The Use of Sage Simulation Software in the Design and Testing
THE USE OF SAGE SIMULATION SOFTWARE IN THE DESIGN AND TESTING
OF SUNPOWER’S PULSE TUBE CRYOCOOLER
A thesis presented to
the faculty of
the Fritz J. and Dolores H. Russ College of Engineering and Technology of
Ohio University
In partial fulfillment
of the requirements for the degree
Master of Science
Kyle B. Wilson
June 2005
This thesis entitled
THE USE OF SAGE SIMULATION SOFTWARE IN THE DESIGN AND TESTING
OF SUNPOWER’S PULSE TUBE CRYOCOOLER
BY
KYLE B. WILSON
has been approved for
the School of Mechanical Engineering
and the Russ College of Engineering and Technology by
Khairul Alam
Moss Professor of Mechanical Engineering
Dennis Irwin
Dean, Fritz J. and Dolores H. Russ College of Engineering and Technology
WILSON, KYLE B. MS. June 2005. Mechanical Engineering
The Use of Sage Simulation Software in the Design and Testing of Sunpower’s Pulse
Tube Cryocooler (130 pp.)
Director of Thesis: Khairul Alam
ABSTRACT
This thesis discusses the development of a pulse tube cryocooler and the use of a
software simulation tool called Sage in this effort. A pulse tube cryocooler is a type of refrigerator that operates in the cryogenic temperature range below 120 K (-153° C). The goal of this thesis spans from a basic understanding of the ideal thermodynamics of the pulse tube cryocooler (PTC) to a grasp of one commercial software package’s ability to model an actual, non-ideal PTC. The history and theory are discussed leading to the state-of-the-art technology. The focus then turns to the specific development efforts of the author at his place of employment, Sunpower, Inc. The specific elements that led to the design of the Sunpower PTC are examined and the use of Sage simulation software is demonstrated. Finally, comparisons between experimental data and Sage predictions show the success of Sage as a simulation tool.
Approved: Khairul Alam
Moss Professor of Mechanical Engineering
ACKNOWLEDGEMENTS
I would first like to thank my wife Wendy and my sons Cody and Brendan for their support in my pursuit of this goal. Without their patience and understanding throughout the last five years I never would have been able to complete this work.
Thanks to Sunpower, Inc., both the company and the employees, for giving me the opportunity to have a career with a company that makes me look forward to work every day (well, maybe not every day). Neill Lane and Gary Wood were the driving forces behind me starting on my MSME and the continual nudging and support of my co- workers has made the long road bearable.
This work would have been impossible without the previous accomplishments of two leaders in the cryocooler industry, David Gedeon and Ray Radebaugh. I have been extremely fortunate to work directly with David Gedeon for the last several years and attempt to absorb just a small amount of his inherent feeling for the underlying physics behind the topic. Ray Radebaugh has shared so much of his experience and knowledge through tireless publications and presentations that allow someone like me to readily find thorough and clear information on the history and theory of pulse tube cryocoolers.
Finally I owe thanks to Dr. Khairul Alam and my thesis committee, Dr. Izzi
Urieli, Dr. John Deno and Gary Wood for not giving up on me in getting this completed and helping to guide me through the process.
5 TABLE OF CONTENTS
ABSTRACT...... 3
ACKNOWLEDGEMENTS ...... 4
LIST OF FIGURES ...... 7
LIST OF TABLES ...... 10
NOMENCLATURE...... 11
Chapter 1. Introduction...... 14 1.1 Overview...... 14 1.2 Background...... 15
Chapter 2. Pulse Tube History ...... 19 2.1 Recent Advances...... 21 2.1.1 Double Inlet Pulse Tube...... 21 2.1.2 Acoustic Streaming...... 23 2.1.3 Inertance...... 23
Chapter 3. Theory...... 24 3.1 Carnot Performance ...... 24 3.2 Stirling Cycle ...... 27 3.3 Gifford-McMahon (GM) Cycle ...... 30 3.4 Basic Pulse Tube...... 31 3.5 Orifice Pulse Tube ...... 34
Chapter 4. Orifice Pulse Tube Cryocooler (OPTC) Analysis...... 36 4.1 Enthalpy and Entropy Flow Model...... 36 4.2 Ideal Efficiency...... 38 4.3 Pulse Tube Losses...... 39 4.4 Phase Shift ...... 40
Chapter 5. Inertance Tube...... 42 5.1 Electrical Analogy ...... 42
Chapter 6. Project Motivation and Initial Design Methodology ...... 45 6.1 Sunpower Linear Compressor...... 47 6.2 Gedeon Associates Sage Software...... 48 6.3 Final Sunpower Pulse Tube Cryocooler Configurations ...... 50
Chapter 7. Modeling the Pulse Tube Cryocooler using Sage...... 52 7.1 Linear Compressor...... 53 7.1.1 Piston/Cylinder and Compression space ...... 53
6 TABLE OF CONTENTS (continued)
7.2 Cold Head ...... 58 7.2.1 Main Rejector, Acceptor, Secondary Rejector (Heat Exchanges) ...... 59 7.2.2 Main Rejector, Acceptor, Secondary Rejector...... 60 7.2.3 Parasitic Warm Source and Cold Sink ...... 61 7.2.4 Regnerator ...... 62 7.2.5 Pulse Tube ...... 63 7.3 Inertance Assembly...... 64 7.4 Pulse Tube Cryocooler Assembly...... 65
Chapter 8. Design and Fabrication of Hardware ...... 67 8.1 Linear Compressor...... 67 8.2 Cold Head ...... 73 8.3 Inertance Assembly...... 77 8.4 Pulse Tube Cryocooler Assembly...... 80
Chapter 9. Sensitivity Analysis ...... 81 9.1 Compressor Swept Volume ...... 81 9.2 Connecting Duct Length...... 83 9.3 Regenerator Porosity...... 84 9.4 Pulse Tube Volume...... 85 9.5 Inertance Tube Length ...... 87 9.6 Reservoir Volume ...... 92
Chapter 10. Subassembly Characterization...... 94 10.1 Linear Compressor Subassembly...... 94 10.2 Regenerator...... 96 10.2.1 Calculation of Regenerator Pressure Drop ...... 99 10.3 Inertance Assembly...... 103
Chapter 11. Cryocooler Testing and Sage Comparison...... 108
Chapter 12. Summary, Discussion and Conclusions ...... 115 12.1 Summary...... 115 12.2 Discussion and Conclusions ...... 116
APPENDIX A. Sage Overview...... 120
APPENDIX B. Sage Boundary Connections...... 123
APPENDIX A. Sage Model Tree ...... 125
REFERENCES...... 127
7 LIST OF FIGURES
Figure 2-1. Schematic of experimental setup used in the basic pulse tube discovery...... 20 Figure 2-2. Orifice pulse tube refrigerator...... 21 Figure 2-3. Double inlet (secondary orifice) pulse tube refrigerator...... 22 Figure 3-1. Energy flow in a refrigeration system...... 24 Figure 3-2. Temperature-entropy diagram for Carnot thermodynamic cycle...... 26 Figure 3-3. Carnot COP as a function of cold temperature, assuming 300K ambient .....27 Figure 3-4. Temperature-entropy (T-S) diagram for ideal Stirling cycle ...... 28 Figure 3-5. Processes of the ideal Stirling cycle in a Sunpower cryocooler...... 29 Figure 3-6. Pressure-volume diagram for the ideal Stirling refrigeration cycle...... 30 Figure 3-7. Diagram of basic pulse tube operation...... 32 Figure 4-1. First Law energy balance for OPTC ...... 37 Figure 4-2. Comparison of ideal Stirling and ideal OPTC efficiencies...... 39 Figure 4-3. Vector diagrams of pressure and mass flow phase relationships for a) OPTC, b) desired phase relationship...... 41 Figure 6-1. General layout of Sunpower linear compressor technology ...... 49 Figure 6-2. Inline PTC configuration...... 50 Figure 6-3. U-tube configuration ...... 51 Figure 7-1. Parent (root)-level model components of the piston/cylinder of a compressor...... 54 Figure 7-2. Child-level components of constrained piston and cylinder composite model ...... 54 Figure 7-3. Child-level components of constrained piston model...... 55 Figure 7-4. Child-level components of generic cylinder model ...... 56 Figure 7.5. Child-level components of cylinder-space gas model...... 57 Figure 7-6. Child level components of connecting tube model...... 58 Figure 7-7. Root level components of cold head model ...... 59 Figure 7-8. Child level component of heat exchanger models ...... 60 Figure 7-9. Child level component of woven screen matrix models ...... 61 Figure 7-10. Child level components of regenerator model ...... 62
8 LIST OF FIGURES (continued)
Figure 7-11. Child level components of random fiber regenerator matrix model...... 63 Figure 7-12. Child level component of compliance tube model...... 64 Figure 7-13. Root level components of inertance assembly model...... 65 Figure 7-14. Inline PTC schematic labeled with corresponding numbers from Sage models ...... 66 Figure 8-1. Piston/magnet/magnet ring/flex rod assembly...... 67 Figure 8-2. Piston and magnet/magnet ring components...... 68 Figure 8-3. Inner lamination/cylinder and outer lamination/wound coil assemblies...... 70 Figure 8-4. Attachment of stationary components to transition ...... 71 Figure 8-5. Connection of moving and stationary components...... 71 Figure 8-6. Test linear compressor assembly ...... 72 Figure 8-7. Main rejector subassembly...... 74 Figure 8-8. Regenerator subassembly...... 74 Figure 8-9. Acceptor heat exchanger...... 76 Figure 8-10. Pulse tube component ...... 77 Figure 8-11. Front and back side of secondary rejector...... 78 Figure 8-12. Assembled cold head...... 78 Figure 8-13. Inertance tube assembly ...... 79 Figure 8-14. Inertance assembly reservoir...... 80 Figure 8-15. Inline pulse tube cryocooler assembly ...... 80 Figure 9-1. Performance as a function of compressor swept volume...... 82 Figure 9-2. Performance as a function of connecting duct length ...... 83 Figure 9-3. Performance as function of regenerator porosity...... 84 Figure 9-4. Pressure drop through regenerator as a function of porosity ...... 85 Figure 9-5. Performance as a function of pulse (compliance) tube length ...... 86 Figure 9-6. Performance as a function of pulse (compliance) tube length, accounting for conduction loss ...... 87 Figure 9-7. Performance as a function of the first inertance tube (IT1) length ...... 88 Figure 9-8. Performance as a function of the second inertance tube (IT2) length...... 89
9 LIST OF FIGURES (continued)
Figure 9-9. Performance as function of pressure-mass flow phase relationship for IT1 ...... 89 Figure 9-10. Performance as function of pressure-mass flow phase relationship for IT2 ...... 90 Figure 9-11. Performance as a function of reservoir volume ...... 92 Figure 9-12. Performance as a function of pressure amplitude ratio...... 93 Figure 10-1. Sage model of linear compressor characterization...... 95 Figure 10-2. Sage model for steady flow pressure drop testing ...... 97 Figure 10-3. Comparison between Sage and experiment for steady flow regenerator pressure drop testing...... 98 Figure 10-4. Comparison between Sage and experiment for steady flow regenerator pressure drop testing after setting Sage friction multiplier to 0.8...... 99 Figure 10-5. Comparison of pressure drop calculations and Sage prediction ...... 102 Figure10-6. Comparison between experiment and Sage prediction for pressure drop through inertance tubes...... 104 Figure 10-7. Sage model of inertance assembly characterization test...... 104 Figure 10-8. Comparison of pressure phasors in compression space between experiment Sage and with Fmult = 0.67 ...... 105 Figure 10-9. Comparison of pressure phasors in compression space between experiment Sage and with Fmult = 1 ...... 107 Figure 11-1. Experimental setup...... 109 Figure 11-2. Experimental setup (with multi-layer insulation) ...... 109 Figure 11-3. Experimental setup (with vacuum vessel)...... 110 Figure 11-4. Tracking between Sage and experiment for cooling power vs. temperature curve with constant input power ...... 113 Figure 11-5. Tracking between Sage and experiment for cooling power vs. input power curve with constant cold end temperature...... 114
10 LIST OF TABLES
Table 5-1. Summary of analogy between electrical and fluid flow circuits ...... 44 Table 9-1. Summary of points from Figures 9-9 and 9-10 ...... 91 Table 10-1. Important Sage inputs for compressor test simulation ...... 96 Table 10-2. Comparison of Sage and experimental results for compressor test...... 96 Table 10-3. Adjusting Sage friction multiplier to achieve experimental pressure drop...... 99 Table 10-4. Comparison of pressure response in compression space and reservoir volume between experiment and Sage with Fmult = 0.67 ...... 105 Table 10-5.Comparison of pressure response in compression space and reservoir volume between experiment and Sage with Fmult = 1 ...... 107 Table 11-1. Comparison of baseline performance point of cryocooler ...... 111 Table 11-2. Comparison of baseline performance with friction multiplier reset to 1 ...... 112
11 NOMENCLATURE Acronyms COP Coefficient Of Performance DIPT Double Inlet Pulse Tube GM Gifford-McMahon IT Inertance Tube OPTC Orifice Pulse Tube Cryocooler PTC Pulse Tube Cryocooler PV Pressure-Volume
Symbols Note: These symbols and their definitions do not apply to Chapter 5, which has its own particular symbols and definitions defined within that chapter.
Time average value A Area [m2] 2 Af Flow area [m ] 2 Afr Frontal area [m ] 2 Aw Wetted area [m ] cp Constant pressure specific heat [J/kg-K] dh Hydraulic diameter [m] dw Wire diameter [m] f Frequency [Hz] or friction factor (specific to Chapter 10) H Total enthalpy [J] k Thermal conductivity [W/m-K] Kp Darcy permeability L Length [m] m Mass [kg] P Presure [Pa] Q Total heat transfer [J] R Gas constant per unit mass [J/kg-K] Red Reynolds number based on representative diameter [dimensionless] s Specific entropy [J/kg-K] S Total entropy [J/K] t Time [s] T Temperature [°C] or [K] u Velocity [m/s] v Specific volume [m3/kg] V Total volume [m3] W Total work [J] x Axial distance [m] X Displacement amplitude [m]
12 NOMENCLATURE (continued)
Greek Symbols β Porosity [%] in reference to regenerators ∆ Change in quantity δk Thermal penetration depth [m] φ Phase angle [deg] or [rad] µ Viscosity [Pa-s] θ Phase angle [deg] or [rad] ρ Density [kg/m3] ω Cyclic frequency [rad/s]
Subscripts 0 Mean or average value 1 Amplitude of oscillation a Ambient (warm) temperature c Cold temperature co Compression d Dynamic e Electrical ex Expansion h Hot temperature (actually the same temperature as ambient (a) but a different designation is used to distinguish between the locations of the heat transfer) max Maximum value min Minimum value p Piston r Regenerator wr Warm end of the regenerator wpt Warm end of pulse tube
Superscripts dot Quantity per unit time
13 NOMENCLATURE (continued)
Sage Boundary Connections (see Appendix B)
Fphsr Force connection Pphsr Pressure connection Qstdy Steady heat flow connection QGx Spatial grid heat flow connection QGxt Space-time heat flow connection m& Gt Gas flow connection ρstdy Density connection
14 Chapter 1. Introduction This thesis discusses the development of a pulse tube cryocooler and the use of a software simulation tool called Sage in the development effort. Chapter 1 begins with a broad overview of the topic and then provides a more detailed background with a preface to the layout of the document. 1.1 Overview A pulse tube cryocooler is a heat pump, or refrigerator, that removes heat from one space at a lower temperature and rejects that heat to another space at a higher temperature. Energy in the form of work must be supplied to the system to continue the process of transferring heat from the cold space to the warm space where the combined energy (both heat and work) is rejected. In the case of a cryocooler the cold temperature reaches less than 120 K (-153° C). The more work that is supplied to the system, the more heat that can be transferred from the cold space to the warm space, thus further reducing the cold space temperature. There is a limit to the coefficient of performance (COP), defined as the amount of heat removed from the cold space divided by the work input, as a function of the cold temperature. As the cold temperature decreases the COP limit also decreases. Thus it is a much more difficult task to achieve temperatures approaching absolute zero. The physical process which creates the cooling is the expansion of helium gas inside the pulse tube cryocooler that oscillates back and forth due to an oscillating piston driven by a linear alternator. The linear alternator and piston are located in a portion of the cryocooler called the linear compressor. Another name for this component is the pressure wave generator which is a more useful term because it describes the fact that the piston creates pressure waves, and consequently mass flow waves. The section of the cryocooler where the cooling occurs is called the cold head and is composed of several heat exchangers and a hollow tube called the pulse tube which is where this type of cooler derives its name. One heat exchanger is located at the cooling section and transfers the heat from the surroundings into the gas within the cooler. The other heat exchanger then transfers that heat along with the work required to drive the cycle (electrical input) to an ambient temperature heat sink. Between these two heat
15 exchangers is another heat exchanger called the regenerator. The purpose of the regenerator is to store and release the energy of the gas as it flows for the warm heat exchanger to the cold heat exchanger and vice versa. There is another section of the pulse tube cryocooler called the inertance assembly. Its function is to create a necessary relationship between the pressure amplitude and the mass flow rate of the gas within the cryocooler to achieve the cooling effect. The inertance assembly is simply made of copper tubes and a large volume. Thus the entire pulse tube cryocooler system as just described has only one moving part, the piston. Some other popular cryocooling technologies use two moving parts which gives the pulse tube cryocooler a potential advantage in life and reliability. There are various applications from military to the medical field which use cryocoolers. Pulse tube cryocoolers are gaining popularity is an efficient and reliable technology to meet the needs of these applications.
1.2 Background Cryogenics is defined as “a branch of physics that deals with the study of the production and effects of very low temperature.”1 By common acceptance this temperature range is established to be below 120 K (-153° C, -243° F) 2. Many common gases have boiling points in this cryogenic range. For example, at atmospheric pressure the boiling point of methane is 111.7 K, air is 78.8 K, hydrogen is 20.4 K, and helium is 4.2 K. There are a wide range of applications for cooling at cryogenic temperatures including3, 4: • Liquefaction of gases such as those mentioned above for industrial uses and also including oxygen for medical treatment. • Cooling infrared detectors for thermal imaging in military and security applications as well as Earth observation. • Superconducting electronic filters for the telecommunications industry. • Superconducting magnets for Magnetic Resonance Imaging (MRI) systems in the medical field. • Cryopumping for high vacuum levels in the semiconductor industry.
