Low Frequency Axial Fluid Acoustic Modes in a Piping System that Forms a Continuous LoopbyEric R. MardernessA Thesis Submitted to the GraduateFaculty of Rensselaer Polytechnic Institutein Partial Fulfillment of theRequirements for the degree ofMASTER OF SCIENCE in MECHANICAL ENGINEERINGApproved:_Dr. Ernesto Gutierrez-Miravete, Thesis AdviserRensselaer Polytechnic InstituteHartford, ConnecticutApril 2012(For Graduation May 2012) Copyright 2012byEric R. MardernessAll Rights ReservedCONTENTSLow Frequency Axial Fluid Acoustic Modes in a Piping System that Forms a Continuous Loop1LIST OF TABLESvLIST OF FIGURESviGLOSSARY OF KEY WORDSixNOMENCLATURExACKNOWLEDGMENTxiiiABSTRACTxiv1.Introduction12.Problem Formulation and Description52.1Fluid Filled Elastic Cylinders62.2System Description, Properties and Assumptions143.Theory and Methodology203.1Waves and Wave Propagation203.2Helmholtz Equation253.3Uniform Loop: Theory323.4Uniform Loop: Transfer Matrix Method363.5Uniform Loop: COMSOL FEA Models454.Discussion484.1Uniform Loop: Elasticity Effects484.2Non-Uniform Loop524.2.1Net Change in Phase around the Loop534.2.2Net Change in Volume544.2.3Continuity564.2.4Acoustic Impedance Changes574.3Non-Uniform Loop: Transfer Matrix Method614.4Non-Uniform Loop: Loop with a Single Cavity644.4.1Phase Changes at Impedance Discontinuities824.5Non-Uniform Loop: Elastic Discontinuity954.6Non-Uniform Loop: Elbows and Pipe Bends984.7Full System1004.7.1Summary of Full System Analysis1164.7.2Implications and Physical Interpretation of Axial Modes1175.Results and Conclusions1196.Areas for Future Work1227.References124Appendix A Derivation of Linear Acoustic Wave EquationA1Appendix B Derivation of Korteweg Lamb CorrectionB1Appendix C Transmission Matrix MethodC1Appendix D TMM Results FiguresD1Appendix E COMSOL FE Model ResultsE1Appendix F Additional Related ReferencesF1LIST OF TABLESTable 1 Example System Material Properties16Table 2 Example System Component Dimensions18Table 3 Axial Modes; Theoretical, Baseline “Loop”35Table 4 Axial Modes; TMM, Baseline “Loop”40Table 5 Axial Modes; COMSOL, Baseline “Loop”46Table 6 Axial Modes; Theoretical, Elastic Baseline “Loop”50Table 7 Axial Modes; TMM & COMSOL, Elastic Uniform Loop51Table 8 Axial Modes; TMM & COMSOL, “Loop” with 8A Cavity65Table 9 Axial Mode Frequencies (Hz); TMM, Effect of Cavity Cross Sectional Area92Table 10 Axial Modes; TMM & COMSOL, Variation in Cavity Elasticity97Table 11 Axial Modes; TMM & COMSOL, Example System101Table 12 Changes in Phase; TMM Example Systems, 1l Mode A and B112Table 13 Changes in Phase; TMM Example System RW, 4l Mode A and B115LIST OF FIGURESFigure 1 Schematic of an Example Fluid Filled Piping System with Components that form a Continuous Loop of Fluid1Figure 2 Example System Piping Dimensions17Figure 3 Example System Elbow Dimensions18Figure 4 Example System Arrangement and Dimensions19Figure 5 Real Component of Radial Pressure Modes: n,0,028Figure 6 Radial Pressure Modes: n,0,028Figure 7 Real Component of Circumferential Pressure Modes: 0,m,029Figure 8 Circumferential Pressure Modes: 0,m,029Figure 9 Real Component of Axial Pressure Modes: 0,0,l30Figure 10 Axial Pressure Modes: 0,0,l30Figure 11 Baseline Loop and an “Unwrapped” Baseline Loop32Figure 12 Baseline Loop, Pressure Modes Shape for 1l Mode A and Mode B35Figure 13 Baseline Loop, Displacement Modes Shape for 1l Mode A and Mode B36Figure 14 TMM Uniform Pipe Element37Figure 15 Baseline Loop, TMM, Characteristic Equation and Roots41Figure 16 Baseline Loop, TMM Pressure Modes Shape for 1l Mode A43Figure 17 Baseline Loop, TMM Pressure Modes Shape for 1l Mode B44Figure 18 Baseline Loop, COMSOL FE Models45Figure 19 Baseline Loop, COMSOL Pressure Mode Shapes, Actual Loop47Figure 20 Characteristic Equation and Roots; TMM, Elastic Uniform Loop50Figure 21 Calculated Change in Volume; 1l Mode A, Rigid Wall Uniform Loop55Figure 22 Non-Dimensional Impedance; 1l Mode A, Rigid Wall Uniform Loop60Figure 23 Schematic of Loop System with a Single Cavity64Figure 24 Characteristic Equation and Roots; TMM, Loop with 1 Cavity65Figure 25 Characteristic Eq.; TMM, Loop w/ 1 Cavity, Larger Frequency Range66Figure 26 Pressure Mode Shape, Loop with 1 Cavity, 1l Mode A, Surface Plot68Figure 27 Pressure Mode Shape, Loop with 1 Cavity, 1l Mode B, Surface Plot68Figure 28 Pressure Mode Shape, Loop with 1 Cavity, 1l Mode A, COMSOL FE69Figure 29 Pressure Mode Shape, Loop with 1 Cavity, 1l Mode B, CO