3.0100E- Ph(U d WallA m2 -158.41 D deg 05 ) 20.921 E Htot W

25.304 F Edot W

139.95 G TBeg K

stainle Solid 300.15 H TEnd K ss type ---6--- SX Warm HX

287

9.5187E+ sameas 4a a Area m2 A p' Pa 04 0.7361 b VolPor -106.9 B Ph(p) deg

3.0000E- 7.3898E- m3/ c Length m C U' 02 04 s 2.8340E- Ph(U d rh m -160.28 D deg 04 ) -20.921 e HeatIn W G 0.000 E Htot W

= 300.00 f SolidT K 20.978 F Edot W 6H 300.13 G GasT K

Solid Solid copper 300.00 H K type T ---7--- IMPEDANCE Orifice

Pa- 2.0000E+ 9.5099E+ a Re(Zs) s/m A p' Pa 05 04 3 Pa- 0.0000 b Im(Zs) s/m -106.83 B Ph(p) deg

3 7.3898E- m3/ C U' 04 s Ph(U -160.28 D deg ) 0.000 E Htot W

20.923 F Edot W

---8--- Inertance DUCT Tube 1.1700E- 1.8518E+ a Area m2 A p' Pa 05 04 1.2125E- Mst b Perim m 102.35 B Ph(p) deg 02 r 8a 8.2263E- m3/ 1.4500 c Length m C U' 04 s 5.0000E- Ph(U d Srough -167.94 D deg 04 ) 0.000 E Htot W

3.8919E- F Edot W 02 288

---9--- Inertance COMPLIANCE Tube 2.0837E- Mst 1.8518E+ a SurfAr m2 A p' Pa 02 r 9b 04 3.1600E- b Volume m3 102.35 B Ph(p) deg 04 2.1384E- m3/ C U' 18 s Ph(U -59.53 D deg ) 0.000 E Htot W

-1.8818E- F Edot W 14

A similar code as the one above is used to design the Mk-II PTR system reported in chapter 4.

289

Vita

Dion Savio Antao was born in Bombay, India on the 31st of October in the year

1985. He did his schooling (up to high school) in Goa, India and received his Bachelor of Technology degree with honours from the National Institute of Technology in

Jamshedpur, India. Dion joined the Mechanical Engineering and Mechanics (MEM) department at Drexel University in the Fall quarter of 2007 and worked on his thesis research in the A. J. Drexel Plasma Institute under the guidance of Prof. Bakhtier Farouk and Prof. Alexander Fridman. He received his Master of Science degree in Mechanical

Engineering in the Summer quarter of 2009 with a concentration in thermal-fluid sciences. He then started his Ph.D. research (Fall quarter of 2009) on the application of the thermoacoustic phenomena to cryogenic refrigeration under the guidance of Prof.

Bakhtier Farouk. He has received the George Hill Jr. Endowed Fellowship, the Best

Student Paper and Presentation award at the American Society of Mechanical Engineers

(ASME) International Mechanical Engineering Congress and Exposition (IMECE) in

2011, the Freshman Design Education Fellowship and the Best Student Poster

Presentation at the Drexel University Annual Research Day in 2010. Dion has also received travel fellowships from the American Physical Society (APS) and the Gordon

Research Conference (GRC). He has worked as both a Research Assistant and a

Teaching Assistant during his time at Drexel University. His research interests are in experimental and computational investigation of non-thermal plasma discharges, thermoacoustics, cryogenics, fluid mechanics and heat transfer, fuel cells and combustion. 290

Journal Publications

1. Antao, D. and Farouk, B., “Wave-shaping of pulse tube cryocooler components for improved performance”, 2013, Cryogenics (in preparation)

2. Farouk, T., Antao, D. and Farouk, B., “Atmospheric pressure direct current H2/CH4 micro-glow discharge simulations: Effects of external circuit”, 2013, Journal of Physics D: Applied Physics (under review)

3. Antao, D. and Farouk, B., “Experimental and numerical characterization of the inertance effect on pulse tube refrigerator performance”, 2013, Journal of Applied Physics (under review)

4. Antao, D. and Farouk, B., “High amplitude non-linear acoustic wave driven flow fields in circular and conical resonators”, 2013, Journal of the Acoustical Society of America (in press)

5. Antao, D. and Farouk, B., “Experimental and Numerical Investigations of an Orifice type Cryogenic Pulse Tube Refrigerator”, 2013, Applied Thermal Engineering, v. 50

6. Antao, D. and Farouk, B., “Numerical Analysis of an OPTR: Optimization for Space Applications”, 2012, Cryogenics, v. 52

7. Antao, D. and Farouk, B., “Numerical Simulations of Transport Processes in a Pulse Tube Cryocooler: Effects of Taper Angle”, 2011, International Journal of Heat and Mass Transfer, v. 54

8. Antao, D. and Farouk, B., “Computational Fluid Dynamics Simulations of an Orifice type Pulse Tube Refrigerator: Effects of Operating Frequency”, 2011, Cryogenics, v. 15

9. Antao, D., Staack, D., Fridman, A. and Farouk, B., “Atmospheric pressure DC corona discharges: operating regimes and potential applications”, 2009, Plasma Sources Science and Technology, v. 18, n. 3