16 Depending on the amount of cooling needed, cryogenics can be achieved by extremely large plants that occupy acres of real estate and consume megawatts of power, or by a small device that can be held in your hand and consume only tens of watts of power. Machines that produce cooling, or refrigeration, in the cryogenic temperature range are commonly referred to as cryocoolers. There are several thermodynamic cycles that are the basis of cryocooler technologies [2]. These cycles operate on the thermodynamic principle that when a real gas expands both its pressure and temperature decrease. Conversely, when a real gas is compressed both its pressure and temperature increase. These cycles can broadly be divided into two categories depending if the operating fluid flows in a continuous loop, called a recuperative cycle, or if the fluid oscillates back and forth within the components of the machine, called a regenerative cycle [2]. The terms recuperative and regenerative more specifically refer to the heat exchangers in the cryocooler. The Rankine cycle which is used in household refrigerators is an example of a non-cryogenic recuperative cycle5. The Joule-Thompson cycle is similar to the Rankine cycle with differences mainly in the working fluid. Other examples of cryogenic recuperative cycles include the Brayton, and Claude cycles. Since the focus of this study is not a type of recuperative cycle, these will not be discussed in detail. The regenerative cycles include Stirling, Gifford-McMahon, and the focus of this study, pulse tube cryocoolers. The Stirling cycle will be covered in detail as a precursor to the pulse tube thermodynamic cycle. The Gifford-McMahon (GM) cycle is similar to the Stirling cycle except that it uses a conventional reciprocating compressor to produce a steady-flow fluid circuit with a high and low pressure side. A switching valve is then used to alternatively connect the cold head of the cryocooler with the high and low pressure sides to create oscillating flow. The GM cycle has disadvantages in large size and poor efficiency and also requires routine maintenance. The pulse tube cryocooler has a thermodynamic cycle very similar to the Stirling cycle except that the function of the mechanical component called the displacer in the Stirling machine has been replaced by a gas-filled tube called a pulse tube.
17 The goal of this effort spans from a basic understanding of the ideal thermodynamics of the pulse tube cryocooler (PTC) to a grasp of one commercial software package’s ability to model an actual, non-ideal PTC. To gain an understanding of the basic thermodynamic model of a pulse tube cryocooler it is beneficial to build from the Carnot cycle, progress to the Stirling cycle and then to the evolution of several PTC configurations. The history will be discussed leading to the state-of-the-art of PTC technology. Next, the focus will turn to the specific development efforts of the author at his place of employment, Sunpower, Inc. The specific elements that led to the design of the Sunpower PTC will be examined. The use of the PTC simulation software Sage, a product of Gedeon Associates, will be demonstrated. A comparison of the performance of the actual hardware versus the simulated performance in Sage will culminate this effort. Chapters 2 through 5 cover the history and theory of the PTC and rely heavily on the publications of Ray Radebaugh [2, 6, 10, 20], Robert Ackerman [3] and J.G. Weisand II [4], among others. The author makes every effort to clearly identify those previous contributions. Chapter 6 discusses the previous experience of Sunpower, Incorporated and Gedeon Associates that led to the effort to develop the subject cryocooler. Chapters 7 and 8 give some detail about the Sage simulation model of the pulse tube cryocooler and its physical construction. Chapter 9 provides a study of the sensitivity of the cooling performance of the cryocooler to a number of input parameters. This section gives an understanding or a “feel” for the behavior of the cryocooler subject to changes of these inputs. Comparisons between Sage simulation predictions and experimental results for components, subassemblies, and finally for the entire cryocooler are presented in Chapters 10 and 11. Finally the effort is summarized with some general conclusions and discussion in Chapter 12. The author has been involved in developing pulse tube cryocoolers for over five years through government funded Small Business Innovative Research (SBIR) programs at Sunpower, Incorporated. Sunpower is a privately owned R&D company located in Athens, Ohio which specializes in the continuing research and product development of free-piston, linear machines. These types of machines include Stirling engines, Stirling
18 cryocoolers, linear compressors, and pulse tube cryocoolers. The development work specific to this thesis was sponsored by NASA Goddard Space Flight Center in Greenbelt, Maryland. The work took place between November of 1999 and November 2001. As a result of this contract with NASA, information contained herein is proprietary for 4 years from the end of the contract in accordance with FAR 52.227- 20.
19 Chapter 2. Pulse Tube History6 In 1963 Professor William Gifford and graduate student Ralph Longsworth were conducting experiments on a Gifford-McMahon cycle cryocooler at Syracuse University7. As the name implies, Gifford was a pioneer in the development of the GM cryocooler as an acceptable, reliable solution to the growing need for mechanical cryogenic refrigeration in the development of thermal detectors and superconducting devices. During testing it was noticed that an approximately 25 mm diameter pipe was hot at the closed end and cooler than room temperature at the end open to the pressure oscillation, which had a frequency around 1 Hz. Figure 2-1 shows a schematic of this experimental setup. Gifford later explained the phenomenon based upon thermoacoustics and a heat pumping effect related to the size of the pipe, the working fluid, and the frequency of oscillation. This initial discovery was dubbed the basic pulse tube and work continued to further investigate a cooling device based upon the new discovery. A minimum cold temperature of 150 K was achieved with this first basic pulse tube cooling device. More theory of the basic pulse tube cryocooler will be presented later. In the early 1980’s at Los Alamos National Laboratories, Wheatley et. al.8 conducted experiments with the basic pulse tube at frequencies from 500 to 1000 Hz. Due to the high frequency of operation it was discovered that closely spaced plates were required within the tube to produce cooling. This was the first instance of a thermoacoustically driven refrigerator and work on another branch of devices was launched. These initial refrigeration experiments achieved a low temperature of 195 K and it was discovered that high frequency thermoacoustically driven refrigerators were more suited for cooling near ambient temperatures. At the Moscow Bauman Technical Institute in 1984 Mikulin et. al.9 placed an orifice inside of the pulse tube near the warm end heat rejector and also added a volume reservoir after the orifice. The purpose for inserting the orifice was to create a certain phase relationship between the pressure oscillation and mass flow rate in order to achieve cooling through pressure-volume (PV) work as in a Stirling cryocooler. This effort resulted in a low temperature of 105 K. Soon after in 1985 Dr. Ray Radebaugh at the
20
Ta Heat Exchanger Compressor Qa
Rotation Pmax Pmin
Flow-reversing valve Pulse Tube Regenerator
Qc
Heat Exchanger
Tc
Figure 2-1. Schematic of experimental setup used in the basic pulse tube discovery [3, p. 208].
National Institute of Standards (NIST) placed the orifice outside of the pulse tube rather than inside the pulse tube as shown in Figure 2-210. This enabled the pulse tube to operate in a more desirable, adiabatic fashion by reducing the turbulent heat transfer created by the entrance effects of an orifice entering a larger diameter tube. In addition, the heat exchanger functioned as a flow straightener for the flow coming from the orifice into the pulse tube. The orifice of this setup was actually a needle valve which allowed for the adjustment of the flow impedance to optimize performance. A new low temperature of 60 K was reached. Fundamental research continued on pulse tube cryocoolers, including the use of valveless compressors rather than GM type traditional reciprocating refrigerator compressors. Models were created for harmonic, time-averaged enthalpy flow within the assumed-adiabatic pulse tube11. These models led to improved performance resulting in a low temperature of 40 K but the efficiency was still inferior to Stirling cryocoolers. One particular design detail that was observed was that the increased void volume of the
21
W
ω Reservoir Volume
Piston Orifice
Qa,Ta Qh,Th Pulse Tube Regenerator
Qc,Tc
Figure 2-2. Orifice pulse tube refrigerator [2, Chapter 7].
pulse tube required large displacement valveless compressors to create sufficient pressure amplitude. Efficiencies comparable with Stirling cryocoolers were finally achieved around 1990.
2.1 Recent Advances This section discusses the most recent advancements made to pulse tube technology that have increased their performance and allowed pulse tube cryocoolers to become competitive with Stirling cryocoolers.
2.1.1 Double Inlet Pulse Tube The large swept volume required for a PTC created a large mass flow rate in the regenerator. This large mass flow rate increased the pressure drop through the regenerator and decreased the amount of pressure swing available in the expansion space. In 1990 Zhu et. al.12 developed the Double Inlet Pulse Tube (DIPT) cryocooler, also called the secondary orifice pulse tube cryocooler and shown in Figure 2-3, to address
22
W Reservoir ω Volume
Piston Orifice
Secondary Q ,T Q ,T a a Orifice h h Pulse Tube Pulse Regenerator
Qc,Tc
Figure 2-3. Double inlet (secondary orifice) pulse tube refrigerator [2, Chapter 7].
this problem. The idea of the DIPT was to divert a small amount of working gas directly from the compression space to the warm end of the pulse tube, bypassing the regenerator and thus increasing the pressure swing available in the expansion space. This modification did lead to improved efficiencies approaching Stirling cryocoolers, especially at higher frequencies. However, it was difficult to repeat performance with the DIPT and sometimes the cold end temperature would oscillate by several degrees. The reason for this phenomenon was suggested to be DC flow, later explained in terms of acoustic power flow by Gedeon13. The theory was that the DC flow was created in the loop around the regenerator, pulse tube and bypass tube as a result of an asymmetric flow impedance. Even a small amount of DC flow carries enthalpy from the warm end to the cold end, reducing performance. Gedeon’s theory was proven by Radebaugh in experiment by changing the direction of the needle valve used as the secondary orifice. Radebaugh suggested that using a tapered tube or using a fluid jet pump could reduce this DC streaming loss.
23 2.1.2 Acoustic Streaming14 There is another intrinsic effect present in a large tube with a constant area cross- section and high frequency oscillating flow. This effect is similar to the DC flow streaming discussed above but in this case the relationship between the pressure and flow oscillations in the viscous boundary layer results in flow from the cold end to the warm end along the pulse tube wall. Due to conservation of mass, flow is required in the opposite direction in the center of the pulse tube. This creates an enthalpy flow from the warm end to the cold end of the pulse tube and degrades the pulse tube performance. This effect can be reduced either by creating the correct phase relationship between the pressure and flow or by using a tapered pulse tube. This is a design feature that has been incorporated into the Sunpower PTC. However, the complexity of the detailed thermoacoustics causing this effect is beyond the scope of this work.
2.1.3 Inertance Around 1996, Godshalk15, who had been working on high-frequency thermoacoustic refrigerators, recognized that an effect called inertance held the possibility of improving efficiency. Others continued to develop and test the theory of inertance which enhances the pressure and mass flow phase relationship in the expansion space, which was the original motivation for introducing the orifice. The theory of inertance deals with the fact that a long thin tube can offer flow impedance not only in the form of resistance (friction) but also with inertance (inertia). Roach and Kashani16 demonstrated an analogy between flow impedance and electrical resistance network, relating flow friction to electrical resistance and fluid inertia to electrical inductance. Later, Marquardt and Radebaugh17 used two different diameter inertance tubes in series to further improve the inertance effect. This area of work is very important to this study because the Sunpower PTC implemented a stepped-diameter inertance tube for phase control. The theory of inertance will be covered in more detail later.
24 Chapter 3. Theory This chapter presents the basic thermodynamic and operational theory of several cycles beginning with the Carnot cycle and working through a progression of cycles that help explain the inertance pulse tube cryocooler.
3.1 Carnot Performance The basic purpose of a cryocooler is to produce refrigeration at cryogenic temperatures (below 120 K). One statement of the second law of thermodynamics is that in order for energy to be transferred from a cold temperature to a warmer temperature work must be put into the system [5, p. 253]. The performance of cryocoolers can be compared to each other based on their efficiency, or Coefficient of Performance (COP). This is a measure of the amount of work needed to produce an amount of refrigeration. There is a limit to the COP that a cryocooler can achieve based on the temperature of the refrigerated space and the temperature of the warm reservoir to which that heat is rejected. This theoretical limit is called the Carnot efficiency which is developed here based on Figure 3-1.
Ta
Qa
W Refrigerator
Qc
Tc
Figure 3-1. Energy flow in a refrigeration system [5, p. 250].
25 In Figure 3-1 an amount of work, W, is input to the system in order to move, or
“lift”, an amount of heat Qc from a cold temperature reservoir at Tc to some warmer temperature reservoir at Ta. Applying the first law energy balance to this system yields
W + Qc = Qa (Eq. 3-1) where Qa is the amount of heat rejected to the warm (ambient) reservoir. The Coefficient of Performance for an ideal, reversible process is Q Q 1 COP = c = c = . (Eq. 3-2) Q W Qa − Qc a −1 Qc For an internally reversible, isothermal heat transfer process the heat transfer ratio can be replaced by the ratio of the absolute temperatures at which the heat transfer occurs [5, p. 268] Q T a = a . (Eq. 3-3) Qc Tc Replacing the heat transfer ratio in equation 3-2 with the temperature ratio according to equation 3-3 and rearranging gives the equation for the limit of performance of refrigeration devices called the Carnot COP
Tc COPCarnot = . (Eq. 3-4a) Ta −Tc The Carnot efficiency can also be demonstrated graphically by first considering the equation relating entropy and heat transfer [4, p. 288] Q = T ⋅ ∆S . (Eq. 3-5) The Carnot cycle is a theoretical thermodynamic cycle containing a sequence of 4 reversible processes as shown on the temperature (T) – entropy (S) diagram in Figure 3- 2. The direction of the processes determines whether the device is operating as an engine or a refrigerator. The direction shown demonstrates the cycle of a refrigerator, whereas reversing the direction of the numbers would define an engine. Process 1 to 2: Reversible and isothermal compression, with heat rejection at the ambient temperature. Process 2 to 3: Reversible and adiabatic (isentropic) expansion of working fluid.
26
T 2 Qa = Ta∆S 1 T a W = area within the cycle
3 4 Tc
Qc = Tc∆S
S
Figure 3-2. Temperature–entropy diagram for Carnot thermodynamic cycle [4, p. 289].
Process 3 to 4: Reversible and isothermal expansion, providing heat lift at the cold temperature. Process 4 to 1: Isentropic compression of the working fluid.
Once again we consider Equation 3-1 above relating the work input to the heat transfer at the cold temperature and the ambient reject temperature. Looking at Figure 3- 2 this is shown as area 1-2-3-4 on the T-S diagram. Using the definition of entropy in equation 3-5 along with the T-S diagram in Figure 3-2 and Equation 3-1 provides an alternative way to arrive at equation 3-4a
Qc Tc ⋅ ∆S Tc COPCarnot = = = . (Eq. 3-4b) Qa − Qc Ta ⋅ ∆S −Tc ⋅ ∆S Ta −Tc
Notice from figure 3-2 that as the difference between Ta and Tc gets larger, the heat lift area under the line 3-4 as a fraction of the total work area within 1-2-3-4 gets smaller. This means that refrigeration at progressively lower temperatures becomes intrinsically less efficient. Figure 3-3 shows the Carnot COP for refrigeration from 120 K to 5 K assuming an ambient temperature of 300 K.
27
0.7
Ta = 300 K 0.6
0.5
0.4
0.3 Carnot COP Carnot
0.2
0.1
0 0 102030405060708090100110120130 Cold Temperature (K)
Figure 3-3. Carnot COP as a function of cold temperature.
3.2 Stirling Cyle The most limiting real-world obstacle to the Carnot cycle is the pressure ratio created by isentropic compression and expansion at cryogenic temperatures [4, p. 289]. The ideal Stirling cycle deviates from the theoretical Carnot cycle by utilizing two reversible isothermal and two reversible isochoric (constant volume) processes. The isochoric processes replace the isentropic processes of the Carnot cycle. This modifies the T-S diagram as shown in Figure 3-4 which shows that the heat lift (the area under line 3-4) is the same between the Stirling cycle and the Carnot cycle shown in Figure 3-2. Through geometry it observed that the total work input to the cycle (area of 1/1’’-2/2’-3- 4) is also the same between the two cycles. Thus the ideal Stirling cycle has the potential to achieve ideal Carnot efficiencies. The regeneration process in Figure 3-4 will be discussed soon.
28
2 2' 1 1' T a 0 0
= = 0 = ν s = 0 d d ν
s d d Regeneration
3 4 Tc
S
Figure 3-4. Temperature-entropy (T-S) diagram for ideal Stirling cycle [4, p. 290].
We now incorporate the theoretical Stirling thermodynamic cycle with the basic moving components of a Stirling cryocooler. Figure 3-5 shows this process with the piston and displacer of a Stirling cryocooler along with the expansion (cold, blue) and compression (ambient, red) volumes. The function of the piston is to support a pressure differential between the workspace (including expansion and compression volumes) which varies in pressure amplitude above and below the mean pressure, and the constant pressure backspace (not shown in the diagram). The purpose of the displacer is to shuttle the gas back and forth between the warm compression space and the cold expansion space while supporting a temperature gradient between the two spaces. Additionally the displacer creates a necessary phase relationship between the pressure amplitude and the mass flow rate which will be discussed in more detail in section 4.4. In the Sunpower Stirling cryocooler the displacer also houses the regenerator. The function of the regenerator is to store the thermal energy of the gas as it flows from the warm compression space to the cold expansion space and then release the thermal energy back to the fluid as it flows from the expansion space to the compression space. The Pressure- Volume (PV) diagram for the ideal Stirling cycle is shown in Figure 3-6.
29
Qa
Wco
Displacer/Regenerator
T c T Ta C Piston
1 2
Qc
Wex
3 4
Figure 3-5. Processes of the ideal Stirling cycle in a Sunpower cryocooler [4, p. 293].
1) Isothermal compression: The piston moves in (toward the workspace), reducing the total volume in the workspace and therefore compressing the gas and increasing the pressure. In order for the process to remain isothermal, as the compression work of
the piston Wc is transferred to the gas it is rejected through a heat exchanger, Qa. 2) Isochoric heating of the regenerator matrix: The displacer then moves out (away from the workspace), shuttling the working gas from the warm compression space to the cold expansion space. As the warm gas passes through the regenerator the gas transfers its heat to the regenerator matrix which stores the heat. The gas that reaches the expansion space is at the cold temperature. Since the majority of the working gas is now in the cold expansion space the system pressure decreases. 3) Isothermal expansion: The piston moves out increasing the workspace volume and therefore expanding the gas and decreasing the pressure. In order for the process to remain isothermal, as the gas is expanded the cold end heat exchanger transfers heat
30 to the gas. This heat transferred from the cold (refrigerated) space to the gas is the refrigeration that the cycle provides. 4) Isochoric cooling of the regenerator matrix: The displacer moves in shuttling the working gas from the cold expansion space toward the warm compression space. As the gas flows through the regenerator it absorbs the heat from the regenerator matrix that was stored during process 2. The gas that reaches the compression space is at the warm temperature. Since the majority of the working gas is now in the warm compression space the system pressure increases.
P 2
I so Qa the rm al T a 1 Regeneration 3 Iso the rm al
Tc
Qc 4
V
Figure 3-6. Pressure-volume diagram for the ideal Stirling refrigeration cycle [5, p. 469].
3.3 Gifford-McMahon (GM) Cycle The Gifford-McMahon cycle is based on the Ericsson thermodynamic cycle [5, p. 468] which consists of two isothermal and two isobaric processes. The two isobaric processes replace the two isentropic processes of the Carnot cycle [4, p. 289]. The Ericsson is the only other ideal thermodynamic cycle besides Stirling to theoretically achieve Carnot COP. However, in hardware, because of the use of reciprocating
31 compressors, oil-removal equipment, and a switching valve in the cold head, the Gifford- McMahon cycle turns out to be very inefficient compared to Stirling cryocoolers.
