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Oscillatory Compressible Flow and Heat Transfer in Porous Media OSCILLATORY COMPRESSIBLE FLOW AND HEAT TRANSFER IN POROUS MEDIA – APPLICATION TO CRYOCOOLER REGENERATORS A Dissertation Presented to The Academic Faculty by Jeremy Paul Harvey In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in Mechanical Engineering Georgia Institute of Technology December 2003 OSCILLATORY COMPRESSIBLE FLOW AND HEAT TRANSFER IN POROUS MEDIA – APPLICATION TO CRYOCOOLER REGENERATORS APPROVAL Prateen V. Desai, Chairman, Mechanical Engineering S. Mostafa Ghiaasiaan, Mechanical Engineering Minami Yoda, Mechanical Engineering Carl S. Kirkconnell, Raytheon Space and Airborne Systems Jeffrey F. Morris, Chemical Engineering Date Submitted: November 25, 2003 ACKNOWLEDGEMENTS I have been blessed to have been involved in this research with so fine a group. And it is indeed the group of individuals whom have assisted me that have made this research so enjoyable and productive. Beginning with those most directly involved in the research, I would like to thank my committee. Dr. Prateen Desai, my thesis advisor, fortunately thought of me when Dr. Carl Kirkconnell was searching out an undergraduate student to perform this research for, at the time, Hughes Aircraft Company. The Woodruff School and the Georgia Tech community have been so fortunate to have Dr. Desai as a dedicated professor, and I as a friend and advisor. Dr. Mostafa Ghiaasiaan quickly took an interest in the research after conversing with me and he now will lead the cryocooler research at Georgia Tech. Drs. Minami Yoda and Jeffrey Morris completed my team of advisors by providing breadth of knowledge in their fields. Of these individuals, Dr. Kirkconnell deserves special thanks for being the catalyst for the research at Hughes Aircraft , and now Raytheon Space and Airborne Systems. I hope that I can provide the same enthusiasm and energy that you have provided me. This research was performed due, in no small part, to the research grants provided by Hughes Aircraft Company and Raytheon Company to the Georgia Tech Foundation. Additional thanks must go to the Hughes and Raytheon Doctoral Fellowship Program and the Georgia Institute of Technology Presidential Scholarship for supporting me directly throughout this program. At Raytheon, special thanks to Mr. Alberto Schroth, Mr. Kenneth Price, Mr. Thomas Pollack, Mr. William Croft, and Mr. David Mc Gorrin. At Georgia Tech, I have iii been so fortunate to have several undergraduate and graduate students work with me in my lab; Mr. Matthieu Coutaudier, Mr. Robert Hon, and Mr. Jeesung “Jeff” Cha, the next torch bearer. I thank my whole family for their support and love, especially my parents who are so far away, but so near my heart, my sister Janelle, who I am so thankful is close, and to my wife’s family who I wish were close. My wife Chrystine has been my rock during this undertaking. She has given me so much hope and now our new life begins. iv TABLE OF CONTENTS Approval ............................................................................................................................. ii Acknowledgements............................................................................................................ iii List of Figures......................................................................................................................x List of Tables ................................................................................................................... xiv List of Tables ................................................................................................................... xiv Nomenclature.....................................................................................................................xv Summary............................................................................................................................xx 1. Introduction and Background ..........................................................................................1 1.1 Introduction ................................................................................................................1 1.2 Background.................................................................................................................4 1.2.1 History of Pulse Tube Cryocoolers......................................................................4 1.2.2 Survey of Efforts to Improve the Regenerator.....................................................7 1.2.3 Development of the Volume Averaging Technique ..........................................15 1.2.4 Overview............................................................................................................16 2. Theoretical Development...............................................................................................19 2.1 Governing Equations and Constitutive Relations.....................................................19 2.1.1 Conservation of Mass Equation .........................................................................20 2.1.2 Balance of Momentum Equations......................................................................20 2.1.3 α-Phase Conservation of Energy Equation........................................................21 2.1.4 β-Phase Conservation of Energy Equation ........................................................22 v 2.1.5 Equations of State ..............................................................................................22 2.1.6 α-Phase Entropy Generation Equation ..............................................................23 2.1.7 β-Phase Entropy Generation Equation...............................................................23 2.1.8 Summary of Equations.......................................................................................24 2.2 Volume-Averaged Equations ...................................................................................25 2.2.1 Volume-Averaged Conservation of Mass Equation ..........................................25 2.2.2 Volume-Averaged Balance of Momentum Equation.........................................26 2.2.3 Volume-Averaged α-Phase Conservation of Energy Equation .........................26 2.2.4 Volume-Averaged β-Phase Conservation of Energy Equation..........................27 2.2.5 Volume-Averaged α-Phase Entropy Generation Equation................................28 2.2.6 Volume-Averaged β-Phase Entropy Generation Equation ................................28 2.3 Simplifying Assumptions .........................................................................................29 2.3.1 Density Spatial Deviation ..................................................................................29 2.3.2 Negligible Mechanical Dispersion.....................................................................31 2.3.3 Thermal Dispersion............................................................................................32 2.3.4 Entropy Generation Due to Thermal Dispersion ...............................................33 2.3.5 Negligible Brinkman Effect ...............................................................................34 2.3.6 One Dimensional Model on the Macroscopic Length Scale..............................34 2.4 Simplified Equations ................................................................................................35 2.4.1 Simplified Conservation of Mass Equation .......................................................35 2.4.2 Simplified Balance of Momentum Equation......................................................36 2.4.3 Simplified α-Phase Conservation of Energy Equation ......................................37 vi 2.4.4 Simplified β-Phase Conservation of Energy Equation ......................................37 2.4.5 Simplified α-Phase Entropy Generation Equation.............................................39 2.4.6 Simplified β-Phase Entropy Generation Equation.............................................39 2.4.7 Summary of Equations.......................................................................................44 2.5 Exact Solutions.........................................................................................................46 2.6 Scale Analysis ..........................................................................................................49 2.6.1 Conservation of Mass Equation Scale Analysis.................................................50 2.6.2 Balance of Momentum Equation Scale Analysis...............................................50 2.6.3 α-Phase Conservation of Energy Equation Scale Analysis ...............................52 2.6.4 β-Phase Conservation of Energy Equation Scale Analysis................................54 2.6.5 Scaled Equation Summary .................................................................................55 2.6.6 Limiting Cases ...................................................................................................59 3. Computational Models...................................................................................................60 3.1 Problem Definition ...................................................................................................60 3.2 Numerical Method....................................................................................................61 3.2.1 Time Integration.................................................................................................62 3.2.2 Spatial Discretization .........................................................................................62
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