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Numerical Simulation of Chemical Reactions Inside a Shock-Tube by the Space-Time Conservation Element and Solution Element Metho

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Numerical Simulation of Chemical Reactions Inside a Shock-Tube by the Space-Time Conservation Element and Solution Element Metho NUMERICAL SIMULATION OF CHEMICAL REACTIONS INSIDE A SHOCK-TUBE BY THE SPACE-TIME CONSERVATION ELEMENT AND SOLUTION ELEMENT METHOD MASTER’S THESIS Presented in Partial Fulfillment of the Requirements for the Degree Masters of Science in the Graduate School of the Ohio State University By, David L. Bilyeu, B.S. ***** The Ohio State University 2008 Thesis Committee: Professor Sheng-Tao John Yu, Advisor Approved by Dr. Douglas Davis Professor Jeffrey A. Sutton __________________________________ Adviser Mechanical Engineering Graduate Program Copyrighted By David L. Bilyeu 2008 ABSTRACT The goal of this master’s project is to develop and test a numerical tool to simulate the fluid physics and chemical reactions inside of a shock tube. The one- dimensional Euler equations for chemically reacting flow are employed to model the fluid flows and detailed finite-rate chemistry models are used to simulate chemical reactions. The CESE method is used to solve the Euler equation. The implicit trapezoidal method with sub-time steps is used to solve the stiff ODEs for chemical reactions. The newly developed code has been validated by (i) zero-dimensional calculations for testing the chemistry solver (ii) Calculation of the ignition delay in a shock tube. The results were compared with the ignition time from shock tube experiments. The new shock-tube tool will be used (i) to study the lean-blow-out in scramjet engines and (ii) to assess the required mesh and time resolution for 2/3- dimensional simulation of chemically reacting flows. ii DEDICATION To My Family iii ACKNOWLEDGMENTS I would like to thank my advisor, Professor Sheng-Tao John Yu, for his knowledge and experience in numerical simulations, his constant encouragement and enthusiasm. I am grateful to Dr. Douglas Davis at for his insight into computational chemistry. I would also like to thank Andrew Naples for his insight into the workings of CHEMKINTM and M.S. Word. I would also like to thank Yung-Yu Chen for setting up and managing the cluster used to do most of the computational work. This research was supported by a fellowship from The Dayton Area Graduate Studies Institute. iv VITA May 04, 1984 . Born in California, USA June 2006 . B.S., Aerospace Engineering, California State Polytechnic University, Pomona, CA, USA 2006 – Present . Graduate Research Assistant, Department of Mechanical Engineering, The Ohio State University, USA FIELDS OF STUDY Major Field: Mechanical Engineering v Table of Contents Abstract .................................................................................................................. ii Dedication ............................................................................................................. iii Acknowledgments ................................................................................................ iv Vita ........................................................................................................................ v List of Figures ....................................................................................................... ix List Of Tables ...................................................................................................... xii Nomenclature ..................................................................................................... xiii Chapter 1 Introduction ........................................................................................... 1 1.1. Objectives .............................................................................................. 1 1.2. Numerical Methods ............................................................................... 2 1.3. Organization .......................................................................................... 3 Chapter 2 Thermodynamics .................................................................................. 5 2.1. General Gas Properties of Mixed Gases ................................................ 5 2.2. Thermodynamic Properties for Mixed Gases ........................................ 9 Chapter 3 Combustion Modeling ........................................................................ 17 3.1. Chemical Rate of Progress .................................................................. 17 3.2. Chemical Reaction Rate ...................................................................... 20 3.3. Numerical Procedure for Zero-Dimensional Combustion .................. 26 3.4. Validation ............................................................................................ 34 3.4.1. The Li H2/O2 Mechanism .................................................................... 39 3.4.2. The GRI Mechanism ........................................................................... 42 vi 3.4.3. The UCSD Mechanism ........................................................................ 46 3.5. Future Improvements ........................................................................... 49 Chapter 4 Model Equation ................................................................................... 51 4.1. One-Dimension Euler Equations ......................................................... 51 4.2. One-Dimension Euler Equations for Reacting Flows ......................... 52 4.3. The Non-Conservative Form ............................................................... 54 4.4. Calculating ∂p/∂x ................................................................................. 56 4.5. The Jacobian Matrix ............................................................................ 60 Chapter 5 The CESE Method .............................................................................. 66 5.1. Introduction ......................................................................................... 66 5.2. CESE Scheme for the Euler Equation ................................................. 67 5.3. CESE Scheme for the Chemically Reacting Euler Equation .............. 72 5.4. Boundary Conditions ........................................................................... 75 Chapter 6 Shock Tube ......................................................................................... 77 6.1. Introduction ......................................................................................... 77 6.2. General Layout .................................................................................... 77 6.3. Analytical Solution .............................................................................. 80 6.4. Numerical Method for the Shock Tube ............................................... 89 vii 6.5. Validation .......................................................................................... 103 Chapter 7 Conclusion ........................................................................................ 111 7.1. Future Improvements ......................................................................... 111 Bibliography ...................................................................................................... 114 viii LIST OF FIGURES Figure 3.1: Numerical procedure for zero dimension combustion analysis .......................... 28 Figure 3.2: The time history of temperature and time steps for constant-volume combustion of stoichiometric CH4-O2 mixture diluted with 76.1% Ar. The initial temperature is 1500 K and the initial pressure is 3.0 atm. ........................ 30 Figure 3.3: Time vs. iteration number for the combustion of constant-volume stoichiometric combustion of CH4-O2 diluted with 76.1% Ar. The initial temperature and pressure are 1500 K and 3.0 atm .............................................. 32 Figure 3.4: Comparison of induction times for constant volume combustion of stoichiometric H2/O2 mixture diluted with 70% Ar. The initial pressure for all cases is 1.3 atm. These results are from an old version of Gaskin..................... 36 Figure 3.5: Comparison of final temperature for constant volume combustion of stoichiometric H2/O2 diluted with 70% Ar. The initial pressure for all cases is 1.3 atm. These results are from a previous version of Gaskin. .......................... 37 Figure 3.6: Induction time calculations given different maximum time steps. Combustion is done at constant volume with a stoichiometric coefficient of 1.5, an initial pressure of 0.5 atm diluted with 76.1% Ar. The triangle corresponds with the left y axis while the square and diamond are associated with the right y axis .... 38 TM Figure 3.7: Induction time percentage difference between Gaskin and CHEMKIN for H2 combustion with 76.1% Ar at various initial temperature and pressures at a stoichiometric coefficient of 1.5. ........................................................................ 41 TM Figure 3.8: Final temperature percent difference between Gaskin and CHEMKIN for H2 /O2 combustion diluted with 76.1% Ar at various initial temperatures and pressures and at an equivalence ratio of 1.5. ...................................................... 42 TM Figure 3.9: Induction time percent difference between Gaskin and CHEMKIN for CH4 combustion diluted with 76.1% Ar at various temperature and pressures and an equivalence ratio of 1.5. ................................................................................. 43 TM Figure 3.10: Final temperature percent difference between Gaskin and CHEMKIN for CH4 combustion with 76.1% Ar at various initial temperatures, pressures and a stoichiometric coefficient of 1.5 ......................................................................... 45
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