3.4 Basic Pulse Tube Upon discovery of the basic pulse tube, the explanation required some extension beyond conventional thermodynamics into the area of oscillating thermodynamics and thermoacoustics [6]. The operating principles of the basic pulse tube are different from the principles governing the orifice and inertance tube pulse tube cryocoolers being developed today. Due to the inherent limitations of the basic pulse tube as will be explained soon, it mainly served as a starting point to more efficient pulse tube cryocoolers operating under different principles. Referring to the experimental setup of Gifford and Longsworth in Figure 2-1, the flow-reversing valve alternatively connects to the high and low pressure sides of the steady flow system to the cold head consisting of a regenerator, cold heat exchanger, pulse tube and warm heat exchanger. When the high pressure side is connected to the cold head, the high-pressure helium flows through the regenerator, giving up its heat to the regenerator matrix. Conversely, when the helium is depressurized by connection to the low pressure side, the gas re-absorbs the heat from the regenerator matrix as it flows out of the cold head. Gifford and Longsworth were able to explain the refrigeration mechanism as a heat-pumping effect. Figure 3-7 helps explain the following process.
Process 1 to 2, adiabatic compression: The cold head is connected to the low pressure side of the compressor at state 1. When the cold head is connected to the high pressure side through the flow-switching valve, the gas in the tube gets compressed and the gas element moves from position 1 to position 2. This adiabatic compression results in a temperature increase. Process 2 to 3, constant high pressure cooling: At position 2 the increased temperature of the gas element due to adiabatic compression creates a temperature difference between the gas element and the wall at that location. The gas element is then cooled by
32
Heat Exchanger Pulse Tube
Tc Ta
142 3
Wall tic ba temperature ia ad gradient
ic
Temperature t ba ia ad
142 3 Position
Figure 3-7. Diagram of basic pulse tube operation [3, p. 208].
transferring heat to the wall. Gas that leaves the pulse tube and reaches the ambient heat exchanger rejects its heat to the surroundings at Ta. During this process, the cold head remains connected to the high pressure side of the compressor and the gas element moves further along the pulse tube to position 3 as more high pressure gas enters the tube. This additional gas flows into the tube to compensate for the increase in density as a result of the gas cooling. Process 3 to 4, adiabatic expansion: The cold head is then connected to the low pressure side of the compressor and the tube is depressurized. This results in an adiabatic
33 expansion of the gas that moves the gas element to position 4 and decreases the temperature of the gas element. Process 4 to 1, constant low pressure heating: At position 4 the decreased temperature of the gas element due to adiabatic expansion creates a temperature difference between the gas element and the wall at that location. The gas element is then heated by absorbing heat from the wall. Gas that leaves the pulse tube and enters the cold heat
exchanger absorbs heat at Tc. During this process, the cold head remains connected to the low pressure side of the compressor and the gas element is moved further along the pulse tube to position 1 as more low pressure gas leaves the tube. This additional gas flows from the tube to compensate for the decrease in density as a result of the gas heating.
The surface heat pumping effect of the basic pulse tube is a different method of producing refrigeration than a Stirling cooler or other types of PTC’s as will be explained soon. The reason for the surface heat pumping phenomenon, also called shuttle heat transfer, is related to the frequency of pressurization and the size of the pulse tube. The parameter “thermal penetration depth” is defined as18
2 ⋅ k δ k = (Eq. 3-6) ω ⋅ ρ ⋅ c p where ω is the cyclic frequency, k is the thermal conductivity, ρ is the density and cp is the constant-pressure specific heat, all relating to the gas. In the basic pulse tube the thermal penetration depth is of the same order as the radius of the pulse tube. Thus there is sufficient time in the heating and cooling process for the entire element of gas in the cross-section of the tube to thermally interact with the wall. A tube of the same diameter operating at a higher operating frequency would not have the same effect. There exists a temperature gradient on the wall at which the temperature profile of the gas matches the wall temperature profile [6]. At this critical temperature gradient there is no heat transfer between the gas and the wall because there is no temperature differential to drive heat transfer. If the actual temperature gradient exceeds this critical temperature gradient then heat will be pumped from the hot end to the cold end of the
34 tube. The critical temperature gradient limits the performance of the basic pulse tube cryocooler to temperatures above the cryogenic range.
3.5 Orifice Pulse Tube The orifice pulse tube shown in Figure 2-2 was developed in an effort to achieve refrigeration from PV power as in a Stirling cycle cryocooler rather than the surface heat pumping effect of the basic pulse tube. The first attempt by Mikulin placed the orifice inside of the pulse tube and added a volume reservoir. It was only after Radebaugh placed the orifice outside of the pulse tube that the orifice PTC achieved temperatures below 100 K. Just as the displacer in a Stirling cycle cryocooler creates the necessary phase relationship between mass flow and pressure to perform PV cooling, this relationship is achieved by the orifice and the reservoir in the orifice PTC. The reservoir volume is large enough that mass flow into and out of the reservoir have negligible effect on the pressure. Thus the reservoir is essentially at constant pressure while it stores and releases gas during various portions of the cycle. Figure 2-2 is shown again here to explain the thermodynamic process of the orifice PTC19.
1) The piston adiabatically compresses the gas in the work space (from the face of the piston up to the orifice). 2) The compressed gas in the pulse tube is at a higher pressure than the reservoir and also is at an elevated temperature due to adiabatic compression. As the heated gas flows through the orifice into the reservoir, it transfers heat to the ambient through the heat exchanger at the warm end of the pulse tube. This flow continues until the pressure is equalized between the work space and the reservoir. 3) The piston adiabatically expands the gas in the working space. 4) The gas in the reservoir is at a higher pressure than the expanded, adiabatically- cooled gas in the work space. As gas flows from the reservoir through the orifice into the work space, the cool gas in the pulse tube flows through the cold heat exchanger and absorbs heat. The flow continues until the pressure is equalized between the reservoir and the work space.
35
W
ω Reservoir Volume
Piston Orifice
Qa,Ta Qh,Th Pulse Tube Regenerator
Qc,Tc
(Repeated) Figure 2-2. Orifice pulse tube refrigerator [2, Chapter 7].
At this point it is appropriate to mention that the preceding process also explains the operation of an inertance tube PTC. As will be explained further in Section 4.4 the inertance tube is simply an alternative method to create the necessary phase shift between pressure and mass flow. The pulse tube itself allows the processes to occur at each end of the pulse tube without interaction. The pulse tube must be long enough that no warm gas reaches the cold end of the pulse tube and vice versa. Therefore the gas in the middle remains within the pulse tube and acts as a gas slug, insulating the two ends through a temperature gradient. Thus turbulence in the pulse tube must be minimized in order for the insulating gas slug to be effective. To summarize the function of the pulse tube, Radebaugh states [19] “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.”
36 Chapter 4. Orifice Pulse Tube Cryocooler (OPTC) Analysis In this chapter an analysis of the enthalpy and entropy flows will lead to the ideal efficiency of an orifice pulse tube cryocooler. Losses within the pulse tube component will be discussed along with the phase shift requirement between pressure and mass flow to achieve useful cooling power. While the name implies that this section applies only to an orifice pulse tube cryocooler as opposed to the subject inertance pulse tube cryocooler, this section is directly applicable to the inertance pulse tube cryocooler as well. The goal is to show the shortcomings of the orifice component and how an inertance assembly can improve on those shortcomings. In fact, the orifice and the inertance components of the description could be lumped together into a more general term of impedance pulse tube cryocooler and in any diagram the orifice or the inertance tube could be replaced with a generic impedance symbol. The difference in the two types of cryocoolers lies in the type of flow impedance they offer and the characteristics of their respective impedances. This chapter makes extensive use of references [6] and [11] by Radebaugh and others.
4.1 Enthalpy and Entropy Flow Model [11] The First and Second Laws of Thermodynamics for an open system are used to derive the refrigeration power of the OPTC. Expressions are time-averaged over one cycle to simplify the oscillating flow. Due to the conservation of mass the time-averaged value of mass flow rate must be zero. However, other quantities such as enthalpy and entropy flows are potentially nonzero. Positive flow is defined to be in the direction from the compressor to the orifice. The First Law energy balance for the cold section is shown in Figure 4-1. Since no work is extracted from the cold end the heat absorbed at the cold end under steady-state conditions is
Q&c = H& − H& r , (Eq. 4-1) where H& is the time-averaged enthalpy flow in the pulse tube, and H& r is the time- averaged enthalpy flow in the regenerator which is zero for a perfect regenerator and an ideal gas. Therefore the maximum gross refrigeration occurs when there is no enthalpy flow into the cold space from the regenerator and this maximum gross refrigeration is
37
Regenerator Pulse Tube
H& r Reservoir H& H& Orifice
Q Qc h
Figure 4-1. First Law energy balance for OPTC [6].
simply equal to the enthalpy flow in the pulse tube. The time-averaged enthalpy flow at any location can be expressed by combining the First and Second Laws for a steady-state oscillating system as
H& = PdV& + T0 S& (Eq. 4-2) where Pd is the dynamic pressure, V& is the volume flow rate, T0 is the average temperature of the gas at the location of interest, and S& is the time-averaged entropy flow. The first term on the right-hand side of Eq. 4-2 is often referred to as acoustic power or hydrodynamic power. It represents the potential for the gas to do reversible work if an isothermal expansion process were to occur at T0 in the gas at that location. In the ideal pulse tube the processes occurring within are adiabatic and reversible (isentropic), and thus S& = 0 . (Ideal case) (Eq. 4-3)
For sinusoidal functions of time, the acoustic power can be written as
PdV& = (1 2) P1V&1 cosθ = (1 2) RT0m&1(P1 P0 )cosθ (Eq. 4-4) where P1 is the amplitude of the sinusoidal pressure oscillation, V&1 is the amplitude of the sinusoidal volume flow rate, θ is the phase angle between the flow and pressure, R is the gas constant per unit mass, T0 is the temperature at the location of interest, P0 is the
38 mean pressure, and m&1 is the amplitude of the sinusoidal mass flow rate. Equations 4-1 through 4-4 can be combined to yield the maximum gross refrigeration power as
Q&max = PdV& = (1 2)RT0m&1(P1 P0 ) . (Eq. 4-5) This is a general result that can also be applied to Stirling and Gifford-McMahon cryocoolers. Equation 4-5 shows that the maximum gross cooling power is equal to the acoustic power in the pulse tube when the pressure and volume flow amplitudes are in perfect phase with one another.
4.2 Ideal Efficiency20 In the ideal OPTC, the expansion through the orifice is a purely resistive, irreversible loss that generates entropy because of the lost work that otherwise could have been recovered. If the working fluid is assumed to be an ideal gas and all other components are assumed to be perfect then the COP for the ideal OPTC is P V& Q&c d c Tc COPideal = = = , (Eq. 4-6) W&0 PdV&h Th where the volumetric flow rate subscripts refer to the rates at the cold and hot ends. The pressure times volumetric flow rate at the warm (hot) end of the regenerator is equal to the PV power of the compressor. Because the regenerator is assumed to be perfect, the acoustic power varies along its length in relation to the gas density, reflected through temperature as previously shown in Eq. 4-4 with the variable T0. Recall that the ideal Stirling COP is equal to that of the Carnot COP shown in Eq.
3-5. Comparing Eq. 4-6 with Eq. 3-5 (Ta and Th referring to the same temperature sink), the Tc term in the denominator of the Carnot (Stirling) COP is missing from the ideal OPTC COP. This term represents the work reversibly recovered at the low temperature and used to help in the compression in the Stirling cycle. From the plot of Eq. 4-6, Figure 4-2 shows that due to the OPTC’s inability to recover work at the cold temperature, its ideal efficiency is intrinsically less than the ideal efficiency of a cryocooler operating on the Carnot (Stirling) cycle. This graph shows that the OPTC cannot compete with the Stirling or vapor-compression cycles at higher temperatures.
39
Carnot COP Ideal OPTC COP
0.7 Ta = 300 K 0.6
0.5
0.4
0.3 Carnot COP Carnot 0.2
0.1
0 0 102030405060708090100110120130 Cold Temperature (K)
Figure 4-2. Comparison of ideal Stirling and ideal OPTC efficiencies.
Cryogenic temperatures are more suitable for comparable performance between ideal OPTC and ideal Stirling, yet even below 100 K the OPTC has a theoretical disadvantage down to about 30 K. Of course, practical implementations of both ideal processes can greatly change the comparison between the two technologies.
4.3 Pulse Tube Losses In actual hardware there will be losses in both the regenerator and the pulse tube which subtract from the gross refrigeration power. The largest loss is the regenerator
enthalpy flow H& r . Entropy generation within the pulse tube also creates significant losses. Entropy can be generated in the pulse tube by a number of effects: • Instantaneous heat transfer between the gas and the tube wall. • Mixing of the hot and cold segments of the pulse tube due to turbulence.
40 • Acoustic streaming (discussed in Section 2.1.2). • End-effect losses associated with transitioning from an adiabatic volume to an isothermal volume (the heat exchangers). The entropy flows for these effects are negative, flowing from the pulse tube toward the compressor. Actually, the entropy flow for the first item is positive at higher temperatures before the critical temperature gradient has been exceeded. Recall that this was the principle of operation for the basic pulse tube. At cryogenic temperatures this critical temperature gradient has been exceeded and thus the entropy flow is negative and the mechanism is a loss. Equation 4-2 shows that these negative entropy flows subtract from the acoustic power in the pulse tube and reduce the enthalpy flow which represents the refrigeration from Equation 4-1.
4.4 Phase Shift [11]
From Equation 4-4, for a given PV power ( PdV& ) and pressure amplitude (P1), the mass flow rate (or volume flow rate V&1 ) is minimized for θ = 0. Regenerator losses such as pressure drop and imperfect heat transfer are mainly dependent on the mass flow rate through regenerator. Thus when the average phase relationship between the mass flow rate and the pressure oscillation in the regenerator is zero the regenerator losses are minimized and system efficiency maximized. This optimum phase relationship should occur in the center of the regenerator. Due to the volume of gas in the regenerator the mass flow at the warm end of the regenerator will lead the pressure but the mass flow at the cold end of the regenerator will lag the pressure. The typical phase shift of the pressure with respect to the mass flow from the warm end to the cold end of the regenerator is typically around a 20 to 30° lag. In the OPTC the phase relationship between the pressure and mass flow is zero at the orifice due to the purely resistive nature of its flow impedance. Again due to the volume of gas in the pulse tube, the mass flow at the cold end of the pulse tube will lead the mass flow at the warm end of the pulse tube. In a correctly designed pulse tube, the mass flow rate will lead the pressure by approximately 30° at the cold end of the pulse
41 tube and regenerator. This means that the mass flow rate at the warm end of the regenerator will lead the pressure by as much as 60°. Because of the poor phase angle the mass flow amplitude will need to be large to achieve a given PV power. This high mass flow rate will lead to regenerator losses. Figure 4-3a shows these phase relationships of the OPTC in a vector diagram21.
In order for the average mass flow at the center of the regenerator, m& r , to be in phase with the pressure, the mass flow at the warm end on the pulse tube, m& wpt , needs to shift by approximately -60°. This is shown in Figure 4-3b with the new phase relationship labeled as m& desired . In Figures 4-3a and 4-3b m& wr is the mass flow at the warm end of the regenerator and m& c is the mass flow rate at the cold end (for both the regenerator and pulse tube). The achievement of this desired phase relationship relies on the introduction of the inertance tube.
m & wr m &r m& wr m& c P m&r P
m& wpt m& c
m& desired
ab
Figure 4-3. Vector diagrams of pressure and mass flow phase relationships for a) OPTC, b) desired phase relationship.
42 Chapter 5. Inertance Tube The inertance tube is a long, thin tube that offers a complex flow impedance as opposed to the simple, purely resistive impedance of the orifice. Analogous to inductance in an electrical circuit, the inertance tube provides a reactive impedance that allows the phase relationship between the pressure and mass flow to be widely adjusted. This flexibility offers the potential to maximize the pressure and mass flow phase relationship and achieve a higher cooling efficiency. This chapter discusses the work of Roach and Kashani [16] that creates a helpful analogy between the fluid flow circuit and an electrical circuit. Due to conflicts with other symbols used throughout this text, this chapter contains its own independent definitions of some symbols. These symbols and definitions are provided within the text near their first use.
5.1 Electrical Analogy The following equations define the relations between electrical current, I, and voltage, V, for the cases of a resistor of resistance, R, an inductor of inductance, L, and a capacitor of capacitance, C, Resistor: V = I ⋅ R (Eq. 5-1) dI Inductor: V = L ⋅ (Eq. 5-2) dt dV I Capacitor: = . (Eq. 5-3) dt C Evaluating the 1-dimensional momentum conservation equation for the flow in a tube of radius, r, similar relations can be found for gas flow (analogous to current) and pressure (analogous to voltage) in the elements of a pulse tube cryocooler ∂u ∂P µ ⋅u ρ = − − , (Eq. 5-4) ∂t ∂x K p where P is the pressure, ρ is the gas density, u is the average velocity in the tube, µ is the viscosity and Kp is the Darcy permeability. Substituting the relationship between volumetric flow rate, U, and velocity
43 U u = , (Eq. 5-5) π ⋅ r 2 and rearranging terms yields ∂P ρ ∂U µ ⋅U − = ⋅ + . (Eq. 5-6) 2 2 ∂x π ⋅r ∂t π ⋅r ⋅ K p If these parameters are independent of the distance, x, along the tube, then the equation can be integrated along the tube for its entire length, λ, yielding: ρ ⋅ λ ∂U µ ⋅ λ ⋅U − ∆P = ⋅ + . (Eq. 5-7) 2 2 π ⋅ r ∂t π ⋅ r ⋅ K p Using the above analogy between pressure-voltage and mass flow-current, then comparing Equations 5-1 and 5-2 with Equation 5-7 shows that the fluid flow analogy of resistance is: µ ⋅ λ R = , (K = r2/8 for laminar flow) (Eq. 5-8) 2 p π ⋅ r ⋅ K p and is in series with the fluid flow analogy of inductance with value: ρ ⋅ λ L = . (Eq. 5-9) π ⋅ r 2 The term “inductance” for the electrical circuit changes to the term “inertance” for the fluid flow case.
From the ideal gas law, for volume flowing into an isothermal volume, Vt, the pressure rise will be: ∂P P ⋅U = av . (Eq. 5-10) ∂t Vt This equation is analogous to Equation 5-3 with the analogous capacitance value of: V C = t . (Eq. 5-11) Pav Table 5-1 summarizes the electrical and fluid flow analogy.
44 Electrical Fluid Flow Voltage V Pressure P Current I Volume flow U µ ⋅ λ Friction Resistance R 2 π ⋅ r ⋅ K p
ρ ⋅ λ Inductance L Inertance π ⋅ r 2 V Capacitance C Volume t Pav
Table 5-1. Summary of analogy between electrical and fluid flow circuits [16].
