THERMODYNAMICS APPLIED TO ENGINEERING
INTRODUCTION WHAT IS COMPRESSIBLE FLOW? WHY COMPRESSIBLE FLOW IS IMPORTANT? HISTORICAL BACKGROUND EARLY DEVELOPMENTS THE SHOCK WAVE PUZZLE CHOKING FLOW EXTERNAL ?OW FILLING AND EVACUATING GASEOUS CHAMBERS BIOGRAPHIES OF MAJOR FIGURES REVIEW OF THERMODYNAMICS BASIC DE?NITIONS MAXIMUM WORK OF EXPANSION PROCESS THE VELOCITY–TEMPERATURE DIAGRAM CONTENTS BASIC OF FLUID MECHANICS FLUID PROPERTIES KINDS OF FLUIDS VISCOSITY KINEMATIC VISCOSITY BULK MODULUS MASS CONSERVATION CONTROL VOLUME CONTINUITY EQUATION REYNOLDS TRANSPORT THEOREM MOMENTUM CONSERVATION MOMENTUM GOVERNING EQUATION CONSERVATION MOMENT OF MOMENTUM ENERGY CONSERVATION APPROXIMATION OF ENERGY EQUATION LIMITATIONS OF INTEGRAL APPROACH DI?ERENTIAL ANALYSIS MOMENTUM EQUATIONS OR N–S EQUATIONS BOUNDARY CONDITIONS AND DRIVING FORCES SPEED OF SOUND MOTIVATION SPEED OF SOUND IN IDEAL AND PERFECT GASES SPEED OF SOUND IN REAL GASES SPEED OF SOUND IN ALMOST INCOMPRESSIBLE LIQUID SOUND IN VARIABLE COMPRESSIBLE LIQUIDS SPEED OF SOUND IN SOLIDS SOUND SPEED IN TWO PHASE MEDIUM THE DIMENSIONAL E?ECT OF THE SPEED OF SOUND DOPPLER E?ECT ACOUSTIC WAVE EQUATION – DERIVATION HEARING AND SOUND WAVES SOUND WAVE IN THREE DIMENSIONS ISENTROPIC FLOW STAGNATION STATE FOR IDEAL GAS MODEL GENERAL RELATIONSHIP RELATIONSHIPS FOR SMALL MACH NUMBER ISENTROPIC CONVERGING-DIVERGING FLOW IN CROSS SECTION THE PROPERTIES IN THE ADIABATIC NOZZLE ISENTROPIC FLOW EXAMPLES MASS FLOW RATE (NUMBER) ISENTROPIC TABLES ISENTROPIC ISOTHERMAL FLOW NOZZLE THE IMPULSE FUNCTION IMPULSE IN ISENTROPIC ADIABATIC NOZZLE THE IMPULSE FUNCTION IN ISOTHERMAL NOZZLE ISOTHERMAL TABLE THE E?ECTS OF REAL GASES ISENTROPIC RELATIONSHIPS FOR REAL GASES NORMAL SHOCK SOLUTION OF THE GOVERNING EQUATIONS INFORMAL MODEL FORMAL MODEL PRANDTL’S CONDITION OPERATING EQUATIONS AND ANALYSIS THE LIMITATIONS OF THE SHOCK WAVE SMALL PERTURBATION SOLUTION SHOCK THICKNESS SHOCK DRAG OR WAVE DRAG THE MOVING SHOCKS SHOCK OR WAVE DRAG RESULT FROM A MOVING SHOCK SHOCK RESULT FROM A SUDDEN AND COMPLETE STOP MOVING SHOCK INTO STATIONARY MEDIUM (SUDDENLY OPEN VALVE) PARTIALLY OPEN VALVE PARTIALLY CLOSED VALVE WORKED–OUT EXAMPLES FOR SHOCK DYNAMICS SHOCK TUBE SPECIAL SHOCK DYNAMICS SHOCK TUBE THERMODYNAMICS CONSIDERATIONS SHOCK WITH REAL GASES SHOCK IN WET STEAM NORMAL SHOCK IN DUCTS ADDITIONAL EXAMPLES FOR MOVING SHOCKS TABLES OF NORMAL SHOCKS, K = 1 4 IDEAL GAS NORMAL SHOCK IN VARIABLE DUCT AREAS NOZZLE E?CIENCY DI?USER E?CIENCY NOZZLE FLOW WITH EXTERNAL FORCES ISENTROPIC NOZZLE (Q = 0) ISOTHERMAL NOZZLE (T = CONSTANT) ISOTHERMAL FLOW THE CONTROL VOLUME ANALYSIS/GOVERNING EQUATIONS DIMENSIONLESS REPRESENTATION THE ENTRANCE LIMITATION OF SUPERSONIC BRANCH COMPARISON WITH INCOMPRESSIBLE FLOW SUPERSONIC BRANCH FIGURES AND TABLES ISOTHERMAL FLOW EXAMPLES UNCHOKED SITUATIONS IN FANNO FLOW REYNOLDS NUMBER E?ECT 10 FANNO FLOW 10 1 INTRODUCTION 10 2 FANNO MODEL 10 3 NON–DIMENSIONALIZATION OF THE EQUATIONS 10 4 THE MECHANICS AND WHY THE FLOW IS CHOKED? 