45 Chapter 6. Project Motivation and Initial Design Methodology The applications for pulse tube cryocoolers have been identified, the history of PTC’s have been discussed and the fundamental theory behind the genealogy of PTC’s has been explained. However, in the modern-day cryocooler market a compact, highly- efficient commercial PTC is not available. The author’s employer, Sunpower, Incorporated manufactures a line of compact, highly-efficient Stirling cryocoolers22. As discussed earlier, the Stirling cryocooler has two moving parts: the piston and displacer. Many references [2, 4, 6, 16] point out the theoretical advantages of the pulse tube cryocooler. Most commercially available pulse tube cryocoolers are Gifford-McMahon- type PTC’s as opposed to those driven by valveless compressors. These PTC’s have the same disadvantages as the GM cryocoolers regarding size and efficiency. Thus a need exists in the marketplace for a highly-efficient, compact, low-cost commercial pulse tube cryocooler. Sunpower decided to pursue that opportunity by responding to a solicitation from NASA Goddard Space Flight Center (GSFC) for high efficiency single and two-stage PTC’s. Sunpower had already established itself as an industry leader in the design and development of efficient linear compressors and Stirling cryocoolers. The model M87 Stirling cycle was being manufactured in low volumes, so the linear compressor technology driving the Stirling cryocoolers had been proven to be cost effective. David Gedeon, a former employee of Sunpower was working independently as a consultant and was recognized throughout the industry as a leader in the design and analysis of both Stirling and pulse tube cryocoolers. Gedeon was also the author of the commercial simulation software entitled Sage. Combining Sunpower’s linear compressor expertise with Gedeon’s experience in design and analysis of PTC’s, an SBIR Phase I (paper study) program was awarded23. The paper study began with a market survey on the applications for PTC’s. Then the major portion of the Phase 1 program involved evaluating design trades of the proposed PTC using Sage simulation software. The first effort was to validate and calibrate the Sage models versus existing Stirling hardware across a wide range of inputs and outputs including frequency, piston amplitude, charge pressure, PV power and
46 cooling capacity. The Sunpower M77 Stirling cryocooler was used to produce the test data for comparison. After successful calibration and validation, a series of optimizations were performed on key design parameters such as pressure, frequency and heat exchanger dimensions by David Gedeon and Sunpower’s Gary Wood. The optimization involved the sequential variation of these parameters to maximize cooling capacity while keeping a reasonable limit on some other variables like input power and dimensional constraints. As general trends were established in the optimizations, parameters were successively locked down and the process repeated until a general layout of the cryocooler was established. Actually, two optimized options were completed and layout drawings of these options were created. The choice of which optimization options to pursue was based on the previously mentioned market survey which was continuing along with the design. The market survey included publication searches along with telephone conversations and email exchanges with experts in the field of cryocooler applications. In addition, consideration was given to linear compressor models previously developed at Sunpower for other applications. This was done to facilitate near-term availability of those linear compressors as the project would move toward commercialization. As a result of this consideration low cooling capacity applications like certain electronic and infrared markets and especially high cooling capacity applications such as superconducting transmission lines and particle accelerators were ruled out of the design options. There were two candidate PTC prototypes. The first was based on the M77 and/or M87 Stirling cryocoolers and was projected to lift 7 W at 77 K with 100 W input power. The other was a two-stage design with simultaneous cooling loads of 10 W at 15 K and 40 W at 60 K. This design was based on the use of the Sunpower 1 kW linear alternator in the linear compressor. Funding was awarded to produce hardware of the machines modeled in the paper study. Before discussing the actual hardware development program, the key components of the partnership between Sunpower and Gedeon Associates will be elaborated upon.
47 6.1 Sunpower Linear Compressor At this point, a distinction is necessary between a linear compressor and the device driving both Stirling and pulse tube cryocoolers. Traditionally a linear compressor describes a device that compresses a working gas from a low pressure to a high pressure in a steady-flow cycle such as the Rankine cycle for food refrigeration. A valve assembly is used to maintain one side of the compressor at the high pressure and the other side at the low pressure. For both the Stirling and pulse tube cryocoolers the linear compressor device is more appropriately termed a pressure wave generator or pressure oscillator. This is because the Stirling and pulse tube thermodynamic cycles do not require valves to create a DC flow as in a conventional linear compressor and the Rankine thermodynamic cycle. The Stirling and pulse tube cold heads operate on pressure oscillation created by the motion of the linear motor piston directly coupled to the workspace. While this is generally referred to as a pressure wave generator or pressure oscillator rather than a compressor, there will be no distinction made between the two in this work. Additionally, this discussion will be directly addressing the linear compressor of a pulse tube cryocooler as opposed to a Sunpower Stirling cryocooler, as there are some differences between the two. The linear compressor technology is based on several years’ experience for applications ranging from household refrigeration24 and air compression to specialty gas compression25 and CPU cooling26. Sunpower linear compressors have been built from 10
We to 2.5 kWe (the subscript “e” refers to electrical power) of input power, with compression ratios of less than 2:1 up to 26:1 in a single compression stage27. Sunpower’s linear compressor is an axi-symmetric device with the piston driven by a linear motor28. The motor consists of permanent magnets attached to a structure called the magnet ring coupled directly to the piston29. The piston/magnet assembly oscillates in an air gap created by two sets of steel laminations and a wound coil carrying alternating current. There is no conversion from rotary to reciprocating motion as in conventional compressors and thus the side loads normally transferred from the piston to the cylinder are virtually eliminated. Elimination of side loads enables the use of a gas bearing system30 and oil-less operation. In the gas bearing system a portion of the high
48 pressure working fluid is directed around the piston to act as a lubricant on the running surfaces between the piston and the cylinder. This system prevents contact between the moving parts offering long-life operation. Typically only 1-2% of the workspace pressure-volume (PV) power is consumed by the gas bearing system. Most often planar springs are attached to the oscillating piston and used in conjunction with the gas spring of the workspace to create a spring-mass system whose resonant frequency is at or near the driving frequency of the system. This reduces the amount of force that the motor needs to apply to the piston for a given amplitude and thus reduces the motor’s size and losses. Since a portion of the total spring is composed of the gas spring in the work space, and the operating conditions in the workspace can vary, the gas spring portion of the total spring can vary. In general, to provide a stable operating condition, the mechanical spring portion should be larger than the gas spring portion. However in certain situations the gas spring can be used without any mechanical spring. The mechanical springs are attached to the piston by a compliant member that reduces the side loads on the gas bearing system that may be caused by misalignment during assembly31. The attachment of the piston to the spring through the compliant member also serves to center the piston axially. The combination of the linear motor, planar springs, gas bearings and a compliant member provides the following design advantages: • Stability • Readily manufactured (in terms of part tolerances and assembly) • Modulatable for varying capacity • Long life (non-contact clearance seals and running surfaces) • High efficiency. The general configuration of Sunpower linear compressor technology is shown in Figure 6-1.
6.2 Gedeon Associates Sage Software32 The Sage Stirling and pulse tube modeling software developed by Gedeon Associates excels in all aspects of Stirling cycle and pulse tube refrigerator modeling, including inertance tube modeling and modeling of the pulse tube itself. Commercially
49
Figure 6-1. General layout of Sunpower linear compressor technology [27].
available for eight years, Sage has been used at several sites. Before that many of its numerical algorithms had been developed in a software package named GLIMPS which served the Stirling industry well for nearly ten years. Sage introduced a drag-and-drop visual interface where a user can assemble complete machines from standard components such as pistons, cylinders, heat exchangers, etc. Sage also introduces an interactive optimization capability built into the visual interface. Perhaps the most important feature of Sage is its attention to detail in the underlying physics of oscillating gas flow in ducts and cylinder spaces. Sage has kept abreast of the latest theoretical and experimental developments in oscillatory flow theory, both for laminar and turbulent flows, including the criterion for laminar-turbulent transition. Sage is probably unique in this regard and, therefore, well suited to design of real pulse tube hardware. Recent advances in Sage include the capability to model tapered pulse tubes and tapered inertance tubes from first principles.
50 6.3 Final Sunpower Pulse Tube Cryocooler Configurations The pulse tube development history to this point presented several options for the particular design to incorporate into Sunpower’s PTC. It was time to decide which design elements would create the optimal PTC for Sunpower not only in terms of efficiency but also in the ability to produce a single-stage PTC that would be cost- competitive with Sunpower’s Stirling M87 cryocooler. For the single-stage PTC, it was decided to first construct an inline configuration. The inline configuration involves the simplest mechanical design and also the highest theoretical performance compared to the alternative u-tube configuration. As shown in Figure 6-2, the inline configuration refers to the fact that the regenerator, acceptor, pulse tube and secondary rejector lie sequentially on the same axis, or in line with one another. The reason that this configuration offers the highest theoretical performance is because the gas flow is mainly one-dimensional. However, a disadvantage of the inline configuration is the fact that there is no convenient end face to attach the load that requires cooling. The load must be attached around the perimeter of the acceptor. Additionally, the package is excessively long and cooling is required at two different locations because of the separation between the main rejector and the secondary rejector. After gaining experience with the inline configuration, a u-tube configuration was constructed. The u-tube configuration trades a loss in performance for a packaging advantage. As shown in Figure 6-3, the acceptor is designed as a turning manifold so that the regenerator, acceptor and pulse tube now form a u-shape which creates the namesake. The u-tube offers a convenient flat end to attach a heat load, creates a more favorable
Linear Main Secondary Compressor Acceptor Rejector Rejector
Reservoir Regenerator Pulse Tube Volume Inertance Tube
Figure 6-2. Inline PTC configuration.
51
Reservoir Volume Linear Compressor
Inertance Tube
Main Secondary Rejector Rejector Regenerator Pulse Tube Pulse
Acceptor
Figure 6-3. U-tube configuration.
package and also bring the heat removal sections into close proximity. However, the gas flow must now make a 180° turn which can introduce flow disturbances in the pulse tube and decrease the performance. Sunpower’s approach to handle this difficulty was to design the turning manifold of the acceptor in an aerodynamic manner to reduce any flow losses from the bend. Additionally, a layer of screens can be placed between the acceptor and pulse tube to act as a flow-straightening device. Finally, a two-stage u-tube PTC was designed and fabricated in order to gain experience in multi-staging. The advantages of multi-stage cryocoolers are similar to the advantages of a multi-stage compressor. In the same manner that it is more efficient to achieve a very high pressure ratio by compressing the gas in two stages, it is more efficient to achieve a very high temperature ratio (very low temperatures) by removing heat at two stages. There will be no discussion or diagram of the two-stage PTC. For this effort, only the inline configuration will be considered.
52 Chapter 7. Modeling the Pulse Tube Cryocooler using Sage This chapter discusses in detail the use of Sage simulation software to model the inline PTC. As mentioned earlier, Sage contains a drag-and-drop interface of standard components found in Stirling and pulse tube cryocoolers. The approach here will be to introduce only those particular components used in the modeling of this inline PTC, therefore many available components from Sage will not be covered. The Sage User’s Guide (2nd Edition)33, Stirling-Cycle Model-Class Reference Guide (2nd Edition)34, and Pulse-Tube Model-Class Reference Guide (2nd Edition)35 will be referenced often. Appendix A provides an overview of Sage directly from the Sage User’s Guide. Appendix B discusses the types of boundary connections available from the Stirling- Cycle Model-Class Reference Guide. At the highest level the PTC can be broken down into three subassemblies: the linear compressor, the cold head, and the inertance assembly. In a little more detail, and in the context of the model components, these subassemblies are: • Linear compressor – creates the pressure oscillation for the thermodynamic cycle, composed mainly of: o Linear motor o Piston o Cylinder • Cold head – the heart of the thermodynamic region, composed mainly of: o Main heat rejector o Regenerator o Acceptor o Pulse tube o Secondary rejector • Inertance assembly – creates the required phase shift between pressure and mass flow oscillations, composed of: o Inertance tube (technically two tubes in series) o Volume reservoir
53 Various connecting tubes, flow diffusion sections and plenum volumes are also incorporated to increase modeling accuracy. All of these subassemblies and components will now be individually covered. In order to gain some physical resemblance between the model and the actual hardware, in general the model is setup to address the components from the left-to-right direction as shown in Figure 6-2. Additionally, when a “positive” or “negative” attachment is encountered, negative refers to the left side of the component and positive refers to the right side of the component as shown in Figure 6-2.
7.1 Linear Compressor The linear motor portion of the linear compressor is not modeled by Sage. Typically the linear motor efficiency is estimated and then applied to the pressure-volume (PV) power determined by Sage to predict the electrical input to the compressor.
7.1.1 Piston/Cylinder and Compression Space The piston/cylinder and compression space model is created in Sage using three available components: a pressure source (charge tank) from the “Basic” tab, a constrained piston and cylinder composite component which already includes the necessary child components from the “Composite” tab, and the generic cylinder (renamed compression space) from the “Basic” tab. These parent level model components are shown in Figure 7-1. Additionally, the tube that connects the compressor working space is shown as part of the piston/cylinder rather than as part of the cold head section. The connecting tube is modeled as a tube bundle from the “Heat Exchangers” tab. The pressure source is
introduced with the built-in density attachment ρstdy. The other components do not contain any attachments and child components must be added to gain these attachments. The piston and cylinder composite component comes with the shell, liner and constrained piston child components shown in Figure 7-2. The “constrained” qualifier on the piston component means that the motion of the piston (amplitude and phase) is dictated by input values and not determined by other forces acting upon it. This is an alternative to the “free” qualifier also available on the piston component which means
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Figure 7-1. Parent (root) level model components of the piston/cylinder and compression space of a compressor.
Figure 7-2. Child level components of constrained piston and cylinder composite model.
its motion will be determined by the forces acting upon the piston. The spring component is added at this level from the “Springs and Dampers” tab to model the attachment between the piston and the planar spring. There are no further child components associated with either the piston shell or cylinder liner.
55 Within the constrained piston child component of the constrained piston and cylinder parent-level component, two attachments are added from the “Mechanical
Attachments” tab. The first attachment is a positive-facing area, Pphsr, which physically represents the face of the compressor piston. This attachment is exported out from this child level up to the top, or root, level (Figure 7-1) for later attachment. The other is the negative-facing attachment, Fphsr, which physically represents the attachment of the piston to the spring through the flex rod (compliant member) as shown in Figure 6-1. This attachment is exported out from this child level up to the constrained piston and cylinder level (Figure 7-2) for later attachment. The only child component added to the spring component is the positive-facing attachment which connects to the negative-facing attachment of the constrained piston at the constrained piston and cylinder level (Figure 7-2). Figure 7-3 shows the child level components of the constrained piston. Next we consider the generic cylinder component of Figure 7-1, which is the model for the compression space gas of the compressor. Figure 7-4 shows the first-child- component level consisting of the added components cylinder-space gas from the “Gas Domain” tab and isothermal surface from the “Cylinder Walls” tab. Each of these components comes with built-in heat flow connections which stay at this child level and model the thermal interaction between the gas and solid boundary. There are several
Figure 7-3. Child level components of constrained piston model.
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Figure 7-4. Child level components of compression space model.
choices available in terms of the thermal solid which interacts with gas spaces, and these choices will be discussed later. There are no child components to be added to the isothermal surface. The child components of the cylinder-space gas are shown in Figure 7-5. The gas charge line component added from the “Charge/Inlets” tab contains the density component ρstdy which is exported to the root level for attachment to the pressure source. This attachment is how the system obtains its mean working pressure. The negative volume displacement attachment, Pphsr, is added and from the “Volume Displacements” tab and exported to the root level for connection to the constrained piston to represent the volume displacement at the boundary between the gas and the face of the piston. While it seems that something is incomplete by attaching a volume displacement to an area, one input of the constrained piston is its amplitude and this combination of piston area and amplitude complete the volume displacement connection. Finally, the positive gas inlet attachment,
m& Gt , also added from the “Charge/Inlets” tab is exported to the root level for attachment. This represents the fact that the variable volume gas space is connected to some other external system rather than simply compressing and expanding in a dead volume.
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Figure 7.5. Child level components of cylinder-space gas model.
The connecting tube has similar child components as the compression space model as shown in Figure 7-6. However, notice in Figure 7-6 that the gas domain is labeled with a “D” as opposed to the gas domain in Figure 7-4 labeled with a “C”. The “D” stands for a duct gas domain and the “C” stands for a cylinder-space gas domain. Chapter 8 of the Stirling-Cycle Model-Class Reference Guide [34] states “a duct gas domain is used within relatively short flow ducts (generally tubes or rectangular channels) with not so tiny hydraulic diameters”. On the other hand “a variable-volume gas domain is used within the piston-cylinder spaces of stirling models. It is typically the volume between the pistons and the cylinders in which they ride.” There is yet another type of gas domain that will be presented later. The three different domains are “distinguished primarily by the method by which they track the onset of turbulence”. In Figure 7-6, just as in the compression space model shown in Figure 7-4, there is a connection between the gas domain and an isothermal surface. In the duct gas domain both a positive and negative gas inlet are included and exported to the root level with connection to the compression space and the rest of the system. Again, it is more appropriate to include the connecting tube along with the compressor rather than the cold head model. The connecting duct could also have been accounted as a portion of the
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Figure 7-6. Child level components of connecting tube model.
workspace volume rather than as a separate component. However, in one extreme case a very long connecting duct would have a substantial amount of flow friction which is dealt with in a different fashion in the duct model as opposed to the generic cylinder model. The preceding Figures 7-1 through 7-6 show the “Models as Interconnected Systems” presentation of the system. However it was pointed out earlier that the model can also be presented as a tree. Appendix C shows the model tree structure of the linear compressor. Notice that some of the components shown in the edit view (the “interconnected systems” approach in Figures 7-1 through 7-6) are not listed as part of the model tree. This is because these components have no input or output data associated with them. Therefore they are used purely in the edit view window to indicate a connection between two other components.
7.2 Cold Head Figure 7-7 shows the major components of the cold head portion of the model. These include the main rejector, regenerator, acceptor, pulse tube, and secondary rejector. Additionally, the void volumes adjacent to some of these components are modeled to
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Figure 7-7. Root level components of cold head model.
increase accuracy. These void volumes are generically called plenum volumes, or plenums. A warm heat source and a cold heat sink are also added to model parasitic heat flows between these temperatures. The model tree structure of the cold head is also presented in Appendix C.
7.2.1 Main Rejector, Acceptor, Secondary Rejector (Heat Exchangers) Each of these components is simply a heat exchanger which transfers heat to or from an external heat source or sink. They are all formed by initially selecting a tubular canister from the “Canisters” tab. Because they are all modeled as isothermal heat exchangers the child components within them are identical, of which the first child level is shown in Figure 7-8. This first level of the child component is simply the matrix of the heat exchanger, in this case a woven screen matrix selected from the “Matrices” tab. As
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Figure 7-8. Child level component of heat exchanger models.
will be discussed later in Chapter 8 each of these heat exchangers is composed of a stack of woven copper screens joined to a copper wall. Below this child level are a gas domain and an isothermal surface as shown in Figure 7-9. Notice the “M” on the gas domain component in Figure 7-9 as opposed to the “C” on the gas domain component in the compression space shown in Figure 7-4 and as opposed to the “D” on the gas domain component in the connecting duct in Figure 7-6. The “M” represents a matrix gas domain rather than the two gas domains discussed earlier. Again from the Chapter 8 of the Stirling-Cycle Model Class Reference Guide [34], “a matrix gas domain is used within a porous matrix or within uniform channels of tiny hydraulic diameter.” Below the gas component are positive and negative gas inlets whose connections are exported up to the root level as can be seen in Figure 7-7.
7.2.2 Main Rejector, Acceptor, Secondary Rejector Plenums As mentioned previously the plenums model void volumes adjacent to the named components. These plenums are modeled primarily to account for their volume in the overall system volume. They are formed by selecting the generic cylinder component
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Figure 7-9. Child level component of woven screen matrix models.
as in the compression space component of the compressor model of Figure 7-1. Thus the child components within the plenums are the same as those under the working space volume, a gas domain and an isothermal surface as shown in Figure 7-4. The plenums have no volume displacement child component due to the fact that they have fixed volume. Within the gas domains are positive and negative gas inlets.
7.2.3 Parasitic Warm Source and Cold Sink Both the parasitic warm source and the parasitic cold sink shown in Figure 7-7 are generically called a point heat source and are thermal solids selected from the “Basic” tab. Although they are considered thermal solids the only property associated with them is a temperature, no area properties or physical properties. As such they are used only to
incorporate steady heat flow connections, Qstdy, which are child components that are exported to the root level. The temperatures are associated with an infinite source or sink which are used to calculate parasitic losses such as solid conduction.
62 7.2.4 Regenerator The regenerator model is constructed by starting with a tubular canister from the “Canisters” tab. The regenerator is also a heat exchanger, however it operates in a much different fashion than the heat exchangers discussed thus far in the cold head. The other heat exchangers contain a solid matrix, are treated as isothermal, and either transfer heat from the gas to some external heat sink or transfer heat from some external heat source to the gas. As Figure 7-10 shows, in the first child component level the regenerator also contains a solid matrix but in the form of random fibers as opposed to woven screens. Also at this level is a heat conductor, or generically called a bar conductor, available from the “Heat Flows” tab. This component models a solid conduction path and has built-in heat flow connectors which are exported to the root level for connection to the heat source and heat sink to model the parasitic conduction losses in the regenerator wall. The child components below the random fiber matrix are shown in Figure 7-11. Notice that it looks very similar to Figure 7-9 with the matrix type gas domain. However, Figure 7-11 shows that the surface is now modeled as a rigorous surface rather than an isothermal surface. A rigorous surface is a type of quasi-adiabatic surface. According to
Figure 7-10. Child level components of regenerator model.
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Figure 7-11. Child level components of random fiber regenerator matrix model.
Chapter 7 of the Stirling-Cycle Model-Class Reference Guide [34] this means that at the interface between the solid and the gas there is a time-varying sinusoidal heat flux with zero or near zero mean value. The heat flow connection between the matrix gas and rigorous surface models this interaction. The rigorous type of quasi-adiabatic surface “presumes the skin thickness d is on the same order as the thermal wavelength Lambda. It is intended for modeling any sort of regenerator matrix or duct wall….” There are no child components of the rigorous surface and the matrix gas contains the usual positive and negative gas inlets.
7.2.5 Pulse Tube What has consistently been referred to in this effort as the pulse tube is also known as the compliance tube and that is the moniker within Sage. The compliance tube is a component specific to the Pulse-Tube Model Class and is chosen from the “Heat Exchangers” tab. Chapter 2 in the Pulse-Tube Model Class Reference Guide [35] states “the compliance tube is a descendant of the stirling-class tube-bundle heat exchanger. The main difference is that it substitutes a compliance-duct gas domain in its toolbox and adds a radiation-transport model component, in case one wants to model radiation
64 transport along the tube. Wall conduction is available in the thick-wall toolbox component already present”. The child components of the compliance tube are shown in Figure 7-12. The compliance-duct gas domain, shown with the “CD” label in Figure 7- 12, descends from the Stirling-Cycle Model-Class duct-type gas domain. However, the compliance-duct gas domain addresses the convective losses in the wall boundary layer caused by the presence of an axial temperature gradient. In Chapter 3 this was introduced as acoustic streaming. In all the compliance-duct gas domain accounts for molecular conduction, turbulent conduction, free convection, boundary convection and streaming convection. The rigorous surface in Figure 7-12 was discussed in the preceding section regarding the regenerator and of course the gas domain contains the positive and negative gas inlets.
7.3 Inertance Assembly The root level components that comprise the inertance assembly are shown in Figure 7-13. The two inertance tubes are connected in series and terminate in a large volume reservoir. Each of the inertance tubes are modeled the same as the connecting duct shown in Figure 7-1 with their child components of the duct gas domain and isothermal surfaces as shown in Figure 7-6. Again, positive and negative gas inlets are
Figure 7-12. Child level components of compliance tube model.
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Figure 7-13. Root level components of inertance assembly model.
child components within the duct gas domain. The reservoir is modeled the same as the cylinder space from Figure 7-1 with a cylinder space gas domain and an isothermal surface. At this point the system ends and thus there is only a negative gas inlet within the cylinder space gas domain as shown in Figure 7-13.
7.4 Pulse Tube Cryocooler Assembly The components and subassemblies, or building blocks, shown in Figures 7-1 through 7-13 are assembled together to create the model of the entire pulse tube cryocooler. There are additionally diffuser components at the main and secondary rejectors in the cryocooler model that have not been discussed. These are modeled as very small compliance tubes with a compliance duct gas. But instead of a rigorous surface as in the pulse tube, or compliance tube component, these contain isothermal surfaces. Figure 7-14 relates the Sage model numbering system in Figures 7-1, 7-7 and 7- 13 to the components shown in the schematic of Figure 6-2. The numbers correspond to the appropriate boundary connections in Sage. Note that not all components shown in the Sage models are shown in the schematic.
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Linear Inertance Compressor Cold Head Assembly Figure 7-1 Figure 7-7 Figure 7-13
4 6 20 11 12 17/18 14 15 3 4 Regenerator Pulse Tube Reservoir Inertance Inertance Volume Tube 1 Tube 2 Secondary Main Acceptor Piston/Cylinder/ Rejector 3/4 7/8 Rejector Compression Space
Figure 7-14. Inline PTC schematic labeled with corresponding numbers from Sage models (Figures 7-1, 7-7 and 7-13).
67 Chapter 8. Design and Fabrication of Hardware This section discusses the effort to design and fabricate the individual components that together comprise the inline PTC. Topics of discussion include material selection, functional and structural requirements. Due to the proprietary nature of Sunpower’s technology some details may not be revealed.
8.1 Linear Compressor The key element of the linear compressor design is the linear motor as described in Chapter 6. The linear motor consists of a number of permanent magnets bonded to a structure called a magnet ring that is coupled directly to the piston. Figure 8-1 shows the front and back side views of this assembly. Also shown in Figure 8-1 is the compliant member, or flex rod, that is used to decouple lateral forces caused by misalignment from loading the gas bearing system. Figure 8-2 shows the individual piston and magnet/magnet ring pieces. Since the running clearance between the piston and cylinder is typically around 25 µm (.025 mm, 25 microns) the OD of the piston must be machined extremely accurately and must also be dimensionally stable over time. The piston is usually turned from 6061-T6 aluminum for the benefits previously mentioned but also to keep the moving mass of the spring-mass system fairly light, thus minimizing the spring
Magnet Segment Flex Rod
Magnet Ring Piston Piston
Figure 8-1. Piston/magnet/magnet ring/flex rod assembly.
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Figure 8-2. Piston and magnet/magnet ring components.
stiffness requirement. The piston is actually composed of two pieces because of the gas bearing system but the details of that arrangement are not shared for proprietary reasons. The piston is driven by its connection to the magnet sleeve whose sole purpose is to couple the piston and the moving magnets of the linear motor. The magnet sleeve is a patented design making use of a very thin cylindrical support structure turned from 300 series stainless steel for its strength and non-magnetic properties. The stresses on the magnet sleeve come from the dynamic motion or inertia of the piston assembly. The flex rod connects the piston/magnet sleeve assembly to the planar spring to make use of the resonant design of the spring-mass system. The flex rod experiences eccentric buckling loads and oscillating stresses leading to fatigue. The purpose of the flex rod, patented via a design methodology called compliance, is to transfer the kinetic energy and stored energy between the piston and the planar spring without introducing excessive side loads that may dominate the gas bearing system. Thus the flex rod is designed as thin as possible to reduce radial stiffness that transmits loads to the gas bearings, yet thick enough to meet buckling and fatigue stress requirements. The material is a 400 series stainless steel that is heat treated to obtain high fatigue strength.
69 The magnets oscillate within an air gap created by inner and outer iron lamination structures. The lamination structures are composed of a large number of thin individual laminations made of lamination steel and the individual inner and outer laminations are laser-cut from a large sheet. The inner laminations (those inside of the magnet) are wrapped around the outer diameter of the piston cylinder and glued in place using end containment rings. The piston cylinder functions both to support the inner lamination assembly and to create the mating running surface for the piston. Again due to the piston/cylinder running clearance the bore of the cylinder must be machined extremely accurately and the cylinder must be positioned within the transition extremely accurately as well. The cylinder must also be dimensionally stable over time to not change shape and lose the delicate fit requirements between the piston and cylinder. Typically the cylinder is turned from 6061-T6 aluminum or a material called Ni-Resist Type 1 which is a non-magnetic cast iron with a coefficient of thermal expansion very similar to aluminum. The outer laminations are welded into 8 stacks which are then glued around the wound coil. The coil is created by winding copper wire around a plastic form called a bobbin. Figure 8-3 shows the inner laminations bonded to the piston cylinder along with a wound coil and also the lamination stacks glued around the wound coil. The air gap is created by attaching the outer lamination/coil assembly and the inner lamination/cylinder assembly to a central structure called the transition. The transition is critical component that serves to anchor both the moving and stationary components of the linear motor as well as serve as a pressure vessel. Because of the structural function of the transition it is a thick, robust component and consequently its stress due to pressure loading is very low compared to its strength. Additionally the transition houses a pressure transducer and thermocouple at the duct connecting the linear compressor and cold head. The transition is made of 304 series stainless steel. Most components in the pulse tube cryocooler that contain pressure are made from 304 stainless steel for its strength and corrosion resistance. Figure 8-4 shows the motor components attached to the transition as well as the spring standoff which is simply a
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Inner Laminations Cylinder
Bobbin Bobbin
Welded Stacks
Wound Coil (covered)
Figure 8-3. Inner lamination/cylinder and outer lamination/wound coil assemblies.
stainless steel component used to position the planar spring which attaches to the flex rod of the piston/magnet assembly as shown in Figure 8-1. Figure 8-5 shows the joining of the moving and stationary parts by the installation of the piston and the attachment between the flex rod, planar spring and spring standoff. The planar spring is a heat-treatable high strength steel alloy that is first laser cut or stamped and then goes through a heat and surface treatment process to increase the fatigue strength. The spiral geometry of the spring beams as shown in Figure 8-5 create a combination of bending and torsional stress in the spring beams. The design to achieve the desired axial stiffness, fundamental frequency and stress level of the planar spring is accomplished through finite element analysis.
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Spring Standoff
Air Gap Welded Stacks
Transition
Figure 8-4. Attachment of stationary components to transition.
Planar Spring
Flex Rod
Figure 8-5. Connection of moving and stationary components.
72 The assembly in Figure 8-5 is then placed inside of a pressure vessel for testing as shown in Figure 8-6. The assembly in Figure 8-6 is considered the complete test linear compressor. The pressure vessel can (outer cylinder) is designed to meet the stress requirements of a cylindrical pressure vessel, however the fact that M8 bolts are used to join the transition, pressure vessel can and pressure vessel back end together make the wall thickness of the pressure vessel can many times greater than required for the design pressure of 25 bar. Ideally the ends of a pressure vessel approach a spherical form for the minimum use of material to withstand stress created by pressure, but the functional requirements of the back plate caused its shape to become flat and extremely thick. The back end houses the Fast Linear Displacement Transducer (FLDT) that is used to sense piston position and also fittings for both a pressure charge line and a connection to a vacuum pump. A vacuum pump is used to create a vacuum in the interior of the PTC prior to the helium charge to eliminate unwanted atmospheric gases, water vapor, and
PV End Plate
PV Can (Cylinder)
Transition
Figure 8-6. Test linear compressor assembly.
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other potential contaminants that may be released in gaseous form over time and condense in the cold head.
8.2 Cold Head The next major subassembly is the cold head. The first component of the cold head is the main rejector which is primarily a heat exchanger. The main rejector is made of a special type of copper designated C145 Tellurium copper. The standard C101 oxygen-free copper is extremely soft and difficult to machine. Tellurium copper has chemical elements added to greatly increase its machinability. While tellurium copper makes up the body or housing of the main rejector the actual heat exchanger is formed by a stack of woven wire copper screens bonded to the housing. Originally the screens were joined to the housing with low-temperature solder. But because of the nature of the way solder works, by capillary forces pulling the molten solder into narrow gaps, the screens acted like a “sponge” to the molten solder and some of the flow passages were blocked with solder. We changed that joining process to one called diffusion bonding in which the woven wire copper screens are press-fit into the housing and then taken to just below the melting temperature of copper in a vacuum furnace. The combination of pressure at the contact interfaces, temperature and time result in the copper molecules diffusing between the screens and the housing forming a high-quality, very clean bond. Also, the main rejector diffuser and plenum as called out in the Sage model are machined into the main rejector housing between the compressor connecting duct and the woven wire copper screen stack. The main rejector is shown in Figure 8-7 which also shows the water cooling jacket that serves as the ultimate heat sink for the cryocooler. The regenerator component is made up of the regenerator housing and the regenerator matrix as shown in Figure 8-8. The housing is turned from 304 stainless steel due to the fact that it’s a pressure wall. While designing the housing wall as a pressure cylinder leads to a tendency to make the wall thick, this wall is also a conduction path between the cold heat acceptor and the warm main rejector which is a parasitic loss for the cryocooler. Thus the housing wall should be as thin as possible while withstanding
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Water cooling jacket Rejector screens
Figure 8-7. Main rejector subassembly.
Containment Screen
Regenerator Disks
Regenerator Housing
Figure 8-8. Regenerator subassembly.
75 the pressure stress load. The regenerator matrix is made up of random fiber 316 stainless steel. The random form is an alternative to the woven screen form. The woven screen theoretically offers more uniform flow because of the uniform woven pattern. However woven wire screens are very expensive. Random fiber is used as a less-expensive alternative that has provided very satisfactory performance results in previous Sunpower machines. The random fiber itself also comes in a number of options regarding its form. One option is to receive strips or a sheet of material of a given wire diameter that has been sintered to a certain height to obtain a specified porosity, β, defined as Void _Volume β = . (Eq. 8-1) Total _Volume In this form the regenerator matrix is created by punching a large number of cylindrical disks (as in Figure 8-8) from the sintered sheet and then stacking these disks within the regenerator with only a minimal amount of additional compression on the disks to produce a completed regenerator matrix with the housing. By this method the final matrix porosity is slightly less than (also meaning slightly more solid than) the individual disks due to the need to provide a small amount of compression on each disk to avoid empty volumes within the regenerator. The other material option is to receive the random fiber in “loose” form meaning it has not been compressed to any specific height or porosity. In this form the matrix is created by placing a certain fraction of the entire regenerator mass into to a container with a diameter slightly larger than the actual regenerator housing and applying a tremendous amount of compression to produce a section of regenerator matrix near the final desired porosity. The remaining numbers of sections are pre-compressed and then they are stacked in the regenerator and given a final compression to the desired porosity. There are advantages and disadvantages to each approach. The pre-sintered disks require handling of a far greater number of pieces to create the regenerator matrix stack than the loose method, on the order of 100 pieces to around 5 pieces, respectively. The pre-sintered disks offer more uniformity from build to build than the loose form. But the loose form has the advantage of being to create a much wider range of porosity whereas the pre-sintered disks result in an assembled porosity very close to that porosity of the individual disks. We constructed regenerators
76 from both methods. After packing the regenerator material in the regenerator housing in either of the forms just discussed a containment screen is tack-welded to the ends of the regenerator housing to keep the matrix in place. The acceptor is very similar in function and fabrication to the main rejector. It also has a primary function of a heat exchanger and thus its housing is turned from tellurium copper and a stack of woven wire copper screens are joined to the housing. The acceptor plenum as called out in the Sage model is machined into the housing. Figure 8- 9 shows the acceptor.
Figure 8-9. Acceptor heat exchanger.
Figure 8-10 shows that the pulse tube is a very simple hollow tube except for the fact it has a tapered cross section to avoid acoustic streaming as discussed in Chapter 2. As with the regenerator housing the pulse tube is both a pressure cylinder and a parasitic conduction path from the cold acceptor to the warm secondary rejector. So the thin-
77
Figure 8-10. Pulse tube component.
walled pressure vessel design is applied to the pulse tube as well and it is turned from 304 stainless steel. The secondary rejector is very similar in function and fabrication to the main rejector. It also has a primary function of a heat exchanger and thus its housing is turned from tellurium copper and a stack of woven wire copper screens are joined to the housing. The secondary rejector plenum and diffuser as called out in the Sage model are machined into the housing. Figure 8-11 shows both sides of the secondary rejector. Figure 8-12 shows the assembled cold head components. Except for a solder joint between the acceptor and the pulse tube on the cold end all connections are made via bolted flanged with O-rings for sealing. In Figure 8-12 some other features are present for testing purposes as discussed in Chapter 11.
8.3 Inertance Assembly The inertance tube assembly is simply two tubes of different inner diameter and length joined together by a solder joint as shown in Figure 8-13. The smaller inner diameter tube is rounded at the joint between the two tubes to minimize transitional flow
78
Copper Screens
Figure 8-11. Front and back side of secondary rejector.
Secondary Rejector
Pulse Tube
Acceptor Regenerator
Main Rejector
Figure 8-12. Assembled cold head.
79 losses. The tubes can be made of basically any material but copper is used for the second, longer tube due to its low cost and availability as a plumbing water supply tube. The spring wrapped around this “second” tube in Figure 8-13 is used to prevent kinking or other non-uniformities in the coiling of the copper tube. The shorter, smaller inner diameter first tube is made of brass due to the availability of the desired inner diameter and the fact that brass is readily joined to copper. The inertance tubes are required to hold the charge pressure but their very small diameters and fairly thick walls make them very strong for the charge pressure of the cryocooler.
Solder Joint
Figure 8-13. Inertance tube assembly.
The reservoir in Figure 8-14 is primarily a pressure vessel and is thus constructed from 304 stainless steel. Two halves composed of a turned cylinder and elliptical dome are welded together to from the pressure vessel. Since the reservoir is entirely at ambient temperature there is no parasitic penalty to having a rather thick wall for safety.
80
Figure 8-14. Inertance assembly reservoir.
8.4 Pulse Tube Cryocooler Assembly Finally, Figure 8-15 shows the entire pulse tube cryocooler assembly. The cold head is covered by a vacuum flange that will be discussed in Chapter 11.