10 5 THE WORKING EQUATIONS 10 6 EXAMPLES OF FANNO FLOW 10 7 SUPERSONIC BRANCH 10 8 MAXIMUM LENGTH FOR THE SUPERSONIC FLOW 10 9 WORKING CONDITIONS 10 9 1 VARIATIONS OF THE TUBE LENGTH ( 4FL D ) E?ECTS 10 9 2 THE PRESSURE RATIO, P2/ P1, E?ECTS 10 9 3 ENTRANCE MACH NUMBER, M1, E?ECTS 10 10PRACTICAL EXAMPLES FOR SUBSONIC FLOW 10 10 1SUBSONIC FANNO FLOW FOR GIVEN 4 F L D AND PRESSURE RATIO 10 10 2SUBSONIC FANNO FLOW FOR A GIVEN M1 AND PRESSURE RATIO 10 11THE APPROXIMATION OF THE FANNO FLOW BY ISOTHERMAL FLOW 10 12 THE TABLE FOR FANNO FLOW 10 13 APPENDIX – REYNOLDS NUMBER E?ECTS 11 RAYLEIGH FLOW 11 1 INTRODUCTION 11 2 GOVERNING EQUATIONS 11 3 RAYLEIGH FLOW TABLES 11 4 EXAMPLES FOR RAYLEIGH FLOW 12 EVACUATING SEMIRIGID CHAMBERS 12 1 GOVERNING EQUATIONS AND ASSUMPTIONS 12 2 GENERAL MODEL AND NON–DIMENSIONALIZATION 12 2 1 ISENTROPIC PROCESS 12 2 2 ISOTHERMAL PROCESS IN THE CHAMBER 12 2 3 A NOTE ON THE ENTRANCE MACH NUMBER 12 3 RIGID TANK WITH A NOZZLE 12 3 1 ADIABATIC ISENTROPIC NOZZLE ATTACHED 12 3 2 ISOTHERMAL NOZZLE ATTACHED 12 4 RAPID EVACUATING OF A RIGID TANK 12 4 1 ASSUMING FANNO FLOW MODEL 12 4 2 FILLING PROCESS 12 4 3 THE ISOTHERMAL PROCESS 12 4 4 SIMPLE SEMI RIGID CHAMBER 12 4 5 THE “SIMPLE” GENERAL CASE 12 5 ADVANCE TOPICS 12 6 REMARK ON REAL GASES 13 EVACUATING UNDER EXTERNAL VOLUME CONTROL 13 1 GENERAL MODEL 13 1 1 RAPID PROCESS 13 1 2 EXAMPLES 13 1 3 DIRECT CONNECTION 13 2 NON–LINEAR FUNCTIONS E?ECTS 13 3 SUMMARY 14 OBLIQUE SHOCK 14 1 PREFACE TO OBLIQUE SHOCK 14 2 INTRODUCTION 14 2 1 INTRODUCTION TO OBLIQUE SHOCK 14 2 2 INTRODUCTION TO PRANDTL–MEYER FUNCTION 14 2 3 INTRODUCTION TO ZERO INCLINATION 14 3 OBLIQUE SHOCK 14 4 SOLUTION OF MACH ANGLE 14 4 1 UPSTREAM MACH NUMBER, M1, AND DE?ECTION ANGLE, ? 14 4 2 WHEN NO OBLIQUE SHOCK EXIST OR THE CASE OF D > 14 4 3 UPSTREAM MACH NUMBER, M1, AND SHOCK ANGLE, ? 14 4 4 GIVEN TWO ANGLES, ? AND ? 14 4 5 FLOW IN A SEMI–2D SHAPE 14 4 6 FLOW IN A SEMI-2D SHAPE 14 4 7 SMALL ? “WEAK OBLIQUE SHOCK” 14 4 8 CLOSE AND FAR VIEWS OF THE OBLIQUE SHOCK 14 4 9 MAXIMUM VALUE OF OBLIQUE SHOCK 14 5 DETACHED SHOCK 14 5 1 ISSUES RELATED TO THE MAXIMUM DE?ECTION ANGLE 14 5 2 OBLIQUE SHOCK EXAMPLES 14 5 3 APPLICATION OF OBLIQUE SHOCK 14 5 4 OPTIMIZATION OF SUCTION SECTION DESIGN 14 5 5 RETOUCH OF SHOCK DRAG OR WAVE DRAG 14 6 SUMMARY 14 7 APPENDIX: OBLIQUE SHOCK STABILITY ANALYSIS 15 PRANDTL–MEYER FUNCTION 15 1 INTRODUCTION 15 2 GEOMETRICAL EXPLANATION 15 2 1 ALTERNATIVE APPROACH TO GOVERNING EQUATIONS 15 2 2 COMPARISON AND LIMITATIONS BETWEEN THE TWO APPROACHES 15 3 THE MAXIMUM TURNING ANGLE 15 4 THE WORKING EQUATIONS FOR THE PRANDTL-MEYER FUNCTION 15 5 D’ALEMBERT’S PARADOX 15 6 FLAT BODY WITH AN ANGLE OF ATTACK 15 7 EXAMPLES FOR PRANDTL–MEYER FUNCTION 15 8 COMBINATION OF THE OBLIQUE SHOCK AND ISENTROPIC EXPANSION