Figure 8-15. Inline pulse tube cryocooler assembly.
81 Chapter 9. Sensitivity Analysis The purpose of this section is to use Sage to vary some selected input parameters that allow us to get a feel for how these parameters affect the operation of the pulse tube cryocooler. While this effort could be exhaustive with many different variables we will focus on several individual parameters that give a bigger overall picture of the pulse tube components and subassemblies. The selected inputs are as follows: • Compressor swept volume • Connecting duct length • Regenerator porosity • Pulse tube volume • Inertance tube length • Reservoir volume. For each of these variables we will start with the input data as previously established from optimization of the PTC model and only vary the selected input. The output used for illustration of the resultant behavior will always be reduced to cooling performance, however the discussion will attempt to relate the details behind the results.
9.1 Compressor Swept Volume The compressor swept volume is the amount of the working fluid that the compressor piston displaces during the piston oscillation. The swept volume can either be expressed as the total, incorporating the entire linear displacement of the piston, also known as peak-to-peak, or swept volume amplitude which is half of the total swept volume. Care must be taken to always establish which expression is being used. For this effort we refer to swept volume amplitude and will leave the piston diameter at the baseline 23 mm, mapping the performance versus swept volume amplitude based on piston amplitude. Figure 9-1 shows the cooling power, PV power and COP versus compressor swept volume amplitude. The negative cooling power at the first point indicates that the design temperature of 77 K cannot be achieved. Remember that Sage does not model the
82
Cooling Power COP PV Power 35 480
30 420
25 360
20 300
15 240
10 180 PV Power (W) PV Power
5 120 Cooling Power (W), COP (%) COP (W), Power Cooling 0 60
-5 0 01234567 Compressor Swept Volume Amplitude (cm3)
Figure 9-1. Performance as a function of compressor swept volume amplitude.
electrical efficiency and only considers the PV input power. Notice that both cooling power and PV power increase with increasing swept volume since, by definition, increasing swept volume increases PV power in the compression space which in turn increases the PV power in the expansion space. The efficiency (coefficient of performance, COP) initially increases and then reaches a fairly steady value across a range of compressor swept volume. Experience with hardware at Sunpower has shown an effect not illustrated in Figure 9-1. At some level of compressor swept volume (PV power) the cooling power reaches a limit and further increasing the compressor swept volume (PV power) cannot produce any more cooling power. This suggests that a given cold head design has some finite capacity. The reason for this limit is the losses in the regenerator. A higher swept volume indicates a high mass flow rate which causes increasing pressure drop (friction losses) in the regenerator. These losses can dominate increased PV power and limit the PV power in the expansion space which is essentially the cooling power. When the cooling power is constant and the PV power increases the COP decreases.
83 9.2 Connecting Duct Length One advantage of pulse tube cryocoolers is that the cold head can be separated from the compressor. This has benefits in the ability to reduce vibration since there are no moving parts in the cold head, and also in packaging options. However, there are consequences to increasing the length of the connecting duct including increased dead volume and increased friction losses (pressure drop) along the length of the tube. Figure 9-2 shows the performance of the PTC with constant swept volume and increasing duct length from 20 mm to 100 mm. The sum of these two effects is discernable in Figure 9- 2. The increased volume means that a given swept volume in the compressor will create less pressure amplitude and also decreases the phase between the piston motion and the pressure amplitude. Since the PV power produced by the compressor is a function of the pressure amplitude times the sine of this phase angle (Eq. 10-1), these two effects compound the decrease in PV power produced by the compressor. Along with the reduction in PV power at the compressor the increased friction from the increased length of the connecting duct means there will be even less PV power in the expansion space for cooling due to more pressure drop. The COP is shown to fall gradually as well.
Cooling Power COP PV Power 6.50 90.0
) 6.25 87.5
6.00 85.0
5.75 82.5
5.50 80.0
5.25 77.5 PV Power (W) PV Power
5.00 75.0 Cooling Power (W), COP (% COP (W), Power Cooling 4.75 72.5
4.50 70.0 0 20 40 60 80 100 120 Connecting Duct Length (mm)
Figure 9-2. Performance as a function of connecting duct length.
84 9.3 Regenerator Porosity The function of the regenerator is to alternatively store and release the thermal energy of the working fluid passing through the regenerator between the compression and expansion spaces. An ideal regenerator has the following characteristics [3, p. 31]: • Infinite thermal conductivity in the radial and circumferential direction • Zero thermal conductivity in the axial direction • Infinitely larger thermal capacity of the solid matrix compared to the working fluid at operating temperature. In the actual design the tradeoff must be considered between adding regenerator material to increase the overall thermal capacity and the fluid-dynamic consequence of increased pressure drop. Figure 9-3 shows the result of this tradeoff which yields some optimal porosity when all other parameters in the cryocooler are held constant. In Figure 9-3 the PV power decreases in a fairly linear fashion. However, the cooling power slowly decreases before dropping rapidly. The combination of these two trends yields an optimal COP as shown.
Cooling Power COP PV Power 8 120
7 105
6 90
5 75
4 60
3 45 PV Power (W) PV Power
2 30
Cooling Power (W), COP (%) COP (W), Power Cooling 1 15
0 0 0.70 0.75 0.80 0.85 0.90 0.95 Regenerator Porosity
Figure 9-3. Performance as function of regenerator porosity.
85 Increasing the amount of solid regenerator (decreasing the porosity) results in an increasing pressure drop through the regenerator which helps to explain the trend shown in Figure 9-3. Figure 9-4 shows the pressure amplitudes in the compression space and expansion space. The difference between the two values at each point is essentially the pressure drop through the regenerator, neglecting pressure drop in the connecting duct and main rejector heat exchanger. Figure 9-4 shows that increasing the amount of solid in the regenerator (decreasing the porosity) loses its advantage at some point because of the increased pressure drop through the regenerator.
Compression Space Expansion Space
3.00E+05
2.50E+05
2.00E+05
1.50E+05
1.00E+05
Pressure Amplitude (Pa) Amplitude Pressure 5.00E+04
0.00E+00 0.70 0.75 0.80 0.85 0.90 0.95 Regenerator Porosity
Figure 9-4. Pressure drop through regenerator as a function of porosity.
9.4 Pulse (Compliance) Tube Volume As stated in Chapter 3 the function of the pulse tube is [19] “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.” The pulse tube must be long enough that no warm gas reaches the cold end
86 of the pulse tube and vice versa. Therefore the gas in the middle remains within the pulse tube and acts as a gas slug, insulating the two ends through a temperature gradient. This summary suggests that increasing the length of the pulse tube is a benefit and furthermore, increasing the length of the pulse tube decreases the conduction loss in the pulse tube wall. But once again the consequence of the increased volume that accompanies increased length must be considered. The pulse tube in this design is tapered to suppress the effects of acoustic streaming. Dealing with the effects of acoustic streaming in detail is beyond the scope of this effort. For this sensitivity analysis we shall consider the pulse tube to be straight with a diameter equal to the average diameter of the design taper and turn off the portion of Sage that calculates the acoustic streaming loss to decouple that effect from the volume effect. Figure 9-5 shows that the performance in terms of COP is actually fairly insensitive to the pulse tube length, which is a surprising finding. On the first analysis this may seem to develop from the decrease in conduction loss through the pulse tube wall as the length increases, offsetting the decrease in gross cooling power due to the
Cooling Power COP PV Power
10 100 9 90 8 80 7 70 6 60 5 50 4 40 PV Power (W) Power PV 3 30 2 20 Cooling Power (W), COP (%) COP (W), Power Cooling 1 10 0 0 45 50 55 60 65 70 75 80 85 Compliance Tube Length (mm)
Figure 9-5. Performance as a function of pulse (compliance) tube length.
87 effect of added volume that reduces PV power. The conduction loss is parasitic and subtracts from the gross cooling power. This conduction loss was next taken out of the equation by adding the Sage-calculated value to the cooling power. Thus by adding the Sage value back to the net cooling power we should be able to decouple that effect from the results. Figure 9-6 shows the results of accounting for the conduction loss. Again, surprisingly, the sensitivity of the performance to the pulse tube volume is small. While the cooling power and COP data in Figure 9-6 have moved up as expected, the general trend of the data is very similar to that in Figure 9-5.
Cooling Power COP PV Power
10 100 9 90 8 80 7 70 6 60 5 50 4 40 PV Power (W) Power PV 3 30 2 20 Cooling Power (W), COP (%) COP (W), Power Cooling 1 10 0 0 45 50 55 60 65 70 75 80 85 Compliance Tube Length (m)
Figure 9-6. Performance as a function of pulse (compliance) tube length, accounting for conduction loss.
9.5 Inertance Tube Length The inertance tube constructed in this program was composed of two individual tubes based on potential advantages proposed by Marquardt and Radebaugh [17]. The first tube connected to the secondary rejector is smaller in diameter and shorter in length
88 than the second tube connected to the reservoir. While a sensitivity analysis could also be performed on the inertance tube diameters, for this effort we will restrict the mapped variable to the lengths of each of the tubes. Figure 9-7 shows the result of holding the length of the second tube constant while varying the length of the first tube. Figure 9-8 shows the result of holding the length of the first tube constant while varying the length of the second tube. Each of these figures show a significant sensitivity to performance versus length of the respective tube. Building from information in Chapter 3, Figures 9-9 and 9-10 attempt to explain the sensitivity. Recall that optimal performance occurs when the pressure amplitude and mass flow rate are in phase in the middle of the regenerator. Figures 9-9 and 9-10 show this phase relationship in the middle of the regenerator for inertance tubes 1 and 2, respectively. An important observation is that it appears that this optimal performance is not only a function of the difference between the phase angles of the pressure amplitude and mass flow rate, but also the absolute phase angle where their difference is minimized. Another way to say this is that it not only depends on the range (difference) between the
Cooling Power COP PV Power
10 100 9 90 8 80 7 70 6 60 5 50 4 40 PV Power (W) Power PV 3 30 2 20 Cooling Power (W), COP (%) COP (W), Power Cooling 1 10 0 0 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 Inertance Tube 1 Length (m)
Figure 9-7. Performance as a function of the first inertance tube (IT1) length.
89
Cooling Power COP PV Power
10 100 9 90 8 80 7 70 6 60 5 50 4 40 PV Power (W) Power PV 3 30 2 20 Cooling Power (W), COP (%) COP (W), Power Cooling 1 10 0 0 1.75 2.00 2.25 2.50 2.75 3.00 3.25 Inertance Tube 2 Length (m)
Figure 9-8. Performance as a function of the second inertance tube (IT2) length.
COP Pressure Amplitude Mass Flow Rate
10 100 9 90 8 80 7 70 6 60 5 50
COP (%)COP 4 40 3 30
2 20 (Degrees) Shift Phase 1 10 0 0 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 Inertance Tube Length (m)
Figure 9-9. Performance as function of pressure-mass flow phase relationship for IT1
90
COP Pressure Amplitude Mass Flow Rate
10 100 9 90 8 80 7 70 6 60 5 50
COP (%)COP 4 40 3 30
2 20 (Degrees) Shift Phase 1 10 0 0 1.75 2.00 2.25 2.50 2.75 3.00 3.25 Inertance Tube Length (m)
Figure 9-10. Performance as function of pressure-mass flow phase relationship for IT2.
two phases, but also the average of the two phases. Table 9-1 demonstrates important values from Figure 9-9 and 9-10. As expected, the same average and difference phase relationships between the two tubes results in about the same COP. A length of 350 mm for IT1 with a phase average of 54.5° and phase difference of 4.1° yields a COP of 6.5% while a length of 2.80 m for IT2 with a phase average of 54.3° and phase difference of 4.6° also yields a COP of 6.5%. Now notice the results for IT2 of lengths 2.00 and 3.20 m. Both have a phase difference of ~30°, yet the 2.00 m length has an average phase of 77.2° with a COP of only 2.4% while the 3.20 m length has an average phase of 50.5° with a better COP of 3.4%. Finally notice the results of the 500 mm IT1 length and the 2.40 m IT2 length. The IT1 has an average phase of 50.3° with a phase difference of 15.4° and a COP of 6.2% while IT2 has a higher average phase of 63.9° with a smaller phase difference of 13.3° yielding a COP of only 5.5%. This data suggests that not only is it important for the pressure amplitude and mass flow rate to be in phase near the middle of the
91 Inertance Tube 1 (Figure 9-9) Length Average Difference COP (mm) θ (Deg.) Θ (Deg.) (%) 50 65.4 12.2 4.8 200 60.1 5.4 5.9 350 54.5 4.1 6.5 500 50.3 15.4 6.2 650 49.4 25.7 4.9 800 48.6 35.1 3.3 950 48.1 44.2 0.9 Inertance Tube 2 (Figure 9-10) Length Average Difference COP (mm) θ (Deg.) Θ (Deg.) (%) 2.00 77.2 30.3 2.4 2.20 69.4 21.2 4.5 2.40 63.9 13.3 5.5 2.60 59.3 5.7 6.1 2.80 54.3 4.6 6.5 3.00 50.5 18.4 5.8 3.20 50.5 30.7 3.4
Table 9-1. Summary of points from Figures 9-9 and 9-10.
regenerator but there is some optimal target phase at which the two should be in phase together. This target is probably specific to many other parameters of the design such as PV amplitude and target temperature. For the design involved here that target seems to be around 55° as can be seen in Figures 9-9 and 9-10. While the comparison of these two phase angles was only investigated for the inertance tube length sensitivity analysis this relationship could be investigated for each of the previous analyses as well. This relationship is the underlying cause of the results shown previously with the parameters discussed in those previous analyses contributing in some fashion to this key relationship. For example, changing regenerator porosity would change this relationship in the middle of the regenerator. A decision was made to intentionally save the demonstration of this relationship on the performance until the inertance tube analysis in order to tie into the theory discussion about the importance of phase relationships.
92 9.6 Reservoir Volume The purpose of the reservoir is to maintain a constant pressure by containing a large mass of gas that does not create a substantial pressure swing as mass flows into and out of the reservoir [19]. Thus ideally the inertance reservoir would be extremely large. Yet practical design requirements of the pulse tube cooler force this reservoir volume to be minimized without affecting the performance. Figure 9-11 shows that smaller reservoir volumes decrease the performance and that as the volume increases the performance levels out and reaches a point of diminishing returns. The explanation for this is found in Figure 9-12 which shows the ratio of the pressure amplitude in the reservoir to the pressure amplitude in the compression space. As the pressure amplitude in the reservoir as a fraction of pressure amplitude in the compression space decreases the performance trend becomes level. This reflects the idea that the reservoir should not experience significant pressure amplitude as a result of mass flowing into and out of the reservoir. Most importantly Figure 9-12 gives a good indication of the minimum reservoir volume without affecting performance.
Cooling Power (W) COP PV Power
10 100 9 90 8 80 7 70 6 60 5 50 4 40 PV Power(W) 3 30 2 20 Cooling Power (W), COP (%) COP (W), Power Cooling 1 10 0 0 0.00E+00 2.00E-04 4.00E-04 6.00E-04 8.00E-04 1.00E-03 1.20E-03 1.40E-03 Reservoir Volume (m3)
Figure 9-11. Performance as a function of reservoir volume.
93
Percent Pressure Amplitude COP
20 10 18 9 16 8 14 7 12 6 10 5
8 4 (%)COP
Amplitude (%) 6 3 4 2 Compression Space Pressure Pressure Space Compression
Reservoir Pressure Amplitude / / Amplitude Pressure Reservoir 2 1 0 0 0.00E+00 2.00E-04 4.00E-04 6.00E-04 8.00E-04 1.00E-03 1.20E-03 1.40E-03 Reservoir Volume (m3)
Figure 9-12. Performance as a function of pressure amplitude ratio.
94 Chapter 10. Subassembly Characterization The Sage simulated predictions of the behavior of components and subassemblies in the cryocooler can be compared and/or correlated to the hardware by conducting tests at the component and subassembly level. This is useful in providing the opportunity to adjust Sage inputs such as fluid friction multipliers to correlate with experimental results at an assembly level that is not overwhelming in terms of the number of input factors that can affect the results. Three components/subassemblies were chosen for characterization in this effort: the linear compressor, the regenerator and the inertance subassembly.
10.1 Linear Compressor Subassembly The first correlation involves attaching the linear compressor directly to an adiabatic reservoir volume and comparing the pressure amplitude created by a certain piston amplitude. A pressure transducer is installed in the compression space and this transducer is used in conjunction with the Fast Linear Displacement Transducer (FLDT) that measures the piston position to provide a calculation of PV power in the compression space. The equation used for the calculation is
PV = P1 ⋅V&1 ⋅cos(φ) = P1 ⋅π ⋅ f ⋅ X P ⋅ AP ⋅sin(θ ) (Eq. 10-1)
where P1 is the measured pressure amplitude, V&1 is the volumetric flow rate, f is the
operating frequency, XP is the measured piston displacement amplitude, AP is the frontal area of the piston, φ is the phase angle between the volumetric flow rate and pressure amplitude, and θ is the phase angle between the piston amplitude and the pressure amplitude (θ = 90° - φ). This effort is actually a comparison of how well Sage predicts the pressure wave created by the linear compressor rather than an attempt to correlate factors in Sage to fit experimental data. In comparing the simulation and experiment we have the option of using either the piston or the PV power as the input datum. This does not mean that these two variables are independent of each other. The PV power is directly a function of piston amplitude. In some testing the pressure wave data was recorded. Thus we can adjust the piston amplitude in Sage until the Sage PV power matches the experimental PV power. But in some of the experimental mapping full data points, including pressure
95 wave data, were not recorded. In these comparisons we can simply use the piston amplitude as the input to match between Sage and experiment. Figure 10-1 shows the Sage model of the experimental setup. Table 10-1 gives the important input parameters for the model and Table 10-2 compares the Sage and experimental results. Table 10-2 shows good comparison between Sage and experiment in predicting pressure amplitude with equal piston amplitude. The pressure phase comparison is not as close. The asterisk beside the pressure phase points out that the value of 4.3° is an average of measuring the phase angle using a Sunpower-built phase meter (3.6°) and by calculating the phase using the cursor function on the time scale of a digital oscilloscope to compare signal offsets (5.0°). As with most instrumentation it would seem that the Sunpower phase meter would lose accuracy in the very low range of the scale from 0° to 180°. The oscilloscope-calculated value is very close to the experimental value. The difference in pressure phase carries over into a difference in PV power. More piston amplitude points should have been mapped to see how experiment tracks Sage but unfortunately only this data point was taken.
Figure 10-1. Sage model of linear compressor characterization.
96
Frequency (Hz) 60 Compressor Test Working Gas Helium 1 Pressure Source Charge Pressure (Pa) 2.51E+06 Moving Shell OD (m) 2.30E-02 Mass (kg) 2.12E-01 2.3 Constrained Piston 2 Constrained Amplitude (m) 6.06E-03 Piston and 2.3.1 Positive- Cylinder 2 4.16E-04 Facing Area (m ) 2.4 Spring Stiffness (N/m) 1.88E+04 Mean-Flow Length (m) 1.00E-02 3 Compression 3 Space Mean Volume (m ) 7.69E-06 Wetted Surface (m2) 1.30E-03 Length (m) 1.98E+01 4 Transition Duct Tube Internal Diameter (m) 6.00E-03 Length (m) 2.28E-02 5 Adapter Tube Internal Diameter (m) 4.80E-03 Length (m) 1.08E-01 6 Connecting Duct Tube Internal Diameter (m) 4.40E-03 Mean-Flow Length (m) 7.50E-02 7 Test Reservoir Mean Volume (m3) 3.82E-05 Wetted Surface (m2) 1.20E-02
Table 10-1. Important Sage inputs for compressor test simulation.
Sage Experiment % Error Piston Amplitude (mm) 5.75 5.75 0.0% Pressure Amplitude (Pa) 1.94E+05 2.00E+05 3.0% Pressure Phase (Deg) 5.2 4.3 20.9% PV Power (W) 8.0 6.8* (5.0/3.6) 17.6%
Table 10-2. Comparison of Sage and experimental results for compressor test.
10.2 Regenerator There are two important characteristics of the regenerator: its heat transfer performance and pressure drop. Evaluating the thermal performance of a regenerator is an extremely difficult task which was attempted on this program with inconclusive results. However the evaluation of the pressure drop is straightforward. For this effort
97 the only instrumentation needed was a pressure gauge and a flow meter. A special transition test piece was fabricated to eliminate the effect of the flow area transition from a small diameter flexible tube to the regenerator entrance. This test piece had the same ID as the regenerator sample but was significantly long so that flow was fully developed at the entrance to the regenerator sample. The pressure gauge was installed at the entrance to the regenerator while the flow meter was installed prior to the flow area transition test piece. The other end of the regenerator was open to atmosphere. A range of pressures and mass flow rates were experimentally recorded. Nitrogen was the gas used for this experiment. For correlation this test setup was modeled in Sage as shown in Figure 10-2. The model consists of a pressure source set to atmospheric pressure, a mass flow pump to flow the gas through the regenerator sample, a generic cylinder volume set to a very high volume to represent the venting of the gas to atmosphere, and a tube bundle return path with a very large flow area simply because Sage cannot model an open flow loop. The gas used in the model was of course nitrogen since that was the gas used in the experiment. The main input to the mass flow pump is the mass flow rate and the pressure drop through the regenerator sample was a model output.
Figure 10-2. Sage model for steady flow pressure drop testing.
98 Figure 10-3 shows the results of the comparison between Sage and experiment. Sage consistently tends to predict the pressure drop to be slightly higher than the experimental result. Again, one reason for performing component testing was to attempt to correlate the Sage model to experiment. Within the matrix gas component of the regenerator model is a friction multiplier, Fmult, which is used as an “empirical multiplier for viscous pressure drop”[34]. The next step was to use Sage’s optimization feature to optimize this friction multiplier within the regenerator pressure drop model to match the experimental pressure drop. The results of this exercise are shown in Table 10-3. The average adjusted friction multiplier was 0.8. The Sage map of the pressure drop was then repeated with a constant friction multiplier of 0.8 for all points. These results are shown in Figure 10-4. This adjustment results in great agreement between Sage and experiment. This friction multiplier will then be maintained in the regenerator portion of the entire cryocooler model.
Experimental Sage
3.0
2.5
2.0
1.5
1.0 Pressure Drop (bar) Drop Pressure 0.5
0.0 0 50 100 150 200 250 300 Gas Mass Flow Rate (g/min)
Figure 10-3. Comparison between Sage and experiment for steady flow regenerator pressure drop testing.
99
Mass Flow Experimental Sage Adjusted (g/min) Pressure Drop (bar) Friction Multiplier 0 0.00 0.00 51 0.41 0.76 101 0.83 0.79 152 1.24 0.81 205 1.65 0.82 258 2.03 0.81
Table 10-3. Adjusting Sage friction multiplier to achieve experimental pressure drop.
Experimental Sage
3.0
2.5
2.0
1.5
1.0
0.5 Pressure Drop w/ f = 0.8 (bar) 0.8 = f w/ Drop Pressure
0.0 0 50 100 150 200 250 300 Gas Mass Flow Rate (g/min)
Figure 10-4. Comparison between Sage and experiment for steady flow regenerator pressure drop testing after setting Sage friction multiplier to 0.8.
10.2.1 Calculation of Regenerator Pressure Drop The analytical calculation of the pressure drop through the regenerator starts with the Fanning pressure drop equation [3, p. 46]
L u2 ∆P = f ⋅ ⋅ ρ ⋅ (Eq. 10-2) dh 2
100
where f is the friction factor, L is the length of the regenerator, dh is the hydraulic diameter, ρ is the average density of the fluid, and u is the fluid velocity. The hydraulic diameter for is generically defined as36
Af Af dh = 4⋅ = 4⋅ (Eq. 10-3) Pw Aw L
where Af is the flow area and Pw is the wetted perimeter which is further expanded using
the wetted area Aw and the regenerator length L. For the case of a porous regenerator
Af = β ⋅ Afr (Eq. 10-4)
where β is the regenerator porosity and Afr is the frontal area of the regenerator. The wetted area of the regenerator is calculated as
Aw = π ⋅dw ⋅ Leqwire (Eq. 10-5)
where dw is the diameter of the regenerator matrix wire and Leqwire is the equivalent length of the matrix wire found from
(1− β )⋅ Afr ⋅ L Leqwire = . (Eq. 10-6) π ⋅d 2 4 w The numerator in Eq. 10-6 is the solid volume of the matrix and the denominator is the cross-sectional area of the wire. Combining the previous four equations and rearranging yields β d = ⋅d . (Eq. 10-7) h (1− β ) w The fluid velocity is related to the mass flow rate through the following relationship m m u = & = & . (Eq. 10-8) ρ ⋅ Af ρ ⋅ β ⋅ Afr The Fanning pressure drop in Eq. 10-2 is a general calculation for flow through any geometry. The determination of the appropriate friction factor, f, available from published data, is a function of the flow section geometry as well as the flow regime. Within the classification of porous media the friction factor is also a function of the matrix form, from packed spheres to wire screens to corrugated metal ribbons. The matrix form used in this hardware was random wire mesh which is similar to but different
101 than woven screens. For this investigation we will compare the pressure drop versus flow rate found by three different methods: using the Fanning pressure drop equation with friction factor published by Kays and London37 for a woven screen matrix, using the Fanning pressure drop equation with friction factor published by Gedeon and Wood for a random fiber matrix, and the results predicted by Sage with a random fiber matrix and the friction multiplier set to 1. The calculation of pressure drop using Eq 10-2 is an iterative calculation due to the presence of density in the equation. The density represents the average density in the matrix which is a function of pressure and temperature (in the hardware steady flow test the temperature was constant at room temperature). Thus an initial guess is made for the average density to calculate the pressure drop. With the pressure drop the pressures at the inlet and exit (atmosphere) of the matrix are now known. The average pressure is then used to calculate an average density, which yields a new pressure drop. This process is repeated until the average density no longer changes. While the regenerator porosity of the actual hardware was 82.5%, the published data from Kays and London is presented in 5% intervals. Thus a porosity of 80% was selected for this comparison. The general form for the Kays and London friction factor is
C1 0.88 f = ( )⋅(1+ C2 ⋅Red ) (Eq. 10-9) Red
where C1 and C2 are constants based on the matrix porosity and Red is the Reynolds number based on the hydraulic diameter, ρ ⋅u ⋅d Re= h (Eq. 10-10) d µ where all variables have already been defined except the fluid viscosity, µ, evaluated at the same conditions as ρ. The other empirical friction factor comes from a NASA Contractor Report by Gedeon and Wood38 in which an oscillating-flow test rig was used to experimentally derive friction factor and Nusselt number relationships for various regenerator forms including random fiber matrices. The testing included the following random wire samples:
102 • 2 mil inconel, wire diameter 50.8 micron (0.002 in), porosity 0.688 • 1.5 mil stainless steel, wire diameter 38.1 micron (0.0015 in), porosity 0.730 and 0.748 • 1 mil stainless steel, wire diameter 25.4 micron (0.001 in), porosity 0.820 • 0.5 mil stainless steel, wire diameter 12.7 micron (0.0005 in), porosity 0.841. The tests were conducted over a Reynolds number range of 0.8 to 1400. The result of these experiments was the following friction factor correlation
192 −0.067 f = + 4.53⋅Red . (Eq. 10-11) Red The comparison of these three methods, including the Sage model, is shown in Figure 10-5. All three correlations are in very close agreement. It is expected that the Sage curve and the G&W curve would be close since Sage makes use of the G&W correlation in its calculations. The K&L correlation was derived from steady flow experiments while the G&W correlation was derived from oscillatory flow experiments.
Sage G&W Correlation K&L Correlation
3.5
3.0
2.5
2.0
1.5
1.0 Pressure Drop (bar) Drop Pressure
0.5
0.0 0 50 100 150 200 250 300 Gas Mass Flow Rate (g/min)
Figure 10-5. Comparison of pressure drop calculations and Sage prediction.
103 The agreement between these two calculations suggests that the correlation traditionally used for woven screens under steady flow conditions is a good approximation to the behavior of random wire under oscillatory flow conditions.
10.3 Inertance Assembly For the inertance assembly two different characterization tests were performed: steady flow pressure drop through the two inertance tubes and dynamic characterization of the inertance assembly, including reservoir, connected directly to the liner compressor. The first test allows us to again correlate the Sage friction multiplier in the two tubes if necessary. Then the friction factor can be carried over into the dynamic testing where the pressure waves in both the compression space and the reservoir are compared to Sage simulations for various piston amplitudes. The pressure drop test was conducted in exactly the same manner and with the same instrumentation as the regenerator pressure drop test. The Sage model was set up in the same manner as in Figure 10-2 with the regenerator sample replaced with the two inertance tubes in series. Figure 10-6 shows the comparison between Sage, with the friction multiplier set to 1, and the experiment. Once again Sage over-predicts the pressure drop at a given flow rate. Following the same Sage optimization procedure as discussed in the regenerator section yields an average Sage friction multiplier of 0.67 to achieve the experimental results. This friction multiplier is applied to each tube. For the final subassembly characterization the inertance tubes and reservoir were attached directly to the linear compressor. The linear compressor was set at a given piston amplitude and the corresponding pressure response both in the compression space and the reservoir were recorded. Figure 10-7 shows the Sage model of this experiment. The comparison of the experimental results and Sage predictions are shown in Table 10-4 and Figure 10-8. Figure 10-8 graphs the pressure response in the compression space and reservoir volume with the comparison presented by breaking the pressure phasor of the form “Amplitude@Phase Angle” into the cosine and sine components. The x-axis is the cosine component (Amplitude*cos(Phase Angle)) of the phasor and the y- axis is the sine component (Amplitude*sin(Phase Angle)). Each data point can be
104
Experimental Sage
2.5
2.0
1.5
1.0 Pressure Drop (bar) Drop Pressure 0.5
0.0 0 20 40 60 80 100 120 140 160 180 Gas Mass Flow Rate (g/min)
Figure 10-6. Comparison between experiment and Sage prediction for pressure drop through inertance tubes.
Figure 10-7. Sage model of inertance assembly characterization test.
105
Compression Space Reservoir Volume
Experiment Sage (Fmult = 0.67) Experiment Sage (Fmult = 0.67)
Xamp Pamp Pphi Pamp Pphi Pamp Pphi Pamp Pphi (mm) (bar) (deg) (bar) (deg) (bar) (deg) (bar) (deg) 1 0.97 154 1.22 157 0.078 -25.1 0.092 -17.8 2 1.72 133 2.20 141 0.140 -34.6 0.162 -30.4 3 2.38 124 3.02 129 0.183 -40.7 0.220 -39.1
Table 10-4. Comparison of pressure response in compression space and reservoir volume between experiment and Sage with Fmult = 0.67.
visualized as the head of an arrow that starts at the origin expressing the pressure phasor of interest as demonstrated on the results in the compression space. The arrows were not included in the results of the reservoir volume to avoid cluttering the graph. Table 10-4 and Figure 10-8 show that the results compare better in the reservoir volume than in the compression space. A general comment is that the pressure response
Experimental Sage Experimental Sage
2.5
2 Compression space 1.5
1 (bar)
0.5
Increasing Xamp Increasing Xamp 0 -2 -1.5 -1 -0.5 0 0.5 Pressure Amplitude*sin(Phase Angle) Amplitude*sin(Phase Pressure Reservoir volume -0.5 Pressure Amplitude*cos(Phase Angle) (bar)
Figure 10-8. Comparison of pressure phasors in compression space between experiment Sage and with Fmult = 0.67.
106 in the reservoir is much smaller than the pressure response in the compression space. This makes sense recalling that the purpose of the reservoir volume is to maintain a constant pressure as mass flows into and out of the volume. The angles of the pressure phasors in the compression space place those arrows in the upper left quadrant while angles of the pressure phasors in the reservoir volume place those arrows (again, left out for clarity) in the bottom right quadrant. The direction of increasing piston amplitude is also called out in the graph. The direct comparison at a given piston amplitude is associated with the color and type of line on the arrow, for example the two solid black arrows are the comparison of the 1 mm piston amplitude results. Overall the correlation between results is not as good as those achieved in the steady flow testing conducted up to this point, especially after adjusting the friction multiplier found from steady flow testing to theoretically improve the agreement. In the compression space the Sage predictions seem to be less accurate as the piston amplitude increases. In the reservoir volume it appears that at least the pressure angle seems to become more accurate as the piston amplitude increases. The question then arises about the validity of using steady flow testing to adjust factors that will be applied to oscillating flow testing. For the exercise, the modeling of this dynamic test was repeated with the Sage friction multipliers within the inertance tubes reset to 1. The results of this are shown in Table 10-5 and Figure 10-9 with the same formatting as Table 10-4 and Figure 10-8. The results are in better agreement with the friction factors reset to 1. This situation with the friction multiplier will be discussed in more detail in the system testing section.
107
Compression Space Reservoir Volume
Experiment Sage (Fmult = 1) Experiment Sage (Fmult = 1)
Xamp Pamp Pphi Pamp Pphi Pamp Pphi Pamp Pphi (mm) (bar) (deg) (bar) (deg) (bar) (deg) (bar) (deg) 1 0.97 154 1.19 150 0.078 -25.1 0.089 -23.4 2 1.72 133 2.10 133 0.140 -34.6 0.150 -35.6 3 2.38 124 2.87 121 0.183 -40.7 0.200 -43.2
Table 10-5. Comparison of pressure response in compression space and reservoir volume between experiment and Sage with Fmult = 1.
Experimental Sage Experimental Sage
2.5
Compression 2 space 1.5
1 (bar)
0.5
Increasing Xamp Increasing Xamp 0 -2 -1.5 -1 -0.5 0 0.5 Pressure Amplitude*sin(Phase Angle) Amplitude*sin(Phase Pressure Reservoir volume -0.5 Pressure Amplitude*cos(Phase Angle) (bar)
Figure 10-9. Comparison of pressure phasors in compression space between experiment Sage and with Fmult = 1.
108 Chapter 11. Cryocooler Testing and Sage Comparison The Sage model of the entire cryocooler was shown previously in Figure 7-14. The friction multipliers determined for the regenerator and inertance tubes are applied to the model of the entire cryocooler. The Sage model was run to match data points from two different performance maps for the cryocooler. The first test maps the cooling power of the cryocooler at different cold temperatures with the input power held constant at 100
We. The second map holds the cold temperature at 77 K and maps the cooling power at different input power. First the testing method and setup of the cryocooler will be discussed. The cooling power is measured by attaching a band resistance heater to the acceptor component. The temperature is measured using a silicon diode. A first-law energy balance on the cold head shows that when the temperature of the cold head reaches a steady value then the power in the resistance heater is equal to the gross cooling power of the cryocooler minus any parasitic heat loads, or the net cooling power. Some parasitic loads can be minimized but others cannot be avoided. The conduction losses from the acceptor at the cold temperature to the main and secondary rejectors at ambient temperature through the regenerator and pulse tube walls are examples of conduction losses that cannot be eliminated. The cold head components are contained within a vacuum chamber attached to a vacuum pump to eliminate condensation of water vapor in the air as a parasitic heat load on the cold head and to eliminate molecular conduction. Several layers of aluminized Mylar, a very highly reflective foil material, are wrapped around the cold head to reduce parasitic radiation losses. After consideration of these types of loads present during testing the resultant experimental cooling power is an accurate indication of the cooling power that will be available to cool an active heat load at a given cold temperature. The test setup is shown in Figures 11-1 through 11-3. The cold head is positioned with the cold end of the pulse tube below the warm end of the pulse tube. This is done to reduce free convection losses within the pulse tube itself. Since the pulse tube is simply a hollow tube containing pressurized helium it is subject to free convection if cold gas is located above warm gas in the presence of gravity. A water cooling loop is used to
109
Temperature Diode Mylar Radiation Shield Band Heater
Figure 11-1. Experimental setup.
Added Mylar Shield
Figure 11-2. Experimental setup (with multi-layer insulation).
110
Vacuum Vessel
Water Cooling Loop
Figure 11-3. Experimental setup (with vacuum vessel).
reject the heat from both the primary and secondary rejector. Various instrumentation are used to measure the following parameters: • Piston amplitude • Pressure amplitude in the compression space • Phase angle of the pressure in the compression space with respect to piston motion • Temperature of the gas in the compression space • Cold end temperature • Electrical power input to cryocooler • Electrical power input to resistance heater (active load) • Phase angle of the electrical current with respect to the piston motion • Pressure amplitude in the reservoir volume
111 • Phase angle of the pressure in the reservoir volume with respect to piston motion • Vacuum level in vacuum chamber. First the baseline test of the cryocooler is established. The design point is 77K cold end temperature and 100 We input. Table 11-1 shows the detailed comparison between the baseline performance point and Sage. Again, the friction multipliers determined in steady flow testing were applied to this model.
Experiment Sage (Xamp) Sage (PV)
Xamp (mm) 5.75 5.75 5.40
Input (We) 100 Not modeled Not modeled
Pamp (bar) 2.00 2.46 2.37
φcs (deg) 54.6 48.4 47.2 PV (W) 73.4 82.8 73.4 Cooling Power (W) 4.9 7.0 5.9
Pres (bar) 0.19 0.17 0.16
φres (deg) -123 -125 -127
Table 11-1. Comparison of baseline performance point of cryocooler with Sage friction multipliers determined from steady flow testing.
The baseline experimental performance is 4.9 W of cooling power with 73.4 W PV power input. Recall that Sage does not account for the efficiency of the alternator and thus PV power is discussed rather than electrical input power. With the piston amplitude matched between Sage and experiment, Sage over-predicts the PV power by 10 W mostly due to over-predicting the pressure amplitude. The cooling power in this case is more than two watts higher in Sage than experiment at 7.0 W with 82.8 W PV power input. In order to have a more realistic comparison the piston amplitude was reduced to achieve the experimental PV power. This comparison is shown in the last column of Table 11-1. Sage still predicts a higher cooling power by one watt, or 20%. The pressure response in the reservoir seems to be fairly close.
112 In the section on the dynamic testing of the inertance assembly we explored comparing the Sage and experimental results after resetting the Sage friction multiplier to 1 rather than the value determined during steady flow testing. That experiment is also worth consideration here. Not only will the friction multiplier for the inertance tubes be reset, but also the friction multiplier for the regenerator. Table 11-2 shows that the results are in closer agreement for all variables except for the reservoir pressure amplitude. All other variables are within 10% of the experimental value. This again suggests that some thought must be placed on the validity of using steady flow tests to correlate models that represent oscillating flow experiments.
Experiment Sage
Xamp (mm) 5.75 5.55
Input (We) 100 Not modeled
Pamp (bar) 2.00 2.22
φcs (deg) 54.6 49.4 PV (W) 73.4 73.3 Cooling Power (W) 4.9 5.4
Pres (bar) 0.19 0.14
φres (deg) -123 -123
Table 11-2. Comparison of baseline performance with friction multiplier reset to 1.
Next the ability of Sage to track the experimental performance of the pulse tube cryocooler will be mapped for two different tests as discussed earlier. As a result of Table 11-2 the friction multipliers will remain at 1 for the rest of the effort. Additionally, the PV power will be the matched input rather than the piston amplitude. The first
mapping involves holding the input power at 100We and measuring the cooling power at various temperatures. During the experiment the PV power was not measured at every point. Therefore the assumption will be made that the same PV/electrical power ratio is maintained for each point. The results of this mapping are shown in Figure 11-4 which indicates that the general experimental trend is tracked well by Sage. The second
113 mapping holds the cold temperature constant at 77 K and measures the cooling power as a function of input power. Once again for the purpose of this comparison the PV/electrical input ratio was assumed to be constant over the range of data points. This comparison is presented in Figure 11-5 which shows that Sage predicts higher performance at higher input power and lower performance at lower input power. In general Figure 11-5 shows fairly good tracking between Sage and experiment but does appear to deviate at higher temperatures. The deviation may be a result of the assumption that the ratio of PV power to electrical power is constant when in fact it may vary over the range of input power.
Experimental Sage
10 100W Input 9 e (73.4W PV) 8 7 6 5 4 3 Cooling Power (W) Power Cooling 2 1 0 40 50 60 70 80 90 100 110 Cold Temperature (K)
Figure 11-4. Tracking between Sage and experiment for cooling power vs. temperature curve with constant input power.
114
Experimental Sage
7 77K Cold Temperature 6
5
4
3
Cooling Power (W) Power Cooling 2
1
0 40 50 60 70 80 90 100 110 120 Input Power (W)
Figure 11-5. Tracking between Sage and experiment for cooling power vs. input power curve with constant cold end temperature.
115 Chapter 12. Summary, Discussion and Conclusions
12.1 Summary A pulse tube cryocooler is a refrigeration device that removes heat from one space and rejects that heat to another space. The term cryocooler defines that the cold temperature achieved by the refrigeration cycle is below 120 K (-153° C). There is a theoretical limit to the efficiency, or coefficient of performance (COP), of the PTC as a function of the cold and ambient temperatures called the Carnot COP. Real hardware typically achieves some percentage of the theoretical Carnot COP. The refrigeration, or cooling power, in a PTC is created by the expansion of helium gas within the cryocooler. The main components of the PTC are: the linear compressor (or pressure wave generator) that creates pressure and volume flow oscillations; the cold head containing heat exchangers and the pulse tube component itself that transfer the heat and/or work to create the cooling cycle; and the inertance assembly (generically called the impedance assembly) that establishes a required phase relationship between the pressure and volume flow within the cold head. Pulse tube cryocoolers have recently gained popularity as an efficient, reliable cooling technology for various applications that have previously been served by Stirling and Gifford-McMahon cryocoolers. An observation during the development of GM cryocoolers in the 1960’s led to the discovery of the first type of pulse tube cryocooler called the basic PTC. Continued research over the next few decades led to the orifice PTC that greatly advanced the state of the art. Developments in the 1990’s including the double inlet PTC and the use of inertance tubes established the technology as a rival to the efficiency of Stirling cryocoolers with potential advantages including higher reliability as a result of a single moving component. In a Stirling cryocooler there is a component called a displacer whose function is twofold: to create the required phase relationship between pressure and volume flow oscillations and to support a temperature gradient between the cold and ambient ends of the cryocooler. In a pulse tube cryocooler the functions of the displacer are replaced by the pulse tube, also called the compliance tube, that supports the temperature gradient and
116 also transmits acoustic power to the inertance assembly that achieves the other function of the displacer, the phase relationship. Also in a Stirling cryocooler the expansion work at the cold end of the gas work space is recovered in the form of boundary pressure- volume (PV) work through the displacer. This creates the ideal efficiency of the Stirling which is the same as the ideal Carnot efficiency as defined in Equation 3-4. Notice the Tc term in the denominator which represents the ideal recovery of the expansion space work as just discussed. In the PTC the expansion space work is not recovered and is purely dissipated as flow friction. Since this expansion space work is not recovered Equation 4-
6 shows that there is no Tc term in the denominator illustrating that the PTC is more suited for lower temperatures in the cryogenic region to compete with the efficiency of the Stirling cryocooler. At higher temperatures the PTC has a disadvantage in ideal efficiency. The employer of the author, Sunpower, Incorporated, has been developing pulse tube technology for over six years. This development has been a partnership between Sunpower and Gedeon Associates. The partnership combines the established commercial linear compressor technology and R&D experience of Sunpower with the pulse tube design and analysis experience of David Gedeon. The key tool in the development of the PTC has been Sage simulation software authored by David Gedeon. Sage offers the capability to create models based on drag-and-drop elements or building blocks that connect to form an operational system. Sunpower and Gedeon Associates have designed, fabricated and tested PTC hardware through awards from NASA Goddard Space Flight Center. This paper highlighted the use of Sage software in this development effort, discussed the construction of the PTC, used Sage to establish the sensitivity of the performance of the PTC to several operational parameters, and finally compared the experimental results of the PTC to the actual performance of the PTC hardware.
12.2 Discussion and Conclusions Figures 11-4 and 11-5 show that Sage software does a very respectable job of predicting the experimental performance of the inline pulse tube cryocooler over a range
of cold temperature and input power. At the design point of 77 K, 100 We Figure 11-4
117 shows that Sage over-predicts the cooling performance by 11%. This seems to be a fairly common agreement in the use of Sage at Sunpower which is generally expressed that Sage predicts performance about 10% higher than experimental results. This applies to other Stirling cryocoolers as well as Stirling engines. One of the main purposes of a simulation program is to allow the designer to explore different design options and develop knowledge of the behavior of a device without the time and expense associated with the building and testing of the device. The value of being able to predict the performance of a machine with relative confidence at the start of the program, long before hardware is built, is tremendous. It is critical to understand that the results provided by any model can only be as accurate as the input information provided to the model. It is extremely difficult to verify every input of every parameter in Sage. Some parameters are straightforward such as dimensions and temperatures which can be measured. Other parameters such as minor pressure losses between components and convection heat transfer multipliers are not as easy to assign a knowledgeable value. Along with the attempt to increase accuracy in the capability of a model such as Sage by trying to account for every possible physical occurrence comes increasing complexity that requires intelligent input to make use of that accuracy. Just as there is always compromise in the design of a complex device such as a pulse tube cryocooler there is also a compromise between the accuracy and the utility of modeling Software. It is easy to get lost in the details of the model and forget about the value of general behavior and trends. Sage seems to offer that compromise. The most valuable exercise for me personally in this effort was the Sensitivity Analysis in Chapter 9 which provides a very nice “feel” for the behavior of the system based on various parameters. The final design parameters were derived by David Gedeon and Gary Wood during the SBIR Phase I program by running a number of optimizations with many free parameters and gradually fixing parameters and further optimizing the remaining variables. If the results of the sensitivity analysis could be compared with these final design parameters, which have not been conspicuously presented in the interest of protecting Sunpower’s efforts, it could be seen that these values are the results of optimization.
118 The results of Figure 9-1 are intuitive that providing more work in the form of more swept volume will provide more cooling power. The performance is quite sensitive to the porosity of the regenerator as shown in Figures 9-3 and 9-4. Some results, as in Figures 9-9 and 9-10, emphasize ideas that are stated often such as the phase relationship between the pressure and mass flow rate. However, it was unexpected that the performance seems to not only be a function of the phase difference between pressure and mass flow in the middle of the regenerator but also possibly a function of the mean phase between the two as well. It would be interesting to investigate the results shown in Figures 9-9 and 9-10 in Stirling cryocoolers and Stirling engines to see if the mean phase between these two parameters seem to be as critical as the phase difference. The central purpose of the inertance assembly is to create flexibility in the phase relationship that can be achieved and Figures 9-9 and 9-10 demonstrate this function well. Some results, as in Figures 9-5 and 9-6 are surprising. I had suspected that the volume of the pulse tube would have a substantial effect on the phase relationship and therefore the performance but these figures suggest that this parameter is not entirely critical. The discovery from Figures 9-9 and 9-10 helps to explain why the performance is not as sensitive to pulse tube length (volume) as I had anticipated. Investigating the phase relationship throughout the range of lengths shows that when the length is at the 50 mm value the phase difference between pressure and mass flow rate is small (~1°) and the mean phase is in the low 50° range. At the 80 mm length value the phase difference is large (~25°) yet the mean phase is again the low 50° range. The COP between these two limits is comparable. This lends more validity to the theory about the importance of the mean phase as well as the phase difference. Figures 9-11 and 9-12 illustrate the function of the reservoir. As the volume of the reservoir increases, the ratio of the pressure amplitude within the reservoir to the pressure amplitude in the work space decreases. This essentially creates a constant- pressure as explained in the theory of Chapter 3. Another important lesson in this effort is the validity of trying to correlate factors within a model using test circumstances that do not fully reflect the circumstances of the
119 system being modeled as demonstrated throughout Chapter 10. It was simply an assumption that using steady flow pressure drop tests at the component and subassembly level for the regenerator and inertance tubes would allow the correlation of factors in Sage that would increase the accuracy of the cryocooler system model. But this turned out to be incorrect. However, the exercise of pursuing that assumption led to this lesson. It is interesting to see that the steady flow friction correlations from Kays and London actually compares well with the experimentally-derived oscillating flow correlations used in Sage as in Figure 10-5. Upon researching this result I found at least one source that suggests that oscillating flow behavior deviates substantially from steady flow fluid dynamics39 while others believe40 that there should not be any difference between steady and oscillating flow. The last several figures in Chapter 10 demonstrate that using the default values for friction within Sage, i.e. leaving the friction multiplier within Sage set at 1, is an accurate starting point. The ultimate validation of the use of Sage in this effort is the fact that the SBIR program was a success. The inline PTC developed by Sunpower and Gedeon Associates created a buzz in the cryocooler industry about the potential for an efficient, compact commercial cryocooler. This success led to the awarding of another SBIR award from NASA Goddard Space flight center to pursue cooling below 10 K and also a program with a Japanese partner to develop a commercial pulse tube cryocooler for the telecommunications industry.
120 APPENDIX A. Sage Overview
An overview of Sage helps to understand the modeling thought process that evolves into model elements and their connections leading to the system model. The following comes directly from the Chapter 2 of the Sage User’s Guide [33]:
“2.1 What is Sage Sage is a graphical interface that supports simulation and optimization of an underlying class of engineering models. The underlying model class represents something like a spring-mass-damper resonant system, a stirling-cycle machine, or anything else that has been properly coded to work with Sage. The model classes of Sage are not just fixed-geometry models. Each may contain an unlimited number of variations or instances. A model instance, or just plain model for short, is a particular collection of component building blocks, connected and assembled in a particular way, with particular data values, forming a complete system representing whatever it is you are trying to simulate. In other words, you don’t just add numerical data values within the confines of a presumed geometry. You may modify the geometry too. Each particular instance of a given model class resides in its own disk file with a unique name but a common file extension (such as .stl for stirling models). Each model class comes with its own executable file for dealing with its own instances. The resonant system model class (gizmo.exe) is common to all Sage distributions. Other model classes (stirling.exe, etc.) are distribution dependent. Running a model-class executable file brings up the common Sage graphical interface which allows you to: • Create new or read existing model files • Enter numerical data • Edit model geometry • Specify optimization problems • Solve, map or optimize the model • View, save or print a listing These functions are all controlled by menu commands.
121 2.2 What are Models Models are more than the sum of their component building blocks. The way the components are organized and connected together is important too. 2.2.1 Models as Trees Within Sage, model components are organized logically in a hierarchical tree structure. For example, the root-model component of a stirling machine contains a number of sub-components representing pistons, heat exchangers, and the like. These sub-components may themselves contain sub-sub-components. And so forth. The natural way to organize this in terms of child (sub) components branching off of their parent components – as trees in computer-science parlance – not unlike the directory structure on your hard drive. The tree-structured point of view is especially convenient for organizing a model’s disk file or output listing. It does not tell us much, however, about the boundary-interconnections among model components, which are critical to understanding the functioning of the model as a whole. 2.2.1 Models as Interconnected Systems An alternate way to present models is through their boundary interconnections, which are the abstractions by which quantities like fluid flow, force, heat flux, etc., pass from one model component to another. A special form, known as the edit form, presents the model from this point of view. In the edit form, each model component is represented by an icon, with sibling components (belonging to a common parent component) grouped on the same page of the form. Boundary connections among components are indicated graphically by matching numbered arrows attached to the individual model components. In this way it is possible to understand the physical connections among components. An analogy would be this: A catalog of parts, even if tree- structured, tells us little about how an automobile works. We also need to know that the wheels are connected to the engine through the clutch, gearbox and differential, before we begin to understand the whole machine. So it is with Sage. To understand your model you must take some time to delve through its interconnections.
122 2.3 Numerical Input and Output Model components are self-contained entities. As such they manage their own inputs and outputs. Continuing with the automobile analogy: If you want to know what a wheel is doing, ask the wheel. In Sage, if you want to specify input data for a model component, you do so directly within that model component. And if you want to find the output for a model component, you look within that same component. One ramification of this is that output listings are organized differently than you may be used to. Instead of finding all similar quantities from the whole model listed together, you find a sequence of component sub-listings following each other in hierarchical order. A table of contents at the beginning makes it easy to navigate through the listing. Once you get the hang of it you will find it quite easy to home in on a particular component of interest and ignore the rest. 2.4 Solving, Mapping and Optimizing An important thing to do with models is solve them. After you modify a model’s numerical inputs, some of its numerical outputs may no longer be valid. This is because models are defined in terms of implicit relationships among variables which must be iteratively solved. Solving is a menu activated process that brings numerical outputs back into sync for the whole model hierarchy simultaneously. You can also map your model, another menu-activated process available after you have selected a number of input variables to be automatically stepped over a range of values. The stepping sequence is that which would be produced by a nested loop structure. After each step, the model is automatically solved and selected outputs are stored in a disk file for later inspection. More details on mapping are available in chapter 6 of the Sage User’s Guide. Yet another menu-activated process is optimization, which is what you do after you have specified an optimization problem – involving optimized variables, constraints and an objective function. Unlike mapping, which is an exhaustive investigation of a broad area, an optimization is more like a logically- guided walk to the top of a hill. At each step of the way the model is solved and selected outputs are stored in a disk file. More details on optimization are in chapter 7 of the Sage User’s Guide.”
123 APPENDIX B. Sage Boundary Connections
There are a number of available boundary connections in Sage as used in Chapter 7. Almost directly from Chapter 3 in the Sage Stirling-Cycle Model-Class Reference Guide (2nd Edition) [34], edited to only include those items relevant to this effort:
“Stirling model components communicate with each other using the following boundary connections. As usual you may only connect together like connectors of opposite sign (opposite-facing arrows).
• Force Connections, Fphsr These represent phasor forces acting on points of attachment. The points of attachment will share the same motion when connected together. Force connections are used primarily for connecting springs and dampers to moving parts.
• Pressure Connections, Pphsr These represent phasor pressure variations acting on area faces. The area faces share the same volume displacement when connected together. They are used primarily for connecting pistons and the like to gas domains.
• Heat Flow Connections, Qstdy, QGx, QGxt These represent either steady, spatial grid, or space-time grid heat flows acting on thermal boundaries. Boundaries share the same temperature when connected together. Steady heat flows are useful for time-averaged parasitic conduction losses. Spatial grid heat flows are useful for steady but distributed heat flows, such as occurs in two-dimensional fins. Space-time heat flows are for connecting thermal solids to gas domains.
• Gas Flow Connections, m& Gt These represent the flow of gas from one gas domain inlet into another. Two inlets conserve mass flow rate, energy and momentum when connected together.
• Density Connections, ρstdy This represents the common mass density between a gas domain and a pressure reservoir. The two share the same mean pressure when connected together. Density connections are used to connect the stirling working gas to a
124 fixed-pressure source in order to establish charge pressure. This connection type is generally used only once per stirling model instance, but its use is critically important.
125 APPENDIX C. Sage Model Tree
Linear Compressor Model Tree 1 ...... constrained piston and cylinder 1.1 ...... cylinder liner 1.2 ...... piston shell 1.3 ...... constrained piston 1.3.2 ...... positive-facing area 1.4 ...... spring 2 ...... generic cylinder 2.1 ...... cylinder-space gas 2.1.3 ...... neg-facing vol displ 2.2 ...... isothermal surface 3 ...... pressure source 4 ...... Connecting tube 4.1 ...... duct gas 4.2 ...... isothermal surface
Cold Head Model Tree 1 ...... parasitic cold sink 2 ...... parasitic warm source 3 ...... main rejector 3.1 ...... woven screen matrix 3.1.1 ...... rejector gas 3.1.2 ...... isothermal surface 4 ...... regenerator 4.1 ...... heat conductor 4.2 ...... random fiber matrix 4.2.1 ...... matrix gas 4.2.2 ...... rigorous surface 5 ...... acceptor 5.1 ...... woven screen matrix 5.1.1 ...... acceptor gas 5.1.2 ...... acceptor wall 6 ...... compliance tube 6.1 ...... compliance-duct gas 6.2 ...... rigorous surface 7 ...... secondary rejector 7.1 ...... woven screen matrix 7.1.1 ...... sec rej gas 7.1.2 ...... acceptor wall 8 ...... 2nd Rejector Plenum 8.1 ...... cylinder-space gas
126 8.2 ...... isothermal surface 9 ...... acceptor plenum 9.1 ...... isothermal surface 9.2 ...... cylinder-space gas 10 ...... main rejector plenum 10.1 ...... isothermal surface 10.2 ...... cylinder-space gas
Inertance Assembly Model Tree 1 ...... inertance tube 1 1.1 ...... duct gas 1.2 ...... isothermal surface 2 ...... inertance tube 2 2.1 ...... duct gas 2.2 ...... isothermal surface 3 ...... reservoir 3.1 ...... cylinder-space gas 3.2 ...... heat source
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